Tag Archives: Multiple Representations

Multiple or Best Reps?

I find it interesting that on the day that we post our most recent lesson, 4 x 4, (sample page), Dan Meyer posts the question: Aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

I understand where Dan is coming from…Why the overkill, when one proper tool solves the problem. I have three quick responses to this.

1) If the goal of the current activity is to apply previously learned skills, then I agree with Mr. Meyer. Students should develop the savvy to choose the most appropriate tool at hand, and implement it properly. When faced with embedding a nail, is there any sense in using both a hammer and a rock?

2) If the goal of the lesson is to build conceptual understanding of the four formal representations of a linear relationship (words, equations, data, graphs), then generating the other three from any given representation develops this insight. How many students can graph a line by plotting the y-intercept and then counting the slope up and over, but have no idea that they just stated the infinite set of points that satisfy the equation?

3) If the goal of the day is to offer a point of access to the students, then the temporary representation will eventually give way to a higher level of abstraction. Look at the banner on Christopher Danielson’s blog. These multi-link cube models can represent the various ways to factor the number 24. Alongside these 3-dimensional arrays, students could be representing factors symbolically, 2 x 2 x 6, 2 x 3 x 4,  3 x 8 etc. In time, the blocks are left behind for a level of abstraction that is far more efficient. Afterall, it is faster to write the factors on paper than it is to build them with the blocks, especially when the students start factoring much larger numbers. So to push back a little bit on the original question: Are we doing kids a disservice by offering training wheels when learning to ride a bike?

The answer to the question of “multiple representations” or “best representation” is, as always, up to the judgement of the teacher at the time.

Ultimate Cosmic Power in an Itty-Bitty Thinking Space

“Give me any combination of two numbers that have a sum of seven,” I said to my students. One person offered, “Two, five,” which I wrote on the board as (2, 5). I asked for a few more and got (5, 2), (1, 6) and (0, 7).

“Good,” I praised, “now give me ALL the combinations of two numbers that have a sum of seven.” They chuckled. “I want them all, and I want you to write them down.” The students were hesitant, because they knew there are an infinite number of pairs that have a sum of seven. So I challenged one of them to a race. “You write them down on your paper, I’ll write them on the board. Nobody goes to lunch, until one of us is done. Ready, Go!” I scribbled on the board x + y = 7. “Done!”

They didn’t buy it. “I have just written ALL the combinations of two numbers that have a sum of seven. Since you don’t believe me, I’ll do it a different way. In fact, I’ll take you all on. All of you write down combinations of numbers, that way you get done in one-thirtieth of the time, and I’ll still woop ya. Ready, Go!” I quickly sketched the graph of x + y = 7. “Done!”

This goofy little exercise was intended to impart the idea that mathematics gives us the ability to represent an infinite number of elements in a finite time with little effort. I spread my arms wide in front of the class and exclaimed “Ultimate cosmic power…” then brought my hands to rest on a student’s head and continued, “…in an itty-bitty thinking space.” (A play on the Genie’s words from the Disney movie Aladdin.) No offense to the student, but our brains are not very big. Yet, we were able to take all the pairs of numbers whose sum is seven, shove them all in our heads at once and think about them all at the same time. The ability to then communicate them to the world outside of our heads using equations and graphs is what makes mathematics the Ultimate Cosmic Power.

However, most people don’t share in our awe of this power. I believe that is because we never initiate them into our mathematical club. We keep students on the other side of the room while they watch us speak club code and give the secret club handshake, but we never let them in. I have proof of this …

Thoughts on Math by the Uninitiated

From an Algebra 2 student who was just kidding, but his joke reflects how many people perceive the purpose of math:

 Algebra would be a lot easier if they just told you what x was.
– Scott, Class of ’94

For a moment, I thought this next one was kind of cute when a student had just simplified 3x + 2x to equal 5x:

Only in math do you put two things together and get a smaller thing.
– Neal, Class of ’99

Then I realized … In math we don’t combine to make a smaller thing; we combine to make an equal thing!

Then there was the English Teacher who stopped me in the hallway one day, visibly irritated, and poking me in the chest:

You math teachers aren’t very good. My whole life you have been asking people to find x. Why can’t you find it yourselves?

I don’t know what set her off on that day, but I do know one thing about the three people who made the statements above. They all see the sole purpose of math as a cognitive Easter egg hunt. They close their eyes, while the teacher hides the variable. Ready, set, go. Praises and smiles when the basket is full.

They have no appreciation for the true beauty and power of Algebra, because we never initiated them into the club. So how do we teach them the secret handshake?

The Initiation Rites

  • Context
  • Multiple Representations
  • Complexity
  • Abstract Generalizations


Too often we jump straight to the naked math problem. For example,  we ask students to graph y = 2x + 1 or evaluate 2 – 5, without offering any kind of context. Context gives the students something to cognitively hold onto while they are grappling with the math concepts. Take for instance, the teaching of negative integers. In the Wallflowers lesson, students are asked to mathematically represent scenarios that they can relate to (a high school dance). While these scenarios are a bit contrived (girls are positive and boys are negative), students can “see” how a balance of positives and negatives equals zero, taking away negatives leaves behind positives, etc. From here it is useful to go to other contexts that are more applicable. The Postman Always Ring Twice relates positive values to checks and negative numbers to bills, showing how truly “subtracting a negative is the same as adding a positive.” By presenting the context first and then asking the students to represent it symbolically, we give them a framework in which to think when eventually presented with the naked math problem.

Multiple Representations

Thoughts on Math by One of the Initiated

I once met a Calculus teacher in Massachusetts who was originally from India. She was very distraught about teaching in America, She said that she kept getting complaints from both students and parents about her teaching style. She said that all she was doing was teaching the way they do in India. When I asked her to characterize the style for me, she said that in India there is a saying:

If you know how to do one problem inside and out, you can do a hundred just like it.
– Seheti, Math Teacher from India

I could easily see the contrast. In India they teach students a hundred ways to do one problem. In America we teach one way to do a hundred problems.

To further make Seheti’s point, I had a conversation with my daughter’s third grade teacher on this very point. She asked me:

Thoughts on Math by the Uninitiated

How would you do this problem?

Of course, she gave me this example because it requires multiple borrowing in a traditional algorithm. I told her, however, that I don’t see it as a subtraction problem; I see it as an addition problem. I have 165 above 3,000 and 13 below 3, 000, therefore, 165 plus 10 plus 3 was 178 … with no borrowing. Her response was, “But why would you go through all that trouble?” I chuckled at the unintentional irony, placed the pencil on the paper and challenged, ” Do your method without picking up the pencil.”

These examples show the strength in teaching students Multiple Representations of a problem, and the weakness of teaching only one method. In the Candy Bars lesson, multi-link cubes are used to demonstrate why we need a common denominator when adding fractions. Fractions, after all, are merely relations to the whole. Operations on fractions can only be done then on the same size whole!

When students are first asked to show one-half of a candy bar with the cubes they put together two cubes of different colors. When asked to “build” two-thirds of a candy bar with the cubes, the construct a stick of three cubes, one being of a different color. When prompted to “show” what fraction of the candy bar they have if they are given portions, they naturally connect the two to make one, which shows two-fifths, which is exactly how they incorrectly complete the algorithm for adding fractions: they add both numerators and denominators.

When corrected and told that the bars must be the same size and still show one-half and two-thirds, the students independently build sticks of 6 cubes each (three of one are colored, two of the other are colored). When then pressed to now tell us how much “of the same size candy bar they now have, they combine the colored cubes, but keep the stick the same 6-cube length and present … five-sixths, which is correct. Publically generating an algorithm that now represents what we do is easy, and the cubes offer a model for students to fall back on in the event that they forget the procedure.


Too often we think that our job is to always make math simple for our students. This initiative then leads us to break problems down to their tiny, separate parts, and we never ask the students to put it all together. This point was made by Tad Watanabe in NCTM’s Mathematics Teacher (Vol 93, No 1, p 31). Mr. Watanabe showed a high school entrance exam from Japan. It had only 7 problems. One of them is displayed on the right. (Click to enlarge the image.)

Take some time to do the above problem. It puts to shame what we expect from our 8th graders: “There are 10 marbles in the box. 3 are red. What is the probability of drawing a red marble?” The Japanese expect there students to do complex problems, therefore, they teach them to do complex problems. We Americans often feel that we have failed if we pose students with difficult problems. One would have to look long and far to find an Algebra final exam in the United States with the level of complexity of the Japanese example above. It is not that Japanese students are more capable than American students and therefore, they can do these kinds of problems. The Japanese students are more capable because they are regularly asked to do these kind of problems!

To further the point on Complexity, I would like to share the story of the American math teacher who visited schools in Bulgaria. When asked to contrast math education between Europe and the U.S., he said he could do so by showing a typical question from both countries:

Typical Geometry Question in Bulgaria: Draw a triangle. Draw a semicircle on each side. Within each semicircle , inscribe the rectangle of the greatest area. Draw the lines that pass through the centers perpendicular to the side of the triangle. Prove that these lines are concurrent.

Typical Geometry Question in America: Draw a triangle.

Abstract Generalizations

Finally, the ultimate in Ultimate Cosmic Power: Abstract Generalizations. In other words, students are asked to model their world mathematically. As when I ask my daughter and her friend when they were in third grade, “Your class has 20 students. How many are boys and how many are girls ? How many boys and girls might be in another class of 20 students?” We went through several scenarios, and then I asked, “If we allow b to represent the number of boys, and g to represent the number of girls in a class of 20 students, what would you say about the the number of boys, girls and the total students?” They said “b plus a equals 20.” I then showed how to write b + g = 20, and they agreed. There were several abstract representations going on with the girls. Words and an equation, and a brief encounter with data.

The Rising Water lesson does the same thing, but more formally. It first poses a context in words (one representation): A swimming pool contains water 10 cm deep. The water is rising 3 cm per minute. The students are to then generate a table of values, an equation and a graph for this scenario (3 more representations). The objective of the lesson is to teach students that all four representations describe the same relationship between two quantities (time and water depth). The students are then asked to generate their own scenarios, with the four representations. The more that students are asked to create their own models, the better capable they will be when they are presented with one.

So our Initiation Rites into our math club are these four components of Ultimate Cosmic Power (Context, Multiple Representations, Complexity and Abstract Generalizations). We will go back to these constantly in our discussions, as well as to the Four and a Half Principles of Quality Math Instruction posted previously. To show that there is hope in teaching in this manner, I share with you a statement from a former student at the end of the year.

Thoughts on Math by One of the Initiated

Poetry is the language of love. Math is the language of everything else.
– John, Class of ’99