What is this Class About?

A Guide to Math 130/131

2003-2004

For most students, Math 130/131 is much different from previous math courses. This guide is designed to help you adjust to and succeed in this new environment. The guide is divided into two parts. Although the first part of this document was written by one of the instructors, all sections reflect a similar philosophy. This part of the guide contains information about the aspects of the course that are similar in all sections; the second part of the guide contains information about policies and procedures that are specific to your instructor. Please keep this handout for reference throughout the year.

Part I: Information for All Math 130/131 Students

by Debbie Borkovitz

A Note to the Nervous:

It's normal for everyone, including the instructor, to be nervous at the beginning of the semester, but this anxiety can be especially acute in Math 130. Some of you might be nervous because this is your first college math class, and you're not sure how you'll do. Some of you might be nervous because you've had bad experiences with previous math courses or math teachers; you might worry that this class will make you feel stupid. Some of you might be looking forward to the challenge of a math class; some of you might feel like you'd rather go to the dentist three times a week. Some of you might feel like you can't possibly do math, no matter what anyone says. Some of you might feel like math is an important subject, others might feel like you don't really need to learn any more math to achieve your professional goals.

Math 130/131 can be an opportunity for you to succeed in mathematics, whether you have succeeded in the past or not. I want everyone in the class to do well, including you. I am not trying to separate out who is "smart" and who is "dumb" or to rank everyone in the class. I work to make the class a safe place, a place where people can make mistakes without being laughed at, where people are not afraid to share ideas that are not completely perfect. I will not intentionally humiliate you or try to make you feel stupid.

When I say that I try to make Math 130/131 safe, that doesn't mean that it is easy. Math 130/131 is a hard course. You will have to work hard, and I will push you. I push you out of respect, not to be mean. It is my job to see your potential, even when you might not be able to. I know that Math 130/131 can be frustrating, but frustration is a normal part of the learning process. Think about when you learned to read or to tie your shoes or to ride a bike -- I'm sure you were frustrated sometimes, but in the end that frustration led to a good thing.

What Math 130/131 is not:

In the U.S. (and many other places) there is often a standard structure for a math class: the teacher reviews homework, then shows students how to do a new topic, the students practice the new topic, then the teacher tells them whether their work is correct or not. If you went to school in the U.S., the chances are pretty high that most, if not all, of your math classes followed this format. There are some serious problems with this format, however:

1) Students often don't learn the material. The format encourages memorization without understanding. After the test, the material is forgotten.

2) The students only learn how to do problems that are very similar to the ones the teacher has showed them how to do. They don't learn how to think through new problems or problems that appear in a new context.

3) It gives an incorrect impression of mathematics. Many students learn to think of math as a bunch of disconnected facts that they have to remember for no apparent reason.

4) Students rely on the teacher to tell them whether they are right or not, instead of developing confidence in their own reasoning skills.

5) It leads to very boring classes.

Although there are a lot of problems with the format I describe, it might be the format with which you are most familiar. Even students who don't like this kind of math class still often think that it is the way math classes are "supposed to be." You might find this format boring, but you understand the rules, and that familiarity can be comforting. In Math 130/131 it can seem like all the rules have changed, and that can be upsetting, even if you didn't like the old rules. This document is designed to help you understand what are the new "rules" in Math 130/131.

What do we do in class?

We don't always do the same things in class, but virtually every Math 130/131 class in an active class. You will have to work and think in class; you won't be able to just zone out while I go on and on. You will work with other people. Often you will use manipulatives, which is the fancy education word for toys that help people understand math concepts. Many activities will include visual and/or kinesthetic (involving motion) components.

Class often runs like this: I will give the class a problem or set of problems. The problem will usually be new, but often related to problems we have already done. I will also give an explanation of why we are doing the problem, how it fits in with our goals for the course and what we have been doing. We will often discuss the problem together to make sure everyone understands what it is asking, and then we will break up into small groups.

I will circulate between the groups, often just observing what is going on. If you stop me to ask a question, I will usually ask you to explain what you've done so far, why you've done it, and how you know it is correct or what you think the problem is. At first, you might find it very difficult to answer my questions, but I will encourage you to try and to practice mathematical communication, which is one of our goals in the class. If you ask me whether you are right or wrong, I often won't tell you, but will encourage you and your group to reason yourself. At various points in the class, depending on what is going on in the groups, I might bring the whole class together, ask two groups to consult with each other, ask groups to report on their work, etc. At the end of class, we will all come together to summarize what we've learned, and I will sometimes add new points or help clarify things. Sometimes activities will take more than one class period.

Other classes take different formats. Sometimes we will have whole class discussions, sometimes we'll have tests, sometimes we'll work on the computer, etc.

Does this sound like teaching?

Some students believe that the teacher is only "teaching" when she is standing at the board showing them what to do. Sometimes Math 130/131 students get mad at me because they think I am not doing my job when I don't show them how to do something or tell them whether they are right.

You might find this surprising, but the way I teach is actually harder than the "traditional" way of teaching math. When I teach I have to listen carefully to students' ideas and respond to each one in the moment. I have to monitor what is going on in the whole class and make decisions about where to go.

Now don't get me wrong, I am not complaining. It is much more fun and much more rewarding to teach this way. I have taught by lecturing, and in general, I find that it is much less interesting and much less effective than the "hands-on, minds-on" way of teaching. Occasionally I do give something resembling a lecture in Math 130/131, but only when absolutely necessary.

Where did you get all this stuff from?

Well, I didn't make it up all by myself. You are not guinea pigs in some weird math teaching experiment. In fact, there is a national (actually, international) movement to change the ways in which mathematics is taught. The "old" way of teaching math is not effective for many students, and for a long time no one cared, as long as a small number of people learned enough math to become engineers, scientists, and mathematicians. As the culture changes, however, there is a need for more people to understand mathematics, and hence a need to find ways of teaching math that work for a wider range of students.

Also, technology changes things. With the availability of calculators and computers, it doesn't necessarily make sense to spend so much time practicing doing exactly what the calculator can do more quickly and accurately. It is more important to be able to estimate, to evaluate whether an answer seems reasonable, and to understand the operations, than to do long division with large numbers. Also, tools like spreadsheets and graphing calculators can be used to help students acquire a deeper understanding of mathematics.

For many students math education was never really about mathematics: it was about learning to sit still and follow instructions, whether the instructions made sense or not. This was a way of training workers for jobs where they were supposed to follow orders without question. Math education is changing to help prepare students to work cooperatively to solve problems, instead of just following orders.

The kinds of things that I require in Math 130/131 are consistent with things that are being required of students all across the country, from pre-school to college. You might or might not have felt the influences of this reform movement in your previous classes (if you used the IMP curriculum in high school, you have definitely felt the influence of this movement; to some extent the MCAS exam in Massachusetts also reflects the reform movement). If you are going into teaching yourself, you will hear more about these ideas when you take education courses, and you will be expected to incorporate them into your own teaching. Since most teachers teach as they were taught, Math 130/131 can be an opportunity for you to experience a new way of running a classroom before you start teaching yourself.

One good place to find out information about math education is from the National Council of Teachers of Mathematics. You might want to check out their web site at www.nctm.org or their journals in the library (Teaching Math to Children, Mathematics Teaching in the Middle School, The Mathematics Teacher).

What if I'm not a "math person?"

In the U.S. there’s a common belief that some people are "math people" and some people aren't. Sure, math comes easier for some people than for others, but generally people who are good at math are good at it because they've worked at it (and in some other countries, the common belief about math is that students who work hard will do well). Some students have the impression that if they don't "see" how to do a problem right away, that they won't ever be able to do it, and that "math people" can just look at any problem and do it instantly. If you believe that math is something that you can't really do, this belief will become self-fulfilling. Part of learning how to do math is learning how to stick with a problem and not give up. In this course, we will work on strategies for solving problems and for getting "unstuck."

Unfortunately, in some math classrooms, the only contributions from students that are valued are correct answers. The students who often provide correct answers are labeled "math people," and other students come to see themselves as bad at math. Correct answers aren't the only things that can help people learn, though. In this class, it is valuable to share your thinking. That doesn't mean that all answers are equally valid, it just means that everyone in the class can learn from different approaches, and that incorrect answers often lead to discussions that help everyone deepen their understanding of the topic.

What kind of math are we going to do in the class?

All sections of the course will focus on the same general topics, although individual instructors will often choose different activities and formats and might vary the order of topics. The first topic for all sections will be “Introduction to Problems Solving, Reasoning, and Communication,” which will focus on all the process goals (emphasizing the three in quotes). We'll look at strategies for solving new problems, practice writing and speaking about mathematics, and work on justifying answers. I will constantly ask you questions such as, “How did you get that answer? How do you know it’s correct? How could you convince a skeptic that your answer is correct? How do you know your pattern will continue?” You might be unfamiliar with answering these types of questions in a mathematics class, but you'll catch on.

The next topic is number theory, which focuses on the positive integers (the “whole numbers”). We'll look at patterns in the multiplication table, study divisibility rules, prime numbers, and other topics. Then we’ll look beyond the integers to rational numbers, focusing on fractions, decimals, and percents -- topics that might have given you trouble in the past. This part of the course can be stressful because it involves topics that everyone has seen before, but that many people do not understand. Sometimes these are the topics that have convinced students that they are bad at math. Our approach to these topics will emphasize understanding and thinking, without any rote memorization. We will use a variety of manipulatives to help you build both physical and mental models of the concepts so that you can deepen your understanding of them. We will apply some of the problem solving skills from the first unit to these new problems in the second unit. As I’ve taught these topics over the past several years, I’ve come to see that the reason that students have trouble with them is that they are genuinely difficult. Often schools do students a disservice by trying to make the topics easier than they really are, thereby preventing students from acquiring a deep understanding of them. If you have trouble with fractions, decimals, and/or percents, rest assured that you are not alone, you are not stupid, and that in this class you will have the opportunity to deepen your understanding of them.

After we look at rational numbers, we will study some probability. Probability is a topic where you can use your understanding of fractions and percents to make sense of problems where the outcome is uncertain. Next, we will study algebraic thinking, which might be quite different than the algebra you studied in high school. We will expand our work from the first topic to look at representing patterns with equations, graphs, tables, pictures, and words and study the relationships between these representations. We will continue with the proportional thinking that is part of studying fractions. We will look at functions.

Why are the class periods for Math 130/131 longer than those of most Wheelock courses that meet three times per week?

The extra meeting time is to give you more contact with your instructor so you can make more progress. At many schools, no credit is given for courses that cover some of the material in Math 130/131. At Wheelock, instead of making you take a no-credit course followed by a for-credit course, we just expanded the for-credit course an extra period.

What are the study groups about?

The study groups are required. They meet for two hours per week throughout the year and are led by a student teaching assistant (a TA). You have to do homework anyway, and this is an opportunity for you to have a scheduled time where you can work with some classmates and have a TA to help you.

You are encouraged to work with others on homework, however, anything you write-up as an individual must be your own. Unless it is some kind of group project, you should not be with classmates when you write-up an assignment that you are turning in. Don’t do the final write-up during study group! Work on the problem together, then go back and write it up yourself. It is important that you yourself understand what is in your paper, and that your paper is in fact your own work. When you have worked with other people on an assignment, please indicate whom you worked with.

How should I prepare for study group? What do the TA's do?

First, you should prepare for study group. Start working on the homework by yourself before your study group meets. At the very least, be sure you understand the problem(s). It wastes time when students haven't even looked at the problem(s) before study group starts.

Your TA is not going to tell you the answers. The TA's have been trained in the kind of teaching that I use in class -- they are there to help you clarify your own reasoning, to encourage you, to suggest strategies, etc. They are not there to show you how to do the problems or to tell you the answers.

The TA's are students like you. Some are Math/Science majors or other students who really like math. Some are people who took Math 130/131 and did well in it. They are not perfect. Ultimately, you are responsible for your own work; the TA's try their best, but sometimes they make mistakes. If there is a mistake on your paper, please don't blame it on your TA.

If you have a problem with your TA, first try to work it out with her or him. If you can't, feel free to see me about it. Ditto for problems with your study group.

What if I get stuck?

Many students are unfamiliar with the kind of homework that we often have in Math 130/131. Lots of time math homework involves finding a similar problem in the examples in a book and then copying the steps. Most of our homework won't be like that. You will often have new problems to try.

Sometimes Math 130/131 students think that if they look at a problem and don't know how to do it right away, that then they can't do it, so they quit and figure that the teacher will show them how to do it next class. You might think that if you can't do a problem right away, then you don't know how to do it, and so you have to ask for help.

Remember, however, that in Math 130/131 one of our main goals is to learn how to approach unfamiliar problems. I expect you to work on your homework, not to give up quickly. In class, we will discuss many problem-solving strategies, here are some things to think about if you feel stuck and want to quit working on a problem:

* Do I really understand the problem? Can I restate it in my own words? If not, what is the confusing part? Is it a word I don't understand, a concept, etc.? Can I find the information I need?

* Can I solve a part of the problem?

* Can I solve a simpler version of the problem? What if I used smaller numbers, a simpler picture, etc.?

* Have I seen any problems like this? How did I solve them?

* Can I draw a picture? Make a table? Make a graph?

* Can I guess a solution, and then check my guess? Can modify my original guess to make a better one?

* Can I work together with a classmate?

* Have I explained myself well enough to convince a skeptic?

These problems are not all or nothing, and the goal isn't just to get the right answer. It is much better to make some progress than to quit.

Why do we have to write so much in a math class?

Lots of times in math class students practice artificial skills. Very few tasks in the world look like doing a worksheet of nearly identical problems and then asking someone to tell you whether they are right or wrong. By writing you practice your writing skills and you practice learning how to communicate mathematics. These are much more authentic skills. Also, your writing can give me a much better sense of what you understand and what you don’t than a multiple choice question ever could. Virtually all mathematics classes that have been influenced by the reform movement in mathematics education include lots of writing.

Why do we have tests? Isn't that a fake skill too?

I have considered not giving tests, but every time I give tests I find that there are some students who surprise me by doing better than I would have expected them to. I think of tests an opportunity for you to show me what you know, as opposed to an opportunity to punish you for what you don't know. This distinction is very important. One of the key tenets of the reform movement in math is to see assessment as part of learning, not separate from it. In many traditional math classrooms, the whole class stops for a test, to see what people have learned or not, and then they move onto something else. In Math 130/131, the tests are also an opportunity for you to learn more

What if I don't test well? What if I can’t finish in time?

Test anxiety is a problem for some students. If you have a lot of trouble with test taking, you might want to talk to me about making reasonable accommodations to help improve the environment for you. I never intend for time to be a factor on tests, I am not testing for speed, so if you cannot come early or stay late during a test, you can schedule it at another time when you have more time. To do this, you must talk to me in advance. Some students find they test better if they sit off to the side, facing the wall, so there are fewer distractions. Others arrange a private room in the academic advising office.

Here are some general strategies for dealing with test anxiety:

* Read all the questions first, and start with the one that looks most familiar to you. Once you start working, you will settle down.

*Prepare for the test by doing problems, not by highlighting notes. Redo old homework problems; try to change them slightly. Do the problems from the review sheet and old tests.

*If you find yourself panicking, interrupt your train of thought. Some students start thinking things like, "I'm going to flunk this test, I'm going to flunk this class, I'll flunk out of school, my life is over, etc." Needless to say, this doesn't reduce anxiety. Try to breathe and tell yourself that you've gotten stuck before and solved the problem. No one test is going to make you flunk out of school or even flunk the class.

*If you get stuck, try to write down your thinking and why you are stuck. This might help you get unstuck, but even if it doesn't, in any case, it'll help me see what you were thinking. Sometimes students leave things blank when they were actually close to getting the answer.

* Feel free to move to a different problem and come back later.

What are the assignments listed on the syllabus?

There are two assignments that will be the same (or very similar) across all sections of Math 130/131. First, at the end of Math 130, you will write a letter or a paper that reflects on what you learned in the course; this letter will also help you get started in Math 131 after winter break.

The second assignment common to all classes is the final portfolio. For your final portfolio, you will select your best work, work that shows your progress in the course, and other work that was important to you in class, and you will explain why you chose each selection. The portfolio is very personal; it’s a way for you to show your work in the way that best reflects your accomplishments and struggles. You will get more guidance about the portfolio later; for now, it’s important that you find an organized way to save all your work from both semesters.

What are the Baseline Proficiencies?

In a few weeks, you will receive descriptions of the “baseline proficiencies.” Some of the baseline proficiencies refer to processes such as problem solving and mathematical writing, others refer to specific math topics. These baseline proficiencies serve as a “bottom line” for everyone in the class. They cover some of the most important parts of the course, things that every college graduate should know or be able to do. In order to pass Math 131 (i.e. the spring semester), you must pass all the baseline proficiencies.

Every one of the baseline proficiencies can be satisfied in a variety of ways. For example, you might satisfy a proficiency by correctly answering some questions on a test, by giving a presentation in class, or by explaining your reasoning to me one-on-one. For every proficiency, you will have many opportunities to pass. When you have passed a proficiency, you have passed it forever. It will take some students longer than others, but it is not a race and not a competition. In the two years that Math 130/131 has existed, virtually all students have passed all the baseline proficiencies.

Although you do not officially have to satisfy the proficiencies until the end of the spring semester, it is critically important that you take them seriously as soon as they are distributed. The list of proficiencies includes information about when in the semester you should get extra help if you haven’t yet passed that proficiency. It is YOUR responsibility to seek out extra help, and to do what you need to do so that you can pass them by the end of the course.

The baseline proficiencies might sound scary, but please remember that I want you to do well on them. I am committed to working with you to help you to understand and to help you find a way to demonstrate your understanding.

What if I'm having trouble in the class?

It is better to take care of trouble sooner than later. You have many resources available to you. You can make an appointment with me or just talk to me before or after class. It is part of my job to meet with students out of class, and I am available for one-on-one work. I have many students, however, so there is a limit to how much time I can work outside of class with you.

You can also get help from your TA or a tutor. Please remember that your TA is also a student and he or she has committed to leading a study group, but not to helping you whenever you want help (even if s/he lives in the same dorm that you do). Please respect your TA's time; if you would like to meet with your TA one-on-one, you should make an appointment. Your TA might not have time. Another alternative is to go to the Office of Academic Advising and ask for a tutor. There are many Math 130/131 tutors signed up with the office. If you have a learning disability or think you might have a learning disability, the Office of Academic Advising also has a specialist who can work with you.

How did I get in this class? What if I think I'm in the Wrong Class?

You were placed in this class based on the test you took when you entered Wheelock, as well as you high school courses and grades, your self-evaluation, and your SAT scores. It is not a perfect system. If you believe that you are in the wrong class, please see me.

What if I have other questions?

Please feel free to ask. I am also interested in feedback on this guide: both at the beginning and at the end of the semester. Remember, communication is one of our goals, and it can be hard for me too. If something here isn't clear to you; it probably isn't clear to others too.

Motto for Math 130/131: Math makes sense! I can do math!

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