Teaching/Learning


It is generally agreed in the modern world, that all students need to learn math. And the fact that our schools require everyone to study it, assumes that every pupil is capable of learning it. Unfortunately, standardized tests show us that vast numbers of pupils are not learning it. What is the problem here? Are the math teachers not doing their job? Most teachers would sincerely and passionately assert, “I taught it to them, but they didn’t learn it!”

The fact that so many middle and high school math students are doing so poorly is a source of great pressure and frustration for everyone involved. Employers—who need well-educated employees to be competitive in a global economy—blame schools, administrators, and teachers for not getting better results. Teachers—who work very long hours in situations of intense pressure—often blame uncooperative, unmotivated, careless, lazy students, and their parents, and society at large for preventing them from doing a better job. And many low-achieving students—forced to deal daily with things that they truly don’t understand—are understandably frustrated, too. Who cannot sympathize with the complaints of any of these aggrieved parties? But after all the blame is spread around, not much changes. The number of under-performing math students is still staggeringly high.

Many students struggle sincerely with their classwork and homework and tests, and often they can master no more than 60% of the material presented. That actually represents a significant amount of learning; but for all their trouble, those students are rewarded with nothing but an “F.” Frustrated teachers and students—who have had more than enough angry scolding from concerned citizens—may justifiably feel that, since they are already doing their honest best and working as hard as they can, it just doesn’t get any better than this. They realize that their best is just not good enough, but are convinced that that’s just the way it is.

However, slumbering at the back of everyone’s mind is the vague, hopeful feeling that there is, there must be, a better way of enabling more students to succeed—if only they knew what it was. The fact that something needs doing implies that there must be a way of getting it done. And that is true. But the achievement of different results requires the use of different methods.

Current math teaching methods and materials are actually quite effective, but only with a limited number of learners—students whose minds work in much the same ways as their teachers’. Many of today’s math teachers and textbook authors went to school at a time when only an elite group of pupils was expected to study college prep math. But today, all students—regardless of their personal background (socio-economic level, home language, parents’ level of education, previous mastery of prerequisite math concepts and skills, etc.)—are expected to master algebra. The problem is that today’s pupils are being taught mainly with methods and materials that were designed for use with the elite students of an earlier period, whose personal background and level of preparation was markedly different.

For more universal access to success, pupils and instructors need a truly different model for teaching and learning—a method that takes into account the mindset of the great mass of students whose minds, for the most part, do not work in the same ways as their teachers’. That is the aim of the new books based on the guided discovery method: to provide a new model for teaching and learning about math. Fortunately, the new model is simple and pleasant, for neither teachers nor students have time or energy to take on a complicated new methodology.

Some of the main features of this new approach are: exploration of physical models as a starting point for taking on new abstract ideas; engaging students’ power to notice relationships when working with concrete, numerical examples; using students’ amount sense to inform their sense of procedure; replacing book/teacher-centered explanations with pupil-centered development of reasoning; providing frequent links to underlying concepts and skills; and developing fluency in a context-derived way.

Many (but not all) college prep students in years gone by were capable of memorizing and successfully applying rules, short-cuts and formulas for concepts that they truly did not understand. Students and teachers alike tolerated the ambivalence of plunging ahead without real comprehension, confidently assuming that understanding would come along in its own sweet time, made possible by the enriched perspectives supposedly provided by successful practice (“Don’t worry, you’ll understand it later”). Sometimes that understanding came, sometimes it didn’t. Some students never did understand what they were doing, but got good grades anyway; they relied on the expediency of rote mimicry to eke out reasonable academic success. The goal they achieved was getting good grades; they did not achieve the goal of increased comprehension.

This approach simply does not work with many math under-achievers. For them, real sustainable progress only flows from real understanding. And the development of real understanding generally requires a slower pace, giving proper time to the thinking of deep new thoughts. This may seem grossly inefficient to many teachers, considering the tremendous pressure they are under to produce quick results. But this worry is misplaced. Just as we must spend money at a good sale in order to save money in the long run, so we must spend time to save time. Students who work their way carefully through guided discovery lessons progressively display increased confidence, accuracy, and speed. And each new concept, solidly understood, provides a sure basis for taking on the next new concept. The students also begin to feel a greater connection between their formal math education and their natural sense of curiosity.

Books based on guided discovery are purposely written in a simple style so that students can just read the directions and follow them—and then what should happen in their minds will happen. However, there are always students who have trouble reading or following directions. When teachers or parents are called upon to assist these students, they will get the best results not by telling them what to do and explaining why and how to do it (that approach has already proven not to work with many low-achievers)—but by asking the students leading questions, guiding them to figure things out for themselves. One of the main reasons that students are required to study high school algebra, after all, is for them to learn to think and reason more effectively. Guided discovery, when properly practiced, makes this achievement possible.

One of my favorite Shrek movie moments is Donkey leaping up and down behind a crowd of people who have just been questioned by Shrek, enthusiastically exclaiming, “I know the answer! Call on me! Call on me!” While used for comic effect, these few seconds of film capture the essence of a much repeated scene in childhood: children vigorously raising and waving their hands in class, desperately eager to catch their teacher’s attention: “I know! I know! Call on me!”

While such noisy, chaotic behavior can sometimes be a problem for teachers in terms of managing the group behavior of younger children, it is nevertheless a wonderful problem to have. There are too many awkward moments of silence in classrooms filled with older children, where very few of them seem interested in raising their hand at all. Younger children love to notice, discover, think, and know things—and when they have even partially mastered a new concept or skill, they love sharing their euphoria, especially with their elders.

I think of our little neighbor Veronica knocking at our front door, asking me to come out and see what she could do. “Watch this!” she called, as she jumped from our front porch all the way down to the sidewalk (a total distance of two steps). I remember her happy, expectant face looking up at me, waiting for words of admiration to cheer her fresh triumph. That scene played itself out again another day, when I was called upon to witness the fact that she could not only ride her bike from clear down the street all the way to my house—but she could do it with only one hand holding on to the handlebar!

Not just young people find the simple act of knowing a fact to be stimulating and delightful and worth sharing with others. Think of all the adult games and TV game shows where the whole point is telling what you know—and beating everyone else by telling it first. Possessing knowledge is pleasurable, whether you are young or old.

Now go back in time and imagine some of these enthusiastic hand-raisers, boys and girls who are whizzes at math in elementary school. Some people naturally find numbers to be more fascinating than others, and these kids are definitely at the top of that list. They always catch on first in math class, and are thoughtfully probing in making new discoveries. They often have their hand in the air, hoping to be called on yet again, eager to demonstrate their mastery of new knowledge. Math learning is not a miserable, thankless task for them, as it is for many other children. They are good at math, and they love it.

Now follow these boys and girls on through middle school and high school. Their enthusiasm never wanes, their mastery is ever-growing, their thirst for new knowledge is never quenched, and their appreciation of the beautiful logic of numbers continually matures and deepens. And their attitude of “I know it! Call on me!” never leaves them. Their school grades are brilliant, and their SAT and college entrance exam scores are breathtaking. They could easily have impressive careers in science, research, or industry. But their love of math is so great that they decide to become high school math teachers. What could be more exciting and satisfying?

So they do just that. Their personal and professional vision is to produce students who are as good at math and as excited about it as they are. And that does happen part of the time with some of their students—the students who are naturally gifted at math, like they are. But then there are the rest of the students. The development of their understanding seems to move at a snail’s pace, compared to what the teachers and the best students can do. But the teachers know that everyone has untapped potential, so they pull out all the stops in a concerted effort to bring out that potential in the “lesser” students. Their lectures are very focused, organized, efficient, and high-energy, they have a rich storehouse of examples and evocative stories to share, they excel at making connections between math and other subject areas, and they vigorously challenge all their students to make their best better.

After some years, though, the teachers become somewhat tired and discouraged. It is true that some of their best students go on to do great things in math. But the low-achievers are draining the fun out of their jobs. The teachers are under great pressure from parents, administrators, and the general public to improve average scores on standardized tests—which are embarrassingly low compared to many other countries. And nowadays, students can’t even get a high school diploma unless they have passed Algebra. The pressure on teachers for all their kids to produce is intense and exhausting.

But the low-achievers seldom break through their shrouds of limitation. Their natural “I know it! Call on me!” reaction is rarely seen to surface. Their curiosity seems stifled. Their mental energy and their ambition are too low. They don’t focus, don’t pay adequate attention, and don’t follow directions well. And the teachers’ sense of hope is often dampened by their perception that there are too many lazy kids who don’t care, too many with sloppy study habits, too many with unsupportive family backgrounds and inadequate proficiency in English, too many with concept and skill gaps that show inadequate preparation by previous math teachers, and too many who are so far behind that their individual needs are very difficult to address—even in typical mixed-level classes.

And perhaps most exasperating to the teachers, these low-achieving students tend not to ask questions in class! They claim not to understand the math topics and problems under consideration (which seems almost impossible, when the teachers have gone to such pains to make everything abundantly clear)—but then they don’t ask intelligent questions in class when they have a chance to. How can the teachers help these students, if they don’t know what it is that they don’t get, and if the students don’t clarify the exact nature of their need by asking good questions? If they’re not going to say “I know it! Call on me!” the least they could do is to raise their hands and say, “I don’t get it. And this is the part that I don’t get. Help me to understand it!” But that doesn’t happen nearly often enough. And yet the teachers are being held accountable for the students’ lack of initiative. For teachers and students alike, it can feel frustrating, maddening, and unfair.

It may seem like a hopeless situation. But are there changes that could be made that would significantly improve the situation? Many of the teachers feel that they do not need to make any changes; they are already performing well, doing all of the things that are expected of a good teacher—and spending more time doing it than they are being paid for. The problem is that they are doing a good job of teaching, but the students are not doing a good job of learning—or so they think. The obvious solution to many teachers is for everyone else to change. The students’ earlier math teachers must do a better job. Lagging students must get private tutoring. Students and their parents must get more serious about getting homework done. Families must place greater importance on education. Society must elevate its values. Etc.

But here’s an important question for teachers to ask themselves. Who is the most able to initiate and sustain the changes that will improve the performance of low-achieving math students? Is it the students who have been struggling with math ever since the first grade? That is not likely. Is it the parents who never got very far in math themselves? Is it the college-educated parents, who don’t know how to help their math-challenged children, because they are so unlike themselves? Or is it the highly intelligent, highly educated teachers themselves who are the most capable of coming up with the effective innovations that will make epidemic low achievement a thing of the past? This author argues that educators have by far the greatest potential in that direction.

But what changes can teachers make? Perhaps the change that is most needed has to do with the “I know it! Call on me!” phenomenon. What do high school math teachers typically do in their classes? They explain new math concepts. They explain the mathematical processes that connect with the new concepts. They give brilliantly concise summaries. They tell the formulas, the rules, the shortcuts, the tricks, and the patterns that the students need to know. And when conscientious pupils raise their hands to ask for help with problems from last night’s homework, the teachers tell step by step exactly how to solve them. And they tell all the answers. In short, it is as if someone had said to the teachers, “How do you do these problems?” and the teachers collectively and vigorously raised their hands and said, “We know it! We know it! Call on us!” And then they proceed to do what they have been so good at from their childhood.

This is something that can be changed. It is the students (including the low-achievers), and not the teachers, who should be yelling out, “I know it! Call on me!” How can this be brought about? The first step is for teachers to decide to set aside their natural desire to demonstrate their mathematical knowledge, deferring that pleasurable excitement to their pupils. To do that, they would have to stop telling the answers, stop telling the explanations, the shortcuts, the formulas, the tricks, and the patterns. Instead, they would guide their students’ development by breaking problems down into more easily digestible component parts, and then use leading questions to stimulate their pupils to figure things out for themselves. A teacher that I admire in this regard uses phrases such as, “How would you work this problem?” “Tell me how you did it. Take me through it; give me some numbers here.” “I’m gonna play stupid. I don’t know how to solve this problem. Somebody help me out.” “What do I do next?” “Is this how you do it (showing a wrong example)?” And so on.

A major benefit of such an approach is that the teachers get to hear what their students are thinking, and get a feel for what they know, how well they know it, and how they feel about it. When the teachers are busy doing all the telling, they know what they have just said, and they know that their class was silently watching and listening. But they don’t have any tangible evidence of what the students are actually thinking. All they know is that, based on what they just did, the students should get it. But they often find out later that they didn’t get it. Asking questions opens a window into the students’ minds.

It is easy for teachers to deceive themselves into thinking that “I taught it, but they didn’t learn it,” when they have told their pupils how to do a new kind of problem. But it could be argued that “If they didn’t learn it, then you didn’t teach it,” or “You didn’t teach it in ways that they could learn it.” This may be a tough pill to swallow, but it is not without logic. We know that there are many styles of learning, and that individuals all learn differently. For teachers to rely solely on telling the rule, the explanation, etc. is a one-size-fits-all kind of solution that is really not appropriate in the arena of learning, where one size most emphatically does not fit all.

“Ask a good question, get a good answer.” But our low-achieving students tend not to ask questions in class for three main reasons: some are too shy, others are afraid to expose their ignorance, and many truly don’t know what to ask or how to ask it. Teachers’ exhortations don’t tend to change that. And teachers’ continued wordy explanations do not provide their students with a model for how to ask useful questions in an orderly manner. If teachers want their students to learn to ask better questions, then they must get better at asking their students questions; there is no possible better model. I once heard it said, in reference to the relative merits of telling vs. asking, “If you want to be a teller, go work in the bank! If you want to be an asker, go work in a school.”

To be qualified, math teachers must know their subject matter. But their actual job is not primarily to demonstrate their mastery of the subject matter; it is to lead their pupils to develop their own mastery of the subject. Nor is the teachers’ main job to disseminate information through explaining and modeling; we already have books that do that. And judging from student performance, it is clear that explaining and modeling is a teaching device that is ineffective for many pupils. Books are designed to teach math. It is the teachers’ job to teach math to kids by stimulating curiosity, creating perplexity, and engaging children’s natural power to notice. And to do that with precision, teachers must be actively aware of the students’ varying levels of readiness. What better way is there to accomplish all of these things than to ask questions?

Math teachers are clever people. When they decide to stop telling their students so much and begin asking them more questions, they will find that an intimate knowledge of exactly what their students really know (as opposed to what they are supposed to know) is not that hard to come by. That will enable them to meet their pupils exactly where they are (instead of where they should be). Task analysis of the material to be learned will then guide them in formulating a series of guided discovery tasks and related questions that will create a series of small successes, which will get the learners from where they are to where they need to be. The right series of tasks and questions will stimulate their pupils to develop and “own” their own unbreakable chain of reasoning. Then they will be the ones exclaiming, “I know it! Call on me!” And that is as it should be.

“How do you teach math to students who can hardly speak English?” That’s what the teachers wanted to know, and I was at their school to train them to do just that. “How can you explain a new concept, when they don’t even understand the words?” Having accomplished that task myself with English-learners many times, I began by suggesting that they would get the best results if they abandoned their usual explanations altogether—after all, didn’t they also have pupils who spoke only English, who often didn’t respond well to the teacher’s explanations? It does appear that some students don’t get it, no matter how you explain it. Or—to put it another way—precisely worded definitions do not function universally to make new concepts vividly clear and memorable. We wish they did and we think they ought to, but they don’t.

I decided that the best way to show the teachers how to make a new math concept clear to students who are not fluent in English (or not fluent in math!) was to teach them something in a language that they didn’t speak. I found out that only one of the teachers had ever studied French, and that she had forgotten most of it, so I set out to teach them a lesson in French—with no English whatsoever to help. If I could make my points clear to them in a language that they didn’t speak—with no reference at all to their home language—then they would know that my pedagogical principles were sound, and that they could use those principles themselves to give all of their students access to greater mathematical success.

I pointed to myself and said the French word for me (“moi”), and pointed to them collectively and said the French word for you (“vous”). I repeated that, and then gestured for them all to stand up, while I gave the command in French (“levez-vous”). They all got up, so it was clear that they got the message. Then I gestured for them all to sit down, while I said “asseyez-vous.” I did not begin the lesson by explaining what I was going to do, telling how I was going to do it, and quoting the appropriate state standard. I just jumped in and started doing it. I did not give them a vocabulary list to memorize, or go over basic pronunciation rules, nor did I urge them to pay attention and concentrate. I just told them what to do, showed with gestures how to do it, and watched them to make sure that they did it right. My meaning was clear, everyone responded, and no one was left behind. They could all see for themselves that a combination of brief actions and brief words creates a context of understanding.

I continued in French, telling one individual (with gestures and brief words, of course) to “take the pencil” and “give me the pencil.”  The person next to her was asked to do the same thing. The next teacher was told to “stand up,” “sit down,” “take the pencil,” “give me the pencil,” “take the pen,” and “give me the pen.”  I continued in this way, testing individual comprehension and developing memory by varying the order of the commands with different people, introducing new vocabulary, and gradually eliminating all gestures. In a few minutes, all the teachers were able to accurately answer questions (still in French, of course) such as “Is this a pencil?”—even when being shown a pen.

Being the recipients of an effective lesson, the teachers were then asked (in English, now) to tell what it was that made the lesson so effective. In their summary, they noted that I used very brief phrases; that the first word of every phrase was a command verb; that every command was introduced with an accompanying gesture; that I varied the order of the commands I gave; that I used a lot of repetition, but without being boring; that my tone of voice was polite, inviting, and expectant; that I gradually eliminated the gestures; that I observed individual responses, and supplied gestures when they were needed; that I gave them immediate reassuring feedback to confirm the accuracy of their responses and maintain a positive feeling of success. I also avoided excess verbiage such as “OK everybody, what I’d like you to do next is…,” and other phrases which are intended to politely soften the tone of commands, but actually serve only to confuse the listener’s ear with too many words (when I gave similarly polite commands in French, they could no longer tell which words went with which gestures!).

I then proceeded to do a math lesson, using exactly the same approach, but in English. Everyone understood the content of the lesson—including the mathematically less well-endowed—and became fluent with the incidental facts, details, and relationships. I was concerned about one thing, however: not all of the teachers from the school were present that day. The teachers agreed with me that those teachers should benefit from learning about this approach to teaching/learning, just as they did. So some of the teachers promised to meet with the absentees and tell them about what we did that day. I suggested that it would be more powerful if they could duplicate the lesson that I just did, rather than just tell their fellow teachers about it. They agreed that would be the best course of action—but that it didn’t necessarily have to be done in French. Any other language would do, as long as their absent colleagues didn’t already speak that language.

We ran into a problem here. Most of the teachers were not fluent enough in any second language in order to reenact the lesson that I had done in French. Then one of the teachers mentioned that she was fluent in Chinese. Everyone thought she would be the perfect candidate to do the presentation, because no one else on the staff spoke a single word of Chinese. She hesitated to volunteer, though. She was not sure that on such short notice, she could effectively duplicate the lesson that I had just done. The teachers assured her that of course, she could, it was simple, all she had to do was follow the summary they made just a while ago—which they then went over again for her benefit. She still looked a little nervous about the whole idea, so I suggested that we practice together right then. We would play the role of the absent teachers, she would do my French lesson in Chinese; and we would coach her, offering constructive criticisms when needed.

So she took a pencil in her hand, moved deliberately and quietly to the front of the classroom, visibly gathered her energy, and made eye contact with us in a very commanding way. We looked at her expectantly, and she looked at us with intensely focused concentration. Suddenly she burst into a torrent of Chinese that sounded to us something like, “最高! の四川料理!と至福の一杯!” There was an uneasy silence in the room. Was she expecting a response from us? Or were we supposed to just listen quietly? We couldn’t tell, because we had no idea what she was saying. She paused briefly to come up for air, wagged the pencil at us in a meaningful way (we just couldn’t tell what meaning), and continued with a further uninterrupted stream of rapid-fire Chinese: “両者の思が奏でるモをで楽む!” She paused again and looked at us as if to say, “Well…?” Stunned, we all looked at each other and suddenly burst out laughing! We didn’t intend any disrespect, but it was just too hilarious for us to contain our mirth.

No one doubted this teacher’s sincerity and earnestness, but it was clear that she had forgotten everything that the class had learned from my earlier presentation—or hadn’t really understood it in the first place. I had failed to make it clear to her. What had happened to brief words + brief action = a context of understanding? Suffice it to say, we had some major coaching to do, helping her to break her presentation down into short commands (beginning with the imperative verb) accompanied by a gesture, and so on. Apparently, in her usual spirit of serious efficiency, she was so focused on summarizing what she was going to do, telling us how she was going to do it, defining her vocabulary terms and so on, that she completely forgot that none of us spoke Chinese. I then realized that must be how she typically taught the immigrant children at her school in English— with voluminous verbiage, devoid of any “sheltered English” accommodations that would make he meaning of her words comprehensible.

And that is exactly how many math teachers work with their students. Rather than defining new concepts and procedures through contexts created by brief words and brief actions (especially student actions), their normal default is to define new thoughts through verbal definition, sometimes accompanied by hand-drawn pictures on the board. While this widely used process may be validated by centuries of academic tradition, it does not compare well with the method employed by toddlers throughout the world, as they educate themselves in their mother tongue. Little ones do not begin teaching themselves a brand-new language by attending classes, consulting dictionaries, memorizing vocabulary lists, and so on. They watch and they listen and they do, with much loving guidance from those around them (more in some homes than in others).

Here’s an example of the insufficiency of verbal definitions to invoke a sense of context and connectedness. Elementary math teachers think that they are making themselves perfectly clear when they define the fraction 3/4 by drawing a picture on the board, and point out that there are three shaded sections, and four sections altogether—which is why it is called three-fourths. While teachers may feel that the foregoing explanation is compellingly logical and clear, they may not realize that there are about eight ways that children typically misconstrue these words and pictures. A frequent teacher reaction to the predictable failure of some of the students to grasp their meaning is “I taught it, but they didn’t learn it,” and “What’s not to get?!”

Here’s a sampling of what is actually going on in the minds of some of these “don’t-explain-it-to-me” learners. They hear the teacher pointing out that three parts of the drawing are shaded. And their minds immediately race ahead at lightning speed (yes, even the slower students think incredibly fast). The students don’t hear the rest of what the teacher says, because they’re busy thinking, intuitively jumping to the wrong conclusion: since we just counted three shaded parts, we will now count the one unshaded part. The teacher’s words can easily take automatic second place to the rapid and preemptive power of thinking and noticing.

A more productive approach is to harness the students’ incredible power to notice and think by engaging them in direct action: “Copy this picture, everyone (make sure they do). Let’s call it a pizza. Count how many slices there are altogether (clarify intent with gestures)? Yes, count all the slices, whether they are shaded or not. How many did you get? Four? So did I. Take your pencil and write ‘4’ on your paper, near the picture you drew (demonstrate on the board; make sure everyone follows directions). Now draw a little line over the 4, like this… (demonstrate; quickly check for students who drew the line under the 4, instead of over it). Now count how many slices are shaded. How many? Three? That’s right. Write that number above the bar. Now point to the 3 on top and say ‘three.’ Point to the 4 on the bottom and say ‘fourths.’ What do we call the fraction…? Good. Let’s do another one.” Carry on the lesson in this way, gradually giving less step-by-step directions, gradually using fewer words, until all the students demonstrate that they can look at a picture of a fraction and figure out what to call it by themselves.

Defining through multi-sensory context gives every student access to understanding, because that is a mode of learning that every human has practiced since infancy. Defining exclusively through verbal or written definition is less universal in its immediate impact; academic vocabulary and style of communication is definitely an acquired taste. Does that mean that only definition through context should be employed in education? No, it means that it should be used first, and then followed by the other. When understanding is already established through context, then students are able to invest the following academic verbiage with greater meaning; the abstract words refer to something that they have already experienced.

A phrase in music education pedagogy makes this hierarchy very clear with three words: “function before nomenclature.” That is exactly the opposite of an introductory lesson on fractions I once observed. The teacher began by saying, “Today we’re going to learn about numerators and denominators.” She then proceeded to have the students carefully pronounce these new words, and then spell them. But they didn’t know what they meant, yet! More predictably positive results follow when students are guided to experience something new first, and then be given its name—or at least learn the name while experiencing the new thing.

This approach, while obviously effective with younger children, still applies even at the high school level. For example, a geometry teacher can challenge his students to draw a triangle, then draw a line down from the top vertex (corner point) to the line below it—in such a way that the two lines meet at a right angle. The students can be guided to explore the possibilities: “Now make another triangle with a different shape, and draw an altitude— that’s what we call the line you just drew— from its top vertex down to the other line. Make sure it meets the other line at a right angle. Now draw another triangle with still another shape. Draw an altitude from the top vertex. Now see if you can draw an altitude from one of the other vertices, going from the vertex to the opposite side, meeting the other side at a right angle—is it possible to do that? I’ll come around to look at what you’ve drawn.”

From a student’s perspective, this is very different from the teacher merely defining the new phenomenon with words, and then supplying a diagram or two on the board. The multi-sensory drawing experience gives students something concrete and personal and specific to associate with the new vocabulary—which can then be individually duplicated later for the purpose of review. What a contrast between this and the vague struggle to fathom the meaning of a few distilled words that are supposed to convey the whole essence of an author’s or teacher’s prior experience!

Some might object that defining through context is inefficient, because it takes longer than defining through verbal definition. But is efficiency only a matter of time spent? Or is effectiveness also important? It’s true that it doesn’t take much space in a book to briefly define a new concept, and it doesn’t take much time for a teacher to do the same thing. But is the goal merely for the definition to be stated, heard, and then regurgitated by rote on command—or is the goal is for the definition to be understood? (A third grade remedial reading student once told me that “Silent e at the end of a word makes the vowel say its name.” But he didn’t actually know what a vowel was, and he didn’t know what “silent” meant, either! He had memorized the words for a phonics rule, but had no idea of the meaning of the words!) If failure to provide a context for understanding a new concept results in the new concept not being understood, then where is the efficiency in that? Compared to the usual I-taught-it-but-they-didn’t-learn-it scenario—with all the re-teaching and endless remediation it requires—defining through context is very efficient indeed. It does take longer in the short-term. But it saves much time and confusion in the long run.

Math Teaching Tip #3: An Example of Informal Assessment

A few years ago I tried out a new kind of informal assessment. I had noticed that some of the high school algebra students were confused by expressions like “5a + 3(a + 7).” I wanted to find out what they really understood about the distributive property and the gathering of like terms. So I took a pencil and a clipboard with two pieces of paper on it, and wrote “8a” on the top piece of paper. Next, I lifted up the top paper and drew a vertical line down the middle of the second page—and then let the top page fall back into place. As the students meandered in before class, I went up to one of them and said quietly, “I’m taking a little survey; mind if I ask you a question?” Then I pointed to the “8a” on my clipboard and asked, “What does this mean to you?” The kid said, “I don’t know.” “OK, thank you,” I said politely, and turned away from the student. I jotted down the kid’s name to the left side of the vertical line on the second page; that was where I wrote the names of students who I would gather later in a small group for remedial work. I turned to another pupil, who said, “It means eight times some number.” I thanked him and wrote his name on the right side of my recording page. The next student said, “It means you’ve got eight a’s.” “And what does that mean—that you’ve got eight a’s?” I asked. “I don’t really know,” she said. I wrote her name on the left side. The passing period between classes is short, but it was enough time to identify a small group of students that needed remediation.

Looking back at this experiment, I think there were certain aspects that made it successful. First, my survey focused on a key underlying concept that was essential for success in the students’ current math work. Second, I selected students that I suspected might not know the answers to my survey question (students who usually scored low on homework and tests, or who rarely spoke up in class). Third, I approached the students in a relaxed, informal manner before class, when the ambient noise and movement would provide the mini-interviews with a pressure-removing cover of privacy. Fourth, I wanted them to feel and know that I was sincerely interested in knowing what they thought; and I wanted them to feel at ease, so they would be inclined to honestly tell me what was on their minds. So I purposely used a conversational tone of voice, rather than an authoritative teacher voice—which could have communicated a feeling of “I’m going to ask you something that you really ought to know, so pay attention, concentrate, and answer correctly!” Fifth, to keep the students feeling at ease and to protect the authenticity of future similar encounters, I didn’t tell them if they were right or wrong; I purposely suspended judgment in favor of just getting honest input.

This worked so well with the first class, I decided to try it again with several students from the next class. As before, there were a few students who did not understand the meaning of the “8a” notation. Then it occurred to me to probe a little farther, so I also wrote “5a + 3a” on my clipboard, and asked the students what that meant. Every single student told me that would equal 8a. And when asked again what 8a meant, they still didn’t know. Now that was interesting—they were fluent with the process of gathering like terms, but did not understand what the terms meant!

I tried this same informal assessment technique another time when I wanted to find out who needed help with squares and square roots. When undertaking the Pythagorean Theorem or quadratic equations, it’s essential for students to know what a2 or x2 stands for—and that they know squares at least up to 152. So this time I wrote 52 on the top page of my clipboard, and asked selected students what it meant to them. Some said “twenty-five,” others said “ten,” others waffled between those two answers, one thought it might be “seven,” some were certain, some were not sure, and some had no idea. To probe further, I also wrote 112 and 142 and √100 and √169 and √225 on the clipboard. I found that some students who knew 52 did not know the larger squares; some knew about the squared numbers, but knew nothing about square roots. I also noticed that some students overheard what other surveyed students near them had said, so I added 72 and 82 to my list of questions to prevent them from merely mimicking their classmates. Once again, this survey process produced an accurate selection of students for small-group remediation later in the class period.

When helping out another teacher with his class, I asked if I could do a similar informal survey with his students. I had recently helped one of his students with some other math work, and discovered that she knew nothing at all about squares and roots. I was hoping to help her and some other students with that on this day, and wanted to gently and accurately identify students who would benefit from a small group remedial lesson. The teacher didn’t like the survey idea, and instead wrote a dozen simple expressions containing squares and roots on the board, explained that he wanted to find out who needed help on this sort of thing, passed out paper, and asked students to write their answers. This approach seemed to be simple and direct, but it had less than the desired effect. Students were immediately troubled: “Why were they being tested on this?! No one said anything about there being a test today! Everyone knew that stuff! Who on earth would need help with that?! Did he think they were stupid?! What a waste of time! Etc.” Finally the class quieted down and wrote their answers. Then they exchanged and corrected papers. Guess who got all the answers right but one? Yes, the girl who didn’t know any of it. She was very proud, very protective of her ignorance—and an absolutely consummate cheater. No way was anyone going to find out that she didn’t know that stuff; she made sure that one way or another she got those answers. I wished that we could have just taken the informal survey approach; even though it was less comprehensive than a written quiz, it was more gentle, more accurate, and less time-consuming.

Judging from the number of low-achieving math students in almost every school, it’s clear that even an excellent teacher can give a splendid lecture to an interested, involved class—and the students’ subsequent class work, homework, and tests can nevertheless be riddled with conceptual and factual mistakes. “I taught it, but they didn’t learn it” is an oft-repeated teacher lament.

Trying to teach students lessons that they are not prepared to learn is an exercise in futility. To be successful, lessons must address the minds of the students exactly where they are—not where they are supposed to be. For that to happen, teachers must be aware of what their students really know. In an effort to put their finger on the pulse of their pupils’ minds, teachers usually utter two obligatory words at the end of every lecture: “Any questions?” But all too often, students do not ask questions when given the opportunity to do so. So the teachers are left to wonder: Who doesn’t know what?

If they knew the answer to that question, they would have a realistic chance to do something about it. But even better than the typical end-of-the lesson query—the before-the-lesson mini-survey is an informal assessment tool that teachers can use to get a more accurate picture of the current state of their students’ mathematical thinking. This awareness can help them to adapt the content, delivery style, and pace of the lesson in ways that fit more comfortably to the developing minds of their students—right where they are, not where they should be. When the teachers know the mathematical contents of their students’ minds half as well as they know the content of the math lesson they’re about to teach, then it will be much more possible for them to proudly proclaim, “I taught it, and they learned it!”

Building on the last two problems, what do students need to notice in the diagram, what shapes, properties, and relationships do they need to consider to solve this problem? Then solve it.

The area of rectangle ABCD is 588 cm². The left side is 21 cm. What is the circumference of the circle? Use Π = 22/7.

1.  What do you know about circles. (All circles have a center, a radius, a diameter, a circumference, and an area.)

2.  What are some things you know about squares. (All squares have 4 equal sides, 4 right angles, a perimeter, and an area.)

3.  How do you calculate the circumference of a circle? What information is needed? (How about the diameter. Pi x D = C.)

4.  How do you calculate the perimeter of a square? What information is needed? (You need the measurement of one side. 4 x a side = P.)

5. Draw a diameter from left to right through the middle of the circle, parallel with the bottom of the square. Is the measurement of the diameter the same as the width of the square?

6. Imagine that the diameter of the circle is 7 meters. What is the circumference? (Use 22/7 for Pi.)

7. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

8. Imagine that the diameter of the circle is 10 meters. What is the circumference? (Use 3.14 for Pi.)

9. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

10. Imagine that the diameter of the circle is “D” meters. What is the circumference? (Use 3.14 for Pi.) (D x 3.14)

11. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle? (D x 4)

This week we have a different type of problem. In order for students to solve a problem in a given lesson, they often need an understanding of and fluency with prior concepts, procedures, definitions, or theorems.  If the needed concepts or procedures are missing, the path can be closed to them. Equally important, in order to support their students’ discovery of a successful path, teachers, parents, and tutors also need to know what prior knowledge and skills are required to solve a particular problem.

So for this week, what concepts and procedures does a student need in order to prove that the distance around the inner object is shorter than the distance around the outer object? We will post some suggestions next week. In the meantime, post some yourself by leaving a comment.

Second-grade classes are usually taught to skip-count by twos, fives, and tens as preparation for learning to multiply and divide. That is, the teachers give instructions with that goal in mind, but typically not all of the students reach the goal. Some of them memorize the words, but fail to understand the idea behind them, which leads to failure with multiplication and division when they are presented. And some do not even memorize the words correctly. Why is that? I have observed several situations which shed light on this question, which I would like to share with you.

Consider the experience of Jose, a seventh-grader who wanted to show his study hall teacher how fast he could count by twos from zero to twenty-four. He enthusiastically began: “Two, four, six, eight, ten…, eleven?, no…, twelve?, no…, thirteen?, no…” He asked if he could begin again. He seemed to think that—like riding a bike up a steep hill—if he could get going really fast, the right words would flow and he would make it to the top (to twenty-four). He started over several times, but never got past ten, where his memory always failed him. He apparently had no idea how those counting words came to be in the first place; so with no concept of what he was doing, he could not extend the pattern. He could, however, quite successfully repeat from memory a limited number of words that had no meaning to him.

Consider the case of a high-school special ed student, who was attempting to count objects that were grouped in fives. She began, “Five, ten…,” and then went blank. The teacher’s aide mouthed the first sound of the next word in the pattern, to cue her next response: “Ffff…” “Ff…four,” said the girl. “No. Five, ten, ffff…,” intoned the adult. “Ff…fourteen?” replied the child. She could not successfully repeat from memory a limited number of words that had no meaning to her.

And there is the case of Robin, a sixth-grader who recognized that an array of objects was arranged in groups of five, and counted them like this: “Five, ten, fifteen, twenty, twenty-two, twenty-four, twenty-six, twenty-eight!” She understood that she should be skip-counting by fives, but did not realize that she had suddenly switched from counting by fives to counting by twos. A similar thing happened with a middle-school boy, who counted, “Five, ten, fifteen, twenty, thirty, forty, etc.” He unknowingly switched from fives to tens.

I had two experiences myself with memorizing as a high-school student. Our cheerleaders came back from a special summer camp all fired up to teach us some new cheers. The first one they taught us was composed entirely of nonsense syllables. It was crazy, funny, and we could say it! What did it mean? Nothing. Not less crazy was the day that my Algebra teacher demanded that our class memorize a formula, which she wrote on the board: “y = mx + b.” I raised my hand and asked what it meant. “Don’t worry about that,” she said. “Just memorize it now, and we’ll learn what it means later.” Like the nonsense cheer that we learned during the pep rally, we could say the formula. And what did it mean? Nothing. It was crazy, but it wasn’t fun.

Memorizing musical words can be quite charming and engaging—although equally nonsensical. Many Beatles fans can still sing from memory the French words to “Michelle,” but they don’t know what they mean. Thousands of elementary school children enthusiastically sing Frère Jacques every year without understanding the lyrics they repeat. They have fluent recall of the words, but cannot use them to develop their understanding of the language. In fact, millions of American children annually memorize the words to the Pledge to the Flag—which is in English—but they don’t know what many of those words mean, either. And when they proudly recite, “…and to the republic for Richard stands,” they don’t think to ask who Richard is.

I once witnessed an entire class of kindergartners perform a feat of instantaneous linguistic mimicry. Their music teacher told them to listen quietly while he sang them a new song. To his surprise, they started singing the new song right along with him—even though they had never heard it before. As the words of the song rolled out of his mouth, they also came out of the children’s mouths! Somehow those little five-year-olds were able to instantaneously imitate both the words and the tune of the new song! Did they also understand the meaning of the words? Not necessarily. Just as reading comprehension does not automatically follow accurate decoding (reading a sentence aloud correctly, without understanding its meaning), the universal talent for linguistic mimicry and memorization does not always connect with the mental realm of curiosity, meaning, and understanding.

What do the stories of Jose, Robin, and the other students have in common? They were all skip-chanting, not skip-counting. Their words were mere regurgitations of memorized linguistic patterns—like Frère Jacques—not indicators of perceived amounts. They recalled them the same way that students recall the words to the Pledge to the Flag, through sheer rote memorization—just as I had memorized the nonsense syllables at my high school pep rally. But without an understanding of the concepts behind them, fluently recalled words do not provide a sure basis for developing mathematical understanding. Unfortunately, classrooms throughout the country regularly engage in whole-class unison skip-chanting. When all the students are saying the right words together, the teachers are too often satisfied with the deceptive evidence that learning has taken place—not realizing that if only one student in the class can count by fours, the rest of the class can simultaneously imitate the words, just as the kindergartners instantaneously mimicked the words of their new song.

However, rote memorization is only one of many kinds of memory. Just as there are many different learning styles or modalities, there are also many ways of developing memory. I have found that the model most applicable to the development of conceptually connected mathematical (not merely linguistic) memory is the way in which we learn our way around a new city. We do not use flash cards, timed tests, rhymes, tricks, and unison chanting to accomplish that task. Instead, we look at maps and travel repeatedly around the city, gradually building a mental map of landmarks, streets, and locations. And as we continue traveling to meaningful destinations, our need for maps and directions gradually diminishes until we know our way around with such familiarity that we can go where we want without even thinking about it.

This everyday approach to memory development is normally referred to as remembering—as opposed to memorizing. Even though they both have to do with memory, they are clearly not the same thing. Memorization is purely verbal, sometimes has no connection with experience or conceptual understanding, and is much more stressful. Remembering, on the other hand, is generally derived from multi-sensory kinesthetic experiences within a spatial/conceptual context, with a connected verbal component, and generates relatively little stress.

Here is an example of how to teach skip-counting by fives in a way which incorporates a mathematical way of remembering, rather than a linguistic way of memorizing. Present the students with their own copy of a vertical array of pea pods, with each pod containing five numbered peas.

[   (1)     (2)     (3)     (4)     (5)  ]

[   (6)     (7)     (8)     (9)    (10) ]

[  (11)   (12)   (13)   (14)   (15) ]

etc.

Giving very brief directives, accompanied by clarifying gestures…

• Have the pupils touch each pea and count them aloud in a quiet voice—except for the last pea on the right in each group, which should be counted in a loud voice:

“One, two, three, four, FIVE;

six, seven, eight, nine, TEN; etc.”

Watch the students carefully, and do not allow the class to go faster than the slowest student can manage.

If the speed accelerates to a pace comfortable only for the fastest students, the slower ones are likely to stop touching-and-counting, and will cope with the situation by reverting to rote verbal imitation of their peers.

When the special ed student mentioned above was given these directions, she followed them easily, then smiled and said “I get it!” It is important that the objects (the peas, in this case) in such an array be numbered; without that visual imprint, some students do not make the connection between the numbers that are saying aloud and the written numbers that teachers hope they are imagining (because they are not imaging them!).

• Next, have the students whisper the counting words as they touch the peas, saying each fifth number in a regular voice:

“One, two, three, four, five;

six, seven, eight, nine, ten; etc.”

When the girl mentioned above followed these directions, she laughed and said “This is easy! I can do it!”

• Then have the pupils touch-and-count the peas again, thinking the first four counting words as they touch the peas, and saying each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The girl’s reaction conveyed the pleasure engendered by understanding: “This is fun! I like learning this way!”

• Then have the pupils slide a finger across each group of peas, smoothly gliding from left to right as they again think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

• Now present the students with another vertical array of pea pods, with each pod containing five unnumbered peas (see graphic above).

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]

etc.

Then have the pupils again slide a finger across each group of peas, as they think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The students are now skip-counting by fives. The same approach can be used to teach skip-counting by other numbers—only different graphics are required. Each child must be given the opportunity to show that they can do the last step (skip-counting groups of unnumberd objects) by themselves, without the possibility of verbally imitating other students. The first three steps should be reviewed (with teacher support) by students who display uncertainty, hesitation, or inaccuracy. (“You sound like you’re not certain about that. Count them for me again so we can see if you’re right.”)

What distinguishes skip-counting from skip-chanting? Skip-counting requires physical interaction with numbered objects; skip-chanting requires no actual counting at all. Skip-counting requires mathematical thinking; skip-chanting can be accomplished with no thinking whatsoever. Many teachers rely on slow-paced counting activities with physically constructed grouped amounts to develop the concept of skip-counting, but then revert to skip-chanting to develop fluent recall of the number-facts. They assume that since their students understood the conceptual part of the lesson, they will automatically invest the following rote verbal exercise with understanding. Experience shows that some students do make that connection. Other students, however, are not rapidly recalling the previously counted groups during the group chanting exercise; for them, skip-chanting is a near trance-like state with only one goal: successful verbal duplication. Verbal imitation does not require any conceptual connection in order to be successfully performed. For that reason, some students make the connection while others do not; during the same group activity, some students can be truly skip-counting, while others are merely skip-chanting. The pea-pod lesson modeled above provides all students with the means to remain conceptually connected during the memory-building phase of the skip-counting lesson.

Mathematics presents students with a mode of thinking/reasoning. It includes observation, attention to detail, analysis, synthesis, relevant question asking, and problem solving.  It involves some valuable traits like the ability to handle sweat, frustration, dead ends, perseverance, and the discovery that there is wonder, joy, and even some exhilaration at the end.  We invite students deeper into or higher up this mode of reasoning year-by-year, subject-by-subject. So what is higher level mathematical reasoning?

A look at some of the approved and adopted texts suggests that a typical answer is, “algebra.” Algebra is generally considered to be higher level math thinking for today’s school students. Constants, variables, coefficients, expressions, equations, quadratic equations, real, rational, and irrational numbers, and combining like terms…  If we can just get upper and even lower elementary students to start thinking about some of this, we believe that there is more of an opportunity for higher level math reasoning.

But what about geometry students who have already passed Algebra I, but still have not mastered basic number sense concepts involving fractions? For example, I tutored a high school geometry student recently who did not realize that if amount A is half as much as amount B, then amount B must be twice as much as amount A. This student had memorized the formula for determining the measure of an inscribed angle (it is 1/2 the measure of its intercepted arc), and had solved many problems correctly. But when asked to find the measure of the arc when given the measure of the angle, the student was stumped. It seems that for this student, thinking about basic fractional relationships was actually higher level mathematical reasoning—higher than the current level of understanding.

Higher level math reasoning for students is simply whatever the next step is from where they are now. The relationship between 1/2 and twice, or that a group can be both one and many, or that a “1” sitting in the tens column has a different value than a “1” in the ones column are all higher level math thinking for students who do not yet understand those concepts. People generally consider algebra more abstract than arithmetic, because it appears to be less concrete—and therefore it must be the flagship of “higher level mathematical reasoning.” But any concept is “abstract” to the student who does not understand it yet!

The critical element is not the level of difficulty of the work, but whether or not the work is being addressed through reasoning. Students who can factor quadratic equations because they have memorized a bunch of rules cannot be said to be applying higher level mathematical reasoning, unless they actually understand why they are doing what they are doing. There is a big difference between “higher level activities” and “higher level mathematical reasoning.” When higher level activities are taught through mere memorization or repetitive activities devoid of real understanding, they do not involve any reasoning at all. When lower level activities are taught in ways that make students really think, then those students are involved with higher level mathematical reasoning. And math teaching need not hang its head and feel inferior to other academic disciplines while focusing on these lower level activities.

Algebra is not the problem in itself. Thinking that it accomplishes the need for higher level reasoning and application is.

Another unfortunate answer to what is higher level mathematical reasoning can be seen in the rush to complicate problem sets in textbooks. The geometry book that the student I tutor is using in school, published by a major publisher and state adopted, has outstanding higher level math reasoning problems to solve. I’m having as much fun with some of them as I’m sure that authors and state committee members had. But my student and many in her class are not. There are precious few problems in any section of this book that allow students to develop a confident understanding of the basic concepts and procedures before “higher level math reasoning” is introduced in the form of clever and complicated levels of application.

Rather than leaping to higher level activities that require fluent reasoning that has not yet been developed, the interests of students would be better served if this book (and others like it) presented step-by-step contexts of problems of graduated difficulty—each problem based on the reasoning developed in the previous problem, and preparing students for the next step of reasoning represented in the following problem. The proper function of a math book is to develop mathematical reasoning, not merely to create problems that require its use. By rushing to over-complicate the problems, textbooks unwittingly exclude many students from success, actually thwarting the development of their reasoning and forcing them to rely on mere memorization to cope with their work.

Yes we need to keep earlier concepts and procedures alive by integrating them into problems in subsequent chapters, and yes students need to explore multiple uses and applications, and yes they need to use all of this to solve mathematical problems and not merely perform arithmetic calculations. I am not arguing against any of this. But enrichment is enriching and higher level mathematical reasoning is only reasoning when students have access to it. We should take as much pride in opening up and developing that next level of higher mathematical reasoning, whatever it may be, as we do in the creative, clever, complicated, and fun problems our mathematical minds conceive. We should remember what it’s like for those who are new to all of this. What is higher level mathematical reasoning for them?

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