“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.

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