# A Handbook for Mathematics Teaching Assistants

### Table of Contents

- Introduction
- Types of TA Assignments: Recitation, Lecture, Grading
- Before You Teach: A Checklist
- Day One
- What Goes On in Recitation?
- What Should be on a Syllabus?
- Lesson Planning: Survivalist Tactics
- Grading Issues
- Cooperative Learning
- Technology
- Writing Assignments
- Making Up Exams and Quizzes
- Using Cognitive Models to Make Appropriate Problems(with Mary Ann Malinchak Rishel)
- The Active Classroom
- "What Was That Question Again?"
- Motivating Students
- How to Solve It
- Course Evaluations
- Get Along with Colleagues
- What is a Professional?
- Teaching Methods for Various Types of Classrooms
- Problems of and with Students
- Student Types: Who is the Audience?
- How to Get Fired
- Advice to International TAs
- Silly Stuff...
- ...And Not So Silly Stuff
- The Semester in Five Minutes
- Jobs, Jobs, Jobs
- Letters of Recommendation
- Mathematical Talks
- Becoming a Faculty Member
- University and College Governance
- What Does an Evaluator Evaluate?
- The Essence of Good Teaching
- Case Studies

### Introduction

This is a text about teaching college mathematics.

At most every juncture in the text, I emphasize nuts and bolts considerations over theory. This is not because I believe that theory does not exist or is not important, but because I think that good teaching starts with seeming trivialities--"talk loudly, write large, prepare carefully, explain a lot, be friendly." Only after we are familiar with such simplicities do we begin to feel comfortable moving into theories of learning. This last is not to say that such theories are never useful or important--otherwise, Mary Ann Malinchak Rishel and I would not have written the long section on how using cognitive methods can lead to better examinations, for instance. However, I do think that you, as a graduate TA or a young faculty member, will profit more and improve faster from short, simple, clear suggestions that have immediate and obvious impact in your day-to-day classroom. If this improvement leads you to decide that you want to think more deeply about your current and future teaching, so much the better. Send me e-mail so we can talk trishel@maa.org.

Finally, let me address a very common view about the discipline of teaching; namely, as I was told again just last week, "Teaching can't be taught." Well, maybe, just maybe, great teaching is lightning in a bottle and can't be explained, but I claim emphatically that good teaching can be taught. Of course, I am biased in my view, if only because I have spent the last twenty years (roughly) trying to achieve this aim. But, in fact, I believe not only that teaching can be taught, but that if mathematics is to progress, it must be taught -- to the bright young people who will carry it on after us. I hope that, by the end of this volume, you will agree with me.

So, let's stop talking and get to work...

### Types of TA Assignments: Recitation, Lecture, Grading

Probably the most common TA assignment in mathematics, and the one with which the majority of the faculty began their careers, is that of recitation instructor. Those of you who have received an undergraduate degree from a large university will be familiar with the lecture-recitation format: a faculty member lectures to a large class of students two or three times a week on an assigned topic from a textbook, after which a graduate student answers questions about the lecture and discusses assigned homework problems. In this format, the lecturer decides which homework to assign, and often determines the structure of the recitation. By this I mean he or she may say: "Don't do all the problems; just the ones that are designated not to be turned in for grading." Alternatively, the lecturer may suggest that you begin each recitation with a couple of "example problems." Generally, however, most instructors will give you little or no advice, except to say something like: "Just do a standard recitation." (For a sample "standard recitation," whatever that may be, see the later section, What Goes On in Recitation.)

Another common assignment for TAs is to be asked to lecture. Schools vary as to when in a graduate student's career this is to be done; at some institutions you are handed an algebra and trigonometry text and told, "Go teach this. Don't mess up!" Other schools wait for a year or two until you have had some less demanding assignments before they ask you to plan lessons, make up your own exams, determine grading policy, and generally deal with the problems of teaching undermotivated freshmen (or worse, undermotivated seniors!) the joys of precalculus.

It is probably worth pointing out here that at some point in your graduate career you should pursue a lecturing assignment, for two basic reasons:

1. A graduate student who has lectured has a real advantage in the job market (see the section, Jobs, Jobs, Jobs).

2. By lecturing before you take a first faculty position, you remove some of the stress over teaching that goes into the tenure-pressure.

A third common TA assignment is that of grading, sometimes in an elementary course, more often in an advanced undergraduate or even a graduate course. Many TAs describe such assignments as "easy" or "boring." While the assignments can be either or both, grading jobs, however, can teach you how far you have come since the days when this coursematerial was a real effort. These assignments can also show you how hard it is to teach others to write clear, concise answers and proofs. A third benefit to a grading job is that you can use it to review the material that may be asked on a graduate comprehensive examination. I will say more about the questions involved in grading papers later on in the section titled Grading Issues.

For now, think about:

Which type of TA assignment appeals to you most now? Is there one that you might never want to do? Do you think that your opinions might change later on in your career, or are they set in stone?

### Before You Teach: A Checklist

- Do you have keys for your office or your classroom?
- Are the classrooms going to be open?
- Will you find chalk and erasers in the classroom?
- Where is your office? What is your office number?
- What is your telephone number? Your email address?
- Can you get a desk copy of the textbook for your course?
- Can you write in the margin of the textbook? Or will you have to return the book at the end of the term?
- Where is the library?
- Where are the restrooms?
- Can you get pencils and paper from the department office? or do you have to buy your own?
- What is the policy on making copies of exams for your class?
- Are the copy room and mail room accessible during evenings and weekends?
- Where can you find the class schedule so you know where you are teaching?
- Can you get old syllabi for your class? How about last year’s exams?
- Can you get an overhead projector and transparencies? Do you even want these things? How about a projector for calculators?
- How long is the semester? Is the exam schedule made up in advance?
- How many students might you expect to see in your class?
- How much grading are you expected to do?
- Can you get your teaching schedule changed easily?
- What is a typical workload for a new TA?
- And, where do you get your paycheck?

### Day One

- Call the roll but don't stop here.
- Hand out, read, and answer questions about the syllabus.
- Explain how you intend to handle classes.
- Discuss the nuts and bolts of homework, exams and grading.
- Offer an overview of the course.

Now let's discuss some aspects of each of the categories above.

Handing out a syllabus is another common first day activity. If you are new to teaching, you will have many questions as to how to construct such a syllabus, some of which can be answered in a later section, What Should be on a Syllabus.

Many instructors assume that students will read what is handed to them; I think this is incorrect. Every time I hand out a document, whether it be a syllabus or a homework assignment, I read it to the students. By reading through the syllabus, I allow students to ask questions that I may not have answered clearly in my text, and I also ensure that, within reason, students know what is required of them. First-time graduate students are often teaching first-time undergraduates. The undergraduates need to know how college operates: "Should I bring my textbook to each class?" "Will you collect homework every day?" "Do you answer questions during class, or do we wait until later?" "Do you grade on attendance?"

More advanced students will have questions, too. Maybe they have never had a mathematics course in college, or more likely, they just want to know what the rules are: "I have lots of job interviews this semester. Do you require attendance?" "Will you have answer sheets in the library, the way they did last semester?" By the way, there is nothing wrong with your answering, "I don't know; I'll check it out let you know next class." Just make sure that you carry out your part of this bargain and give them a definite answer at the next class. As to more specific comments about how class is to be handled, we will return to this topic in the section, Types of Assignments. Students want to know whether and how often homework are going to be collected. Will you grade each problem, or only some? How will they know which? Do you have an idea of how you'll assign grades to the homework? For instance, will you use a numerical system where each problem is worth, say, from zero to five points? If you know what system you or the course leader is using, now is a time you can tell the students.

Similarly, you can describe when you will give exams, and whether they will occur in class or in the evening. You can also describe where the exams will be given, for instance, in a large lecture hall with 400 students, or in the classroom. You can also tell your class that "You will have ninety minute exams, and I will show you some old exams for review."

Then you can explain what you know of the final exam and grading policies. Is the final cumulative? Does it have the same length as the other exams? Does it count for more points than the earlier exams?

There are other bits of information you also should give: The names of the texts for the course, your office hours, and any supplemental texts or materials you will use.

Now that you have spent about twenty minutes on the nuts and bolts of the course, it is time to turn your attention to content. What are the topics your students will be learning? How do those topics relate to other subjects they may be studying? In what ways will the material be useful in the real world?

Let's be more specific about details; many of you will start teaching with a first semester calculus course. You may want to say something like this:

Calculus is usually split into two types: differential and integral. Differential calculus deals with instantaneous rates of change: how things change right now, not over six years or ten miles (those are average rates of change), not over six seconds or six one-hundredth of a second, but right now, this instant. We will be learning about this instantaneous change this so-called derivative, how to find it, how to manipulate it, and how to use it in problems from physics and chemistry to business and economics. For instance, if the instantaneous change takes place over time, then this derivative is the velocity of the object that is moving, and this concept is of special interest to physicists and engineers; it is one of their tools for explaining the physical world. When Isaac Newton wrote F = ma, for instance, he was saying that forces are related to acceleration, and acceleration is a derivative, a rate of change.

Scientists are not the only people interested in calculus. Economists and business people also use the subject; for instance, the cost of doing business changes essentially instantaneously over time; this change of cost is called marginal cost. Monitoring marginal cost allows businesses to track their changes today, not over the last twenty weeks or twenty months.

Then you might go on to explain how taking a derivative requires having a function to work with; thus you will begin with a review of some continuous and not-so-continuous functions. After that, you can say that you will go on to talk about various methods of taking derivatives of more and more involved functions, and then you will discuss some applications of derivatives, such as how to maximize and minimize profits, say, or maybe velocities, or areas of land.

At this point, I will leave as an exercise for you can decide what you might want to say about integral and/or differential equations. Meanwhile, let's shut the door on this first day calculus class, and move down the hall to the precalculus class, where a more "activist" discussion has begun:

Instructor (I):"... and we'll also talk about functions. Maybe some of you have seen some functions, like, say, polynomials. Can you name some functions that are polynomials?"

Two students together (S1and S2): S1: "Sure. y = ax^{n} + bx^{n-1 }+..."

S2: "Unh maybe x^{2}?"

I: "O.K. y = x^{2} works. It's a polynomial. Any others?"

S_{1}: "x^{3}?"

S_{2}: "How about y = x^{2}+ x + 1?"

I:"Yes." [Writing both polynomials on the board.] "Anything harder?"

S_{3}: "How about the square root of x?"

I: [Writing y = x = x^{1/2} on the board.] "That one doesn't work. Does anyone know why? "

[Silence. Then]

S_{1}:" Cuz one-half is wrong."

I: "Good. One-half doesn't work as a power, right? I mean, y = (1/2)x^{2} is a polynomial, right? [Pause] So, *this* 1/2 points to the power in x^{1/2} doesn't work--I mean, it's not 'legal' for being a polynomial, although it *is* 'legal' for being some kind of function, yes? (This [points] is called a power, by the way, and the other is a coefficient of the polynomial. We'll define these terms pretty carefully during the course..."

[A couple of minutes later.]

I: "How about some other kinds of functions? Have any of you heard of trig functions? Can you name some?"

S_{1}: "Sure. y= sin x."

I: "Yep, sine works. We'll study it, and the others, like cosine and tangent and why they're all different from polynomials. 'Sine's' picture, by the way,is, sin (x) right? And, it comes up in spring and pulley mechanisms, and electrical stuff, and things like that, and..."

Let's tiptoe away now, we get the idea.

This last instructor can teach us a lot about managing the classroom. Notice how she accepted the answer she *needed* to her first question, rather than going with the seemingly more complete response from Student 1, who obviously knows a good deal of the material she may be spending the semester teaching to the others in the class. She also did a good job adapting to the incorrect answer y = x^{1/2} suggested by Student 3. She did so without emphasizing the student's wrong answer; in fact, she turned a common mistake into a learning experience for the entire class.

There are many good points to the classroom discussion we have just witnessed, but in the interests of keeping the discussion short, let's just say the following: Most people say that teaching precalculus is boring, boring, boring, but this particular instructor doesn't make it seem so.

**Exercises:**

Which of the two methodologies described above for a first-day discussion of course material would you be more comfortable with? Fill in the details of what you would say to a first semester calculus class about the topics of integration and differentiation. (Your answers may be nothing, of course, but you should then have an explanation based on the syllabus.)

### What Goes On in Recitation?

Why do you want to prepare meticulously when you know this stuff so well? Because:

In this section, I have emphasized the importance of being prepared in teaching recitations. Preparation is important, but it isn't the only thing. For more advanced advice, see the sections on The Active Classroom and Motivating Students.

**Exercise:**

Name some of the topics you think I have slighted or ignored in the above discussion. How essential do you think they are to good recitations?

### What Should be on a Syllabus?

Enough said about how to find old syllabi; now, what should *yours* describe?

Also on the syllabus, discuss homework, exams and grading *in general*; if you try to get too specific about requirements, students will come back to tell you how you have "*changed your syllabus"--"unfairly,*" of course. (For more details about grading schemes themselves, see the section entitled Grading Issues. For now, we will stick with what goes into the syllabus.) Will you be assigning homework by the class? By the week? The month? The entire semester? Will you collect and grade all the homework, or just some? If just some, will you be announcing in advance which ones you will grade? When will you collect these problems, e.g., "Right at the start of class each Monday"? Do you want the homework written out in any particular format?

Then there are exams. Do you know when they are to occur? If so, put that information into the syllabus, along with any other details you may have, such as how long the exams will be and where they will take place.

As far as grading is concerned, offer a general statement like, "...three equal exams, along with a comprehensive final exam counting double [alternatively, one-and-one-half?] the value of the exams. Homework and class participation will also count about ten percent of the total grade." In this way, you offer the students a framework, while at the same time allowing yourself some leeway-- "what is class participation" for instance, and how do you propose to measure it? Yet, it's logical to suggest that such participation is worth something, and you do want to have a mechanism for rewarding students who make an extra effort.

At this point, syllabi often diverge, depending on course, material and style. For instance, I have seen a few-- *very* few, actually -- faculty put a short description of their academic credentials in the syllabus. Others, especially those who are teaching in a fairly nontraditional way, will feel the need to describe the classroom situation as they see it happening. For instance, they might describe how their "project-oriented" calculus sections will work, what kinds of writing assignments they will offer in their geometry class, or how they will handle group work in their precalculus class.

Regardless of what you put into your syllabus, it would be well to remember that this document takes on the character of a contract with the students; you are telling them what you plan to do, and in turn what you expect them from them. Thus it behooves you to take a little care with what you write. You might consider passing it by an older, wiser faculty member for approval.

Courses often require unplanned or unexpected changes in midstream. Most of these are acceptable to students. On occasion, however, some adjustments you understand to be minimal or benign will elicit an unexpected outburst "Why are you canceling exam three? I was counting on that one to boost my grade! You can't do this unless the entire class agrees, you know!" And then, heaven forbid, if you decide to "take a vote" on the question, those students with higher grades plus those who just don't want to take an exam along with those who just want to get the course over with will simply outvote the three really angry ones who want the exam. The ultimate outcome is that you end up giving an exam you hoped not to give, while you have lost the respect of, and authority over, your students.

No one can avoid all difficulties or see all the possible problems about to appear. However, you need to think carefully about your syllabus before you start the semester. That and getting input from colleagues is a strategy that will make for a more coordinated course. The outcome of such planning will then be better for you and for the students, and will make your course less work in the long run.

**Exercises:**

What goes into your course? What would you add to the above syllabus? Is there anything that you think should be subtracted from the syllabus, and if so, why? How would you resolve the problem discussed in this section of the student who wants to take the third exam? Is he being unfair? Are you wrong for suggesting that the exam be dropped?

### Lesson Planning: Survivalist Tactics

In this section, let's consider the most basic aspects of lecturing. Later, in sections called The Active Classroom, and Motivating Students, we will look at more refined aspects of making such plans.

Once, some years ago when I was a graduate student teaching a night course in third-semester calculus, I got the twenty-four hour flu about an hour before class started. My office mate, being a very kind person, offered to substitute for me. "Just tell me which section you were supposed to do," he said.

The next day, after I had sufficiently recovered, I asked my officemate how things had gone. "Fine," he replied, and went on to tell me how far he had gotten in the material. "But how did you prepare?" I asked. "Easy. I went in to class, announced that I was substituting for you, asked to borrow a copy of the text, and gave the students a five minute break while I looked over the author's approach to the material. Then I made up three examples of varying difficulty, and I went with it. By the way, how are you feeling?"

So there it is; a basic plan for lesson plans ("a plan for plans"), courtesy of my officemate.

Start by finding out what today's topic is supposed to be.

Peruse the text to see how the author approaches the topic; this helps you preserve the same notation as the text, among other things.

Prepare an intuitive explanation (a "heuristic argument") as to why the topic is important, useful, and relevant.

Next, prepare a few homework-style problems of increasing difficulty to illustrate to the students the main concepts of the section of the text.

Allot remaining class time to answering questions or doing old homework problems.

Of course, this methodology doesn't solve all problems. If it did, teaching would be trivial. So, let's discuss some of the issues raised in the above outline more fully.

One complaint often voiced is "But I don't like the way the author does this section. Why should I encourage bad mathematics?"

Fair enough. Even though we may not have had a choice in the textbook, the students will still be using it for explanations, exercises and homework. We can offer alternative proofs or better methods, but if the students are getting their homework from the text, they would rather not have to keep "translating" from our language and symbolism to the author's. Thus, we owe it to the students to at least say, "Here's how the author approaches... An easier [more common, better, more useful, more sophisticated] way is as follows. On the homework and tests, use whichever method you like best. I don't care as long as you get the right answer and can explain your method.

Another common objection is that we should not use "heuristic argument" rather than an "honest, direct, complete proof."

This suggestion may simply be a function of audience level (discussed more fully in the section Student Types). Clearly, if you are teaching the intermediate value theorem in real analysis or topology, you will want to consider the roles compactness ands connectedness play in the discussion. But, for a freshman English or biology major, some pictures of continuous and discontinuous functions that have positive y-values at x = 1 and negative y-values when x = 3 will be much more convincing than an unintelligible, unmotivated "formal proof."

This last is also not to say that you can't be lucky and draw a class of eager students in an enriched calculus program for potential mathematics majors but now we are back to the Student Types question.

A third objection is, "Why do examples? They're right there in the book."

You're right; there are worked out examples in the textbook. But, first of all, many, if not most, students donÃ?t read the book. Second, not every detail of the examples is spelled out in the author's exposition. Further, it isn't always necessary to choose the examples in the text; many instructors I know don't choose the author's exercises. Instead, they opt for a few problems "near" the assigned homework problems, telling the students, "If you understand how to do these examples I'm showing you, you'll have a great start on tonight's assignment." The underlying message is the "great motivator": "It's worth watching me do these problems, because they're like the ones you'll be trying soon."

One more objection to the proposed lesson plan is often brought up: Is the suggested allotment of time for a lecture correct? That is, how can you leave so much time for questions and homework? Don't you need all that class time to explain the details of the current topic?

This, too, is a reasonable objection. Different instructors find that they take different amounts of time to explain details of a lesson. Still, I try to find ways to leave time for student questions; otherwise, how do I know whether they are absorbing the material I claim to be teaching them? The best way to find out if my lecture is being received is to give the students a chance to tell me what is still bothering them. I will return to this topic again in the Active Classroomsection.

### Grading Issues

A zero to five scale is probably better:

"0"-- didn't even try the problem,

Uniformity and fairness are related to one another. You may be a harder grader than your officemate, but if you can defend your methodology to other TAs and students, they will "generally" accept it. (Note that last generally. Some may not; see the section on being a good colleague.)

Most TAs see the "speed" part of "grading with speed" as only being of benefit to themselves--"I want to get this pile of papers done and out of here!" But speed with accuracy also benefits students, because they get to have their problems back while they still remember what the questions were.

To aid in speediness, try some of the following:

- Grade problems "backwards" if the answer is correct, you can scan the earlier parts to see if the details are there.
- If a few students have a unique, strange method of solving a problem (this happens maybe five percent of the time), put these papers aside for a while until you can let your subconscious work on where the isolation may have come from.
- Do not write long comments on the examination papers; use the advice given earlier in this section.
- Make an answer sheet to hand out to all the students. Go over the answer sheet on the day on which you hand back the examination copies. If students feel that they did not get graded uniformly, you can make adjustments right after class.
- Don't fight with students over problems that were obviously graded incorrectly; at the same time, don't capitulate over every request for a regrade just because it was asked for.

If you are grading a full class of exams--thirty students, the entire set of exam questions, say--grade problem one for each student, then go on to problem two, etc. In this way, you will ensure more uniformity. Also, try to grade each individual problem in one sitting; take a break only after you have seen all the unique, exotic methodologies the students can come up with. (By the way, I find that I can sometimes bribe myself into grading by promising that I'll take a break as soon as I finish these last eight copies of problem 3. Not being very bright, I'm usually able to use that argument to convince myself to work ten more minutes.)

Uniformity has other benefits. Among them: It leads to fewer re-grades, which take a lot of time. It also makes for more defensible scores, so that students consider the grading (and the grader) fairer.

After you have graded as many homework, quizzes and exams as you can stand, you will have to assign final grades. Each department of each university and college seems to have allowed its own system to evolve and each of these systems is like each other, but not quite. For the bare-bones description of one such system, check out the grading section of What Should be on a Syllabus. Note, however, that this section is not completely forthcoming as to how allocation of final grades is done in an standard class. Well, let us lift the veil.

I am occasionally in charge of a large number of calculus sections, for instance 26 sections averaging 20 students each. Thus, by the end of the semester, we calculus instructors have in the range of 500 grades to assign. Assume that we give three exams during the term (these are called prelims where I come from), each worth 100 points. We also administer a 150 point final exam (yes, it is called a final); and additional materials, such as homework and quizzes, add up to 50 more points. Thus students can earn a total of 500 points. If the exam is scheduled for a Tuesday morning, we will spend that afternoon, Wednesday and Thursday grading. By late Thursday afternoon or Friday morning at the latest, each instructor will have collected finals, recorded grades, and totaled raw scores. (Of course, there are always one or two instructors who have failed to do the above; they should read the section Get Along with Colleagues.) We then have a meeting at which instructors put up the raw scores of their students. This we do in ten point intervals, from 500-491 to 210-201. Numbers below that fit into the 200-0 category.

We find the median grade (not the mean), and assign to its ten-point interval the set of last B- grades. Working up and down the intervals, we then assign an A range, a C level, and a D range.

Interval | Total | Sum |

491-500 | 3 | 3 |

481-490 | 8 | 11 |

471-480 | 14 | 25 |

461-470 | 20 | 45 |

451-460 | 16 | 61 |

441-450 | 20 | 81 |

431-440 | 16 | 97 |

421-430 | 22 | 119 |

411-420 | 28 | 147 |

401-410 etc. |

Having put together a curve based on class scores, we now assign letter grades to each student in each section. We are not done, however. In each section there are grades that are anomalous. Some students have one grade that is much lower than the rest, say. Others have a rising set of scores, e.g., 49, 62, 87, and 130/150 on the final, showing that they maybe have caught on later than others. Occasionally, students will have personal problem. I discuss each such anomaly about fifteen- percent of the total -- with the individual instructor, and we come to some sort of consensus. We seem to end up raising about half the grades, but no single grade ever goes up more than one level, e.g., from C+ to B-.

One of my general feelings about grading is that students always do less than or equal to their best on individual exams; but still, there must be two or three exams where they perform to expectations theirs or mine. Further, good homework and classroom questions may show interest, but they are a precursor of good exam performance, not a substitute for that performance. And finally, I have a thing about the grade of A+; I will never raise a total below 490 points out of 500 to an A+. To my mind, an A-natural is a perfectly wonderful grade, and I won't apologize for giving it.

### Cooperative Learning

### Technology

**Rule Two:** Make sure your examples justify the technology you are using.

**Rule Three:** Make sure that your overheads and displays fit.

**Rule Four:** Realize that teaching with technology is not the same as lecturing.

**Rule Five:** Be prepared for total system meltdown.

### Writing Assignments

A variety of small writing assignments are usable in the mathematics classroom.

### Making Up Exams and Quizzes

So what are some of the steps you can use to avoid pitfalls?

Suppose you have decided on a five question exam, based on the fact that there were seven major topics since the last test, and one of those is easy enough to skip, while a second can be embedded in a later, more important topic. Make about forty percent do-able by anyone who stayed awake long enough to watch you show some examples on the board. (This is one of the reasons I don't want to pass students who cannot get a 40% average on my exams. See the section on Grading for details.) The forty percent do not have to be just like trivial homework, by the way; you might split some of your five problems into easy, moderate and difficult sections, thereby spreading the easy stuff around the exam.

Now you have sixty percent of the test left for more challenging material. Half or a bit more of that can be similar to some of the more interesting examples and homework problems the types of problems that make students think, but this group had a chance to do that thinking last week while they were doing their homework exercises. I basically never give the students assigned homework problems on the exams, by the way, although I do know some people who do. I just feel that using old exam questions as homework problems often makes the students feel that the instructor didn't really put an effort into the preparation.

Assuming that about twenty percent of the exam is still to be constructed, it's now time for you to think of a more challenging question -- or parts of questions. Now is the time to think, "What is the essence of the material I have been teaching for the last four weeks, and how can I ask the students to show whether they have absorbed that essence?" This does not necessarily mean asking them to formulate a proof; rather it should indicate that you could quiz them about some fundamental points that you have been making repeatedly during your excellent lectures. One effect of putting such questions on the exam is to increase attendance at the rest of your excellent lectures--"Wow! If I go to class, it might help my grade on the next prelim!"

For many more details on how to make examination questions that hit the mark, try the section Using Cognitive Levels to Make Appropriate Problems.

Let me say a bit more about finding challenging problems. Early in my career, I used to expend real energy trying to fashion a problem that would force students to use current knowledge to discover something new. For instance, I might be inclined to write, "You've seen exponential growth. Well then, now I'll ask you to find out about logistic growth all on your own." These well-meaning attempts almost always turned out very badly. The "numbers" would turn out to be too messy, and the concepts were too far from the students current awareness. Further, thirty minutes or so was simply too little time for serious thought. So, eventually, I came to the realization that at best I could formulate a couple of problems that proceed from easy to difficult, with the difficult part counting maybe only five points. ("You couldn't get that part? Well, good thing it was only worth five points. OK, let me show you how...")

How do I do this? Glad you asked. Let's go back to the exponential growth into logistic growth problem. We split it into four parts, each worth five points:

In problem four, you found the rate of growth of a strain of bacteria. Now let's suppose that the bacteria are growing in a lab on a circular Petri dish whose area is 5 cm^{2}. Thus it is fair to assume that the area, A(t), covered by bacteria in the tube at any time t is governed by the equation

a) dA/dt = k(5 A(t)). If you know that A(0) = 1 and dA/dt = 0.2 at

t = 0, what value do you get for k? Is this k value positive or negative, and what does it tell you about dA/dt?

Now writing b) dA/dt = k dt

5 A(t)

for your value of k,

solve this equation for A(t). Your solution in part b) will have an arbitrary constant in it. Calling that constant D, find its exact value. Using your final solution to part c), make a reasonable argument that A(t) is never larger than 5. What is your reasoning for this?

Notice that the above is still not an easy problem it wasn't supposed to be. However, the first part should be manageable for any student who understands what you have taught about exponential growth and decay. Part b) is harder, of course, except that you have already separated the variables in the equation the students have to solve. Even if students found a wrong answer to b), you can still grade part c) as if b) was correct. Thus they can still receive credit for part c) without getting very many points for b) at all although they do have to get some kind of reasonable answer for b). That leaves part d). It's not easy, "but hey, at least it's only a five-pointer, right?" If you now design one more problem with a hard five-pointer as part d) and you are done. On this exam, it'll be easy to get at least 40%, the average should be around 70 or 75%, and more than 90% shows that the students have worked.

Before moving on, let me make a comment about quizzes: I tend to make them relatively easy. For instance, if I lecture on the chain rule on Friday, and maybe I've shown the students how to find the derivative of sin^{2}(3x), I might then ask them to use the last five minutes of class to find the derivative of cos^{3}(2x). Once the students see that the quizzes are reasonably easy, they have incentive to come to class and listen carefully to what I am teaching. Further, the quiz is then easy for me; I can sometimes finish grading in the fifteen minutes between classes, if no one stops me to ask questions.

### Using Cognitive Models to Make Appropriate Problems

Let us now aim for higher levels of questions by going to the next level in Bloom's model.

2. An Example from Infinite Series.

Find the limit of the series: 3/2+3/8+3/32+3/32+

To gauge the students at the level of comprehension, I propose that they:Find lim[1 - 4 - 6 + Î£(3/2)(1/4)^{n}]

The following qualifies as an analysis question in a freshman class:

Find all a, r such that Î£ar^{n }converges.

This last is a possibility for either a longer quiz question or an examination problem.

The next question, a "synthesis" problem, was used as a major part of a final exam:

**3. Levels of Cognition and Tutorials.**

Once we understand levels of cognition, we see that we have been using them in everyday situations:

Further, you can address synthetic and evaluative questions:

Return to Top **5. Some Final Considerations**

Although we have chosen Bloom's vertical model of cognition, there are others that can be useful in the processes described in this paper. For instance, see Chaffee [7], who offers a horizontal model based more on student writing strategies. Another model, more linguistic in format, is offered by Vygotsky [26]. See also Piaget [21] for a discussion of cognitive strategies in young learners.

Another simple cognitive model from composition courses, somewhat overlapping with Bloom's, consists of just three classifications of writing: personal, informational and argumentative. One personal paper might be "I chose to do mathematics because I found it as creative as art"; another is "My mind is more inclined to algebra than geometry." Most mathematical papers fall into the informational category: the "large-scale geometry of the universe can be partially explained by curvature in two and three dimensions"; or "Messages can be sent with (almost) complete security." Thesis statements formed around business applications of linear programming and fractals and fractals dimensions can also fall into this category. Argument papers, as much rhetorical as mathematical, have either soft or hard theses. A soft thesis requires only information to prove its claim through a lower level of cognition; a hard thesis, unless it employs sophisticated statistics, must be supported by more analysis and interpretation. Examples of soft theses might be the following: "Calculus students are ill-prepared by their high school experience" or "Writing assignments provide an effective form of evaluation in the mathematics classroom." Some examples of hard theses, depending on how they are approached, are: "Statistical analysis shows that cancer rates highly correlate with cigarette smoking" and "Better socialization in middle school leads to higher retention rates of women in mathematics and science programs."

By using cognitive models as guides in our mathematical teaching in ways that our writing colleagues have long done in composition, we can move students to deeper levels of understanding of mathematics. In the classroom, where students often ask, "What's it good for," the use of cognitive techniques can help them, and us, find answers to this kind of question.

References.

Bloom, Benjamin, ed. *Taxonomy of Educational Objectives*. New York: Longmans, Green, 1952.

Chaffee, John. *Thinking Critically*. Boston: Houghton Mifflin, 1997.

Piaget, J. *The Child's Conception of Number*. London: Routledge and Paul, 1952.

Rishel, T. *Writing in the Math Classroom, Math in the Writing Class*; or How I Spent My Summer Vacation *Using Writing to Teach Mathematics*, ed. A. Sterrett. MAA Math Notes 16. Washington: The Mathematical Association of America, 1992.

Vygotsky, L. *Thought and Language*. Cambridge: MIT Press, 1986.

### The Active Classroom

### "What Was That Question Again?"

**The question that makes no sense.**

Don't make a big deal of it. Act as if it's an honest question. Answer it quickly, then move on.

Alternatively, you can ask someone else to try to rephrase the question.

**The "challenge to your authority" question.**

Questions and answers are an integral part of learning. Our method of handling them is important to our effectiveness in our teaching, and ultimately in our careers. It behooves us to get used to them, to think about them, to encourage them, and to enjoy what they can teach us about ourselves. Sometimes we will even be surprised at how much we actually know!

Return to Top

### Motivating Students

To me, motivating means addressing the history, culture, and usefulness of mathematics.

### How to Solve It

- Read the problem.
- Read the problem again.
- Draw a picture or diagram.
- Find and label the unknowns what are you looking for.
- Find and label the known quantities.
- Write down all the formulas and relations between the known and unknown.
- Solve the problem.
- Check the answer.

And here I might add a suggestion:

Think about how you might generalize the problem.

- Make people comfortable don't make time an enemy.
- Suggest special cases to try.
- Praise--not inordinately, but praise--those who get the answers.
- If appropriate, ask the students how the students might generalize the problems.

### Course Evaluations

In view of all the above skepticism, what is the function of the student evaluation process?

### Get Along with Colleagues

### What is a Professional?

A professional is one who speaks for and has responsibilities to the discipline he or she is teaching and to the other practitioners of that discipline. Some of these responsibilities have been described in the section called Get Along with Colleagues, but not all. There is more to being a professional than speaking courteously to an officemate, as important as that is.

You have responsibilities to students:

- Don't discuss their individual grades in public, and don't compare the students to each other. It is one thing to say, "You're a very strong student"; quite another to comment, "I thought that Joe would be better than you [or vice versa], but..."
- We have all met people who are very likable, but favoring them with "hints" or "extra help" that others don't get is not fair.
- Socializing can lead to difficulties, even in the most benign situation--See Case Study V for an example.) So, if you know deep down that you are not going into a "benign" situation, do not participate. A night of binge drinking with your undergraduate class is "definitely contraindicated," as a friend once said to me.
- If you are not sure how much fraternization to have with students (after all, we don't want to be totally standoffish), ask trustworthy colleagues and faculty for their advice.
- Be careful the kinds of jokes and comments you make in front of students, who can be sensitive in very unusual ways. For instance, I once teased a student who knew an arcane fact about Galois theory that he "must be reading the same kind of weird stuff that I am." When he obviously bristled, I had to apologize to him for my comment.

You will also have responsibilities to the faculty and other TAs:

- Do not insult or belittle others' teaching styles, or their approach to research. For instance, in a discussion of methodology, "Here's how I teach word problems" is clearly more tactful and better received than "Students tell me they don't like the way you do that topic. " And, you don't need to tell your officemate that Professor Jones "can't be very competent, since he's still writing papers on..."
- If you have an honest disagreement with a colleague, keep it on a professional level: "I really think that problem might be too hard for these students," said directly to Professor Jones, is a professional comment. You may be right, you may be wrong, but at least you have had your say. The alternative of going to your office mate to gripe that "Ole Jonesy's just trying to nail as many freshmen as possible," maybe true, but not a professional alternative.
- Pitting your class against every other instructor's is not professional. The fact that "My class had a 73% average, but Joe's was 67%" does not make you a better instructor. You may have overlooked the information that your class met at 11 a.m., while Joe's was right after lunch. Then there was also the fact that you asked two students with low averages to switch out of your section after the first exam.
- Similarly, you don't sit in the department lounge bragging about how much better your course evaluations are than others--or how yours, bad as they were, at least beat out Joe's. If someone wants to make an honest comparison of his or her evaluations with yours, you certainly don't need to lie, but you, as a professional, know that there are many factors involved in various ratings of classes, students and even TAs. For instance, is it really true that you passed out donuts on evaluation day and then told the class that your job was on the line? Well, that method seems to have worked!

Most importantly, you have responsibilities to mathematics itself:

- Prepare the material. Read up on it (Yes, even "precalculus" has a history). You needn't be a cheerleader, but you should be ready to make an honest reply to "Why do we need to know this?"
- Show some interest in your teaching assignment, and in mathematics in general. If you can't find any reason for teaching that is more compelling than drawing a paycheck, is this really the way you want to spend the next forty years of your life?VJust as no one can ever know all of mathematics, no one ever knows all it takes to be a professional. But, through a combination of talking to trusted colleagues, thinking before acting, and using common sense, we can avoid most pitfalls. If it feels wrong and sounds wrong, act carefully, because it likely
*is*wrong.

### Teaching Methodologies for Various Types of Classrooms

### Lesson Planning

- As a Teaching Assistant
- As an instructor in, say, calculus, differential equations or topology
- In a "discussion" situation

While doing homework before class, think about:

- Where is the "trick," and will it recur in other problems?
- How does the problem fit the topic being taught?
- Will a similar, perhaps extended, problem be coming up later?
- Have the students seen a similar problem or method before?
- Is there some other "point of interest" in the problem?

Also as a TA, you may also be called on to discuss:

- "Here's a three minute synopsis, without proofs or additives, of the topic we're working on."
- A review of topics since the last exam, or possibly for the next prelim.
- Some shortcuts or related methods, if relevant.

Some pitfalls TA's can experience:

- Running recitation as if it were a "better lecture than the lecture."
- Putting down certain problems or answers as "too easy."
- Introducing a competition with other sections, where your students are yours, and others are "the enemy," real or perceived.
- Not coordinating with the lecturer, so that students feel that they are being asked different things by you than by the instructor.
- Showing open distaste in class or office hours for some of the topics and methodologies used in lecture.

### Problems of, and with Students

The vast majority of student problems are not so serious, and you can deal with them fairly easily:

- "My goldfish died. Do I have to take the exam?"
- "Well, I can understand that you can get attached to a pet, and I guess I can give you a makeup, but that exam would have to be harder because you would have had a chance to see tonight's exam. Since you already did the studying for the test, maybe it'd just be easier to take it?"
- "My grandmother died. Do I have to take the exam?"
- "Well, you'll probably have to prepare to go home and all that. I can certainly give you a makeup after you return. If you think it might be easier to take the exam tonight, you can just show up, and if you don't come tonight, I'll just assume that you'll be taking the makeup at a later date. I'll do whatever's best for you, and I'm sorry to hear about your loss."

On a lighter note, students are apt to ask some really rather bizarre questions:

- Do I have to come to class? ("Only if you want to pass.")
- Will I have to buy the book?
- Will I have to
*bring*the book? - Are you sure you're the instructor? You're too young to be teaching us. (Or, as another one said, because professors dress better than that.)
- I wasn't here yesterday. Did we do anything important?
- Could I copy your notes after class? I don't have time to take my own.
- Could you slow down? I can't write that fast.

Don't worry too much; you'll do fine.

### Student Types: Who is the Audience?

### How to Get Fired

- Act as if you don't have a "real job." It's just a TA.
- Don't show up for class. Several times. Without an excuse.
- Be insolent to faculty (especially your advisor), the students and the staff members.
- Call in sick every Monday. Leave for the city on Thursday afternoon.
- Never plan what you are going to do in class, this stuff is too easy, anyway.
- Make clear that research is everything; you are going to solve a great problem and join one of the top departments where you will only teach one graduate course a term.
- Skip office hours. Your officemates can take care of any of the students who'll come by.
- Make it clear from the start that you don't intend to do anything extra--in fact, you won't do anything that you don't get a salary for. And, you only do that under duress.
- End all your classes early. Can't do this, you say? Too many questions? Simple to solve; just belittle the students who ask them--that'll ease up on class time.
- Leave for vacations, breaks, and end of term early. Aruba awaits!
- Don't hand in grades on time. Got some graduating seniors? So what! They didn't like you anyway.
- Offer "grades for favors"--only "jokingly," of course.
- Show up at undergrad parties. They're so much more fun, anyway. Drink a lot. Leave at 2 a.m. with one of your students.
- Tick off the TA supervisor. He isn't a real mathematician, anyway. He stopped doing research about the time you were born.

### Advice to International TAs

**A First TA Assignment--What to Do, How to Cope**

Some questions to consider about your TA assignment:

Sorry, but let me tell one more story from my past:

As usual, I have some suggestions. If you find, as I do, that students should be able to ask questions, but in a more mannerly way than by just yelling out, "That's not right! The answer should be five," then I propose that you tell them on day one that "I am open to questions. But, please raise your hand, so that I know who is asking the question, and so that I can finish my thought before answering you." (For more suggestions as to how to make your classroom more active, look at such sections as The Active Classroom and Motivating Students.) If, on the other hand, you believe that students should hear what you have to say before they start asking questions, ask them to hold their questions until you have fully explained the topic, the example or the exercise you are working on. In short, remember that the culture may not be yours, but the classroom is yours, and it is your right to decide, *within reason,* what is the best way for you to get the material across.

A short list of thoughts about teaching and living in an alien culture:

- Although it is usually easier to live with, room with, and deal with people from your own country and culture, you will find yourself learning so much more if you make the effort to spend a fair amount of time as you can in the "native culture."
- There are a number of advantages to TAing in the U.S. You will learn the language, which can help you in your studies as well as in you teaching. Further, if you decide to stay in the States after graduation, you will need all the language skills you can get for your job.
- If you go back to your home country, your knowledge of English will also be helpful there in your job and your research.
- If you find that some classroom situations are bothering you, talk them over with your mentor from your own country. The problem may turn out to be easily solvable, or it may really reflect some cultural differences. In either case, you can get advice on dealing with the situation.
- You do not have to accept what you perceive to be improper behavior on the part of your class. If you find the students acting in a way you think is unacceptable, talk it over with your U.S. mentor. In this way, you can gain some valuable insight into how standards vary between your own culture and the one in the United States.
- And finally, enjoy your teaching experience. View it as an opportunity to learn about another culture. Try to develop a classroom atmosphere that is in keeping with your personality, while at the same time is relaxed. Why not have some fun while you are getting a degree?

### Some Silly Stuff...

Where do you get keys to the buildings and offices? Do you have to pay a deposit for the keys?

Ask where and when you get your paycheck. When is the first payday? The last?

Find out whether your keys to the building work on evenings and weekends.

Who decides where your desk is located?

Where do you turn in grades at the end of the semester?

Are there syllabus files? Or files of old exams? Who can use them?

What do all the staff members do? How much of it "concerns you?"

When students ask you advising questions you don't know the answer to, to whom should you send them?

Who is your immediate supervisor, and how often should you report to him or her, if at all?

Do you have to go to the calculus lecture you've been assigned to TA for?

### ...And Not So Silly Stuff

Well, "solution one" didn't work too well. Is there another?

The four stages of student development are:

Basic duality,

Multiplicity,

Relativism, and

Commitment.

### The Semester in Five Minutes

**Week Two**

Two new students arrive. They want you to tutor them on what they missed.

After the exam is over, you spend the next two days grading over five hundred copies of one problem.

### Jobs, Jobs, Jobs

### Letters of Recommendation

Start your letter with: "[Student] has asked me for a letter of recommendation for [your job]."

Then: "I have known [student] since [date] in [capacity]."

Finish with a comment as to the student's potential.

### Mathematical Talks

### Becoming a Faculty Member

### University and College Governance

So now we can figure out how you got a paycheck:

### What Does an Evaluator Evaluate?

### The Essence of Good Teaching

Maybe it's because I have been trying to teach people how to teach for over fifteen years now, but I do believe that some aspects of instruction are definitely teachable. For instance, if you go back to a previous section, What Does an Evaluator Evaluate you will see that I talk there about levels one, two and three of teaching. Certainly, such level one aspects as speaking loudly, writing clearly, not standing in front of the material on the board, coming to class prepared, and treating students with respect are all teachable. In fact, many people would say that they are such common sense that they need not be taught--but we have all seen too many instructors who seem to have skipped that lesson. My belief is that all the level one aspects can be taught during a one-week TA training session, and then reinforced throughout the first semester's teaching so that they become close to second nature.

Further, I would continue that level two aspects can also be taught. When I look at these qualities, however, I do not see "common sense" principles, but rather teaching traits that must be developed over a period of time, through teaching itself, but also through mentoring, peer suggestion, and perhaps also through taking some teaching courses. Unfortunately, college teaching courses seem to be a rarity these days--we must hope that they will grow in number.

So simple teaching, "good teaching," I would claim, is teachable. Any graduate student who uses his or her time in graduate school can become a better than adequate college-level instructor. There, I said it, and I'll say it again: Good teaching *is* teachable.

"Great" teaching; now that's something else. Although to me *some* aspects of great teaching are approachable by us mere mortals, there is also a sense in which teaching is an expression of personality, and just as some of us don't really want to be stock brokers, others simply aren't geared up for teaching. Is this bad? I don't think so, unless we find ourselves having to teach in order to live; in that case, I still think we should "give it our best, and not apologize for our supposed shortcomings."

I know that I myself was an incredibly shy child who never wanted to be called on to recite, and at times I still have more-than-normal problems with the concept of standing in front of an audience. Yet, I have managed to teach classes of up to five hundred undergraduate students and give serious mathematical lectures to working research mathematicians (usually the latter is easier than the former). But I think that I can do so only by wrapping myself completely in the mathematics. Early in my career, I used to try to memorize every detail of my lecture, hoping, I guess, to fool people into thinking that I knew all about the material. That would work only as long as I never had to look at my notes; once that happened, it was all over for any "quality exposition." Later on, I realized that I could use transparencies, notes, even full text, whatever it takes to get my point across, and that, by trying to use my memory, I was often depriving my audience of the gist of my talks.

Some people have told me that it is possible to find a model of good teaching in those who have taught us well in the past. Well, maybe. I recall some people who taught me well; while they have definitely shown me many things about teaching, and about their fields of study, they didn't seem to conform to any single mold.

One of the first college instructors I had, and who I thought was "gifted," broke most of the rules I might try to enunciate. I won't give you his name or his school, for reasons that will be clear from my description. First, Professor E., an English professor, would drink his lunch, as we used to say. Then he would start a seventy-five minute class at 1:30 in the afternoon with a thirty- to forty-minute standup comedy routine with no basis in the classroom readings or discussion. At some point in the routine, he would stop, sigh, pull out some old, yellowed papers from a severely beaten up briefcase, and say to the assembled multitude, Well, I guess I have to say something about Hawthorne. Don't feel you have to listen; you can go to sleep now, if you wish. Then he would proceed to offer a careful, lucid analysis of *The Scarlet Letter *and its implications for Hawthorne's life and the sociology of early New England. In case you think that Professor E. was uniquely sensitive to students or the community's concerns, I'll just point out that he told us one day in class that he stayed in our backwater town only because, "as I have told the faculty many times, this is a place to which culture is coming. Although when, I don't know."

Another professor at a different school, a mathematician, was incredibly shy; when he wasn't teaching, he seemed incapable of conversation. Yet, when in class, he gave the kind of mathematical talk that made every student sure that he or she understood every detail--until we tried to do the exercises. When we would come to ask him about the problems we were stuck on, he would say, "Oh yes, that one got me for a while, too. Let's see if we can figure it out again." Professor M. was an incredible motivator who allowed us into his mind. He took apart proofs as if they were watches and then put each piece back together exactly where it should go.

A third instructor was a stickler for proofs in an engineering calculus course. Somehow, he was able to convince the engineering students that "You need proofs to understand why things work; otherwise your bridges won't stand up!" And, he had the force of personality to make his opinion stick. Thus, when he took the class through the difference between a hypothesis and a conclusion, when he showed us by examples how each hypothesis was necessary to the proof, when he counted hypotheses in his proofs, we listened, and listened carefully--and not because the material was going to be on the exam.

What have I learned from these instructors? Well, definitely not that I should drink to excess before going to class. Nor do I do standup comedy for my class--although I do sometimes exhibit a sense of humor, I can't remember, let alone tell a canned joke in any circumstance. While I am shy, I don't walk around hoping that people will not talk to me, nor do I try to convince engineers that proofs are "where it's at."

I guess what I have learned from all this is that great teaching comes in all forms, but that mainly it comes from the delicate interaction between two personalities: That of the instructor, who somehow conveys a love of learning, and the student, who comes ready to absorb and apply what the instructor has to give--no matter how imperfect that instructor may be outside his or her domain of expertise.

### Case Studies

Exactly what is going on in class, and how do you handle it?

The TA coordinator calls you in anyway to discuss the situation. What do you do?

**References**

- Angelo, Thomas and K. Cross,
*Classroom Assessment Techniques.*San Francisco: Jossey-Bass, 1993. - Belenky, M.F., B. Clenchy, N. Goldberger and J. Torule.
*Women's Ways of Knowing: The Development of Self, Voice and Mind.*New York: Basic Books, 1986. - Bloom, Benjamin, ed.
*Taxonomy of Educational Objectives.*New York: Longmans, Green, 1952. - Bonwell, Charles and J. Eison.
*Active Learning: Creating Excitement in the Classroom.*Washington: George Washington University, 1991. - Boyer, C. and Uta Merzbach.
*A History of Mathematics.*New York: Wiley, 1989. - Boyer, Ernest.
*Scholarship Reconsidered.*San Francisco: Jossey-Bass, 1990. - Chaffee, John.
*Thinking Critically.*Boston: Houghton Mifflin, 1997. - Cohen, Marcus et al.
*Student Research Projects in Calculus.*Washington: The Mathematical Association of America, 1991. - Countryman, J.
*Writing to Learn Mathematics.*Portsmouth, NH: Heinemann, 1992. - Culver, R. S. and J. Hackos.
*"Perry's model of intellectual development,"*Engr. Educ. 73, 221. - DeNeef, Leigh and C. Goodwin, eds.
*The Academic's Handbook.*Durham: Duke, 1995. - Eble, K.
*The Craft of Teaching.*San Francisco: Jossey-Bass, 1988. Edwards, C.*The Historical Development of the Calculus.*New York: Springer-Verlag, 1979. - Highet, G.
*The Art of Teaching.*New York: Vintage, 1977. - Hilbert, S. et al.
*Calculus: An Active Approach with Projects.*New York: Wiley, 1994. - Klein, Morris.
*Mathematical Thought from Ancient to Modern Times.*New York: Oxford, 1972. - Lewin, M. and Thomas Rishel. "Support Systems in Beginning Calculus."
*PRIMUS.*V(3). 275-86. - McKeachie, W.
*Teaching Tips.*Lexington, MA: D.C. Heath, 1994. - Meier, J. and Thomas Rishel,
*Writing in the Teaching and Learning of Mathematics,*MAA Math Notes 48. Washington: The Mathematical Association of America, 1998. - Perry, William.
*Forms of Intellectual and Ethical Development in the College Years: A Scheme.*New York: Holt, Rinehart and Winston, 1970. - Piaget, J.
*The Child's Conception of Number.*London: Routledge and Paul, 1952. - PÃ³lya, G.
*How to Solve It.*Princeton, 1945. - Rishel, T. "Writing in the Math Classroom, Math in the Writing Class; or How I Spent My Summer Vacation."
*Using Writing to Teach Mathematics,*ed. A. Sterrett. MAA Math Notes 16. Washington: The Mathematical Association of America, 1992. - Stewart, J.
*Calculus: Concepts and Contexts.*Pacific Grove, CA: ITP, 1997. - Thomas, G. and Ross Finney.
*Calculus and Analytic Geometry.*Reading, MA: Addison-Wesley, 1996. - Vygotsky, L.
*Thought and Language.*Cambridge: MIT Press, 1986.

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