Having become a student again, I recently got this urge to teach, and have been applying around for a teaching position. If I were to conduct a class in math (or a subject that uses math extensively), I’d keep reminding myself of the following, which all came to me sitting through lectures:

1. **I’d choose a good textbook. **Students need very good/excellent texts, so that they can study at their own pace. Every lecture should be covered by text that the student can go back to if they miss a point or don’t understand things at first hearing. Lectures are a bad way to introduce hard concepts. Bullet-point presentations are even worse, because the teacher can now go faster through the prepared text. Distributing bullet-point presentations as course materials is unsatisfactory. They should be treated only as course outlines (see next point). There is no substitute for a course textbook. Spend time and effort choosing it (them).

**2. I’d be very specific about the course curriculum. **Students who want to read ahead or even finish the course materials earlier should be able to do so. The students should always be informed in advance about the coverage of the next class session, so that those who want to read ahead, or who need to miss class, may do so without missing the lessons.

**3. I’d make students spend their class and off-class hours learning course concepts by reading through the text carefully and applying course concepts by solving problems/exercises. **The only way to learn is to accumulate “flying hours”, i.e. hours of practice. Here are some ways to encourage stepping through examples and problems: a) give easy, graded take home exams after each class session, which call help pull up the students’ average; b) compile a set of, say 100-200 problems that will cover the entire course curriculum, with 3 or more questions per topic, and distribute this problem set at the start of the course; c) announce to the class in advance that all exams (midterms, finals, others) will be picked at random from this problem set (i.e., if they take the time to learn to solve the entire set, they will surely get excellent marks.)

**4. I’d encourage students to ask questions. **Most students are shy about asking questions for fear of sounding stupid. To encourage them, make them write their questions on a piece of paper, signed with an alias (so the teacher may follow the progress of the alias.)

**5. I’d help clear away obstacles to the students’ self-learning. **In particular, the teacher should be immediately available to help them get out of dead-ends. This occurs when a concept is so intractable to them that they can not grasp it without help, and they are unable to make any further advance until this learning block is cleared away. Thus, in class, students should either be working on problems or asking the teacher’s help in clearing away such blocks. The teacher can use the questions and the written answers to exercises to identify which concepts are most difficult for students and to conduct special lectures about these concepts.

**6. I’d conduct a math class like a swimming class.** The coach stays dry, *standing by the pool side*. The students are *in the water*, learning to swim. In math, the pool is the chalk and blackboard, pencil and paper.

**7. I’d give exams that are fair. **Stick closely to the text as much as possible. Do not ambush students with surprise topics or trick questions. As a rule: ask many easy questions, a few hard ones. Give students a second chance; if most did not get a mid-terms problem, ask it again in the finals. Think of it as a bonus question.

## 5 Comments

It sounds like you have some great ideas for teaching math. You have really thought through how to help students who struggle in math and you have come up with some great ideas. I’m not sure what age students you are talking about, but you have to consider that you may have some students in the class who will master the material very quickly or who already know the material. These students are going to drown at point 3a. The very thing that will help a struggling student will be very boring and even overwhelming to an advanced student. In point 2 you say that students who want to work ahead should be allowed to, but then what? If it is a college student, will you let them take the final early and be done with the course? If it is a high school (or younger) student, what will they do after they have mastered the material? Will you give them more? Some students see this as a punishment for being smart, others love math so much that they crave more work. Some schools don’t want students to work too far ahead because they will run out of courses to take.

In point 7 you talk about easy and hard questions. What makes a question easy or hard? If a student really understands the material, all of the questions should be easy. On the other hand, some multi-step questions can take a long time to solve, even if it is an easy question, and some short questions can require you to look at it in a different way and that is hard for some people.

Just some food for thought.

Thanks for the comment. Obviously, I have to work out many of the details yet. I was thinking both of high school and college students.

I get your point about advanced students. If it were up to me, any student can request an early exam and be done with it. I’m not sure if schools allow that though, probably due to fears of leakage, which become moot if exams are randomly taken from a set of 100-200 problems announced at the start of the course. Once advance students have worked out the answers (or think they can, without going through the actual solutions, they can always stop attending classes and then take the finals later with the rest, if the school doesn’t allow early exams.

What to do with advanced students who finish the course early is a separate issue, and I need to think more about it. It seems less of a method issue and more of a policy one: let them join more advanced classes? let them graduate early?

Greetings from Manila,

Roberto

wow!

Great ideas! I can think of two ways to engage advanced students: 1) enlist them in teaching parts of future lessons and 2) get them to help you create problem sets.

It’s one thing to be able to do some problems but it takes a whole new level of mastery to teach a lesson. Again, solving the problem is a lot easier than creating a problem to solve.

I group my class into 5-6 students each, making sure that each group has its share of high and low scorers. And I encourage groupmates to help each other. I haven’t tried asking advanced students to create problem sets. I’ll think about it. My attitude so far has been to focus on the low scorers because the high scorers can take care of themselves. I’ll reconsider this attitude.