Math Teaching Tip #1 – Remembering What It’s Like Not to Know

I had been to London a few times before, so I knew my way around pretty well. Nevertheless, I always carried a map. So I felt sure that I would not have any trouble finding my way to my appointment with a local education official-especially since he had given such good directions: “Take the Northern Line; get off at the Elephant and Castle; go straight out the door and cross to the other side of the road; go up the first street a couple hundred meters; our office is on the left, just before the park. You can’t miss it.”
That sounded pretty easy. I had ridden the Northern Line of the Underground dozens of times, although I had never gotten off at the Elephant and Castle. So I got off the Tube at the correct stop and went up the escalator, thinking to head straight out the door. That’s when my troubles began. When I got to the top of the escalator, there was not a door to go straight out from-there were five doors, all distributed around the circumference of a circular-shaped exit/entry area! The official hadn’t mentioned that. I had no idea which direction to exit. So much for “go straight out the door!”
But all was not lost. I had my trusty map, and I knew the name of the street I was headed for, so I just headed out the nearest door to look for the street sign. As I emerged, I discovered that the tube stop was a round island surrounded by several wide lanes of swirling traffic, with streets radiating out in several directions. The street signs in London are embedded in the walls of the buildings, and none of them could be seen from where I stood. (What is the practicality, I wondered, of street signs that are only visible once you’ve actually turned onto the street? Do they serve to offer reassurance to people who already know where they’re going?!)
It took a long time for me to wander around that circus (well, that is what they call it) until I finally found the right street. I finally arrived at my appointment somewhat late and rather perturbed. But the experience was not lost on me. The man had given me directions that described exactly what he did every day. But he failed to take into account that I had never been there before. And the fact that he did not remember what it was like to be there for the first time caused him to omit important information, which rendered his directions meaningless to me. They would only make sense to a person who had already been there! “You can’t miss it,” indeed.
It struck me as I left my appointment that this was a perfect metaphor for what often goes wrong with math education. I once heard a teacher introduce fractions to his class by pronouncing “numerator,” and “denominator,” writing them on the board, quizzing his pupils on the correct spelling of the words, and then verbally defining their meaning. While his presentation was technically correct, and was an accurate description of how he thought of fractions every day, the lesson was meaningless to many of his students because it provided no connection to physical or visual experience. The instructor had forgotten what it was like never to have seen or considered a picture of a fraction before, or to have divided an object or groups of objects into fractional parts. He had forgotten what it was like to not know about fractions. As a consequence, his instructions would make sense mainly to students who already knew about fractions; but the lesson would go right over the heads of other students, even when they’re diligently paying attention.
Fortunately, most teachers now know better than to present a fraction lesson like that-although that style of presentation is still pretty much the norm in algebra classes! To introduce fractions, it is more typical for the teacher to begin by drawing a circle on the blackboard, drawing vertical and horizontal diameters through it, shading three of the four resulting parts-and then proceed to tell the students that since there are four parts altogether, and three of them are shaded, we call this “three fourths.” A few teachers might consider this one illustration sufficient to define all fractions. But most teachers would provide several pictures of different fractions, and then ask volunteer students to name them properly. They then consider their introduction complete.
This type of presentation seems to many teachers to cover all the bases, so they are surprised and dismayed to discover later that a couple of their students still have no understanding of basic fractions whatsoever. Naturally, teachers feel a need to account for this “I taught it-but they didn’t learn it” situation. In days gone by, teachers would simply label those students as stupid, lazy, and incompetent; they weren’t paying attention, they weren’t following directions, they weren’t trying hard enough, they weren’t focused, they didn’t care. Nowadays, a different label is invoked: the students didn’t learn the lesson because they have learning disabilities.
But there are other reasons why this seemingly effective presentation is very much like telling a first-time visitor to London to get off at the Elephant and Castle and go “straight out the door.” If the teacher is doing all the drawing on the board, the teacher owns the drawings, not the students. Some pupils make better sense of the teacher’s drawings when they copy them onto their own paper. For them, feeling the information through their own fingers is more effective than merely looking at someone else’s pictured thought. But even when the lesson requires students to copy the teacher’s drawings, some students copy the drawings incorrectly, because they fail to notice important details, or fall behind and become confused or flustered. So they still don’t learn the lesson that is supposedly being taught.
Even if their drawings are perfect, pupils can still fail to connect the pictures to the fraction nomenclature voiced by the teacher. While the teacher is proclaiming ” that’s why we call it three fourths…” some students are busy studying the picture, noticing that three sections are shaded and one is not. While their minds are completely occupied with taking in this visual information, they may not even hear the teacher’s voice at all. It is easy for teachers to assume that because they said something, everyone heard and understood what was said-forgetting how many times a day their students fail to respond to the sound of their voice telling them to put their books away, or to put their pencils down, or to be quiet. Even if the students do hear what is said, the teacher’s words can sometimes provoke nothing but confusion: “Why is he calling it three fourths, when one part is white and three parts are shaded? That doesn’t make sense!”
And there is still more that can go wrong, even when the students understand that they should count how many parts there are altogether, and how many of that total are shaded. When writing the fraction, the learners may write the total number of parts on top, and the number of shaded parts on the bottom. Or they may write the fraction correctly, but read it from the bottom up, instead of from the top down. Or they may use the ordinal number terminology incorrectly: “third fourth,” “three fours,” “thirds four,” etc. There really are five doors you can go out at the Elephant and Castle-and even more ways to misconstrue a simple introductory lesson on basic fraction identification.
One imprtant key to avoiding these instruction land-mines is for the teacher to remember what it’s like not to know. What is potentially confusing about the subject? What can go wrong? What steps of learning are prerequisite to other steps? It is helpful for the teacher to adopt the attitude of an actress in a stage play. Before the first performance, the actress rehearses her part thoroughly-and naturally, she knows how the play ends. But when it comes time to perform Act I, Scene I, she acts as if she didn’t already know the outcome of the play. She acts in a way that is appropriate for the beginning of the play.
So the math teacher should guide her students at the beginning of the lesson with the attitude of someone who doesn’t already know what it all means. In guiding her students’ exploration of the subject, the teacher’s words should give voice to the questions that are emerging in the students’ mind-or that ought to be. The students’ attention must be skillfully directed with simple commands and questions. Here is an example of how to do this with a lesson that introduces fractions.
The teacher hands every student a copy of a page that has many pictures of fractions (there are many ways to do this, but pictures of “pizzas” will do for now). Each pizza has only one shaded slice, no matter how many slices there are altogether. The first pizza is a picture of “one fourth.” The teacher says, “Everybody touch the first pizza on your page. Count all the slices. Yes, count the shaded slice, too. How many slices are there altogether? Write that number on a piece of scratch paper.” The teacher writes the number on the board and looks to make sure that everyone has followed the directions precisely. “Now draw a little line over the four.” The teacher models his instruction on the board, and quickly inspects the students’ work, offering guidance to students who have somehow managed to draw their line under the four instead of over it. “Now count how many slices are shaded… Yes, just one. Now write that number above the line you drew. Everybody touch the top number and say ‘one.’ Now touch the bottom number and say ‘fourth.’ What do we call this fraction? That’s right: ‘one fourth.’ Good. Now let’s look at the next pizza.”
[By having the students count all the parts first and then the shaded part, the teacher has shown how to determine the denominator and the numerator-even though the specific nomenclature has not yet been introduced. If the students had counted the non-shaded part first, some of them-in spite of verbal instructions-would have automatically counted the shaded ones next, rather than the total amount. Task order is important in shaping the direction that the students’ thinking takes.]
Continuing the lesson, the teacher gives exactly the same directions for the next four or five pizzas. Then he tells the students, “Now turn your pencil around so it looks like you’re going to write with your eraser. Count all the slices on the next pizza. Pretend to write that number on your scratch paper. Now draw an imaginary line over the number. How many slices are shaded? Then write an imaginary ‘one’ over the line. What is this fraction called?” Two or three similar examples follow.
“Now put your pencils down. Count how many slices there are altogether on the next pizza. Pretend to write that number with your finger, and draw a line over it. How many are shaded? Pretend to write that number above. What is the name of this fraction?”
“Now I have a challenge for you. Who can name the first five fractions?” The teacher calls on a volunteer. Then another volunteer names the next five fractions. “Now I need two volunteers who will act as partners.” The teacher hands an answer key to one of the partners and says to the other partner, “Name each fraction. Your partner will check your accuracy with the answer key. When you answer correctly, she will say ‘Yes.’ When you are wrong she will say, ‘Try again,’ and you will have to figure out the right answer.” After the partners model the new activity, the teacher gives an answer key to each pair of students, and together they practice proving their mastery of the new lesson.
A lesson such as this uses commands and questions to engage students’ natural ability to notice. And the noticing is directed in such a way as to avoid potential points of confusion. The strategies are simple and learner-friendly: What do you count? What do you call it? Supervised practice is undertaken immediately, providing the teacher with almost instant assessment-and it involves every single student, rather than a few vocal volunteers. Practice is safeguarded by immediate peer feedback, which demands immediate student self-correction. A lesson such as this makes sure that every student finds their way out the right exit at the Elephant and Castle.

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