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~~\\[2in]
{\Huge \bf Math 105 Teachers Guide \\[2ex]
Preliminary release dated 24 June, 2003}
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\section{What is Math 105?}

Math 105 is the first in a sequence of mathematics courses 
for future elementary and middle school teachers.  Its primary goal is to
educate mathematically.  We cannot expect schoolchildren to learn anything
beyond the rote execution of algorithms if the first ten years of their
education is at the hands of teachers who can't themselves go beyond this.
By providing elementary educators with a good mathematical foundation, we
hope to reverse the downward spiral into which many perceive mathematics 
education to have fallen.  A secondary goal is to introduce students to
the cooperative learning environment.  Keep in mind, though, that the course 
is a content course, not a pedagogy course.  

The content covered in Math 105 has a {\em how} and a {\em what} component.
Students will be taught how to think and speak mathematically, many for
the first time in their lives.  They will solve problems they have not 
been shown how to do, they will learn to put their ideas into precise
language and to prove their assertions when possible.  They will learn
to break down problems into smaller problems, use trial and error, 
generalize when appropriate, and to harness their (atrophied) common sense.
It is the {\em how} content that determines the format of the course, 
necessitating the problem-solving theme and the socratic style of
discussion.  

The {\em what} content, i.e., the choice of topics, deserves some 
explanation as well.  It is guided by the work of scholars such as
Liping Ma as well as by the demands placed on elementary school
mathematics teachers.  The topics of place value, arithmetic
algorithms, fractions and decimals need no justification.  In addition,
we believe that a solid mathematical base at roughly the high school
level is esssential even for teachers who will be teaching children
who are four to eight years old.  These teachers will be given a great
deal of independence in designing lessons and choosing aspects of the
curriculum.  Thus in addition to the topics listed above, we include
three additional major topics.  These are laced throughout the course:
problem-solving, algebra and logic.  

All three of these are in line with
the 2000 NCTM Standards.  The problem-solving curriculum is designed
to get students thinking on their own and explaining coherently what
they have done.  The algebra content has a major and a minor focus,
neither of these being algebraic manipulation.  The major focus is
word problems.  In particular, these problems teach students how
to assign variables and to write equations expressing relations
between them.  The minor focus is on algebra as a means of
formulating and expressing generalizations.  Finally, we expect students
to learn to justify their solutions with some rigor.  We assume
this class is the first place they have done this, so we go easy
on them at first, but we constantly harp on the level of rigor of 
their arguments (or ours) and urge them forwards.

Some specifics about the student body here at OSU are worth noting.
College algebra (Math 148) is {\em not} a prerequisite for this course, 
which means that along with the problem solving will by necessity
come some algebra review, and indeed a continual reinforcing of the 
use and meaning of algebraic notation.  There
may be some resistance on the part of some students to the way the class 
is run, though we have found the small group format by and large to
be popular; resistance is more likely to come from the emphasis on
problem-solving and lack of answers provided.  The classroom format 
turns traditional norms upside down by making students take responsibility 
and authority for determining whether a problem has been solved 
correctly\footnote{We do not, in the end, withhold information and 
techniques they need to know or judgment on what is correct, but we do 
insist that they develop their own judgment and we train them to do so.}.  

Teaching a course by means of cooperative learning, Socratic discourse, 
and the like involves decentralizing yourself (the instructor) as the focus 
for learning and authority.  This is a skill which takes some time to develop, 
and it's not always easy.  This is why it's best to start preparing to teach a
course like this well in advance.  Don't worry, though --- you're not on your
own.  If you're expecting to teach in this sequence, you should already have
looked into observing others' classes, and the rest of the informal instructor
preparation program.  Part of this is the extensive set of notes included in
Section~3 of this guide.  Reading and discussing them with folks who have had
experience in the sequence will help defray a lot of the anxiety you may have.

This guide is designed to answer any questions you may have, large or small,
about this class (although of course conferring with other people who have
taught it is an irreplaceable resource), and to provide all the help you need
with the details until you feel comfortable enough to stand on your own.  Even
once you've come into your own, it's a good idea to refer to the guide
throughout your first time teaching the course, as we have collected the
potential pitfalls of several semesters' worth of teaching it, and it's always
best to be prepared.  Section~2 gives a detailed blow-by-blow of each of the
activities in the course pack (including solutions), along with suggested times
and a couple of sample syllabi.  Section~3 discusses how to teach this kind of
class.  Section~4 goes into detail on the write-ups: which activities have
traditionally had write-ups associated with them, how often to assign them,
which ones are important to assign, how to make the assignments in class, and
specifically what to look for in critiquing them.  
Section~5 discusses exams and evaluation, including a large bank of
sample exams from a variety of past instructors.  Finally, Section~6 details
the organization of the directories and LaTeX files used to develop this guide
and the course pack, so that you can take things you like and adapt them for
your own handouts, exams and worksheets.


\pagebreak
\section{The activities}

In this section we discuss each of the activities presented in the
coursepack.  We include the main point of the activity and the solution, 
as well as notes on how the worksheet has gone in the past and what
directions one might want to take the discussion or what sorts of 
leading questions might be appropriate for groups that are stuck.
If extra equipment is needed we include a reminder to that effect.
An estimate of the number of (48-minute) hours is sometimes included
as well for the longer worksheets.  The notes for the early activities 
are more detailed since instructors may need more guidance when 
getting acclimated to the workshop format.

The July 1, 2003 version of the coursepack and guide has been edited
to reflect the fact that no one in the past has ever made it through
all the worksheets in the coursepack.  This is because often an activity
takes longer than anticipated, and even more often the instructor finds
it necessary to insert a background reading or worksheet because the 
main worksheet was too difficult.  Building on experience, we have 
added in as many of these background readings and worksheets as we
felt necessary, and removed a few of the main worksheets to bring the
total time back to 45 class sessions.  Of course actual mileage may
vary, so we've included the worksheet outtakes in the instructors guide.
This provides instructors the flexibility to add them in as necessary,
without having to used an untested activity.  Still, instructors shouldn't
hesitate to make up new worksheets or readings -- all of the ones in 
the coursepack and guide were once new!  

One final note on the use of Readings.  The first version of this course
was entirely worksheet based, with no auxilliary readings.  We have
kept the text-free approach as much as possible for various reasons.
We have found that plunging into the workshop style is psychologically
difficult for many students and that removing the text helps to put them
in the right frame of mind.  Also, removing the text emphasizes primacy
of thinking, reasoning, understanding and communicating over remembering
and absorbing.  As mentioned above, some textual material became necessary.
In Math 106 this takes the form of a long glossary.  In Math 105, while
definitions are in a way still the focus of the textual materials, students
need to see these spelled out in lengthy examples.  The present compromise
is a set of Readings (Section~5 of the coursepack) that precede and 
accompany the worksheets.  These are meant to be assigned as strictly 
out-of-class reading assignments.  There are self-check questions in each
Reading, which are meant to be answered at the time the reading homework is
done.  We have often found it necessary in the past to require that the
answers to the self-check questions be out on the students' desks at the
beginning of class, and that credit be given if the questions have been
answered, correctly or incorrectly.  This way, it is possible to spend a
small amount of class time discussing their questions on the Reading and
their answers to the self-checks without having to allot class time to 
a first pass at the material in the Reading.  The worksheets themselves
are marked with the names of any Readings that are meant to be done before
the worksheet is started.  Since the students may not know what day a 
particular worksheet will be begun it is important for the instructor to
{\bf look ahead to see when a Reading will be needed and assign it at least
a few days in advance of that.}  

\pb
\subsection{Problem-solving and abstract thinking}

Several readings should be assigned on the first day of class, to be read
ASAP.  These include Sections~6.1,~6.2 and~6.3.  The most important of
these is Section~6.1, ``Paradigm for abstraction and generalization''.
Instead of spending time discussing this, you will want to show them,
on every worksheet, how they are stepping through this process.  Several
reminders occur in the activity notes to do this.  

\subsubsection{Poison}

[Materials: counters or chips for use in play; time 2 hours] 

This is a great first activity in basically every way. 
(1) The final and most general solution is somewhat sophisticated and 
will challenge the students' explanatory powers.  On the other hand, it 
is easy to start solving the problem simply by playing the game, and
almost every student will find that s/he gains understanding by doing this.
(2) The solution will typically emerge piece by piece.  Students will 
generate several theories, some correct and some not, and the ensuing 
discussions will be good prototypes for how to argue mathematics.
(3) The ultimate authority lies in being able to win the game.  
The teacher cannot win against a student who thoroughly understands  
the game.  It is a tremendous confidence builder for students to see that
something they have figured out is correct no matter what the teacher says.
When the point is reached where several students understand completely
how to win, teachers often play it up by issuing a challenge: 
``OK, so-and-so thinks she can win against me, no matter what I do
--- I am going to try my hardest to beat her; the rest of you vote on
who you think will win.''  This can be done at several stages: when
they think they can win the 10 counter game with you going first, 
when they think they can win the $n$ counter game if they get to say
who goes first, and when they (erroneously) think they can win the
10 or $n$ counter game no matter who goes first.
Poison should be the basis for the first write-up, and you should insist 
on continuing the discussion of the solution until everyone claims to 
understand why the pattern works.

{\em Solution:}\/ The winning strategy for the general $n$ counter game is: if
$n$ is one more than a multiple of three, choose to go second and always do the
opposite of what the first player does.  If $n$ is two more than a multiple of
three, go first and take one counter, then always do the opposite of what the
other player does.  If $n$ is a multiple of three, go first and take two
counters, then always do the opposite of what the other player does.  Here's
why it works.  In any of these cases, you first force the other person to move
when the number is one more than a multiple of three.  By doing the opposite of
what the other person does, you will together have removed a total of three
counters, thus arriving at one more than another multiple of three (the next
smallest multiple).  Thus you can force the other player to move when the
number of remaining counters is 4 and finally 1, meaning that the other person
ends up with the last counter.

Remember not to give any of this away in class!  Old habits die hard, 
especially when the students become frustrated, but the whole point of 
the activity is that if the students struggle long enough they {\em will} 
discover the answer!  Here is a list of typical milestones they will
come to and blocks they will encounter.

Many students will fixate on odd-even parity and try to argue that the
strategy depends on parity, even in the face of evidence to the contrary.
It is hard to discover the $3n+1$ pattern, since the number 3 does not
appear in the game description and since the pattern $3n+1$ is harder
to recognize than the patterns $3n$ or $2n+1$.  Initially they all play 
10 counter games, and the first thing they discover is that leaving the
opponent with 4 is a winning move.  Some will now say they have found the 
strategy.  This opens the door to a discussion of what a winning strategy
is, and why ``leave 4'' is not a complete strategy.  Some students will think 
that a strategy can include requiring the opponent to make certain moves;
this can also come up in the discussion of what a strategy is.  

In general, the students will have a
very hard time at first saying exactly what they mean, and being precise (you
will get $3x+1$ vs. $3(x+1)$ confusions, for example).  Don't get too
frustrated by this; just encourage them, and try to help them come up by
themselves with a clear formulation of their thoughts.  You may want to
differentiate explicitly between understanding how to win and being able 
to explain how to win.  A student understands if they can beat a perfectly
smart opponent (you) every time.  A student can explain how to win if
they can give you a set of instructions which you will win if you follow,
even if you play dumb and try hard to misinterpret them.  Depending on
what they come up with, it can be fun to demonstrate this.

{\em Prompts:}\/ If groups get stuck right away, tell them just to try playing
the game a few times.  If they say they don't know how, tell them not to worry
about winning or losing, just try it.  If they don't seem to be recording the
play-by-play, ask if they see any overall pattern --- this should lead to a
request to see (or hear) specific sets of moves from previous games.  If a
group appears to have solved the problem much faster than the others, ask to
hear what their solution is.  If there's a flaw in it, ask a question that will
bring this flaw out.  If it only works for the ten-counter game, tell them to
move on to an eleven-counter game, or even down to a nine-counter game.  If
they've really hit on the solution without having played the game before, ask
them for detailed justifications to ``convince'' you their solution is right
(don't tell them they're right!).  You might want to bring a few copies
of ``Rat Poison'' (from the Outtakes) so that if a group finishes early,
having obtained a complete solution they can justify, you can tell them 
to move on to ``Rat Poison''.   

Also, at each step, as you move around, when you ask where a given group is,
also ask if everyone in the group believes (or understands, or agrees with) the
response you receive.  It's important that the group members help each other
keep up, and it probably won't yet be obvious to you which students are the
quiet ones.

{\em Large group discussion:}\/ When it comes time to discussing the solution
in large group, remember that this is the first time for everyone, and students
will not be used to explaining themselves (or convincing each other, or even
speaking up at all).  You will have to keep prompting them with, "Why?"  Also
try to get as many as possible to come to the board to write even a partial
strategy, or a table of recorded moves, etc. --- the first activity sets the
tone for the semester, and you want them to be used to writing on the board,
and talking to each other rather than to you.  Encourage the notion that
ideas can be road-tested: if two students claim winning strategies, one
going first and the other going second, play them against each other.
Same if two students suggest different moves from a given position.  
This is also a good time to start working on their verbal skills.
When a student says something essentially correct but poorly stated,
write it on the board.  If the student immediately wishes to restate it, 
good.  If not, the class as a whole can comment on it and refine it.

One question you should ask at some point (not necessarily right away) is
``What does it mean to have a winning strategy?''  The answer should include
things like: you are in control of the game, you can always force your opponent
to lose provided you have choice of going first or second, etc.

The inevitable question ``Why are we doing it this way?'' may come up as soon
as this first large-group discussion, so be prepared to offer a rational,
non-confrontational explanation of why ``we'' (and you) believe in it.  Use the
problem they've just solved as an example, and try to get some people to admit
it gave them a little self-confidence to see that they could eventually come up
with the answer by themselves.

{\em Reminder:} At the end of Poison, show them how they have 
used problem-solving heuristics and have stepped through
the Paradigm for abstraction and generalization.  The problem-solving
was to try a few examples (playing the game a few times) and observe
that whoever was left with 4 counters would lose.  Getting to the
next step usually required making this observation explicit and precise.
This gave them a template for the next observation, which was that
the player left with 7 counters also loses.  From here, they could 
conjecture and verify the pattern: $4, 7, 10, \ldots$.  Having 
solved 10-counter poison, they could attempt to conjecture and verify
patterns for larger games.

The Paradigm steps were as follows.  (i) Solve 10-counter Poison.
(ii) Observe the nature of the solution: the pattern $4, 7, 10, \ldots$.
(iii) State this observation in words.  (iv) State this observation
mathematically: you lose if you must play when the number of counters
is $3n+1$ for some integer $n$; the winning strategy is to take~2 counters
if your opponent takes~1 and 1 if she takes~2.  (v) There are a variety 
of means of proving this.  (vi) One possible generalization is: if players
are allowed to pick from 1 to $k$ counters each time ($k=2$ in standard
Poison) then you lose if you are forced to move and the number of
counters is $(k+1) n + 1$ for an integer $n$; the correct strategy is
to take $k+1-x$ counters if your opponent takes $x$ counters.  

\subsubsection{Time to weigh the hippos}

[Time: 2 hours]

This starts out looking like a
hard but otherwise ordinary algebra problem (here's your first shot at
identifying their abilities with algebra) but ends up confusing almost
everybody, because the problem has two solutions.  Getting the groups to
discover this fact, and to understand what it means in the context of the
problem, is tricky but very important.

{\em Solution:}\/  The first part of solving the problem is identifying
variables, and this takes more work than you would expect.  For one thing,
students tend to assign variables randomly to hippos without thinking about
what they mean (in the end, we really want to assign variables to ``the weight
of the lightest hippo'', ``the weight of the second lightest hippo'', etc.).
And even when the subject of assigning such meaning to the variables came up,
several students were not sure if this was allowable.  Also, the students
usually fail to distinguish between the hippo itself and the hippo's weight.

The second, even more problematic, aspect of the solution is writing down
equations.  It is easy enough to assign random pairs of letters to each of the
five given numbers, and unfortunately one of the simplest ways of doing so
results in one of the two solutions.  In fact, you have to be careful assigning
summed pairs of variables to pair-masses, because doing so makes implicit
claims about specific hippos.  If Hippo $X$'s weight appears in the equation
with 312 kg and also the one whose right-hand side is unknown, for instance, we
have a problem, because this says that Hippo $X$ is simultaneously one of the
two lightest hippos and one of the two heaviest hippos, which is impossible.
It may take a surprisingly long time for the students to realize the trouble
inherent in this method.

If we choose $A$ to represent the mass of the lightest hippo (in kg), $B$ the
second lightest hippo's mass, $C$ the second heaviest hippo's mass and $D$ the
heaviest hippo's mass (try to let the students come up with variable names),
then we know that $A+B=312$, $A+C=356$, $B+D=466$, and $C+D$ is the unknown
mass.  Since we don't know which of $A+D$ and $B+C$ is heavier, we don't know
which of the two is 378 kg, and which is 444 kg.  Thus we should try each of
the two possible assignments, and see the result of each assumption.  As it
happens, both lead to meaningful solutions: If $A+D=378$, then (after some
algebra) we find $A=112$, $B=200$, $C=244$, $D=266$, and $C+D=510$.  This is
the solution groups are most likely to find.  On the other hand, if we let
$A+D=444$, then we find $A=145$, $B=167$, $C=211$, $D=299$, and $C+D=510$.
In the context of the problem, this is bad, because we cannot know what the
individual hippo masses are, although we do know that the last pair mass is 510
kg regardless.

{\em Prompts:}\/  If they have trouble just getting started, ask them what
they're interested in finding out, and tell them to identify it mathematically.
(Even so, at this point you may have to push them to make them think of
variables.)  Tell them to try to quantify the information given to them in the
problem.  If a group gets as far as the equations, you should probably make
sure that they understand what they're doing, although of course it's your call
as to whether or not to make them go back and re-examine their assumptions in
claiming that ``these are the equations'' (they may only get confused, shaky as
their progress is already).  If they get stuck doing the algebra, you can
either try to remind them of a particular technique (like substitution) for
solving systems of linear equations, or tell them to sit tight with that (if
the large group discussion is coming up) and think instead about whether
there's any other way to write the equations.  Finally, you're likely to have a
``good'' group or two which struggles through, finds one solution, and contents
itself with it.  If there's sufficient time remaining before the large group
discussion, ask them to go back and write down the assumptions they made early
on.

{\em Large-group discussion:}\/ At least one group should have a solution by
the time you go into large group.  With luck, each of the solutions will have
been found by at least one group.  If this is so, have the two solutions put up
on the board, and then ask the class to critique them, or say why one is right
and one is wrong.  Once they see that the algebra is right in both cases,
someone may decide that there are an infinite number of solutions.  Even if
this doesn't happen, eventually you have to cement the notion of how to assign
variable pairs to the pair masses.  Preferably, of course, the students do the
cementing, and as seen above, the crux is the assignment of the intermediate
pair masses to variable pairs (following an ordering assumption along the lines
of $A<B<C<D$).

The question of whether random assignment of letter pairs to pair masses always
generates a real solution will come up, and it will take a while to separate
the difference between a real permutation of pair assignments and one which
effectively does a global variable replacement:

\bigskip

\begin{tabular}{ccc}
Original & Equivalent & Fundamentally \\
assignment & assignment & different \\
& & \\
$A+B=312$ & $D+C=312$ & $A+B=312$ \\
$A+C=356$ & $D+B=356$ & $A+D=356$ \\
$A+D=378$ & $D+A=378$ & $A+C=378$ \\
$B+C=444$ & $C+B=444$ & $B+C=444$ \\
$B+D=466$ & $C+A=466$ & $B+D=466$ \\
$C+D=???$ & $B+A=???$ & $C+D=???$ \\
\end{tabular}

\bigskip

Once everybody agrees that there are exactly two solutions (and why), be sure
to spend time discussing what this means in the context of the problem.  Is it
good or bad to have two solutions?  How could you avoid having more than one?
In what other context might it be good?  If you carefully marked down the 
sequence of pairs of hippos and their weights, can you eliminate one of
the solutions?

\subsubsection{Dots and patterns}

[Recommend: out of class assignment]

This is taken from the old 105 lab book.  It is easy enough, 
at least at the begining, to serve as their first out-of-class 
assignment.  Students will answer question~1~(a) by various
methods, so you may want to take the opportunity to discuss
shortcuts other than the explicit formula.  In fact the explicit
formula of $n (n+1) / 2$ is hard to find if you haven't seen it before.
Question~1~(b) is a classic example of when step~4 of the paradigm
can be hard: just because a student has been able to state an 
observation doesn't mean that student has the facility to state 
it algebraically.  The same goes for the question after problem~2,
relating it to problem~1.  By the way, some students will make
a mistake and overlook the fact that $n$ on problem~1 corresonds to
$n+1$ on problem~2.  The best way to point this out is to try
plugging in a small number for $n$ in the formula from~1~(b),
then counting subsets to check that this correctly answer the 
number of 2-element subsets of a set of size~$n$, and when it doesn't,
trying to track down the error and correct it.

\subsubsection{Glicks, Glucks and Dr. Seuss}

The original version of this worksheet had Glacks and Glocks as well.
In all, there were four logically different kinds of statement, and students 
had some difficulties (most, but not all, of which are ironed out on
their own) distinguishing between them.  On the present worksheet, the
forms of the statements are ``if Glick and A, then B'' and 
``if A or B, then Gluck''.  Some of the questions will engender no 
discussion, but at least some should spark conversation.

In the next version of the coursepack and guide, the Glacks are 
going to be put back in.  They may be found in the ``Outtakes''
section of this guide.  The reason for re-including them is that 
the statement about Glacks is a defining statement, so is different
from the others which are simply statments known to be true.  It is
instructive for students to stumble on this and to debate the 
interpretation of the Glacks statement.  

{\em Solutions:}\/  1. (a) true (b) can't tell (c) can't tell (d) can't tell
(e) can't tell (f) true  2.  (a) can't tell (b) true (c) can't tell (d) false
(e) can't tell (f) false  3. (a) true (b) can't tell (c) can't
tell (d) can't tell (e) can't tell.


\subsubsection{Gotta hand it to you}

This one might fit in under an hour.  When groups get started, 
go around and check that they understand the statement.  Ask them
how many handshakes took place before the judge arrived if only
4 people were on the float.  

Many will want to use the formula they learned in Dots and Patterns.
They will find that ignoring the unique role of the judge and
solving for a non-integral number of people on the float to give 
1625 handshakes doesn't work.  They will be stuck solving or trying
to solve this quadratic for hours if you let them!  Nudge them
by getting them to interpret the computations they are doing.
E.g., ``Does the number $57.51\ldots$ I see on your paper represent
the number of people on the float?''  

When they find an answer, make sure they realize that they must
find all answers.  As it happens, the problem is neatly set up
so that for any integer number of handshakes there is precisely
one solution.  It would be nice to bring the discussion to this
point.  This may or may not work, but certainly you can get somewhere
by insisting that they have found a solution other than the
one you had in mind.  This is plausible to many of them, especially
to those who found it unnatural to introduce integer constraints 
when they had a formula that looked as if it worked for all real numbers.
They will be wondering whether finding a nearby integer and then letting
the leftover handshakes go to the judge can always be done in exactly
one way.  They should find it possible to convince you that for these
particular numbers there is a unique solution.  See if you can push
them at least to formulate the statement that for any positive integer 
number of handshakes in place of 1625, there is a solution and only one
solution.

How far you get through the paradigm may vary on this one.  If you make
it to the generalization stage, it is worth pointing out explicitly
what they have done: taken the uniqueness statement, with its justification,
and generalized this to any number of handshakes. 

\subsubsection{Banned book survey}

[Time: 2 hours]

This problem, is most easily solved via Venn diagrams and sets.  Placed
as it is, before the readings on sets, it serves to introduce the
utility of Venn diagrams and iron out some misconceptions as to their
conventional meaning\footnote{At this point, we find it philosphically
better to allow nonconventional use as long as they are aware of the
convention; later, in ``Describign sets of number (Section~2.4), they 
will need to start adhering to the convention.}.  
In particular, the idea that a number drawn where
it is inside just one circle should count how many objects have that
property but no other (rather than counting the total number of objects
with the given property) is usually forgotten by students who might
intuitively know it if asked.  Students who agree on a meaning for their
diagrams, conventional or not, will usually get pretty far.  So the
first thing the instructor needs to do is get the students to be clear
on what their diagrams mean, without tipping off that there may be a
problem.  Ways to do this include asking one group member what their
groupmate meant by writing a particular number in a particular location.

Since sets and Venn diagrams have not formally been studied, some students 
will probably use algebra with no diagram.  A similar principle applies:
those who are clear what their variables represent will have won half
the battle.  

{\em Solution:}\/ The most straightforward solution is to draw a Venn diagram
with three intersecting circles, each one representing one of the books.  If we
label each of the eight regions so created, we can solve the resulting linear
equations without (one hopes) too much trouble.  If we let the number of {\em
Huckleberry Finn}\/ readers be $A+B+E+F$, the number of Bobbsey Twins readers
be $D+E+F+G$, and the number of {\em Lady Chatterly's Lover}\/ readers be
$B+C+F+G$, then the information given in the problem can be written
$A+B+E+F=.53$, $D+E+F+G=.42$, $B+C+F+G=.36$, $E+F=.25$, $F+G=.14$, $B+F=.10$,
and finally $H=.12$, so that (using $A+B+C+D+E+F+G+H=1$) $A+B+C+D+E+F+G=.88$.
The solution (in percentages) is $A=.24$, $B=.04$, $C=.18$, $D=.09$, $E=.19$,
$F=.06$, $G=.08$.  Thus 6\% (F) of the people surveyed had read all three
books.  In general, with linear equations we need 8 equations to solve 8
unknowns, and that is what we have here.  Thus the last piece of information
was necessary.

{\em Further notes:}
This worksheet also serves to introduce inclusion-exclusion, though it
is really a more complicated instance than would normally be used
to introduce this topic.  There is a reading which is assigned by
the start of this worksheet, namely "Counting revisited''.  It introduces
inclusion-exclusion (limited to 2-deep nesting).  After discussing that
Reading, students should know that you can count $A \cup B$ by adding
$|A| + |B|$ and then subtracting $|A \cap B|$ for the overcount.
(In fact they will have seen that $|A \cup B \cup C| = 
|A| + |B| + |C| - |A \cap B \cap C|$ when all the pairwise intersections
are the same, and will interpret that as subtracting off two of
the three counts for the triplecounted items.)

Thus another solution, which some might try, is to add $|A| + |B| + |C|$,
then subtract off $|A \cap B|$, $|B \cap C|$ and $|A \cap C|$, which
correctly counts everything except the items in $A \cap B \cap C$.  When
they see that these have been counted with weight exactly $3 - 3 = 0$,
they can add this number back in.  Warning: in the past some groups have
come close to this solution by voodoo.  That is, they add $A| + |B| + |C|$
and describe this incorrectly as ``total number who have read one book''.
They then subtract $|A \cap B| + |B \cap C| + |A \cap C|$ for no particular
reason (perhpas they see this will put htem in the right ballpark), and 
start playing with numbers to see how close they are.  It's hard to do
anything useful with that, which is why we have sometimes introduce the
alternative inclusion-exclusion-based solution: it demonstrates that
adding $|A| + |B| + |C|$ and then subtracting off the pairwise
intersections really does count something, and that you just have to
think it through to see what. 

\subsubsection{Chickens, rabbits, temperature, driving, ages and headaches!}

Everyone know that the hardest of all things to teach (from grade school
through differential equations) is word problems.  In fact the focus
of the algebra in this course is entirely away from algebra skills 
and onto word problem skills: algebra as a means of describing and
analyzing a problem (and to some extent, algebra as a means of
stating generalizations).  The problems on this sheet progress from
straightforward to very difficult.  It is a good idea to try and elicit
at least two solutions to each, one more {\em a priori} algebraic, and one 
based more on trial and error.  The trial and error one must
be presented first or no one will take it seriously, so you need to
snoop on the groups and make sure the first one you call on in the 
whole-class discussion uses the most elementary method.

When you are going around to get groups unstuck, keep in mind the
variety of paths to a solution.  Rather than prejudging what will
work for them, try to see if there is anywhere that their present
dead-end can usefully lead.  For students having trouble getting
anywhere, help them try trial and error, and then help them
``algebreify'' it (see the notes on parts~2 and~3 of the
``Word problem review'' worksheet).  

1.  The equations $C + R = 50$ and $2C + 4R = 120$ are pretty easiy to
find and to solve, yielding $C = 40, R = 10$.  If a group has trouble
formulating or solving the equations, encourage them to solve the problem
by trial and error: try a number for $C$, see what one equation says for
$R$, see if that works in the other equation; if not adjust the number
and see if the adjustment helped; pick new numbers, attempting to 
move in the right direction until you hit a solution, and to make
your guesses better educated as you go.  The point of this is to give
ownership of the problem to students who need it most.  This is not an
algebra skills class, so you don't need to get them through the annointed
method.  Rather, they need to learn that they can get through it themselves
if they keep their head clear.  Later, in the whole-class discussion, they
will see the algebra, and will understand it much better for having
several times plugged in the numbers that the algebra represents.

2.  You can give them the conversion formula if you wish, or you can
tell them that $0^\circ$C = $32^\circ$F and $100^\circ$C = $212^\circ$F 
and let them work it out.  The first difficulty for them will be in 
understanding that two variables are necessary, namely the temperature
measurement in degrees Celsius and the measurement in degrees Fahrenheit.
Encourage them to use something other than $F$ and $C$ for the variable 
names, since the tendency is already for students to confuse the object
with its measurement (e.g., temperature, which is the same no matter 
how it is measured, with number of degrees).  

A second difficulty is seeing the two rather different equations: the
equation $X = 2Y$ comes from the problem, while the equation 
$X = 32 + (9/5) Y$ is the conversion formula.  In addition, some students
may get the formula $X = 2Y$ reversed.  This brings to mind the problem
of expressing the relation ``There are 12 students for every professor''
as an equation.  The problem is famous, having been used in research
studies.  The wrong answer of $12 S = P$ is very common.  This is probably
because a professor is (informally) equated with 12 students.  So if you
think of $S$ as ``students'' rather than ``the numer of students'', you will
get the equation backwards.  If you see many students reversing equations
on this worksheet, you might want to take a break and have a whole-class
discussion where you ask them quickly to write down (each on their own)
an anser to the student-professor problem, then compare notes.  Another 
reason to bring up this problem is if you need more motivation for them
to be pedantic with their interpretations of variables.  Distinguishing
between ``students'' and ``number of students'' may seem silly to them 
if they have not tripped up on a problem of this sort.

3.   The main difficulties again lie in assignment of variables and
in finding hidden equations.  They can see that in addition to the speeds,
the distance is unknown.  The clearest way to solve this is with
three variables: slow speed, fast speed and distance.  They then need
to relate time, distance and speed in the standard way to get
$d = 5 f, d = 6s, f = s+13$, which has the solution 
$s = 65, f=78, d=390$.  Some students may use only two variables, 
$s$ and $f$, in which, in addition to the equation $f = s + 13$, 
they will need to observe that $65 s = 78 f$ because the distance
is the same no matter how fast you drive.  It can of course be done
as well with only one variable.  This type of problem is difficult
enough that we added the ``Word problem review'' worksheet to reinforce 
what they have learned (originally called ``Word problems out the wazoo'').  

4.  This one is really hard!  Most people see it as having four relevant
variables: Donna's and Janlen's ages now ($D$ and $J$), and Donna's and
Jalen's ages then ($d$ and $j$) whenever that was! It takes a very 
clear head to see that the information may be encoded as
\begin{eqnarray*}
D & = & 2 j \\
d & = & J + 30 \\
D-d & = & J-j
\end{eqnarray*}
To get the first two equations, one must let go of wanting to know 
how long ago was the past being referenced, and simply state the 
relations between one present age and the other past age.  For the third, 
one must recognize that the past begin referred to is equally long ago 
for both people (this specifically gets reinforced in the first problem 
of ``Word problem review'').  And to top it off, the system is 
underdetermined!  To be honest, I have never gotten all the way through 
this problem.  Usually the farthest I get is to have the class be
able to tell, if I tell them two possible ages now, whether these
are consistent with the statement of the problem.  

5.  Sometimes we've skipped this for time reasons, and sometimes not.
They don't appear to learn all that much from it, but on the other hand,
we have been stepping up the emphasis on their ability to design problems
(see the worksheets ``Why? part II'' and ``The sense of it'').  To be
good teachers, they will need to be able to invent problems for specific
purposes.  This is an easy place to start, though not as instructive 
as inventing a problem to illustrate a specific point.

\subsubsection{Comparing without counting}

They have already read ``Counting revisited'', so they are quite
well prepared and this worksheet should go quickly.  That Reading
purposely refrained from discussing correspondence as a means
for determining the direction of an inequality, in order that
problem~1 on the present worksheet not be trivial.  So the main
point of problem~1 here is to point out that you can test
$|A| < |B|$ by matching all elements of $A$ one-to-one with
a proper subset of $B$.  This is a case, where if the students
do not quickly come up with the answer, there is probably not much
use lingering.  Tell them the observation yourself and go onto
the next problem.

Problem~2 is a good way of reinforcing some of what they have
learned by reading ``Intorduction to sets''.  You will now get
to see whether they know what a subset is.  They have studied
two-element subsets at length in ``Dots and patterns'', which
has in the past led some to believe that all subsets have 
two elements.  Even though they
have seen set notation used, they may have trouble writing down
a new set.  They may also need explicitly to discuss the extension
property: a set written in any order is the same set, and there
is no need to write one twice, since an object is either in the
set or it's not.  The main point of the problem is to see that
there are one-to-one correspondences between the subsets of
a set $S$ and each of two halves of the subsets of a set $S \cup \{ x \}$.
They then pass to a recursive understanding that adding an element
doubles the number of subsets, and from there to there being $2^n$ 
subsets of an $n$ element set.  This last observation is not an
answer to any of the stated questions, but the whose-class discussion
there as a generalization and crystallization of the foregoing discussion.

This gives a nice example of the paradigm, which you point out
to them explicitly once the whole-class discussion is complete.
Steps they ought to pass through in this last part
of the discussion are: finding the number of subsets of a $n$-element 
set for $n = 1, 2, 3, 4$; observing the pattern recursively as doubling;
stating this in English; stating this in mathematical language, leading
to the statement of an explicit formula $2^n$ for the $n^{th}$ size;
justification of the doubing observation, which was done when arguing
problem~2~(d), and of the explicit formula -- here they may not
feel it necessary to justify that a repeatedly doubled sequence 
yields $2^n$, but you want at least to point out that an initial value
is needed for this to follow -- also, try checking the degenerate
case $n=0$; lastly, is it possible to generalize? here they might
want to u8se what they know about Cartesian products to write 
a formula for the number of sequences of length $n$ of elements from
a $k$-element set, which generalizes counting subsets when you view
a subset as a sequence of elements from ${\rm \{ yes , no \} }$ indexed
by elements of the original set.  

\subsubsection{Word problem review}

[Recommend: out of class assignment]

A solution is given in the on-line materials, so the notes here are
restricted to pedagogical points.  The way we have used this worksheet
in the past is as out-of-class homework.  They are supposed to have learned
how to do this kind of problem on their own, but typically need a lot
of practice.  The worksheet is intentionally composed of two sets of
similar problems (1 and 4 are similar and 2 and 3 are similar).  If you 
have some students who are particularly bad at this, you can meet with them
individually, walk them through 1 and 2, and have them solve 3 and 4 in
front of you, with the knowledge that 3 is like 2 and 4 is like 1.  

Problems 1 and 4 are probably more straightforward and emphasize 
variable naming.  In particular, they require that a quantity that
changes not be named the same thing at two different values.
On problem~1, if you choose the four-variable approach, the equations 
pretty much find themselves.  If you choose the two-variable approach,
then you must be clear that only present ages are named as variables,
and that past ages are referenced by present ages minus 1~year.  
Problem~4 is similar, with a now-and-then theme and a known relation.
It is harder because the words ``2~to~1 majority'' and ``1~seat'' majority
are unfamiliar to some and refer to different types of discrepancy 
(additive versus multiplicative).  

Problems~2 and~3 emphasize how to assign variables and find equations
in more difficult problems when you are having trouble getting started.
The idea is to make guess and check methodical enough that it can
lead reliably to algebra.  The student must, as she goes through this,
learn on her own how to check a particular numerical guess and see
what other numers it leads to and whether they are consistent.  Having
done this one or more times, the student then tries to replace the
guess with a variable.  Where before it was hard to say what the
other quantities were in terms of this variable, now the student can
refer to her own explicit computations to see how she got the next number.
This leads eventually to more and more complicated expressions, one of which
may finally be set equal to something, and an equation is derived!  This
method produces equations with reliatively few variables, since each time
all expressions are computed in terms of the input data.  While more
variables are conceptually easier in a purely algebraic approach, this
approach may be the only hope for students who can't seem to
get to square~1.

Problem~2 and~3 relate best to the Paradigm for abstraction 
and generalization.  They begin with trial and error, observe
how one input leads to determination of other quantities, they
put into words how they made this inference, and they write it as 
an equation.  The justification and further generalization steps
are not present.


\pb
\subsection{Whole numbers and their properties}

This section has two distinct curricula.  The first covers primes, factors,
and the GCD and LCM.  Our aim here is to give teachers familiarity with
basic multiplicative properties of whole numbers.  In this area, more
than in any other, it will be important to have whole-class discussions
that crystallize what the students have discovered on the worksheets. 
Major concepts such as unique prime factorization and the significance
of the GCD are never addressed in the readings nor explicitly stated
in the worksheets.  Rather, the worksheets run the students through 
activities meant to prepare them for a deeper understanding of these
ideas, and this preparation will be wasted if the discussions are not
brought to fruition.  The weekly summaries may be used to emphasize 
for which of the ideas the students will be held accountable; one might
even include statements of important theorems in the weekly summary or an
online handout (but not before the relevant activities are completed!).

The second curriculum in this section concerns properties of the whole
number system and operations in the context of abstract properties
of operations on sets.  The dangers of too much abstration with too little
motivation have been well recognized since the failure of the New Math
in the 1960's.  Nevertheless, to be effective teachers, the Math 105 
students need to gain a little sophistication about the objects they
are dealing with.  It is important, for instance, to understand
commutativity, since learning that addition and multiplication
are commutative, and learning why, are recognizable cognitive steps
in the education of young children.  Learning the names of these
formal properties helps prospective teachers to better recognize,
categorize and understand these properties.  More importantly, teachers
need to go through the exercise of imagining ``what if not'' with these
properties.  It is difficult to teach a child to see that $7 + 4 = 4 + 7$
without ever having tried to see why one might think it not to be true.

\subsubsection{All in the timing}

This worksheet will take 1.5 to 2 days, depending on whether you 
cut any of it (many instructors don't work all the way through \# 5)
or assign any of it for homework.  It will probably become evident that
all or most of the students are familiar with the idea of making a factor
tree.  They are a little less clear on exactly what a factor tree 
accomplishes and to what degree factor trees are unique.  It is best
to have this discussion a little later, when they have all tried 
problems~1--3, since then the discussion can take full aim at the 
meaning of unique prime factorization.  

Set aside the first few minutes to discuss questions from the reading,
because there will surely be some.  In particular, remind them that
they will be responsible henceforth for the vocabulary (prime, composite,
relatively prime, GCD, LCM) as well as the divisor notation.  

Problem~1 starts them off with something they can get their hands on.
Many will use their calculators to see which are divisible by~7, but
they will still need to work out factorizations.  They are meant by
the end to have decided for themselves that divisibility by~7 entails
finding a~7 in the prime factorization but divisibility by 24 is not
the same as finding a~24, or even a~4 and a~6.  It may be touch and go 
whether at this point you can get to divisibility by~24 being the same
as finding three~2's and a~3 in the prime factorization.

As pointed out in the discussion below of problem~2, it is important
to have several passes at this discussion.  First, the students must
have ample time to explore.  A brief whole-class discussion may be needed
in order to get everyone on the same wavelength as to what a factor
tree is.  This or a subsequent discussion can get them thinking about
the degree to which these are unique.  They tend to understand that 
writing things in a different order is not an essential difference,
so you can tease them by pretending to believe that $49 \times 2$ is
a very different first step from $2 \times 49$ and will lead to a radically
different factor tree; they will then claim that the trees won't be
too different -- you can just swap the two halves.  Then you can 
pretend to be convinced that the factor trees are always the same.
If they don't see $14 times 7$ as leading to a distinct factor tree,
try one of the other numbers such as 144, where the factor trees can
have very different shapes.  Once they have seen that shape can vary,
it is more meaningful for them to discover what is always constant, 
namely the multiset of prime factors at the leaves of the tree.  
We are aiming here at a deeper understanding of the Fundamental 
Theorem of Arithmetic (unique prime factorization) by giving them
at least a fleeting though that it is not true.  

Many students will attack problem~2 with a calculator.  Most calculators
will show $7^{29}$ and $13^{22}$ in scientific notation and will have
them differing in the third digit after the decimal point.  This leads
them to believe the numbers are different.  If this belief is too firm,
you might need to shake it a little by having one student use a 
calculator to compute the decimal expansion of, say, $2^{1/1234}$, 
and then having another student key in (so as not to take advantage
of digits stroed but not displayed) this number to the power 1234.
Explain (if necessary) that this should be 2; the result should
be off after about five decimal places on any caluclator with seven or
eight digits of accuracy.  The idea of course is that, whether or not
a calculator can distinguish these, there is a quick and easy way
to say ``yes, these must be the same number'' or ``no, these numbers
must be different''.  This discussion will be the first and simplest
test of whether they have understood unique prime factorization, so
the time sequence is important.  They must try problems~1 and 2, then
discuss them both in that order.  

Problem~3 introduces the idea that the {\em number} of factors of a
number depends only on the form of its prime factorization -- all the 
powers of the primes, but not which primes they are.  There are 
many possible followup questions, or intermediate questions one might
prepare in case they are slow here.  Examples would be (1) to give them
two factor trees, one being of $p^2 \cdot q$ for some new
primes $p$ and $q$, and one being of $2^5 \times 3$, and asking 
them which will also have exactly 6 factors (intermediate question),
or (2) to ask them to come up with another number that has exactly~6 
factors (followup question -- having them say how they came up with
their answer should be interesting!).  

Problem~4 is really an exercise in understanding the question -- the
verbal quantifier here is quite difficult for them.  Unless you're
ahead of schedule, either skip this, or get through the semantic
portion of it, but there is no real need to spend enough time to
actually arrive at an answer.  Questions such as these are good to 
throw at faster students who have completed the remaining problems
on the worksheet.  Problem~5 is similar.

Problem~6 introduces a physical significance to the LCM.  The second
half of the problem is eminently skippable, serving only to highlight
the difference between two possible questions one might ask.  If left
to their own devices, students are unlikely to mention LCM explicitly
in their solution to problem~6, so you might want to lead the discussion
there.  

In problem~7, again it is the first question that is important and 
highlights the significance of the LCM.  By the end of this problem,
you need to have tackled the connection to the LCM explicitly, since
they need fair warning that the LCM is important and that they are expected
to understand its physical significance.  The LCM appears as an object
of more theoretical study in the next worksheet.

\subsubsection{Counting factors}

This worksheet explores in greater detail what was explored by problem~3
of the previous worksheet: how the number of factors is related to the
form of the prime factorization and several special cases thereof.

Problem~1 shows that primes are characterized by having exactly
two factors.

Problem~2 shows that having a number of factors which is a power of~2 is 
equivalent to having distinct prime factors.  In the past students have
caught onto this pretty well, but have struggled to put it into words;
take several passes at verbalization if necessary.

Problem~3 sometimes stalls them for a while.  First they can't think
of any, then they incorrectly generalize from 4 and 9 to all squares.
You must let them discover their own error if they do this, but then
nudge them so they don't give up altogetherwhen they are so close.

Problem~4 takes the previous worksheet's problem~3 one step farther
on the Paradigm -- they must put their verbal understanding into 
more succinct notation.  

Problem~5 is impossible unless they have begun to have a 
glimmer that if you have a number with $a$ factors and one with
$b$ factors, then sometimes (i.e., if the proper factors are distinct) 
you can mutliply the numbers and get a number with $a \cdot b$ factors.
You can lead them a little if necessary by asking them to list the
factors of a product of two numbers, each of whose factor list
is written on the board (and conveniently distinct); if they still
don't see it, try it with the number 10 and 43 and ask them to list
it in an order that will best convince you that you've gotten them all.
If necessary, prompt them about what they know about Cartesian products
(the ``Counting revisited'' reading and ``Comparing without counting'' 
worksheet).

Problem~6 extends problem~5 one or two more steps along the Paradigm.
Point this out to them explicitly!  

Problem~7 reinforces the limitation on the Cartesian product method.

\subsubsection{Stamps}

[Time: 2 hours]

This problem is, in the present version of the course, the only
introduction the students will have to the significance of the GCD.
For that reason, we felt it was worth including, even though past
experience has shown progress on this problem can be frustratingly slow.

It revolves around relative primeness, although it will be some 
time before the students get any notion of this.  The 4-cent and 
9-cent problem eventually (beyond a certain point) allows one to
make any amount of postage, while the 9-cent and 21-cent problem does not, for
the simple reason that 9 and 21 are both multiples of 3, so only multiples of 3
can be made.  It usually takes the students at least an hour to get even 
this far.

The crux of the problem in the beginning appears to be the way one keeps track
of the amounts ``makable'' and ``unmakable''; more than any other problem in
this set of materials, the ease of solution depends directly on the method of
representation chosen.  Making two lists, one of each type of amount, gives
little insight into the underlying structure.  The optimal representation
appears to be a grid-style list of all numbers (starting with 0 or 1), with
each row of length one of the two stamp denominations (e.g., 9).  One can then
circle each of the ``makable'' amounts, and see that once a number in a given
column is circled, all the numbers [physically] below that number in the same
column are also circled, as they are multiples of the row length added to a
``makable'' number.  An intermediate strategy is to make several lists,
organized by one of the denominations, e.g., ``multiples of 9'', ``4 plus
multiples of 9'', ``8 plus multiples of 9'', etc.  (This latter can also
manifest as a grid, which seems to be more helpful for the students.)  There
are, of course, many possible strategies, but students may choose one which
gives them little insight for the problem and eventually get stuck.  In this
case it is your decision whether or not to give them a push.  One instructor
suggested a grid representation but let the students decide the width; another
gave no suggestion, with the result that discussion took an extra hour before
students came to an agreement.

Try to keep the students from getting discouraged, and if they tell you they
cannot make any generalizations (as asked in the problem) after solving the
first two questions, tell them to (a) look for the differences between the two
pairs of numbers, or (b) make up some more pairs of small numbers and try
answering the same question with them.  (Ideally these ideas would occur to
them naturally, but their problem solving skills are not likely to be honed
this sharp yet.)  In any case, realize that you will not get as far as a
complete solution with explanation in class, and take what you can get (don't
spend more than a week on this, interesting though it is).  You want to get to
the point where students recognize that the important thing is the relative
primeness of the denominations involved, and why.

{\em Solution:}\/  In complete generality, the answer is that given stamps of
denomination $a$ and $b$, one can make amounts which are multiples of
GCF($a$,$b$), and the last such multiple which cannot be made is
LCM($a$,$b$)$-a-b$.  Thus if $a$ and $b$ are relatively prime, one can make
every amount greater than $ab-a-b$.  (Note: Every part of the preceding
solution was originally discovered by a student!)  Less generally,
given denominations 4 and 9, one can make 4, 8, 9, 12, 13, 16, 17, 18, 20, 21,
22, and everything beyond (but not including) 23.  Given denominations 9 and
21, one can make all multiples of 3 except 3, 6, 12, 15, 24 and 33.  It should
seem completely reasonable (though the students will probably not have thought
of it in these terms) that since 9 and 21 are multiples of 3, you can only hope
to make multiples of 3 using them.

As short a proof as I can find of why $ab-a-b$ is the last ``unmakable'' number
given $a$ and $b$ relatively prime:  Make a table of numbers in rows of $b$
entries each:

\bigskip

\begin{tabular}{rrcrr}
     1 &     2  & ... &  $b-1$ &  $b$ \\
 $b+1$ &  $b+2$ & ... & $2b-1$ & $2b$ \\
$2b+1$ & $2b+2$ & ... & $3b-1$ & $3b$ \\
   ... &    ... & ... & ...    &  ... \\
$(a-1)b+1$ & $(a-1)b+2$ & ... & $ab-1$ & $ab$ \\
\end{tabular}

\bigskip

Note that numbers in the same column differ by a multiple of $b$.  Thus, the
first (smallest, or highest-up in the table) multiple of $a$ in each column
will be the first "makable" amount in that column (by using, say, $m$
$a$-stamps).  Every number in the column below $ma$ can be made by using $m$
$a$-stamps and the same number of $b$-stamps as there are rows separating $ma$
and the number in question (e.g., the number three rows below $ma$ is $ma+3b$
and can be made with $m$ $a$-stamps and 3 $b$-stamps).

Thus the question of being able to make all numbers is equivalent to finding
a multiple of $a$ in each of the $b$ columns (except for the last one,
if you like, since they're all multiples of $b$ in that column).
So we need at least the first $b$ multiples of $a$ to hit all the columns
(i.e., one multiple of $a$ for each of the $b$ columns).
Suppose the first $b$ multiples of $a$ {\em aren't} all in different columns,
i.e., suppose there's one column with two multiples of $a$ in it.
Since the numbers are both multiples of $a$, their difference is a multiple of
$a$.  Since the numbers are in the same column, their difference is a multiple
of $b$.  Since $a$ and $b$ are relatively prime, their difference must be a
multiple of $ab$.  Thus the greater number (lower down in the column) is at
least $b$ multiples of $a$ after the smaller number, and hence is not in the
first $b$ multiples of $a$.
So the first $b$ multiples of $a$ do it, and eventually we can make anything.

Note also that the $b$th multiple of $a$ is, as previously mentioned, not
especially important because it hits the $b$th column, which were all multiples
of $b$ anyway.  So the last column to be hit is the one hit by the $(b-1)$th
multiple of $a$, $(b-1)a$.  Thus the last number that cannot be made is the one
directly above $(b-1)a$: $(b-1)a - b$, or $ab-a-b$.

There are several other similar ``tabular'' ways to look at this problem
which you will undoubtedly encounter from your students.  In this activity
more than most, it's vital to keep your mind open to solutions which never
would have occured to you (but still have some validity).

There are several levels of understanding to this problem:  Level 1.  With 4
and 9 you can eventually make everything; with 9 and 21 you can't.  Level 2.
The difference is due to 4 and 9 being relatively prime; with 9 and 21 you can
only make multiples of 3.  Level 3.  In general you can eventually make every
number precisely when the two numbers are relatively prime.  Level 4.  In
general you can make every number beyond $ab-a-b$ if $a$ and $b$ are relatively
prime; otherwise you can eventually make multiples of GCF($a$,$b$).  Level 5.
Proving $ab-a-b$ is the last number one cannot make; or knowing the generalized
formula (above).  Level 6.  Proving the generalized formula.

You should try to get the class discussion to Level 3.  Some individuals will
only get to Level 2; others will get as far as Level 4.  One or two may get to
Level 5.  I myself have not bothered getting to Level 6.

\subsubsection{Describing sets of numbers}

Warning: some of the pdf versions of the coursepack have unusable versions
of the figures for this one!

This worksheet uses their newly acquired number theoretic understanding
as fodder for a basic discussion of sets and Venn diagrams.  They have
had some experience with diagrams in Banned Books, but only now begin
a more formal study.  In the Banned Books worksheet you stressed
that any diagram-writing convention is OK as long as you are
clear and consistent about it.  Now they need to learn what the
conventional meaning of a Venn diagram is and to stop using it in
nonconventional ways.  As far as we know, each diagram is satisfied
by only one letter from~(a) to~(j).    

\subsubsection{The locker problem}

[Time: 1 to 1.5 hours]

Like the Stamps problem, this is a good problem for finding and
recognizing patterns.  It can come before the stamps problem if
desired.  The bottom line is that squares are the only numbers with an odd
number of factors; if students have recently solved Related Problem 1 at the
beginning of the unit, they may have made this observation and remember it (not
likely, but just a warning).  Here as with the stamps problem, there will be a
multitude of ways of keeping track of which lockers are open and which closed,
some of which will prove more useful than others.  Here, however, students are
usually able to find a representation that serves sufficiently well.

{\em Solution:}\/  The number of times the door of a given locker is moved is
equal to the number of factors of the locker number.  All numbers have pairs of
factors (and hence an even number of factors) and thus end up closed, except
for the ten squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which have a square
root which only counts once (it's paired with itself).  Students are likely to
start off modeling just the first twenty (e.g.) lockers and going through and
making the twenty first passes, and then trying to see the pattern.  Eventually
they will see that only squares end up open, but by the end of the large group
discussion they should be able to say why square-numbered lockers end up open
and the rest don't.

\subsubsection{Clock arithmetic}

The real content in this worksheet is probably in the Reading.  
The ``bow-tie'' operation will freak them out.  Go over this to 
make sure they become comfortable with the abstraction template here.
We recommend that you have them put their self-check answers on their 
desks at the start of class (if you don't always do this) and that you
cruise around to see how they did before starting the whole-class
discussion and question session on it.  Once you get to the actual
worksheet, it should be pretty quick.  

\subsubsection{Tarzan}

This is an exercise in suspension of belief.  In order to appreciate
their implicit understanding of the properties of addition, they
must consider in turn what might happen if each failed.  By doing 
this, they reinforce their understanding of the abstract properties
while having a few laughs.  The properties of addition to which the 
parts of question~1 refer are: (a) closure; (b) that~0 is the identity
element; (c) commutativity; (d) associativity.  

Problem~2 demonstrates a method for adding one-digit numbers with carry.
The explicit point of the problem is to practice justifying this in
terms of known formal properties of addition.  That's a good exercise, 
but the method analyzed is important as well.  This addition technique
is one commonly invented by young children on their own, and one which
is often not discussed or even explicitly recognized by teachers 
in the U.S.\ (compare to curricula of other countries, where this 
is taught to teachers as a standard cognitive step in attaining fluency
in mental addition).  It is crucial in the discussion to mention 
this explicitly, as this is the only place in the curriculum to 
address this important skill.  Such a discussion also paves the way
for later worksheets where the students must play detective and figure
out what a hypothetical child has done by looking at the scratchwork.

Problem~3 is a straightforward reinforcement of the definition of
closure.  Closure, being more obvious than commutativity for addition,
and being almost a foregone conclusion in many contexts, is harder 
to come to grips with than commutativity, which is why we devoted
a problem just to closure.  

\subsubsection{Negative numbers and the question ``Why?''}

There is more in this worksheet than you need to do.  The important part
is to get across to the students one or more models for negative numbers.
This worksheet is almost an extended reading, but since the reading
on models of operations was already quite extensive, we fashioned it
as a worksheet.  There is a mini-lecture on arithmetic and models that
you might want to give now, or perhaps later when they first tackle
place value and its relation to addition.  The content is as follows.
\begin{itemize}
\item Primitive math begins with no formal arithmetic 
(no numeration system, no place value, no concept of addition, etc.) 
but only actual problems to be solved.  I have 3 sheep, you have 8.  
How many will I have if I kill you and take all your sheep?  

\item Next comes the recognition that this problem comes out 
the same regardless of whether we are discussing sheep, apples 
or some other objects.  We invent numeration to stand for the 
numbers involved and operations to stand for the combining we are doing.

\item Next we come up with rules for doing this operation (for adding
numbers in our numeration system).  This rules are built on what we
know from observations, but are ultimately simply self-consistent rules.
The relation to future observations depends on our faith that the
connections we observed in the past will extend analogously in 
the future.  At this point we have moved to formal addition, along
with a physical model that we {\em believe} will always correspond to it.

\item Lastly, we start using the formal system to solve 
problems previously too hard to solve.  For example, we
might need to know in a hurry how many sheep we'd have
if we combined tribal holdings of 378 sheep and 491 sheep.
\end{itemize}

On this worksheet, the point is that ``why'' questions about 
negative numbers and other such entities must be answered in
terms of the physical systems they were meant to model.  One
can define formal systems of addition with negative numbers
any way one wants, but to be useful they must be informed by
outcomes of physical problems in which negative numbers arose.
Here, we discuss three possible modeling origins of negative
numbers, and see why each leads to the rule: {\bf subtracting a 
negative number is the same as adding the corresponding positive number}.
The second (charges) model is the most hands-on and most appropriate
for a sliimmed down version of the worksheet.  It comes from the
old Math 105 lab book and requires only minimal equipment.  But
the comparison as well to the vector model and to a formal model
(in which the main reason to invent negative numbers was so that
all numbers could have additive inverses) is instructive if you 
can afford the time.

\subsubsection{``Why?'', part II}

Except possibly on the ``Chickens'' worksheet, this is their first chance
to invent a story problem for a specific purpose.  We make it easy
on them by making it multiple choice.  Still, they find it tricky.
Story~2 is the easiest to rule out, since it is clearly an illustration
of how subtraction of a negative is like adding a positive.  Nevertheless,
when we have taught this in the past there have been some groups 
who chose it because it was simple and correct, even though it didn't
model the right thing.  Needless to say, they need the implicit reprimand
this worksheet provides!  

Ruling out story~1 is trickier.  It appears to
be a justification of why the product of two negatives is a positive.
Further examination shows that it really justifies that the product
of numbers with any signs is positive!  More precisely, it shows that
area is an unsigned unit, so is the product of unsigned lengths, so 
interpreting a product as area requires that you take absolute values
before (or after) multiplying.  This is a more subtle point, but still
crucial to anyone who will ever have to anwer children's questions
``Why?".  

The last choice is the correct one.  The only trick here is to see that
the problem can be modeled as multplication of two negatives or of
a negative and a positive, depending on whether you change over to 
upstream measurements before operating on the numbers.  The point to
be made here is that the latter is not incorrect: it is just that only
one of the correct models for this problem illustratees the point you
want to make.

\subsubsection{Stupid number tricks}

This worksheet could have been included a little later in the section 
on place value.  It is placed where it is for two reasons: (1) so that
the students are not tipped off as to the correct approach, and (2)
in order that the idea of place value be motivated and exemplified
before the in-depth study has begun.  

Students will struggle to understand this.  In fact, effective
understanding is almost impossible to attain without formal 
understanding.  Groups that represent this algebraically can usually
figure out why it works.  For groups that don't, I have suggested
doing the computation with several different numbers (each group
member does one or two) and tabulating the results so that each
computation, with all its intermediate results, occupies a column.
This helps them to see that at a given stage, they have a number depending
somehow algebraically on what they started with.  If you ask them to
use 10 and 100 as inputs, the algebraic dependence should be even clearer.

To explain their solution, they will need the fact that $10 x$ looks 
like the integer $x$ with a zero appended.  Allow them to assume this
without proof for now, but point it out as something we would like to
see a proof of shortly, when we study place value.

\subsubsection{Tarzan II}

This extension of Tarzan I goes very quickly.  The properties Tarzan
is lacking are: (1) distributivity of multiplication over addition; 
(2) commutativity of multiplication (viewing taxation and discounting
as multiplication by $1 + x/100$ and $1/2$ respectively); (3) reversal
of all signs when subtracting a quantity in parentheses, so that
if $\sum_{i=1}^n x_i = 100$ then $T - x_1 - \ldots - x_n = T - 100$.


\pb
\subsection{Place value}

We, the teachers of Math 105, all know that the use of other bases
is often necessary to give students the perspective they need to 
understand their own base.  This is the not the only reason to
explore other bases, but it is the only one that many of the 
students will accept.  With many populations of students, it is 
important to stress this at the beginning and throughout.  We have
also in the past stressed that once they have used other bases 
to understand base ten, the other based {\em per se} pretty much
vanish from the curriculum (but the will still be tested on them!).

Some instructors find it useful to be very consistent about using
spelled out words for numbers when using them in meta-numeration 
contexts, such as when specifiying a base for numeration.  We have
tried to make the worksheets consistent with this.  

\subsubsection{Throwing yourself off base}

The name of this worksheet is a pun meant to reinforce the reason
they are being made to study other bases.  

Problem~1 is a straightforward exercise using a multiple-base 
monetary system that everyone in England had to use not too 
many generations ago.  

Problem~2 is a counting problem.  It really has more to do with 
Cartesian products than bases.  Make sure everyone agrees on 
the definition of what constitutes a three-digit number.  

Problem~3 is an introduction to divisibility tests.  While the 
notion of divisibility has come up already, the means of testing
has not.  You will find that most students know how to test
divisibility by 2, 5 and 10, and some by 4, 3 and 9.
It is safe to forecast that no one will be able to justify any 
of these, and certainly not the~3 or~9 test.  At this point,
we are mainly looking for coherent and correct answers, not
for justification.  Ask for the justification, as always, but
be content if they leave it as ``believe, but unable to justify''.
Still, parts~(c) and~(d) ought to give enough data that they can
come up with something interesting for part~(e).

Problem~4 is a useful prelude to the rest of their formal study
of bases.  They should be told after they have done this that it
is important and that they will be expected to know how to
do this.  

\subsubsection{Arithmetic in other bases}

This is really an extended self-check for the Introduction to Bases.
In the past we have assigned it to be done in class onoy because at
home they tend to get stuck early on and therefore don't do any of it.
In class, you can ease them through a few problems untili they get
they idea.  By all means, don't feel compelled to spend more than 10 or
15 minutes on it.  If they do a few of these, they've probably gotten
all there is to get from the worksheet.  One thing to watch out for is 
whether they are comfortable with what digits are allowed in each base,
and particularly with what digits we choose (arbitrarily) to add 
to bases higher than ten.  

\subsubsection{Justify!}

Our lofty goal for Math 105 was originally to have them understand 
complete justifications for all four basic arithmetic algorithms 
in base ten and to have them write these using correct mathematical 
language and grammar.  We have pulled back substantially from this,
not because this understanding is a bad goal but because it is hard
to reach in any way other than having them essentially copy and
not really learn the justifications.  

This worksheet does a limited version for addition of 2-digit numbers
of what we would have liked for all the operations.  Still there
are quite a few steps.  To add $ab$ and $cd$, they must
\begin{itemize}
\item Write these as $(10 a + b) + (10 c + d)$;
\item Re-arrange (use associativity and commutativity)
to get $10 a + 10 c + (b + d)$;
\item Separate into cases $b+d \leq 9$ and $b + d \geq 10$;
\item in the hard case, $b + d \geq 10$, re-arrange to get
$(10 a + 10 c + 10) + (b + d - 10)$;
\item In either case, factor out the 10 to get $10 (a + c + {\rm carry})
+ (b + d - 10 {\rm ~if~applicable})$;
\item Prove a lemma that $10 x$ is written as $x0$ even if $x > 9$
(see Stupid Number Tricks) and use this to justify writing down
$10 (a + c + {\rm carry})$ next to $b + d$ or $b + d - 10$ as 
applicable.
\end{itemize}

Whether or not they can do this in groups, you need to get through 
this as a whole class and possibly even make a handout of it afterwards.

\subsubsection{A base four lesson}

This one is optional.  Advantages are that it emphasizes how to teach
a regroup-and-carry approach to adding, which nicely complements the
difficult algebraic derivation they have gone through.  The manipulatives
involved are ones they might actually have to use in classrooms.  
Drawbacks are that it will be a little vague what the goals are 
and whether they have been met, and that some students tend to 
spend oodles of time making this kind of assignment pretty
(using colored pencils, graph paper, etc.) which will not enhance
their absorption of mathematical content.

\subsubsection{Big and little}

Some of the content having to do with large numbers and 
exponentiation has been moved to Math 107, but a basic 
understanding of scientific notation and exposure to ideas 
of estimation remain in Math 105.  

Problem~1 is there solely for exposure.  They need to see that
the reality of a billion is different from what they might guess
as a shot in the dark.  Those who think the most important aspect 
of mathematical preparation of teachers is along the lines of
Paulos' ``Innumeracy'' will be disappointed that there aren't
more worksheets like this one.  

Problem~2 is meant to convey one specific idea: that it is easier 
to come up with valid estimates for quantities for which we have
an observational basis for comparison, e.g., rate of hair growth
per month rather than per second, and that often we may effectively
estimate other quantities by breaking them down into multiplicative
factors of the kind we can more easily estimate.  Blades of grass 
per square inch, for example, is easier to picture mentally and count 
than blades per acre or square mile, and the size of the oval might
be easier to estimate by estimating walking time and walking speed
and multiplying.  Problem~3 just puts this into words.

Problem~4 illustrates that when adding quantities of different
magnitudes, one may ignore the smaller one entirely.  Problem~5
points out that this is not true when multiplying!  And problem~6 
points out that, when a number is weritten in scientific notation,
the exponent is a lot more important than the other number (what's
it called?).  

\subsubsection{Negative place value}

This could go before the worksheet on scientific notation (since
the introduction of scinetific notation necessitates and understanding
of negative powers of ten) or could go later, when the curriculum turns
to decimals.  In fact, thorough exploration of exponentiation is
delayed to Math 107.  

Here all that is needed is an understanding that $10^{-n} = 1 / 10^n$
to get the studetns started generalizing to other bases and to
the general decimal representation.  Students who don't know this
fact explicitly may still be able to manufacture a procedural
understanding sufficient to generalize decimal notation to other
bases, going through the route of fractions, for instance, rather
than decimals.  It is best to take a pictorial, number line approach to 
problem~1 (e-f), since the visual depiction of subdivided intervals
is a crucial complement to the students' verbal understanding of
decimal representation.  

Problem~2 makes sense only if the places for both positive and 
negative place value have already been explicitly identified.  
Thus you should make sure the discussion of Problem~1 at some 
point begins indexing the columns to the left of the decimals
point as $0, 1, 2, \ldots$ and to the right as $-1, -2, -3, \ldots$.
On problem~3, many students want to answer ``Because the right way to 
do it is this, not that.''  Whether or not a complete and sophisticated
answer is obtained here, be certain to point out that the response
quoted above does not answer the question.  To answer why $X$ is wrong,
it is seldom adequate to answer that $Y$ is right: you need to show
at what point the reasoning leading to $X$ goes astray.  


\pb
\subsection{Division and fractions}

This section of the curriculum is in a way the most straightforward,
in that there is less ``de-familiarization'' to do.  With fractions,
as with whole number arithemtic, the students' manipulation skills
are way ahead of their justification ability and modeling 
ability.  On the other hand, whereas with whole number arithmetic is
was difficult to find challenging problems on the basic operations
(the most challenging problems were on the more novel number theoretic 
curriculum), it is easier with fractions to find challenging problems 
that bear directly on fractions, decimals, percentages and the associated
algorithms.  Thus this section is rather straightforward.

\subsubsection{Ratio and Proportion problems}

In problem~1, the main difficulty seems to be seeing that the
natural answer to ``how much less'' is in terms of a proportion
of the original amount rather than an additive (subtractive) 
displacement as the termm ``less'' seems to imply.  Also, the
total amount of pizza is not given, nor the number of slices per pizza,
which throws many students for a loop.  If this is troubling them,
encourage them to make up numbers, solve the problem, then make
up new numbers, solve again, compare a few such solutions and see 
what can be said in general.

Problem~2 is a straightforward proportion problem.  This is a good
point to make sure answers are convincingly delivered, units not 
forgotten, and so forth.

I am baffled by why problem~3 is hard for a lot of students.
They have too many choices as to which substance will be the reference
measure and which will be measured in proportion to proportion to
the reference.  Because of this, some reverse sweeter versus less sweet 
and do not know how to make the adjustment. 

\subsubsection{The sense of it}

This is the only worksheet to emphasize physical units.  On problem~1,
try to contrast one story that adds inches to inches with another that
adds feet to inches.  The moral is, feet plus inches does make sense,
but to compute a total, one must choose one unit and convert everything
to that unit.

On problem~2, you will find that many students are not familiar 
with the most common unit in this vein, namely man-hours or
person-hours.  I think it is worth mentioning that this unit is
in common use in the business as a unit of labor.  

Problem~3 is meant to get at the unit of ``unitless''.  They will
learn more about this in Math 106, but for now what is important 
is that they see that it is invariant under changes of scale.
If I weigh twice as much as you in pounds, then I weigh twice as much
as you in kilograms.  

Problem~4 is there mainly because this division of units tends to
come up a lot in word problems and makes some students uneasy as to
whether it is automatically justified.  Here, we want to consider
a simple such problem, solve it in a convincing way, notice that 
the units do cancel in the expected way (mi / MPH = hrs), and 
remember this in the future.

Problem~5 should illustrate that different units cannot be meaningfully
added or subtracted.  Don't let them spend too long on this one!

\subsubsection{The cider press and the condominium}

These are two problems where fractions must be understood to
represent a proportion of a whole.  

Students who are struggling with problem~1 need to be asked what 
happens when an apple is run through twice -- how much of the possible
juice will have been squeezed out of it?  Once they understand that,
it is simple to continue and compute the remaining juice after
successive squeezes to be the powers of $2/3$: $2/3, 4/9, 8/27, 
16/81, \ldots$ and since $16/81 < 1/5$, this is sufficient to give the 
answer.  Optional: to proceed along the paradigm, have them state in
algebraic notation that the amount of juice left after $n$ times
through is $(2/3)^n$ of the original juice.

Problem~2 requires them either to find the hidden equation 
$$(2/3) M = (3/5) W$$
or to see that the relevant way to compare $2/3$ and $3/5$ is to
form a common numerator.  If we rewrite the fractions as
$6/9$ and $6/10$, then we can marry off six out of every nine men 
to six out of every ten women.  Many students instead spend time
writing the fractions as $10/15$ and $9/15$ and then don't know where 
to go from there, or spend time worrying about how many men or
women there are in total.  We usually tell students having the latter
problem just to pick total numbers as they please, solve the problem,
and then try to determine whether the answer depends on the numbers
they put in.  In this particular problem numbers of men and women
can't be assigned independently but there is one degree of freedom 
in choosing a scale, so either the total number of men or the 
total number of women may be arbitrarily assigned.  It is amazing 
how many students prefer to work with 100 men, even though this
immediately forces $66 2/3$ of them to be married.

\subsubsection{Law and order}

This worksheet deals with the number-line representation of fractions.
I think it is a weak worksheet because the number-line representation
is predicated on a linear unit, so it makes sense to add, subtract and
compare these but not to multiply or divide them, at least if the
answer is going to be placed back no the number line.  It made the
final cut primarily because, with this being the calculator era, 
there are a lot of students who have no idea of the relative sizes 
of anything not in decimal form.  If you choose to do this
worksheet, you should point out the abuse of units.

\subsubsection{Lynna's arithmetic}

Lynna is dividing 17 into 544 and correctly computing a quotient of 32.
The justification is straightforward and there isn't much to write up.
The hardest part is figuring out what she's doing.  On this worksheet,
the learining is in the process.  Once a student has figured out what
problem Lynna is doing, the student is much better positioned to 
understand the conventional long division algorithm.  The conventional
long division algorithm never appears in Math 105, though perhaps 
after this worksheet a brief discussion would be in order.

\subsubsection{Visual operations}

This worksheet is a good antidote to the abuse of units in Law and Order.
It deals both with number-line representation and length representation.
The point is that $x+y$ and $x-y$ can be drawn even when reference is
permitted only to lengths already drawn.  The ratio $x \div y$ can
be stated as a number, though cannot be drawn since the is no reference
unit.  The product $x \cdot y$ cannot be drawn as a length but can be
drawn as an area.  On the number line, even less is possible without
a zero reference point.  

\subsubsection{Fractions!}

Problem~1: Students don't have too much trouble coming up with 
pictures of eighths subdivided into $24^{th}$s.  I find the
arrangement of squares in a $3 \times 8$ rectangle a particularly
useful diagram, since it is obvious when drawn this way that each
eighth is a strip of three twenty-fourths and that $x/8 = (3x) / 24$.
But some will draw a line of eight intervals, each subdivided in three
equal parts, or a pie-shaped chart, all of which are effective diagrams.

Problem~2: They all know how to comp are by cross-multiplying, so for
some (not all!) the statement is easy.  You need to demand a good 
proof here.  The easiest route to this is to find a common denominator
algebraically.  Don't let them waste too much time trying algebraically
to find the least common denominator.  

Problems~3 and~5: These make sure they understand the why as well as
the how when it comes to multiplication of mixed fractions via the
distributive law.  In both cases, the problem forces them not to
convert to improper fractions.  Those who can do~\#5 will enjoy 
explaining this seemingly daunting feat.  Many get the idea of 
using the distributive law on a suitable truncation of~\#3, but
then don't realize they can multiply $1 \, 1/2$ by $8604$ in their heads.

Problem~4: This one is meant simply to require an explanation for the
ubiquitout ``common denominator'' method of adding fractions.  They must
also address conversion between mixed and improper fractions, since
adding the integer and fractional parts separately might end up with
a mixed fraction with fractional part greater than 1.

Problems~6 and~7: Logically problem~7 comes before problem~6, but it
works better to have them attack~\#6 and put it on hold for~\#7 if they
get stuck.  The idea is for them to understand the ``flip and multiply''
algorithm for division of fractions.  It is easiest when dividing
an integer by a unit fraction, as in problem~7, since one can argue that 
$1/n$ fits $n$ times into~1, thus $kn$ times into $k$.  They then need to
fill in a few steps in order to give a convincing explanation for~\#6.

Problem~8: This takes them through one more step of the Paradigm, 
expressing in algebraic notation a procedure they should understand 
pretty thoroughly by now.

\subsubsection{Decimals}

The most basic problem on this worksheet is problem~2.  We assume they
will not have too much trouble with it, but have placed it there so they
will have a more complete record of what they know how to do.  

Problem~1 makes them gather some data for an assault on the question
of which decimals repeat.  The only one with a long period is $13/23$,
which is the one they are meant to give up on.  Every one other than this 
will terminate or repeat soon enough for them to see it repeats.  Some
students will probably do the manual computation on $13/23$ long enough to
see that it repeats.  This should make the discussion interesting, as
you can challenge whether they did this long computation correctly (look
over their shoulders from time to time to make sure they do!) and 
try to elicit from them some kind of argument that it must eventually 
repeat.  Perhaps you will save this part of the discussion until problem~4.

Problem~3 can be answered succinctly: fractions whose denominators have
prime factorizations with only 2's and 5's.  They should be able to
formulate this eventually.  The proof may be within their grasp as well:
from problem~2 they know that the decimal for $x$ terminates if and only
if $x = y/10^n$ for some integers $y$ and $n$.  If $x = p/q$ in lowest 
terms, that means that $q | 10^n$ for some $n$; but $10^n$ factors as
$2^n 5^n$, so this is equivalent to $q$ containing only 2 and 5 
as prime factors.

Problem~4 challenges them to prove that all fractions must terminate
or repeat as decimals.  The idea is for them to see that the long division
algorithm must get caught in a loop.  Based on their data gathering 
in problem~1 and their further efforts on $5/17$, they should have a glimmer
of this already.  If they don't yet get it, try working collectively 
on $351/487$ and express doubt that it ever repeats.  Perhaps even say
that you go on forever getting different remainders.  Surely someone will
point out that you get a rerun eventually.  Then you can say, ``Yes I see
you have to get a remainder eventually that has come up before, but 
most likely you get a completely different sequence after that.'' 

Problem~5 is really a wimpy comromise between doing a worksheet on
how to convert repeating decimals to fractions and skipping the topic
altogether.  As it stands it is not sufficient to teach students how
to do this conversion and should probably be skipped.  Perhaps the
result could be discussed as something they will be accountable for 
knowing but will not be justified in class.  Even better, there's
always someone who wants to do a brief presentation for extra credit...

\subsubsection{Percentages}

The most natural way to answer problem~1 is to pick a variable $P$ 
to be the price and to write the discounted price as
$P - P \cdot \frac{35}{100}$.  You can of course reduce this to
$P \cdot (1 - .35)$ or even $(.65) P$, which is the least number of 
operations.  The point here is for them to see that \% means times
$1/100$ and that discounting by some percentage means subtracting a 
portion of the original, or multiplying by (1 - something).  
One subtlety it is good to convey is that percentages such as 35\% 
are unitless.  In some sense, they are in units of ``percents'', which 
are just $1/100$'s, but these are still just numbers, in other words,
percents themselves are unitless.

Problem~2 is a straightforward problem in algebraic expression.  We simply
want them to find a description equivalent to ``multiply by $(1 + x/100)$.

\pb
\subsection{Outtakes}

We include here ten worksheets that have been used at one time 
or another in this course and its precursors.  Take a look at them
now so you will know what is available should you want to add an
extra worksheet in a topic area or substitute one.  Three of them 
(Rat Poison, Glacks and Glocks, and Sets: extra problems) are   
sequels to existing worksheets, so if any of these activities
(Poison, Glicks and Glucks, or Sets) seems to be going well and
you want more material, it's there.  Quarto Publishing, Frame Up
and Four Cuts are general problem-solving activities, Whiz Kids
is on place value, Septobasiland is on base seven, What {\em are} they
doing is on deciphering children's arithmetic and $1/3$ and $2/5$ is
on different meanings for fractions.

\pb
\subsubsection{Rat poison}

\medskip

\centerline{\em To win you have to be as sneaky as a rat!}

\bigskip
\bigskip

RAT POISON has the same rules as POISON except that in RAT POISON you may take
1, 3, or 4 counters; you {\em cannot} take 2 counters.  The team that takes the
last counter is still the loser.

\vfill

\noindent \underline{Questions:} 

\begin{enumerate}
\item 1. If RAT POISON starts with 10 counters, is there
a winning strategy?  What is it?

\vfill

\item
2.  Can you answer question 1 when the game starts with 20 counters?  
With any larger number of counters?

\end{enumerate}
\vfill

\pb
\subsubsection{Quarto publishing}

In Quarto publishing, four book pages are printed on each side of
the large sheets of paper actually used.  One side of a quarto sheet
is shown in the illustration.  Once the sheets are printed on both sides,
they are stacked up, and the stack is folded twice: first from top
to bottom on the dashed horizontal line, and then from side to side
on the dashed vertical line.  The tops of the pages are cut apart.
Thus every quarto page produces eight book pages; see the illustration.

\vfill

1.  How many quarto sheets are needed for a book of 1240 pages?

\vfill

2.  When the pages of this book are stacked and ready to be folded,
what does the top sheet look like?

\vfill

3.  Draw a picture that shows where on the sheet the
page numbers appear, and which are upside down.

\vfill

4.  What other page numbers on on the same quarto sheet as page 13?

\vfill

\pagebreak

\begin{figure}[h!]
\includegraphics[scale=0.7]{figures/quarto} 
\end{figure}

\pb
\subsubsection{A frame-up}

I painted a picture that is 4 inches wider than it is high.  If I put a 
3-inch frame all the way around my picture, the area increases by 
48 square inches.  What are the dimensions of my picture?

\pb
\subsubsection{Four cuts}

If you were to take a block of cheese and make a cut all the way through,
you would end up with two pieces.  If you were then to make another cut 
without moving either of those pieces, you should be able to do it in such
a way that you end up with four pieces.  What is the maximum number of
pieces you can have after three cuts (remember you can't move any of 
the pieces)?  What about after four cuts?  You should be ready to explain 
how to make the cuts so as to obtain the maximum possible number 
of pieces.

\pb
\subsubsection{Glacks and Glocks}

\bigskip

\noindent Read the following statement and indicate TRUE, FALSE,
or CAN'T TELL for each of the conclusions listed.

\bigskip

\begin{enumerate}

\item A {\bf Glack} is an odd number that either is less than 29 or is
a divisor of 48.

\medskip

\noindent\begin{tabular}{|l||c|c|c|}
\hline
{\sc Statement} & {\sc True?} & {\sc False?} & {\sc Can't tell?} \\
\hline
a.   All Glack numbers less than 15 are odd. & & & \\
\hline
b.   14 is a Glack. & & & \\
\hline
c.   All odd numbers less than 21 are Glacks. & & & \\
\hline
d.   16 is a Glack. & & & \\
\hline
e.   No odd number greater than 27 is a Glack. & & & \\
\hline
f.   Nine Glacks are prime. & & & \\
\hline
\end{tabular}

\vfill

\item All {\bf Glock} numbers are divisible by 2 and are multiples of 13.

\medskip

\noindent\begin{tabular}{|l||c|c|c|}
\hline
{\sc Statement} & {\sc True?} & {\sc False?} & {\sc Can't tell?} \\
\hline
a.   169 is a Glock. & & & \\
\hline
b.   No prime number is a Glock. & & & \\
\hline
c.   52 is a Glock. & & & \\
\hline
d.   All Glocks are even. & & & \\
\hline
e.   All Glocks are divisible by 26. & & & \\
\hline
f.   All numbers divisible by 26 are Glocks. & & & \\
\hline
\end{tabular}

\end{enumerate}

\renewcommand{\baselinestretch}{1}
\small\normalsize

\vfill

\pb
\subsubsection{Sets: extra problems}
%
\begin{enumerate}
\item In a special ed classroom where there are to be 20 children, 9 of
the children will need new desks high enough to accommodate wheelchairs, and
seven of the students in the classroom will need desks fitted with electrical
power strips for special equipment.  If eight of the children can use already
existing (lower) desks without power strips, how many of the new, higher desks
need to be fitted with power strips?
%
\item In a given class of 25 students, there are a total of 11 boys,
and a total of 7 left-handed children.  If 9 of the boys are right-handed, how
many of the girls are right-handed?
%
%\item In a class of 35 students, 20 have a last name beginning with a
%consonant.  25 students have a last name with more than one syllable.  What can
%you say about the number of students with a multisyllabic last name that begins
%with a vowel?  What about the number of students with a multisyllabic last name
%that begins with a consonant?
%
\item For each Venn diagram below, shade the numbered regions
indicated, and find a way to express the region in terms of $A$,
$B$ and $C$ using unions, intersections, and complements.
\end{enumerate}

\begin{figure}[h!]
\hspace{.5in}\includegraphics[scale=0.8]{figures/setextras} 
\end{figure}

\pb
\subsubsection{Whiz kids}

\begin{enumerate}

\item Fletcher, age 10, can immediately tell you the square of
any two-digit number ending in 5.  He does NOT have them memorized.
How does he do it?

\vfill

\item In order to stay a step ahead of Fletcher, his mother Diana
has found a new calculating trick.  If you ask her to multiply
74 by 76, she will pause just a second or two and then tell you
the answer.  How does she do it?  She can do the same for
$24 \times 26$, or $44 \times 46$.
\end{enumerate}

\vfill

\pb
\subsubsection{Driving in Septobasiland}

\bigskip

The king of Septobasiland has proclaimed that the digits 7, 8 and 9 are never
to be used in Septobasiland upon pain of death.  As a result, the little wheels
on the odometers of all the cars in Septobasiland have just the digits 0, 1, 2,
3, 4, 5, 6 on them.  Thus, if the odometer registers 0006 and you drive one
more mile, the odometer will register 0010, and if the odometer registers 0016
and you drive one more mile, it will register 0020.

\bigskip

\begin{enumerate}

\item If the odometer registers 0066 and you drive one more mile, what
will the odometer register?

\vfill

\item If the odometer registers 0325, how many miles has the car gone?

\vfill

\item How many miles will the car have gone when the odometer turns
over to all zeroes again?

\vfill

\item After a car has gone nine hundred miles, what will its odometer
register?

\vfill

\item You and your companions go on a trip in Septobasiland, traveling
together in two cars.  The first car, which is new, starts with its odometer
reading 0000, and the second starts with the odometer reading 1435.  At the end
of the trip, the odometer of the first car registers 0324.  What will the
odometer of the second car register?

\vfill

\item Suppose again that the odometers of the two cars register 0000
and 1435 at the beginning of the trip.  But this time at the end of the trip
the odometer of the second car registers 2153.  What will the odometer of the
first car register?
\end{enumerate}

\vfill

\pb 
\subsubsection{What {\em are} they doing?}

For each of the following students, try to figure out how they 
are computing.  Then, try their algorithm on the given problems 
and describe why their algorithm works.

\vfill

\noindent{\bf Adrian}

\begin{tabular}{rrrr}
& 5 & 2 & 8 \\
- & 2 & 8 & 3 \\
\hline
&&&
\end{tabular}
\hspace{.5in}becomes\hspace{.5in}
\begin{tabular}{rrrr}
& 5 & 12 & 8 \\
- & 3 & 8 & 3 \\
\hline
& 2 & 4 & 5 
\end{tabular}

\begin{tabular}{rrrrr}
& 2 & 0 & 0 & 3 \\
- & & 8 & 9 & 6 \\
\hline
&&&&
\end{tabular}
\hspace{.5in}becomes\hspace{.5in}
\begin{tabular}{rrrrr}
& 2 & 10 & 10 & 3 \\
- & 1 & 9 & 10 & 6 \\
\hline
& 1 & 1 & 0 & 7 
\end{tabular}

\vfill

Now use Adrian's algorithm to compute: 

(a) \hspace{.1in} \begin{tabular}{rrrr}
& 8 & 1 & 2 \\ - & 4 & 5 & 9 \\ \hline &&&
\end{tabular}
\hspace{.5in}(b) \hspace{.1in} \begin{tabular}{rrrrr}
& 6 & 2 & 2 & 1 \\ - & 4 & 2 & 2 & 7 \\ \hline &&&&
\end{tabular}
\hspace{.5in}(c) \hspace{.1in} \begin{tabular}{rrr}
& 4 & $2_{five}$ \\ - & 1 & $3_{five}$ \\ \hline &&
\end{tabular}

\vfill 

\pb
\noindent{\bf Bill}

$27 \times 34$ becomes
\begin{figure}[h!]
\includegraphics{figures/bill} 
\end{figure}
thus, $27 \times 34 = 918$.  

\vfill

Now compute the following products using this algorithm:
$$(a) \; \; 38 \times 74 \hspace{2in} (b) \; \; 125 \times 35$$

\vfill
\vfill


\pb
\subsubsection{$1/3$ and $2/5$}

\bigskip

The following problems all involve the fractions $\frac13$ and $\frac25$;
however, each has different answer.  Explain what different roles
the fractions are playing (use diagrams when appropriate) and what
fraction operations are represented in the stories.

\bigskip
\bigskip

\begin{enumerate}

\item In your fourth grade class, $\frac13$ of the students voted in the
class elections.  $\frac25$ of the votes in your class were for Tammy. 
What proportion of the vote cast were for Tammy? 

\bigskip

\item In the same class, $\frac13$ of the boys get free lunch and
on $\frac25$ of the days, the only vegetable in the free lunch is catsup. 
What proportion of the lunches (at least) have catsup as the vegetable?

\bigskip

\item In the same class, $\frac13$ of the students have a TI 82 and
$\frac25$ of them have a TI 86.  Nobody has both, and nobody has any
other graphing calculator.  What proportion of the class has a 
graphing calculator?

\bigskip

\item In your class, $\frac13$ of the students remembered to turn in their
field trip permission slips.  In the other fourth grade class, $\frac25$
returned theirs.  If both classes have the same number of students, what
proportion of the fourth graders remembered to turn in their
permission slips? 

\bigskip

\item In chess club, 2 out of the 5 boys wear pocket protectors, and
1 out the 3 girls does.  What is the proportion of chess club members
that wear pocket protectors?

\end{enumerate}


\pagebreak
\section{How to teach in a cooperative classroom}

This section contains a great deal of information and advice regarding how to
manage a cooperative classroom, on all levels.  In fact, the amount of reading
is rather overwhelming, so we suggest that at first you read just the first two
sections, on the basics and on the beginning of the semester.  The remaining
sections, on small group dynamics, large group dynamics, and organizing
yourself, can then be used as reference either as day one nears, or once the
class has met a couple of times and you have some more specific questions.
As with the rest of this guide, the recommended procedure is to browse first,
then come back and zero in on the topics of greatest interest to you.

This guide was written for the Math 130-131-132 sequence at the University 
of Wisconsin.  There are a few references that are specific to that 
situation, though the bulk of what is in here is general advice for
anyone teaching in a group-learning, socratic classroom.

\bigskip

\noindent{\Large \bf DISCLAIMER}

\bigskip

In training one's self or others to teach a Socratic/Cooperative style
class (henceforth SC), it seems no amount of preparation or advice can
substitute for a certain kind of on-the-job training.  The essential
ingredients of on-the-job training are criticism and imitation.
Experienced instructors visit the classrooms of new instructors, taking
extensive notes on what they see: what went right, what went wrong, 
what might have worked better, and so on.   New instructors also visit 
the classrooms of experienced instructors, taking equally careful notes
on what went well or poorly, what might have worked better, and on
what techniques they see used that they would like to use themselves.
Some of this type of work can be done beforehand, via visits in the
previous semester or videotaped classes, but it tends to be more
valuable when it comes after the new instructor has had a chance to
try teaching a class or two first.  Currently in 130-131-132 we are
trying for one visit each direction in the first two weeks, another
in the second two weeks, plus at least one more during the term.  

If you are not going to adhere to a plan as outlined above, then {\bf
don't expect the notes that follow to do much for you}.  You can't
learn to play the piano by reading books about it or by talking about
technique with Rubenstein, so don't try to learn a significant new
teaching skill without practice and guidance either.  On the plus side,
all the interested instructors that have taught SC classes here
(admittedly a self-selected sample) have become pretty good at it, so
you can be pretty confident that you'll be able to step right in and 
teach effectively this way even if you're not a virtuoso.  

\subsection{Basics}

\subsubsection{The philosophy}

Unless you've been on Mars, you're probably aware of the age-old 
battle between ``skills'' advocates and ``process'' advocates.  
I'm from the ``process'' camp but I hope to avoid a lot of the 
partisanship that is prevalent in discussions of pedagogy and
stick mainly to points both sides agree on.  We want students to
come away from (lower level) math classes with certain skills and
attitudes.  In particular we want them to reason and prove, to 
translate between words and symbols, to perform algebraic manipulations
correctly and with understanding of the justification, and to 
solve problems other than clones of problems they have been shown
how to do.  Whether or not you believe specific skills and knowledge
to be paramount (arithmetic of negative numbers, solution of quadratic
equations, propositional logic, summation of common series), you
undoubtedly want them to know these things in the ways mentioned 
above: verbally and symbolically, with justification and proof, 
well enough to apply to new situations.  

The tenet underlying SC classes is that students need to learn that
they can think for themselves, and that they will be able to learn
properly if and only if they are forced or enticed to continue
thinking things through on their own terms.  Realize please that this
does not apply to students who already have this skill.  I don't think
we need SC classes at the advanced undergraduate level, and they
become increasingly inefficient at higher levels.  If college
admissions standards (or high school graduation standards) were 
what we'd like them to be, we wouldn't need SC classes in college at all.
The students who benefit from SC classes are ones I would term
remedial: those taking pre-calculus, and those in the Ed program who
are required to take what is essentially junior high school mathematics.

Our philosophy with these students is to do anything we can to get them
to think and speak mathematics, and then once we have them going, to 
exact from them a quality of mathematical reasoning that is higher
than anything ever asked of them in a traditional course, thus ensuring
that they learn the course content in a useful and permanent way.  The 
meta-skill we emphasize is for the students to {\bf know when they know
something}, versus when they are just guessing or are confused.
In the time-span of the course or sequence of courses we move from a
``process is everything, choice of content matters little'' approach
to a stage where we cover the traditional content and expect students 
to focus on these topics and skills while applying the critical thinking 
they have learned in the first phase.  The first phase is the harder
phase for most instructors, since we have to be psychologists --- and sometimes
mind-readers --- in addition to being mathematicians.  These notes 
concentrate on this phase, though they apply to the other as well.

Perhaps the most controversial part of this approach is our unwillingness
to tell students the answer.  Some skills can only be acquired this way,
and one can be overly dogmatic on this point.  The basis for this is that
much of this material is accessible to them, with a little help from us,
and that our habitually providing answers will cause their problem-solving
ability to atrophy, though it may increase their rate of skill acquisition
(though we argue probably it won't).  Thus we make every attempt when
discussing an attempted solution in class not to tip off whether it
is right or wrong until the whole class has had a chance to criticize
it or register comprehension and agreement.  Depending on the context,
we do or do not in the end provide model solutions.  


\subsubsection{Typical classroom mechanics}

A usual 50 minute class consists of two kinds of time: some time
when the students are working in groups of 3 or 4 on a problem
or worksheet (set of problems) and some time when the entire
class is discussing the problem set.  Some instructors enjoy
keeping to a familiar rhythm, spending the initial 25 to 30 minutes
each day working in small groups, then spending the latter part of
the lesson in a large group discussion detailing what the various 
small groups found, where they got stuck, and so forth.  Often
there are parts of the worksheet that are not covered in this
phase; some of these are assigned for homework and some are discarded.
Other instructors prefer to go back and forth a little more, 
starting in a small group, then convening the large group when
most small groups are done with the first problem, discussing it
a bit, then remanding the class into small groups, and so on.  When
a class meets for 75 minutes twice a week instead of 50 minutes
three times a week, it is usually necessary to go back and forth
this way, and it is also often convenient to continue a large group
discussion from the end of one class at the beginning of the next.  

During the small group working time, the instructor circulates among
the groups, answering questions when necessary, doling out encouragement
when necessary, challenging the students to justify what they claim
to have figured out or to explain their half-baked ideas.  Often
the mere presence of the instructor encourages a renewed attack on
a problem.  

The large group discussion begins with the students explaining what they 
have done.  Other students are required to listen carefully and to 
register agreement, disagreement or incomprehension.  Once the
explanation is comprehensible, those in disagreement are encouraged
to justify their disagreement, with the aim of a resolution or
synthesis.  The instructor plays moderator as long as fruitful
ideas are being produced, but slips into the role of leader when
needed.  For example, if no one challenges something wrong, or if
there is a disagreement but it is too inarticulate to produce a 
good synthesis, then the instructor may rephrase what has been
said so as to sharpen the contradiction or caricature a wrong
approach, in a way that forces a light to dawn for at least some students.

Note also that in order for SC discussions to work, the class size
must be limited, to no more than twenty or thirty maximum.  The point is to
have the class participate as a whole in the same conversation, but if you
think back to social gatherings you've attended, even if everyone knows each
other, it is difficult to keep everyone involved in the same conversation once
you get more than a dozen or so people involved.


\subsubsection{Highly recommended procedures} 

Before getting into specifics of classroom technique, here are a few 
simple procedures that make a large difference.  

\begin{itemize}
\item {\em Nametags.}  Have the students wear nametags each day until
you know their names (in my case 2 or 3 weeks).  When calling on
students, call on them by name and in general attempt to use their
names frequently.  This serves two purposes.  First, by attempting to
learn their names, you create a separate mental category for each
student, which helps you pay attention to how each student is doing
and to their individual needs.  Secondly, there seems to be a psychological
advantage to students hearing their names.  Coming around to a small
group and asking ``Amy, can you tell me what progress your group has
made?'' or asking in a large group discussion whether ``Sam's objection 
to Cindy's idea'' holds water elicits more of a response than the same 
questions without the names identified.  Somehow, students are more
prone to take their own beliefs seriously when names are attached.

\item{\em Randomize groups.}  I usually assign groups using playing cards
randomly dealt: all the aces for a group, the twos form another, and so on.
I re-form groups twice or thrice during the semester.  Preventing 
students from choosing their buddies for a group helps them form 
connections, subject their ideas to the intellectual marketplace,
and treat all the others in their groups fairly.  The playing cards
themselves lend an air of intrigue as students await the results of
the lottery.
If you decide not to use playing cards to change groups, you can assign the
students the task of creating new groups such that no two ``new'' teammates
have worked with each other before.  (This can be done easily once, and with
considerably more effort a second time.)

\item{\em Get enough sleep.}  Alertness is required on the part of
the instructor.  You'd be surprised what a difference this makes.
You can fake it when you're lecturing, but try playing moderator
when you can't concentrate or respond quickly, and you'll see what I mean.

\item{\em Start the semester with a bang.}  That is, don't spend
the first day on administrative stuff and the second on some
kind of review.  Jump into an absorbing problem on day 1, preferably
a tried-and-true chestnut, and fill in the administrative details 
later, when they've gotten the idea of what the class will be about.
The tone of the first day's discussion sets an example that's hard
to erase, so make sure it is as lively as possible.

\item{\em Minimize the time you spend talking.}  In particular, never talk for
more than five minutes at a time --- monologues shift students back into
passive learning mode, and monologues of announcements tune students out.
Of course, you may have to do a lot of talking in leading large-group
discussions, but if you're talking by yourself for more than five minutes at a
time without at least one student speaking up, you're not doing SC --- you're
lecturing.

\item{\em Be yourself}\/ at least up to a point.  If you're the goofy
type, be goofy; if you're serious and intense, let them feel the
intensity; if you're understated and direct, be that way.  You are
on stage, and want to use your charisma, but don't pretend to be
someone you're not.
\end{itemize}


\subsection{Beginning the semester}

\subsubsection{The first day}

In many ways, the first day can be one of the most important of the semester.
It sets tone and precedent for the days that will follow, and lets the students
know what to expect.  Students will be anxious to know how their grades will be
determined, and perhaps anxious in general to be in a math classroom.  Your job
that first day is to sell them on the class, to help them start to get
comfortable.  This job has two main parts: dealing quickly with administrative
details and getting the students engaged in problem solving.  Suggestions
follow to help you with both of these parts.

You should already know all the administrative details about the class when you
walk in the door the first day, and the quickest fair treatment for
communicating these details is in the form of a handout.  You may want to write
your name and the class name, number, section, etc. on the blackboard before
class starts, to make sure people are in the right place, but don't spend time
in class writing all the details on the board.  Rather, have them collected on
a handout, so that you can skim the most important details (components of the
grade, exam dates) and leave the students to peruse the rest at their leisure,
{\em outside of class}.

The coursepack contains a psat syllabus which can serve as a sample
handout -- in fact you need to remove this and insert your own, or
the students will be mightily confused!  

Different instructors will have different ways of building the atmosphere they
desire.  Some instructors do best by jumping right into the first activity
(``Poison'') in Unit One.  They then reserve a fraction of the second day's
class to discuss the nature of the course, and why we do things the way we do,
using the Poison lesson as an example.  Other instructors are more comfortable
devoting some time to ``warm fuzzy'' introductory activities.  They will use a
little class time to introduce everyone.  To take care of any reticence on
students' part to talk about themselves, try having everyone pair up with
someone s/he doesn't know, and spend five minutes getting to know one another.
(This also gets them used to talking in class!)  Then go around the room having
each student introduce his/her partner.  Some instructors also have students
fill out index cards with a few facts about the student (name, major, hometown,
feelings about math, something unusual about the person, and perhaps suggested
office hours); these can serve as backup in case someone forgets his/her
partner's name.  You can also add a couple of icebreakers such as ``favorite
cartoon character'' and ``least favorite vegetable''.

If you spend time on introductions, then you will want to choose a short
warm-up problem-solving activity that can be completed in the remaining time.
Here's one (Pascal's Triangle) that has worked for us.  First, make sure
everyone has a nametag, and then put them into small groups.  Put up on the
board, or on a handout if you prefer, the first five or six rows:

\bigskip

{\tt
\centerline{1}

\centerline{1 1}

\centerline{1 2 1}

\centerline{1 3 3 1}

\centerline{1 4 6 4 1}

\centerline{1 5 {\small 10} {\small 10} 5 1}

\centerline{. . .}
}

\bigskip

\noindent and ask the following questions:  Can you find the rule used to
generate Pascal's Triangle?  What do the next two rows look like?  What
patterns do you recognize in it?

The groups will discover the rule quickly enough, and the last question is
open-ended enough to allow discussion to continue until the bell rings.

\subsubsection{Encouraging Good Habits from Day One}

Encourage good habits early, to set precedents for the entire semester:
Encourage people to talk, to offer ideas without worrying about whether they're
``right'' or not, to talk to each other (instead of to you) as much as
possible, to write on the blackboard, to be precise about what they mean, and
to get used to giving explanations and justifications to back up things they
really believe.  One way to encourage the latter three things is to ``play
dumb'' and keep saying you're not sure what they mean --- even if you do --- as
long as you think someone else in the class might not (which is usually true).
Of course, you should give a disclaimer the first day of class explaining that
you're not going to be giving them straight answers or telling them whether or
not a solution is correct (though you should never let the book be definitively
closed on a subject without making sure that there is closure, and the everyone
understands that the solution on which the class has agreed is indeed correct).
Part of this disclaimer will include the fact that you reserve the right to
``play dumb'' as long as you think something might not be clear to someone in
the class.

Another habit to encourage from the beginning is to keep a running list of
different problem solving strategies and techniques they have used.  By the
time you finish Unit One, they should already have a half dozen or more, and it
might be worth taking a few minutes in class at that point to have them compare
lists.  Chapter 2 mentions another discussion to have around the end of Unit
One, regarding group dynamics.

\subsubsection{Resistance and ``Why are we studying this?''}

One of the questions which you as a teacher in the Math 130 sequence will hear
over and over again is ``Why are we studying this?''  Why are we studying these
topics which we will never be teaching?  Why are we taking this course?  These
questions deserve an answer, and again this is something best discussed openly
and early (although no doubt the question will recur).  Among other things,
elementary school teachers must lay the groundwork for their students' later
experiences.  In math, arithmetic is the foundation of algebra, and teachers
need to know what's coming up in the years ahead, so that they can teach in a
way that will later allow their students to look back and see the connections.
Mathematics is all connected, and the sequence of K-12 math classes students
take should be a natural progression, not an unrelated set of topics.
Teachers also have to be able to anticipate precocious questions.  Equally
importantly, they must be comfortable and familiar enough with every subject
they teach that their students will not all be turned off.  Many, if not most,
of the students in the Math 130 sequence have had bad experiences with math in
the past, and very often this can be traced to a teacher or teachers who
passed on their apathy or disinterest for the subject.

These past experiences will, as noted above and to varying extents, lead to
resistance by the students to the way Math 130 forces them to shoulder
responsibility for their mathematics.  Here again, you as their instructor
should bring this subject out in the open, early and as often as needed
(although it can, at times, threaten to take up a lot of in-class time --- try
not to let this happen).  Point out the first day that the reason for their
doing this is not only to provide a model for them to use later is getting
their own students to think for themselves, but that someday soon they will
have to be the mathematical authority for a classroom full of inquisitive
children, and that before they can be someone else's mathematical authority,
they must be confident in being their own authority.  There will invariably be
one or two students in a section who do not ``buy into'' this philosophy, and
in the end there may be nothing more you can do for them but to tell them that
the class is designed this way because people who have had a lot of experience
teaching math believe in it, and they'll have to play along for now and see
what results come of it in the end.

Some students, especially the poorer (for want of a better term) ones, will be
vocally and continually concerned for their grades.  The most you can do is
try, from the beginning, to make grade assignment rules as explicit as possible
(but don't feel any obligation to explain down to the point level your grading
of particular reports, any more than a composition teacher would), and try to
reassure them.  Don't let their angst transfer to you.


\subsection{Class Composition and Small Group Dynamics}

\subsubsection{Doing the rounds}

When you first assign a problem to work on in small groups, there may
not be much you have to do.  No one is stuck yet; no one needs your
help.  There is a lot you can accomplish in this time.  First, you can
quickly take attendance group by group --- after you know the names this
takes less than half a minute.  After this, you will want to quickly
``do the rounds''.  Visit each group once just to check that they
have gotten down to work.  On the first round, look for any trouble 
with the wording of the problem that may be holding people up.  If 
it's part of their job to decipher it, encourage them to do so.  If it's
a mistake, or if you need to supply a definition, then make a quick 
announcement.  On your second round you can linger longer.  This is
a good time to make a mental note of which groups are going faster
than the others.  It helps, during the subsequent large group discussion,
to have a good idea of who has gotten how far.  If a group has quickly
and incorrectly or incompletely answered a problem and gone on to
another, this is a good time to ask (innocently) for one of them to
summarize for you what they found.  The correct question on your part
can cause them to re-examine what they've done without feeling that
you've invalidated their answer (point of philosophy: you want them
to be able to criticize their own work, realizing that mathematics will
determine whether they are right, and that what they discover about
this cannot be overruled by the teacher).  

Example:  A group has used an incorrect manipulation: $(a + b)^2 = a^2 + b^2$.
You can ``explain for those who are rusty on this'', that this means
``when you compute, e.g., $(7 + 3)^2$, in case you can't tell at a glance
that 7 + 3 = 10, squared = 100, you can use this nifty rule, getting
$7^2 = 49$, plus $3^2 = 9$, which by the laws of algebra must also 
give you 100.'' 
If this doesn't elicit an objection, you can line up the sum as you speak:
$$\begin{array}{lr} & 49 \\ + & 9 \\ = & 100 \end{array}$$
When you do get an objection, ask them to pinpoint what went wrong, leading
to, ``Oh, you mean $(a+b)^2 = a^2 + b^2$ is not a universal rule of algebra?''

A reminder: Lurk early, lurk often.  While students are in small groups,
eavesdrop constantly.  The students should get used to having you listen in to
their conversations.  This keeps you abreast of both (a) how far the class in
general is getting and (b) which groups in particular are having difficulties,
and where.  If one group seems too conscious of your presence, then stand near
another group nearby and appear to be watching them when in reality you're
paying attention to the discussion in the other group.  This part of the class
is an important one for you as instructor and (presently) discussion moderator.


\subsubsection{``Help, we're stuck''}

The twin dangers here are that lazy students will say they're stuck so you
do the work for them, while students who are truly stuck will lose morale
if they have to sit idly during class.  Asking if they have any ideas on
what they might try will prove embarrassing if the answer is no.  
Sometimes I do this anyway.  Sometimes I replace the problem with a smaller
one: if you knew that $A = 15$ could you do the problem?  Can you do
the problem if you aren't required to make the number of cows and chickens
the same?  Sometimes I guess why they're stuck: So the problem is you
don't really know the definition of average speed?  An example of what
might happen here is that they did know this but didn't think of going
back to definitions as a way to proceed.  Now when they say no that's
not the problem, we know the definition of average speed, I say, 
``Oh, then you must be saying you don't have any way of determining 
the quantities defining speed, such as the time or the distance.''
They then say how they will proceed on this and I can smile and leave.
In other cases, they are stuck because they don't really understand
what's being asked.  You can ask them to rephrase it, or ask them
how they would check if someone else's answer (here you specify it)
was right.  It helps to have snooped enough so you have a good guess
of where they're stuck.  If not, you can ask them but won't always
get reliable information.  

\subsubsection{Getting Groups to Work Together}

The pre-service teachers tend to work well together.  The pre-calculus
population is less accustomed to this, and groups will degenerate into
a lot of individuals ignoring each other, or occasionally explaining to
each other.  You may at times need to tell them explicitly ``Janet
has found what she thinks is an answer but Steve and Brenda apparently
don't understand what she did, so Janet, you're going to have to 
explain it and see if you can convince Steve and Brenda.''  However,
you should try other things first, before being this explicit.  Ask
Brenda what her group has found so far, and don't let anyone else
answer for her.  If she says she's stuck, ask if her whole group is
stuck, and if not, tell her you'll come back in 3 minutes and ask her
again for a summary of what her group has done.  {\bf Make sure groups
are sitting in a tight circle}, not a line or a disarrayed cluster.

Sometimes you can give them specific tasks: Jason, finish the calculation
you're doing; meanwhile Ann will add up Rob's numbers and will then
check to see if they agree with yours; if not, it's up to all of you to 
judge which method if either was correct and why.  If a group really
has bad chemistry, change it.  I've had students say to me: I just can't
work with Judy --- she won't listen and hogs the discussion.  In that
case, put Judy in with someone smarter or more aggressive than she.
When re-forming the groups, I usually randomize again, but if there
are trouble students, sometimes I stack the deck so that they get
put with students who can handle them.  


\subsubsection{Free Riders}

There are always lazy students who are content to let others do the work.
If it's just laziness, I don't hesitate to reprimand them explicitly:
``Adam --- you're just staring into space and letting the other three figure
out the problem; if the problems aren't challenging enough, then I 
can let you work faster in a group by yourself, but judging from your 
homework that's not the issue.''  On the other hand, if it's a student
with a confidence problem who needs some nurturing, it's probably better
to make a note of it and to continue to ask that student to explain 
what their group has done whenever you come around (you have to ask
other students sometimes, or it gets too obvious).  Involving the student
as much as possible, with questions that are at their level but not
patronizing, will often cure this.  Also impress upon the others that
it's their job to make sure the free rider is keeping up with the group
since that person (you've just decided) will be in charge of the first
group writeup.

\subsubsection{Staying on Task}

The less you have to reprimand here, the better.  Make sure that when
you tell them to get into groups they know exactly what they're supposed
to start working on.  Make the rounds quickly at the beginning so they
don't start chatting, and keep an ear out for it later on.  If a group
continues to be bad about this, you can watch them, visit them more,
chastise them humorously, make sure they don't get grouped together
next time, etc., but if the whole class is bad, you should examine what 
you're doing that promotes doing something other than the math.  You could
be leaving them in small groups too long, while the slow groups finish.
You could be joking with them too much during class time.  I do also
use reprimands, but sparingly --- on day 3 this year I was in an impatient
mood so I reprimanded the two groups that took more than a minute to
form their groups and start working.  It was something like ``Sounds
interesting whatever you're talking about, but you've got to get started
on problem 1 --- time's short this week!''

\subsubsection{Students who are behind}

First, it's a good idea to know what you do for them and what you can't.
You can do a lot for these students in office hours, but it's not
realistic to be able to spend more than (or even as much as) half an
hour a week outside of class with any one student.  So if they need
help on a regular basis, suggest that they arrange for tutoring.  You
should find out before the semester what such resources are available.  If
they're doing passing work, but still underconfident, point out to them
that if they continue to work at this level, they'll pass the class.
Don't, however, make promises you can't keep about their grades; it's
best not to prognosticate about their grades before they've taken
an exam.  

All that being said, you have to make sure they get the most out of their
group work and don't drag down the group (they're as much afraid of 
this as you are).  Here are some ways to build their confidence and
make them more likely to participate to advantage.  Spend some time 
around their group and be ready to pounce on those times when the lagging 
student, ``Lenny'', comes up with a good idea.  Assign credit: if Jocelyn 
figures out how to do something with Lenny's idea, then it's 
``Lenny \& Jocelyn's method''.  Try to give them constructive comments
on their homework (I try to do this for everyone, but when time does
not permit this, concentrate on students who need it).  Making sure
they are keeping up with their group is delicate --- asking them to 
explain to you where their group is will help if they're not too 
far lost.  Once they are far behind, consider placing them next time
with a group of students that talk a lot, don't go all that fast, and 
are as kind as possible.  At worst, you may have to settle for Lenny 
participating minimally during class and trying to catch up on his own 
at home.  

Select Lenny to give a presentation, either solo or on behalf of his group.  
On a one-time basis you can invest a little extra time to help him with 
this to make sure it goes over well (have him practice it on you till 
he's confident enough to field questions).  

Avoid asking Lenny really easy questions.  This will make him feel like
he must be dumb.  These questions are scary in general since the reward
for a correct answer is almost zero and the penalty for an incorrect
answer is large.  At your discretion, you might choose Lenny to answer
questions that are not black and white, asked of the class at large:
which of these problems was the hardest?, what kinds of thinking did
you think were necessary on this problem?, and so on.


\subsubsection{Students who are ahead}

Having such a student can be a real boon if they are gifted teachers as
well.  If they have a good feel for how to explain things and help others,
they will make your class run more smoothly than you can, on their own.
Even in this case, {\bf avoid treating them in front of the class as a reliable 
source for right answers}.  You don't want to create a situation where
calling on them is tantamount to telling the class something yourself.
It is OK though, to treat them as a reliable source for intelligent 
commentary.  

If a student is obviously head and shoulders above the rest from day 1,
you may consider exempting them from the course.  That is, if they 
can do the worksheets on their own then they can probably pass the
exemption exam, go on to the next course, and leave you with a more
homogeneous class.  Recommend that they see the course coordinator for
an exemption exam; everyone will benefit from this.  Later in the 
semester this is less of a good idea, though I've done it.  

Assuming ``Einstein'' stays in the class and is not self-policing, 
you need to keep an eye on Einstein's group to make sure that
Einstein is not explaining things to the others before they have a 
chance to figure it out themselves.  Let Einstein explain things at
the board in situations where you know there will be some wrong or unclear
stuff in the explanation.  Make sure though, that you give Einstein as
much encouragement for what was right and clear as you would another
student.  If Einstein is a loner and tends to work fast but not share
with the others, that will probably work out fine.  You can explicitly
designate Einstein as a group of 1 next time, or simply allow a 
{\em de facto} group of 1 to form.  Sometimes, you can try asking 
Einstein explicitly to figure out a hint to give the rest of the group
as to how to proceed but that won't completely solve the problem for them.
It will make Einstein summon up teaching skills that are worthwhile in
general, so it's worth a try, but be aware that Einstein may not be
capable of this.  In any case, don't let students disparage themselves
in comparison to Einstein.  You can say, ``I see, because Einstein 
solved this problem in 5 minutes and you can't, you're going to give up?''


\subsection{Managing Socratic Discussions}

This is the hardest part, and the part of teaching SC classes that improves
the most with practice.  When observing someone else's Socratic
discussions, try to imagine what would have happened had they made
different choices (told or not told the students something, came up 
with a good counter-question, decided to pursue or not to pursue a 
student's line of reasoning).

\subsubsection{Dead Ends}

When an idea is proposed, the instructor will usually know right away
whether it will lead anywhere.  If it won't, there is a strong temptation
to discourage the students from pursuing the idea.  This may be a 
mistake.  Probably the best thing that can happen in a Socratic discussion
is a {\em flaming dead end}, meaning that an incorrect line of reasoning
leads to a consequence so patently false that the students are compelled
to re-examine the road that got them there.  If you see your students
headed for one of these, then all you really need to do is encourage
them to get there without undue delay.  Some things you might want to
do are: get them to explicitly reaffirm the wrong assumption, so that
they will remember it later and be able to pinpoint it; shut down any
further sidetracks that branch off of this one (e.g., ``OK, that's a 
good idea, but first we'll finish pursuing this one''); hasten their
demise by keeping the pace brisk, perhaps doing some of the arithmetic
for them or providing clarifying paraphrases.  

You can influence how flaming a dead end will be by making the issue more 
concrete: ask them to illustrate their result with actual numbers (e.g., 
``so if the initial weight was 250 grams, then we see the final 
weight of $400 - 2 w$ comes out to be what?  Oh, I see, $-100$ grams... 
rather on the light side.''). 

Perhaps you will need to summarize their findings, juxtaposing two
findings that are contradictory, or in the case that they have 
contradicted some of the given information, you may need to restate
the givens by saying, e.g., ``so you have now proved that the only whole number
between 100 and 500 having no two digits the same and satisfying blah blah
blah is 337.''  To further rub it in, it often works well to insist that
you believed their method and there must be some other mistake: ``Ah, you've
proved that the long division we did was wrong --- can anyone find where?''
(of course it is actually correct), or ``Ah, you've proved that when you
use variables with subscripts, the usual method of solving linear equations
doesn't work'' (if you can trust them to fight back on this one).  

Other kinds of dead end are less useful.  Perhaps they are following 
a reasonable line of inquiry but it doesn't get them anywhere: looking
for a nonexistent pattern, introducing too many variables, classifying
according to an ineffective scheme.  A reasonable goal in this case is to
get them to figure out that they're stuck.  If you tell them (or indicate
in any of 1000 nonverbal ways) that their idea won't work, they will learn
to look to you for validation of their ideas, whereas if they reach a
dead end themselves and consciously decide to look for another approach,
they have learned something valuable about problem-solving.  That being
said, there are ways to reduce the amount of time spent following a 
dead end.  One trick is to decide after hearing a suggested approach
whether to follow it immediately or whether to treat it as one of many
to be written on the board before the class decides which to follow.
If Jenny reports finding a pattern starting $4, 6, 8, 12$ and reports
her reasoning as to what is likely to come next, you get to choose between
(a) getting the whole class involved in speculating about the next number
or (b) writing on the board ``Idea: look for a pattern''.  The key feature
of this trick is that you're not giving anything away.  Approach (a) is
reasonable in some contexts, where the discussion of pattern promises to
have some depth, and more importantly, approach (b) is something you 
sometimes use when the approach offered is correct.  In fact you should
make sure to use (b) on occasions where there were multiple interesting
approaches but the first one offered happens to be the best: you catalogue
every group's approach before asking the class to pick one and follow it.

Another way to expedite matters is to insist that the goals be well
defined.  Often when a bad approach is put into words, it comes out
sounding discouraging: we thought we'd name as many variables as possible
and then hope that inspiration struck; we decided that if $n$ was
equal to 5 the solution was obvious but we don't know how to do it
for any other value.  Sometimes mild discouragement doesn't work. I
remember a worksheet designed to get them to invent the binary number
system by asking them to come up with a scheme for representing all
numbers with 1's and 0's.  This cost a full day of discussion of the
relative merits of various schemes, none of which had anything to do
with binary.  The instructor was very successful that semester, and
in my opinion the investment of days such as that one paid off when students
continued to work hard throughout the course because they didn't feel 
that the instructor was going to provide the answer for them.  This
takes guts, and doesn't work too well if the instructor conveys a 
growing uneasiness about the whole project.  So if you're not up to
following the wind, you're probably better off treating it the same as 
the situation in the next paragraph.

The least promising dead end is a total lack of ideas.  Probably it's best
not to convene a discussion at this point but to continue working in
small groups where you can ask questions that elicit further work
and break the impasse.  But suppose a class discussion on a certain 
problem fizzles out midway.  This might be a good time to drop it.  If
it's not essential that they end up knowing how to do the problem, and they
don't have a realistic shot at finishing it for homework, perhaps make it
into an extra credit assignment.  If it is essential, consider dropping
it for now and writing a worksheet for the next class that will lead them
to it in more manageable steps.  You probably need more time to solve this
problem than you have on the spot.  

\subsubsection{How to Listen}

You need to listen to students and they need to listen to each other.
Tom Lester once told me of a study showing that the average amount of
time between when a teacher asks a student a question and when the
teacher prompts the student or gives up on them is 2 seconds.  Two
seconds is longer than it sounds, but nowhere near long enough to
formulate a coherent thought unless you were already thinking it before
the question was asked.  There may not be anything you can do about
the sound-bite trend in TV reporting, but there's a lot you can do about
it in your classroom.  The first thing to try is waiting.  Don't nod yes
or no, or say uh huh, or give the student any feedback at all until they
have finished saying what they wanted to say.  Then wait five or ten more
seconds.  The odds are that the student will, after pausing for breath, 
realize that they are not finished and continue.  If not, at least the
other students will have had a chance to think about what they just heard.
If you're uncomfortable with this long a pause, try pacing or holding
eye contact with the respondent as if you expect them to continue, or 
act as if you're trying to digest what they've just told you.  In fact
often you really will need time to think.  If they said something that
was wrong in a puzzling way, see if you can figure out what they really
meant.  Students will only listen to each other if you set an example,
so make sure you don't respond without having really heard.  

Students are also more likely to listen to each other if they feel that
they are responsible for having understood it.  In small groups they are
more likely to feel this, but at a ratio of twenty or thirty to one, many
feel that they can just take notes and sort it out later or not at all.
They may also feel they have no right to interrupt since everyone else
obviously understands.  You can counter this by demonstrating an
expectation that each student understands what each other student has
said.  After one student says something the slightest bit unclear, ask 
another to repeat it in her own words.  This is a good time to pick on
students rather than have them raise their hand to volunteer a paraphrase.
If student B can't paraphrase what student A said, it's not necessarily 
student B's fault.   Student B can ask student A to clarify if necessary, 
or ask for volunteers for someone else to clarify.  Make sure to go 
back and find out whether student C's clarification of student A's remark
did in fact help student B.  After a little experience you'll know better
when to go through this routine.  If student B simply wasn't listening, they
might feel reprimanded, but that's OK.  It doesn't really work when the
remark was clear in the first place, although it doesn't hurt to get a
quick affirmation from the whole class that it is clear so far.  The
basic standard you are setting is that the discussion involves the whole
class and is not a collection of one-on-one dialogues between the teacher
and individual students.  

\subsubsection{Staging}

Your expectations of the nature of a class discussion are conveyed
nonverbally as much or more than verbally.  Most instructors ask the
students to rearrange their desks into a large U shape for any but
the briefest class discussions.  A subtler but important technique is
to put as much of the class as possible between you and the respondent.
If you call on a student on the left side of the room, walk over to the
right side as you're doing so.  As the words flow between you and the
respondent, the almost physical presence of a stream flowing between the
two of you will wash over the students in between.  You and the respondent
also keep eye contact with the rest of the class this way.  

It is often a good idea to get students to come up to the blackboard.
Students will give longer monologues at the blackboard, so be prepared
to be a more active moderator if the student is losing the rest of
the class.  Be careful not to make having the right solution a 
pre-requisite for coming to the board, lest the students stop
thinking critically and accept any blackboard demonstration as a 
surrogate for your telling them something.  When a student is at
the board, I try to take up a position in the back or on the perimeter
of the room.  Sometimes I sit in the student's seat.  This has the effect
of including the rest of the class, as above, and also gives me a new
vantage --- you'd be surprised at what you see this way.  

Other body language to be aware of is whether you are passing judgment 
on what you hear.  Do your eyes flit impatiently with wrong answers?  
Do you gesture in agreement with right answers?  Do you angle your body
to the board as if to write down something correct, then pause if
it's not what you wanted?  If you've chosen to teach SC style classes,
it's because you want the students to develop their own judgment, so
avoid this kind of tip-off.  

A related topic is the use of intentional errors.  These are a hit with
kids, in a slapstick sort of way, but adult students tend to feel
patronized.  Instead, I substitute the mischievous lie.  If a student
tells me the found all five regular polyhedra I may say, ``Ah, so you
haven't found the other two then?''  They can often sense that I'm putting
them on, but will still take the bait and try to prove that there aren't
any more.  On a problem best solved by assigning a variable to a certain
quantity, I once told a group of frustrated students that I'd tell them
the value of one quantity for free if they could decide which quantity
they wanted to know.  I planned to lie and tell them it was 10 when it was
really 6.  In fact I had tried this previously with success: the students
figured out that the assumption of 10 led to a contradiction and were 
able to figure out the unique value that didn't lead to a contradiction.
This time it was even more successful.  By the time I came back to give
them their free question, they had figured out what quantity they wanted
to be told, had put in $x$ for this, and had gotten the solution (well
ahead of the rest of the class).  

\subsubsection{Asking the Right Questions}

When my teaching is evaluated by my students, they often say that I never
answer questions, or answer them with another question.  I take it as a 
compliment even though it isn't meant as one.  The most common such 
interchanges are
\begin{quote}
Student: Is this right? \\
Instructor: I don't know.  Does it sound right to you?  Can you 
elaborate? \\[2ex]
%
Student: What should we do from here? \\
Instructor: What do you think?  Does anyone have any ideas?   \\[2ex]
%
Student: Can we say blah blah blah? \\
Instructor: I don't know, can you? 
\end{quote}
There is a certain amount of this you can get away with, depending on
your personality and theirs.  If you start sounding like ``Eliza''\footnote{The
computer program imitating a nondirectional therapist, an early (and
crude) approximation to something that could pass the Turing test.},
you won't get good results.  Instead, try to ask them useful questions
related to the specifics of what they've said.  In the three above
scenarios, try respectively
\begin{quote}
Are you asking if your computation is correct, or if it will prove 
useful? \\[2ex]
Is there a problem-solving strategy that you know that might work here?  
Why don't these equations tell you what $x$ is?  \\[2ex]
If you're wondering whether you can assign the variable $z$ to be the
average of all the prices, the answer is yes, but you haven't yet
said whether we know anything about $z$.  
\end{quote}
When observing other classes, this is where you should let your imagination
run free.  Imagine what questions they might have asked.  Your hindsight
now will be your foresight tomorrow.  

The question ``do you understand?'' is the most often abused.  (Notice that
this is virtually the only question in the repertoire of the conventional
lecturer and rarely elicits an honest response.)  Some better variants
are: can you say that in your own words? could you do what John just
said with different numbers? do you agree or disagree? in what way is
this similar to what so-and-so did?  These are all comprehension questions,
testing whether the respondent comprehends instead of asking point blank
for a Yes/No as to whether the respondent comprehends.  

Good questioning can help to reach flaming dead ends.  Ask what happens
when $x = 5$, or whether their purported method works for all starting
data and not just what was given.  If a student gives a vague definition,
find a borderline case and ask how their definition applies in that case.
Try also questions that goad by disingenuity.  If their method is
more general than they realize, ask how they got lucky enough to try 
their method on a square rather than a pentagon or hexagon for which it
``probably wouldn't have worked''.  

\subsubsection{Order versus Chaos}

Ideally your students will be eager to answer your questions and discuss
their ideas, but will listen patiently and attentively to each other 
and to you.  If students are not willing to speak up and discuss their
ideas, you need to loosen them up.  It is a bad sign, for example, if
the students are not happily chatting away when you enter the room five
minutes before class, and are sitting in silence or whispering.  In this
case, you have probably done too well at eliminating chaos.  Try assigning
an activity in small groups where different groups are doing different
things and they need to walk across the room to share information with 
each other.  For example, there's a worksheet in Math 112 on infinite
series where they approximate numerically some infinite sums and try
to form conjectures based on each other's conclusions.  Assigning a group
project where they have to work outside of class together can tighten the
bonds and make people feel more comfortable talking.  When leading class
discussions, be freer and more willing to follow the students' ideas 
wherever they lead.  Dispensing with hand-raising and having students
just call out can quicken the pace.  

A classroom that's too chaotic is a problem also.  If you have to call
the students to attention more than once before they pipe down and listen,
or if there is crosstalk during class discussions, you probably ought to
do something about it.  You can address this explicitly, asking the
students to pay attention to you and to each other; you have to be
consistent about this or they won't believe you mean it.  Indirect 
methods of dealing with this are, however, usually more effective and 
should be tried first or at least in parallel.  Insist on an orderly 
formation of desks into a U shape before a class discussion, rather
than having them minimally perturb the small-group seating arrangements.
It takes an extra minute, but it's worth it.  In fact tell them you're
pressed for time so they have to rearrange the desks in 30 seconds.  A
snappy set change will set the tone for what follows.  When you observe
crosstalk, try to get one of the crosstalkers up to the board to explain
something, or to comment on what's just been said.  By maximally involving 
that student in the lesson, you'll eliminate most of the off-task
crosstalk, and the on-task crosstalk can probably be lived with.  Another
chaos reduction technique is to give them a more rigid idea of the
structure of each class.  Say you're going to spend 16 minutes in groups
before convening a class discussion and then stick to it with absurd 
precision.  The more aware they are of the structure of the lesson, the
more they will stick to the tasks at hand.  

The main point of this section is that you should make a conscious effort
to optimize your position on the order-chaos axis, and that increasing
order or chaos in the physical arrangements or chronological structure
or types of assigned activities can help you change the balance in your
class discussions.


\subsection{Organization (yours)}

SC style classes have more inherent disorganization than traditional
classes, which is why you need to pay particular attention to organization.

\subsubsection{Your Records}

Students, at Wisconsin more than at other places I've been, react extremely
positively to the appearance of organization.  Probably the pre-service
teachers are particularly impressed by this since they are consciously 
judging you on your pedagogical techniques.  {\bf Be a compulsive
record-keeper.}  I find it very helpful to reserve the 20 minutes after
each class for writing a short summary of what happened in class that day.
That way, if I've told a student I'd find them an extra-credit assignment,
or if I've promised to bring something to class next time or promised that
the next class would begin with a discussion of something, I can write this
down along with other notes as to what I have in mind for the next class.
In an SC style class there is a greater opportunity for unexpected things
to happen, and therefore a greater need to write down what did happen.

Keeping date records of the homework you've assigned, both due dates and
the date it was assigned, is essential.  Whether or not you accept late
homework is up to you, but it is certainly better to have students ask
solicitously in advance for you to accept their late papers, which you
will probably grant, than for students to assume it's OK and be upset if
you don't grant them an extension after the fact.  They are more likely to
do the latter if they get the idea that you yourself don't remember when
the homework was due.  In fact, if I arrive at class early, I often take
the opportunity to write up a reminder of what is due when.  

Organization (and the appearance thereof) also helps students perceive you
as self-confident, a quality which is crucial for an instructor in such a
comparatively free-form classroom.  Of course, one goal of the course is for
them to assume that self-confidence themselves by the time they leave your
classroom.

\subsubsection{Grading}

The message here is the same as in the previous section.  Include in your
course packet a clearly defined grading policy, specifying what portions
of the grade are from homework, exams, quizzes, projects, group work,
attendance, or whatever else you grade on.  Give it enough thought so
that you remember it easily, and can answer their questions immediately.
If historical grade distribution data is available, you would be well
advised to stick to it, since this will help to allay fears that the
unusual pedagogical style will adversely affect their grade.  

\subsubsection{The Bell}

When the bell rings at the end of class, everything becomes exponentially 
harder.  My advice is to watch the clock like a hawk, so that you can
make sure to wrap up the discussion at 2 minutes before the end.  The
discussion usually leaks over an extra minute, giving you one minute to
say any summary comments or give instructions on homework, etc.  This
can be an important routine even when you have little to say: it makes
you seem organized and on top of things.

If you can tell you are going to want to go overtime, because of a red-hot
discussion you want to complete or something that's necessary so they can
do their homework, announce to them 5 or 10 minutes ahead of time that you
will probably be going overtime.  It's best if you've let them go a minute
or two early once before and have mentioned at the time that you're banking
those minutes for such an occasion as this.  If they're working in small
groups at the end of class, it's less crucial but you still may want to
halt them 1 or 2 minutes before the bell for closing remarks --- it's even
harder to get their attention after the bell when you don't already have
the stage.  

In short: {\bf don't ever be surprised by the bell}.


\pagebreak
\section{Writeups}

The write-ups (also called problem reports) are an important part of the
course, because they force the student to communicate his/her knowledge about
the problem.  One consequence is that students cannot hide shortcomings in
their understanding of the problem; another, more to the point, is that the
students will develop the ability to give clear explanations on paper.
Theoretically, a write-up should explain a problem clearly enough from
beginning to end that a student could hand it to a colleague at that same
school, and the colleague would be able to understand the whole problem without
consulting anyone or anything else.  It may well take a while for some students
to develop good written communication skills, but you should be able to
convince them that it is well worth the effort (and indeed most students have
indicated at course's end that they believe it was) --- after all, if a clear
written explanation is harder to give than a clear verbal explanation, then
they should come out well-prepared to explain to their own students.

This section of the guide discusses how to assign write-ups, which write-ups to
assign, and how to grade them --- although these are, in the end, only
guidelines.  At the end are two sample handouts you could give students to help
them get used to writing math, and writing problem reports.

\subsection{How, and how often?}

The latter question is perhaps answered more quickly than the former.  In
general, you might want to aim for one write-up per week.  More than this will
cause you either to spend long hours grading (q.v.) or to fall behind in your
grading; fewer than one every two weeks will not give the students the practice
they need in developing their communication skills, and will also make you skip
some important write-ups.  Overall, you probably want a pretty even balance
among individual write-ups, reflections, and other written work (including
group write-ups and other homework).  You may want to skip assigning a write-up
the week of an exam.

As far as {\em how}\/ to assign the write-ups:

{\bf 1.}  Set some guidelines on the first day of the semester (see also
Chapter 3 for this).  Among other things, set a length of time between the
date an assignment is made and the date it is due.  These write-ups take a lot
of time for the students, so they should have five to seven days in general
(possibly less for the reflections, which are treated in Chapter 5).  You
probably don't want to set a single official length for write-ups, but between
two and six pages is probably the norm.  Do set rules for format, e.g., will
you accept reports handwritten in pen? in pencil?  Of course, also mention what
you will be looking for in general, though more detailed specifics will come
later, when you make assignments (see 3. below).  [Part, if not all, of your
guidelines may come in the form of a handout.  See the example first day
handout in Chapter 3, as well as the one given in the last part of this
section.]

{\bf 2.}  The write-up should almost always be on a problem that you have just
finished discussing in class.  It is a bad idea to assign a write-up on an
activity on which you ran out of time, saying that the students should finish
up on their own for the write-up.  If the class discussion stops short of
a full analysis of some aspect of the problem and you expect the students
to address it in the writeup, it should be something you expected to leave
for them to think about, not something you ran out of time for.  Some suggested
write-ups, such as Squares \& Paths at the end of Unit One, are specifically
designed as out-of-class problems, but you should make such assignments very
sparingly.

{\bf 3.}  You should discuss the write-up with the class for five minutes or so
when you assign it, to make sure everyone knows what you want from them.  This
is especially important in the beginning, when students are unsure how to
explain themselves, and in what generality they should present the solution.
One way to do this is to ask the students, ``What do you think are the
important things about this problem, which we should include in the write-up?''
Then write on the board the suggestions students make, adding your own if
necessary once the students finish responding.  (Of course, this means you have
to decide what you want in the write-up before class!)  Be willing to let the
students argue for or against including certain items, and be careful not to
make them include {\em too}\/ much --- you don't want to read a bunch of
ten-page write-ups!

{\bf 4.}  Most of the write-ups you will want to assign as individual
write-ups: this is one of your big opportunities to evaluate an individual as
opposed to his/her small group.  However, assigning group write-ups can reduce
work on both ends (the students' in writing and yours in grading), as well as
being a nice change of pace sometimes.  You might want to be especially
observant the first time you do this, to see that each group member makes a
more or less equal contribution.  Again, use these in moderation.

\subsection{Which ones?}

Problems on which write-ups are assigned should be significant ones, where
there is some complexity to the solution, and consequently a story to tell,
both in the finding and in the explanation of what was found.  You will often
have spent more than one full day on the problem in class.

Here is a list of the problems on which write-ups were assigned in
the Spring quarter of 2002, not including one problem which has been
removed from the coursepack.  Rubrics have been developed for all of these
and are available in PostScript or pdf.  

\bigskip

\begin{enumerate}
\item Photo Layouts (introductory practice writeup, graded but not counted)
\item Pizzas
\item Geoboards, part (f) only
\item Picture Proofs
\item Length of a Square
\item Surface Area
\item Scaling
\item Changing Units
\item Trisection
\item Rigidity
\item Proofs
\item First Constructions
\item Rigid Motions
\end{enumerate}

\bigskip

\noindent Of course, you may feel free to add others which generated good
in-class discussions, or omit one or two of those mentioned above.

\subsection{Grading write-ups}

You should make it clear to your students both before and after grading a given
assignment what you were looking for, and how you determined the grade in
general.  However, unlike a calculus exam, you should {\em not}\/ feel any
obligation to give them an explicit point-by-point explanation of their grade,
any more than a composition teacher has to explain the accumulation of good and
bad points that resulted in giving a paper a B.  Instructors in this course
have historically used numerical scores rather than letter grades in grading
write-ups, but you should do whatever feels comfortable.

Everyone has a different grading style, but one way to go about it is first to
make a checklist of things you're looking for, or sections of the write-up, and
then read/mark only that one section of all of the papers, to ensure that your
comments will be consistent.  Then, after you've examined each part of the
write-up this way, go back and read each paper individually, skimming where
you've made comments, and get an overall impression from which you can assign a
total grade.  When all papers have been graded, sort them by grade and flip
through them to see if any appear to be ``out of sequence'', in which case you
might want to take another look at those.  Again, don't get too bogged down at
the point level.  The time required to grade write-ups properly increases more
than linearly with the number of papers you have to grade, so you'll be looking
for a middle ground in which each paper gets a fair reading but you don't spend
fifteen or twenty hours grading.

{\em Grading criteria:}\/ Here again, it's up to you.  However, if we follow
the ideas expressed in the sample handouts below, you'll be looking, in
general, for:

\bigskip

1.  A clear paraphrase of the problem description

2.  An account of the method(s) used to solve the problem, including any major
milestones or blocks, as well as ``dead ends'' which nevertheless proved
enlightening in some way

3.  A clear statement of the solution, as complete and general as appropriate,
including an interpretation within the problem context

4.  A clear explanation of why the solution works (or is the only, or complete,
solution)

\bigskip

\noindent Since the nature of different problems can be quite different
(compare Time to Weigh the Hippos with A Base Four Lesson), these elements may
take on different forms.

Problem descriptions should be in the writer's own words, and give a complete
enough description that the reader need not consult anything else (like the
course materials) to understand it.  Also, give credit for creativity, both in
problem-solving methods and in writing up the solution.

\bigskip

One important element of the evaluation process is giving your students good
feedback when you return their papers, both in written comments on their papers
and verbally in class (in more generality).  Encourage them; point out where,
on the whole, the write-ups were strongest, and where they were weakest.  It
may help to give a handout back along with the very first problem report, so
that they can see some of the elements of a good write-up.  If you decide to
make one, try to include good (but not bad) elements from specific papers
(without naming names).  {\em Don't}\/ hand them a complete ideal problem
report, as it tends to give them the impression there is/was a unique ideal
which they were all supposed to guess.  (It also gives students who take this
course in later semesters an easy out for the first assignment.)

Here, for example, is an excerpt from a handout giving feedback on the problem
report on ``Poison'' from Math 105.  (This is not the entire handout!)

\bigskip

\noindent{\bf The winning strategy}

\begin{quote}
``The second player's strategy for playing with ten pennies is to watch what
the first player does and do the opposite.  For instance, if the first player
chooses 2 pennies, then the second player chooses 1 and vice versa.''
\end{quote}

\noindent{\sf It doesn't take a lot of space to do this.  Note that the writer
makes it clear that this is the ten penny strategy, not the solution for 11 or
12 or 210.}

\bigskip

\noindent{\bf The reason it works}

\begin{quote}
``If you pick opposites on every move, the total number [of] pennies taken by
both of us on one turn is 3.  There are 3 sets of 3 in ten.  (3x3=9)  However,
you have to add 1, the \underline{poison} penny, to make ten.  That explains
why when you get down to 4, if you pick first, you will have to lose.  In 4
there is one set of 3 and the poison 1....

\bigskip

By taking opposites, you take three pennies in each turn.  Since 3x3 is 9, plus
the poison \#1 equals 10.  For example, in my Group, if [my opponent] takes 1,
I will take two.  Now 7 are left.  7 is 3x2 plus the poison \#1.  [My opponent]
takes 1, then I take two.  Now there are 4 left.  4 is 3x1 plus the poison
\#1.  She takes one, then I take two.  She is left with the poison \#1.
Therefore, the person moving second, which was me, \underline{wins}!''
\end{quote}

\noindent{\sf Okay, this could have been better.  Neither explanation is
completely clear (and this writer did lose some points for clarity), although
together they tell the story well.  Again, writing out the example game helps.}

\bigskip

\noindent{\bf The general solution}

\begin{quote}
``In order to win the 210 penny game, you would want to move first and take out
two pennies.  This will leave you with 208 pennies still on the desk.  208
divided by 3 gives you 69 with a remainder of 1, the poison penny.  Just as 10
divided by 3 gives you 3 with a remainder of one.  From here on, the team that
moved first would take the opposite of the second team to win the game.''
\end{quote}

\noindent{\sf This was from a different paper than the previous, so the
explanation of why the 10 penny strategy was a winner was somewhat different.
But both of these students, by focusing on why the ``opposites'' strategy was a
winner in the 10 penny game, figured out how to win the big game.  I don't
think it's coincidence that the two best explanations of why the ``opposites''
strategy works accompanied the two best general solutions.}


\pagebreak
\section{Exams}

\subsection{What should they be like?}

An obvious question to ask, given the cooperative nature of the sequence, is:
Can the exams be given in groups?  In fact, students will most likely have
become comfortable working in groups by the time of the first exam, 
and you can be certain that a chorus will rise up to ask this question.  
My reason for not giving group exams is that I want to make sure that
a reasonable portion of the student's grade is based on work they did 
without help.  There is some research showing that students perform better
in cooperative learning classrooms when exams are individual\footnote{
Slavin, R.E.  Research for the future: Research on Cooperative learning and
achievement, what we know, what we don't know.  {\em Contemporary Educational
Psychology}\/ {\bf 21} (1996): 43-49.}, but given the group nature of most
of the rest of the work, I think individual accountability is a much
stronger reason for sticking to individual exams.  You may wish to point
out that the students will not have their groupmates with them during 
their careers as teachers.

Exams, of course, invariably cause anxiety in many students, and you may want
to make at least one of your exams open-notes (perhaps the final, if the
midterm is closed-notes).  This option also allows you to ask more detailed
questions than you might be able to otherwise: questions about particular
activities, or variations on problems assigned as write-ups.  In either case,
do be careful to write an exam that is at the level of the students: On one
hand, it's easy to get carried away and overestimate what your students can
handle.  On the other, sometimes your students will surprise you (pleasantly).

One more note:  You should probably not give more than one midterm plus the
final, as time is precious, and using cooperative learning means a sacrifice on
the content-versus-time scale.  Exams can disrupt the flow of a class, and it's
already difficult to cover as much as you'd like.

\bigskip

\subsection{Final grades}

The course grade will typically have at least one component --- attendance and
participation --- which has historically not been present in traditional math
classes.  It is important to include this, however, as the process and
experiences that take place inside the classroom are the most significant part
of the course, and a student cannot really learn what we want him/her to learn
without being present.  Furthermore, students in the 105-106 sequence
respond to attendance grading with a more than 90% attendance rate, which
helps keep the groups stable.  Typically you should take attendance every day, 
which can be done unobtrusively, by counting heads five minutes or so into
the period, when everyone is working in small groups.  After a couple of weeks
you'll be able to tell more or less at a glance who's missing.  You should set
clear guidelines for attendance at the beginning of the term and stick by them
(see, for example, the sample first-day handout in Chapter 3).

Measuring class participation is more subjective.  By halfway through the
semester, you should be able to tell who is pulling his/her weight in the small
groups, and should encourage those who are not to participate more.  For large
group participation, you may want to keep track of who says something, and mark
it down after class (don't do it in class --- students don't like the idea that
you're taking notes on what they do or say).  This method requires some good
memory on your part, but after a few weeks you'll see who's speaking up and
who's not, and can encourage those who don't to participate more.  One way of
doing so without putting pressure on to know the answer to a question is to
call a student up to the blackboard to serve as scribe in writing down others'
suggestions during a brainstorming type of discussion.  However you decide to
do it, you really will have to be diligent about it from the first day.

As a help in deciding how to break down the course grade into components, here
is the breakdown used by the most recent instructors in 105 and 106.

\begin{itemize}
\item Attendance: 15\%
\item Written work, including quizzes: 45\%
\item Midterm exam: 15\%
\item Final exam: 25\%
\end{itemize}

\pagebreak

\subsection{Exam and Study Guide Database}

\begin{center}
{\Large \bf Midterm Exam, Math 105, Au02}
\end{center}

\begin{flushright} {\large \bf NAME:} \hspace{2in} \end{flushright}

\begin{enumerate}
\item (2,2,1) Compute the LCM of each pair of numbers.  (You may state
the answer in the same form that the numbers are given.) 
\begin{enumerate}
  \item 68 and 72
\vfill
  \item $2^3 \times 3^4 \times 7^2 \times 11^2$ and $3 \times 5 \times 7^3$
\vfill
  \item $x$ and $x^2$ where $x$ is a positive integer
\vfill
\end{enumerate}

\vfill
\vfill

\item (5) A gear with 72 teeth turns a gear with 68 teeth.  How many 
rotations must each gear make before both gears are simultaneously back 
in their starting position?  

\vfill
\vfill \pagebreak

\item (1,2,3,2,2) Let $S$ be the set of all subsets of the set $\{ a , b \}$.
\begin{enumerate}
  \item List the elements of $S$.
\vfill
  \item Make a table for the binary operation $\cap$ on the set $S$.
\vfill
  \item Is $S$ closed under $\cap$?  Is this operation commutative?   
associative?  
\vfill
  \item Does $S$ have an identity element for the operation $\cap$
and if so, which element?  
\vfill
  \item Is there an inverse element for $\{ a \}$ with the operation $\cap$,
and if so, what element?
\vfill
\end{enumerate}

\vfill \pagebreak

\item (6) In a room, there are three students for every professor.  
A professor walks in; now there are 2.5 students for every professor.  
Write equations to represent your knowledge and determine how many
people are in the room now.  [Only 2 points on this problem will be
given for a solution by trial and error solution with no equations.]

\vfill
\vfill
\vfill

\item (1,1,1,1) Mark each statement True, False, or Can't Tell, assuming
you know that all Glyck numbers over 100 are even or prime:
\begin{enumerate}
  \item 105 is a Glyck
\vfill
  \item 75 is a Glyck
\vfill
  \item There are no Glycks
\vfill
  \item There are no three-digit Glycks ending in a 5
\vfill
\end{enumerate}

\vfill

\end{enumerate}

\pagebreak

\begin{center}
{\Large \bf Final Exam, Math 105, Au02}
\end{center}

\begin{flushright} {\large \bf NAME:} \hspace{2in} \end{flushright}
\vspace{.2in}

\begin{enumerate}
\item (2,2,2,2)
\begin{enumerate}
  \item Find the prime factorizations of these two numbers: 
    \begin{itemize} \item 777,000 \item 74,000,000 \end{itemize}
  \vfill
  \vfill
  \item Compute the GCD of these two numbers.
  \vfill
  \item How many factors does 74,000,000 have in all?  
  \vfill
  \item Write down a number with exactly twice this many factors.
\end{enumerate}
\vfill
\vfill

\item (4) Order these numbers from greatest to least and give 
one-sentence explanations for why the greatest is the one you chose 
and why the least is the one you chose.
\begin{enumerate}
  \item $.0000000000012345 + .000000000009876$
  \item $.0000000000012345 \times .000000000009876$ 
  \item $.0000000000012345 \div .000000000009876$
\end{enumerate}
\vfill
\vfill

\pagebreak

\item (6) All 25 cars recently sold at Vic's Used Car Lot have either anti-lock
brakes or four-wheel drive.  16 of the cars have anti-lock brakes and
13 have four-wheel drive.
\begin{enumerate}
  \item How many cars have both anti-lock brakes and four-wheel drive?
  \vfill
  \item Draw a Venn diagram illustrating the situation
\end{enumerate}

\vfill
\vfill

\item (4) Find two prime numbers that differ by 7, or prove that 
there aren't any two such numbers.

\vfill
\vfill

\pagebreak

\item (3) Two room-mates are in a grocery line.  The first buys milk and eggs
and the second buys beer.  The next two customers are also room-mates.
The first buys milk and the second buys eggs and beer.  Tarzan, 
the cashier, is surprised to see that both households spent the 
same total amount.  Of which law is Tarzan apparently unaware?
\begin{enumerate}
\item The commutative law for multiplication
\item Order of operations
\item The associative law for addition
\item The commutative law for addition
\item That multiplication distributes over addition?
\end{enumerate}
No explanation is needed -- just circle the correct letter.

\vfill

\item (3) Find a fraction lying between ${\displaystyle \frac{22}{7}}$
and ${\displaystyle \frac{355}{113}}$.  Give a one-sentence argument
that your fraction does indeed lie between the other two.

\vfill

\pagebreak

\item (2) How many license plates are possible in a state whose plates have
two letters followed by four numbers?

\vfill

\item (4) A student comes across the fraction  ${\displaystyle \frac{120}{190}}$
and simplifies it to ${\displaystyle \frac{12}{19}}$ by ``cancelling the
zeros''.  Later, the student comes across ${\displaystyle \frac{113}{263}}$
and simplifies it to ${\displaystyle \frac{11}{26}}$ by ``cancelling the
threes''.  Your job is to explain to the student why cancelling the zeros
is OK but cancelling the threes is not.  Your explanation must be general,
not simply showing it does or does not work for one or several examples.

\vfill
\vfill

\pagebreak

\item (4) It is often said that 10\% of the world's population owns 90\%
of the wealth.  Suppose that each of the wealthy 10\% is equally wealthy
and each of the poor 90\% is equally poor.  How many times wealthier is 
a wealthy person than a poor person?

\vfill

\item (2*) {\em Extra Credit Problem}: Describe which fractions become
terminating decimals in base six (decimals in base six are properly 
called ``heximals'').  

\vfill

\end{enumerate}

\pagebreak


\section{Materials and how to use them}

\subsection{On-line handouts}

Files are included for several types of on-line handouts
we developed as the quarter went along.  They may be found under
the ``more stuff'' link.

First, every week we gave out a summary of what they were supposed to
have gotten out of that week's activities.  We did this partly because
the material is difficult enough for some students that they run out
of gas before getting to the end of a problem, let alone having time
to think it over and assimilate the ideas.  They are liable, even when
looking back later, to miss the point.  Also, there is little redundancy
in the material so it helps for them to have a list of general ideas 
exemplified by each worksheet.  They find this useful when studying
for exams.  It is also a way of conveying to them what they are responsible
for learning.  

Each instructor needs to put out their own such summary of course,
since at the very least, the pace will vary from class to class.
We provide the html file in the hope that it will be useful for
cutting and pasting, or for prompting the rushed instructor to
notice all the ideas embedded in the worksheets and point them out
to the students.  {\bf At present, only the first six weeks
of summaries are available.}  

The second electronic hadout concerned guidelines.  We made this   
mandatory reading on Day~1.  This concerned proper etiquette for group
discussions and for writeups.   The third concerned the proper format
for writeups.

In addition, consider making the Readings in Sections~6.1,~6.2 and~6.3
available online or bringing them as handouts the first day,
as they are important and students may not get coursepacks
immediately after the first class.

\subsection{Solutions and such}

During the course of the quarter, we handed out model solutions
at various times.  These are collected in the Solutions subdirectory,
and of course, are accessible only by permission.  In some cases,
we gave out contrasting bad solutions -- these are included for what 
they're worth.  The good and bad solutions were contrastingly 
annotated in pen, but this annotation is not on the online versions.

Additionally, scoring rubrics were posted religiously after each 
assignment was due.  We tried to make sure the rubric was posted 
before the re-do was due, since we felt that students doing the re-do
were the ones who missed substantial points (except at first when they
all want to do re-do's) and therefore they needed to be hit over the
head with what our expectations were and why they missed points. 
Since improving their communication skills is the paramount mission
in this course, and since the rubric highlights deficiencies here,
we made sure that papers were graded in such a way that they could
see exactly where they lost points on the rubrics: scores would
be written as $(3+1+2) + (2+2) + 6 + 2$ if the rubric identified
four scoring categories, with three subcategories in the first and 
two in the second.  All the rubrics are included in the Rubrics
file in the Solutions directory.  Source code is in the LaTeX directory.
{\bf At present, rubrics are only available for nine of the worksheets
that were assigned for writeup.}

\subsection{LaTeX materials}

If you want to use any of the worksheets, solutions, and so forth, 
in some form other than exactly what is provided, you will probably
want the LaTeX source files.  
The LaTeX directory has source files for the coursepack and for all
the auxilliary materials you might want to use -- exams, solutions and
so forth.  You may download them as needed and adapt them at will.
You will need to obtain permission to access the LaTeX directory.

{\bf At the very least, even if you don't know how to mess with LaTeX,
you will need to remove the two-page sample syllabus that's in
the courspack and substitute your own syllabus.}

Figures were made using xfig and exported both in eps and pdf.
All figures are in the figures subdirectory of the LaTeX directory.

All contributions of new materials will be gratefully accepted by e-mail to
pemantle@math.upenn.edu.  More grist for the exam problem mill would
be particularly useful.  

\end{document}

