## Introduction

Teaching mathematics less than full time has allowed me to get involved in curriculum development and teacher training, and also to think about my practice as an educator. Moreover, working at a small independent school, with wonderful colleagues and supportive administrators, has made it possible to try many new ideas. Even though I am not an exceptionally great teacher, time, collaboration, support, and experimentation have allowed me to come up with some ideas about teaching which I hope will be useful beyond the walls of my classroom.

I have shared some of my thinking in articles that can be found on this Web site. In particular, The Art of Teaching worksheet is a professional assessment and development tool for teachers. It is based on the belief that **there is no one way**: on each of the questions we face daily as teachers, there are many valid answers, and we need to learn to practice as many as possible. Each of us has a certain profile as a teacher, a combination of default behaviors we have within each of the categories shown on the worksheet. Professional growth is in large part the process of complementing one's natural tendencies with learned techniques.

For example, on the "forward motion / review" spectrum, my natural tendency is certainly "forward motion", but over the years, I have learned that even though it doesn't come naturally to me, I can help those students who need more time to absorb a concept by providing ample review. In fact, doing this is also helpful to quick learners, who can use a chance to slow down and get more depth of understanding of techniques they may have picked up quickly but superficially. Conversely, the kind-hearted teachers whose tendency is to eternally review need to balance this out with forward motion, lest their students remain mired in a very limited range of skills and understandings.

This is one of the many dualities outlined with no comments in the worksheet. A few others:

- Educational fashions tend to swing between extremes, as in the example of "teaching to standards" vs. "differentiation", but good teaching must include both of those.
- The realities of everyday classroom challenges counterposes "responding to the realities of the class" with "covering the material", and we constantly negotiate that contradiction.
- Being "over-prepared" is often necessary, but what teacher has not had an extraordinarily effective lesson on a day when they were "winging it"?
- We hope that in the end our students will be driven by "intrinsic motivation" -- curiosity, love of the subject matter -- but in the meantime many will need the "extrinsic" motivation of pleasing a parent or teacher, or getting a better grade.

And so on, down the list. It is never true that the best teaching is always at one end or the other of the spectrum, or for that matter in a carefully worked out compromise. Teaching is an art more than a science, and good teaching requires us to navigate the whole spectrum in each category, and to develop an instinct for when to do what. Given how many categories there are on and off this worksheet, pursuing this balance is a lifetime goal.

In this article, I will present some of my answers to a few of these conundrums, focusing especially on ones where my approaches lie outside the conventional wisdom.

## Teaching for Understanding

During the "Math Wars" of the 1990's, an absurd debate developed. A minority of reformers, overreacting to the mindless memorization at the center of much traditional math education, deliberately de-emphasized skill development. A backlash ensued, charging that the entire reform movement was abandoning skill-building, and that understanding could only come after skills had been mastered. In fact, both sides are wrong: understanding cannot be divorced from the acquisition of skills. Without understanding, it is hard to develop an interest in the skills, or to retain them; without skills, understanding is out of reach. Good teaching requires a skillful weaving of those two strands.

What complicates this discussion is the advent of new technology in the form of increasingly powerful calculators and increasingly accessible math software. The implications of this development deserve a fuller treatment than I can give them here, but you can see examples of my use of technology elsewhere on this site: my calculator pages, my function diagram applets, my views on teaching programming. For the purposes of this article, I will state my two main conclusions about technology's impact on math education:

- Speed and accuracy in computation are no longer legitimate priorities in math education, as no amount of drill and practice can make a student more efficient than an electronic device.
- Teaching for understanding is now more important than ever, as understanding is necessary to make good use of the technology.

In fact, teaching for understanding is the most important part of our jobs. The problem is that understanding cannot be easily conferred by explanations. (A naive opponent of educational reform once suggested that it was easy to teach about variables: patiently explain to the students that variables behave just like numbers! Would that it were that easy.) Moreover, understanding is unfortunately not always valued by students, parents, and administrators, many of whom believe that everything would be so much more straightforward if we could just have the students memorize facts and algorithms. Finally, understanding is difficult to assess, and difficult to encapsulate in a checklist.

Nevertheless, I will attempt to do just that! A student who understands a concept can:

- Explain it -- we should ask students to write explanations, even though they don't enjoy doing that
- Reverse processes associated with it -- for example a student does not fully understand the distributive law if they cannot factor anything
- Flexibly use alternative approaches -- for example, in addition to the usual "do the same thing to both sides" for solving linear equations, I also teach the cover-up method, trial and error, the use of graphs and tables with or without technology, and sometimes the use of manipulatives
- Successfully handle non-rote assessments -- not much understanding is required to merely reproduce a memorized algorithm
- Transfer it to different contexts and navigate between multiple representations of it

Thanks to the calculus reform movement, the idea of multiple representations is widely understood and implemented, mostly in reference to the "big three": graphs, tables, symbols. To a great extent this is a consequence of the advent of graphing technology, and to the related fact that much curriculum development along these lines was easy to get funded. I fully support this advance, and in fact I have taken this idea further, and developed much curriculum in a tool-rich mode, incorporating not only graphing technology, but also low-tech tools of various kinds, including manipulatives. In particular, I have made an effort to use as many geometric and other visual representations as possible. (On this subject, see especially A New Algebra and A Proposal for Early Mathematics.)

In addition to their essential use in deepening understanding, multiple representations provide multiple entry points to various concepts, thus enabling us to reach a broader range of students. Different tools and representations allow us to preview and review concepts and help us extend exposure to a given idea while keeping things varied.

## Sequencing

### Overall

Another consequence of the new technological landscape is that some topics can be taught to younger students than was formerly possible. For example, there is no reason to wait until Algebra 2 to introduce exponential functions. In general, the sequencing of topics is heavily dictated by tradition, and enforced by the decisions of textbook authors and publishers. Math teachers would do well to open up to new ideas about sequencing. Working in a small, innovative department that freed itself many years ago from over-reliance on textbooks, we have come up with a number of ideas to make the sequencing of lessons better suited to teaching and learning.

For example, if a topic's placement is too early, in that students do not have the maturity to really understand it, we teach it later. One blatant example is the quadratic formula, which is traditionally deemed to be an Algebra 1 topic. In part because it is taught too early, many more students can sing it than can understand it or its derivation. At my school, we have gone from teaching it effectively to perhaps 10% of our students in Algebra 1, to teaching it effectively to perhaps 80% in Algebra 2. (We use the approach outlined in my book *Algebra Lab Gear: Algebra 1*, whose title was chosen by the publisher...)

[More information about our Algebra 2 course.]

If a topic's placement is such that it doesn't give students enough time to absorb it, and if that topic is important, we spread it over more than one course. This is what we do with linear, quadratic, and exponential functions. We have also spread our teaching of trigonometry over three courses, and no longer have a Trig course.

### Within a Course

Within a given course, we have found it effective to move traditionally late topics to an earlier spot. For example, we teach inscribed angles at the beginning of Geometry, because it provides an interesting context for applying basic facts about angles. Our less-visual students need to do a fair amount of work with angles at the beginning of the course, in order to be able to make sense of what comes along later. Inscribed angles provide a way to give them more practice in a context that is interesting to their more geometrically-inclined peers. More generally, we try to approach important and difficult topics earlier, so as to have time to get them across -- May is too late to learn anything major.

Another useful and virtually unknown trick in sequencing is to separate related topics. This is counter-intuitive, but it is surprisingly effective. For example, we introduce the tangent ratio in the first term of Geometry, based on the concept of slope, and sine and cosine in the second term. In Algebra 2, we separate quadratic equations from quadratic functions, series from sequences, logs from exponents. This makes it possible to have a built-in review of the earlier topic when approaching the later one, and thus to extend the all-important exposure time.

### Within a Topic

Finally, here are some within-a-topic sequencing guidelines. Those of course do not apply without exception, but they do provide a useful way to think about how to introduce new ideas. For most guidelines, I give examples that can be found on this Web site.

- Concepts first, vocabulary and notation later -- examples: the trig ratios, logarithms
- Discrete first, continuous later -- example for this guideline and the next: the Pythagorean theorem, inscribed angles
- Numerical examples first, generalization with variables later
- Natural numbers to real numbers -- almost any new idea is more accessible if you start with whole number examples -- for example, `sqrt(4) ยท sqrt(9) = sqrt(36)`.
- Concrete to abstract -- example: algebra manipulatives

At the risk of appearing to contradict the above, I'll add that it is best to start any new topic with examples that are rather challenging, so that students not develop a complacent attitude about the topic. Then one can go to easier examples, then back to more challenging, etc. In fact, most of the above guidelines are best implemented as a back and forth motion: for example, after introducing vocabulary and notation, one needs to re-introduce the concepts. Likewise for most of these guidelines.

## Discovery vs. Direct Instruction

When I was younger, I was under the impression that anything my students "discovered", they would certainly understand and remember. Over time, I learned that in fact, most student discovery requires considerable guidance from the teacher, and moreover, that it guarantees neither understanding nor retention. The opposite point of view (that direct instruction is the only way to go) is equally flawed: all of us have had the recurrent experience of "explaining" something with exquisite clarity, only to find out that our students apparently failed to hear a single word we said. In fact, effective teaching requires a complicated mix of discovery and direct instruction.

In France, math educators talk about "institutionalization": after students have explored a concept, we need to bring them into the vast international institution of mathematics, which has its own language, notation, and criteria to determine the validity of ideas. To do this well, I try to

- Make key concepts explicit,
- Clarify what is important and worth remembering, and thus worth writing down, and
- Make connections with other representations and previous knowledge.

However students cannot hear answers to questions they do not have. If we want all this direct instruction to be meaningful, it is best to seed questions in students' minds through active involvement in doing appropriately selected mathematical activities. I have developed a type of rich activities which are particularly well-suited to the phase of exploratory learning, and which can be revisited further along the learning curve.

A rich activity

- Has significant mathematical content
- Is accessible
- Is challenging
- Is engaging
- Is student-centered
- Is student-run
- Has many paths through it
- Is suitable for group work
- Enhances discourse
- Asks simple questions (often reversing traditional questions)
- Provides a good context for discussion, reflection, and generalization

Of course, this is asking a lot of a single activity, and few will meet all these criteria. Still, one of my missions in life has been to collect and create rich activities. Here are some of my favorites: Polyomino Perimeter, Angles Around a Point, Make These Designs, Nine Function Diagrams, Superscientific Notation. (Note that all of these activities depend on visual representations or on the use of learning tools.)

## Class discussion

Once the stage has been set by a rich activity, or by a not-so-rich activity, it is usually helpful to have some class discussion. Individual and group work are usually not sufficient to get enough ideas flowing, even if the activities are top-notch. A good class discussion can reach many (though unfortunately not all) students with key concepts. A discussion is quite different from the direct instruction I recommended above. Many teachers ask questions of their students as a technique for keeping the students engaged and attentive to their lecture. This is fine, but should not be confused with actual discussion, where students venture ideas, voice disagreements, and grapple with each others' contributions.

For students to participate at that level, we as teachers have to make sure it is a safe environment. I do not tolerate students making fun of each other, even if they're "just joking", and I (usually) refrain from sarcasm. I praise participation and risk-taking, rather than correct answers. It is tempting to praise correct answers, but in the long run it is counter-productive. Correct answers are their own rewards, while a brave attempt to answer a difficult question, even incorrectly, needs explicit support from the teacher. In fact, over time, praising correct answers has a chilling effect, as students will hold back from speaking up until they are sure they have a correct answer. For some students, that certainty never comes.

A crucial way to make an environment safe for discussion is to have effective techniques for handling wrong answers. The message to the students has to be that it's OK to make mistakes, and that in fact it is the only way to learn math. "There is no royal road to geometry." Some teachers are very nervous about ever telling a student their answer is wrong. This is of course problematic, as it makes it very difficult to conduct a discussion, and all this tension only increases student anxiety. I strive for a classroom where incorrect answers are voiced frequently, and discussed as learning opportunities, thus making it possible to discuss mistakes openly when necessary.

But nevertheless, students feel bad when they make a mistake in front of everyone. One way I reduce the sting is by keeping a poker face and making lists of answers, for later discussion. Once several answers are on the board, no one remembers who said what. Another very effective technique (if it's already clear the answer is incorrect, or if the student seems unsure) is to ask the student to "choose someone to help you". They immediately bask in the power I gave them and the mistake is no longer important. I also make sure I myself make frequent "mistakes" as starting points for discussion, and I talk about the "mistakes hall of fame" which houses classics such as `(x + 5)^2 = x^2 + 25`.

A few other techniques I use to get kids to participate in class discussion:

- I'll ask a question, and then ask the students to "tell your neighbor" the answers. I am always amazed at how much conversation (of the good kind) this generates. Afterwards, it's much easier for a student to repeat what they said or heard to the whole class.
- "Can you restate what x said" is also a powerful prompt, in that it offers a way for a student to speak up with little risk. Doing this regularly also encourages students to listen to each other -- not just to the teacher.
- To avoid losing touch with what's going on in their heads, I often ask for feedback from all students in the class. This can be done through votes (thumbs up if you agree, down if you disagree, sideways if you don't know), by asking to "show me the answer on your fingers" (if it's a small natural number), by asking for "air graphing" and other gestural responses ("point to the right if the answer is positive, to the left if it's negative"). The most frequent way to get feedback from all is to ask them to write down their answer, and walk around the room to see what they wrote. All these techniques help avoid the problems of relying on the two or three most vocal kids to answer all the questions, which is a sure way to lose contact with the majority.

Finally, I try to keep classroom discussion varied by using humor and stories, by physically moving around the room, and once in a while by teaching in total silence. I also vary the ways to display ideas (mostly the board, but sometimes I project a calculator or computer screen or use manipulatives on the overhead). Finally, instead of "going over" a problem, I often use a problem that is slightly different, so that the students who successfully completed it in homework or class work have some reason to pay attention.

## Homework and Extended Exposure

I give homework daily, but I keep it reasonable, because most learning happens at school. As much as possible, I keep it separate from class work. When I don't, some students tend to rush through the class work in order to get to the homework and do it in class. This undermines the collaborative and reflective atmosphere I'd rather have in the classroom. Finally, and this is the most useful hint I have on this subject, I try to "lag" the homework. In other words, when possible, a given day's homework is about material that was learned a day or a week earlier. This has two benefits: it means that if something goes wrong in today's lesson, the students still have homework to do, and I have another chance at teaching today's concept tomorrow. Moreover, it extends the duration of the exposure to the concept.

As you must have noticed by now, the idea of extended exposure to concepts is fundamental to my teaching. This is because while all students can learn, they don't all learn at the same pace. It is not too difficult to organize a course so as to provide extended exposure for the students who need it, and at the same time forward motion for the student who needs that. The oversimplified chart below shows how something that might happen in one week (topic A) can be extended to four weeks without stopping the forward motion of the course. Say I am teaching topics X, Y, Z, A, B, C, and D, in that order:

Week Number 1 2 3 4 Class work A B C D Homework Z A B C Quiz or Test Y Z A B Quiz Recycle X Y Z A

Topic A happens in class in week 1, as homework in week 2, on a quiz in week 3, and in a quiz "recycle" (corrections done as homework) in week 4. Some students will need all four weeks to get it.

## Nothing Works

I hope that the suggestions I made can complement what you do now, and that you will consider adding them to your repertoire. However I'll repeat the disclaimer I stated at the beginning of this article: **there is no one way**. I have no illusions that I have found the ultimate approach to math teaching -- there is no such thing. In reality, nothing works. Nothing works for every student, every class, every teacher, every day. Thus, I need to know many things. I am increasingly skeptical of inflated claims, whether from traditionalists or reformers. As I learn new techniques, I don't throw away or rule out any old ones. Eclecticism is the philosophy of education that serves me best!