Tutoring my friends' son in algebra reminds me of my math teaching experience and what I learned about making things make sense…

Sense-making should be a priority when we write, present, or program (for those of us who write software for a living). Successful communication, that is, all the way through to understanding, is not an accident. Producing work that other people can make sense of takes extra care.

People are capable of muddling through unnecessary complexity and unclear explanations, but when we reduce the cognitive load imposed by distraction and confusion, we save other people time or free them to think about the more interesting and important dimensions of a problem — possibly a shared problem.

Tutoring my friends' son reminds me of this. In a high school algebra course that seems to have been reduced to printed worksheets, I noticed the following prompt and thought, "Finally, something interesting!" The course presents arithmetic sequences in the context of a unit on linear equations in two dimensions.

If you use a screen reader, please skip all code blocks. I provide accessible descriptions after introducing each of the two patterns.

|  Figure Number:       |
|    2     3      4     |
|  ▌▌▌▌  ▌▌▌▌▌  ▌▌▌▌▌▌  |
|    ▌▌    ▌▌▌    ▌▌▌▌  |
|    ▌     ▌      ▌     |
Accessible description of the original pattern

Shapes consisting of tiles on a grid…

Figure 0 has 2 tiles in the top row, then 1 empty space. The middle row is empty. The bottom row has 2 empty spaces, then 1 tile. Figure 0 has 3 tiles in total.

Figure 1 has 3 tiles in the top row. Both the middle and bottom rows have 2 empty spaces, then 1 tile. Figure 1 has 5 tiles in total. For the next figure, add 1 tile to the top row and 1 tile to the middle row. From Figure 2 onward, the shape is a boxy uppercase "T", with the horizontal line twice as thick on the right side and getting longer.

Drawing Figure 1 is the first exercise. Immediately, the author could increase the chances of a correct and meaningful result by leaving space to the left of Figure 2 instead of having students draw Figure 1 someplace below. It's a sequence, after all.

Printing a faint grid (cf. Dr. Edward Tufte's "grey grid") would improve the quality of the result. Adolescents are still developing their fine motor skills. With cell phones, tablets, and laptops occupying much of the day, young people now need to do less physical writing and drawing than at any time since the era of the one-room schoolhouse, when each student had a slate and a piece of chalk.

We recognize the pattern instantly. Well, we adults recognize it. Are we certain that students see it? That all students see it?

Writing the formula for the number of tiles in Figure n is the next exercise. Peeking ahead, students will have to count the tiles in Figure 8.

The author could support humans' natural pattern-seeking by asking students to shade or circle the new tiles in each figure of this increasing sequence. As soon as a student does that, the student will distinguish what looks like a constant part, 5 tiles, and a variable part, which grows by 2 tiles each time.

Figure
Number:             2     3      4
                  ▌▌▌▌  ▌▌▌▌░  ▌▌▌▌▌░
                    ▌▌    ▌▌░    ▌▌▌░
                    ▌     ▌      ▌
Change:                  +2     +2
Tiles:              7     9     11


Figure
Number,
n:             1    2     3      4
             ▌▌▌  ▌▌▌▌  ▌▌▌▌▌  ▌▌▌▌▌▌
               ▌    ▌▌    ▌▌▌    ▌▌▌▌
Tiles,         ▌    ▌     ▌      ▌
t(n):          5    7     9     11

A plausible initial answer would be t(n) ≟ 5 + 2n

I won't say too much here about notational distraction. To an adolescent, the letters t and n both resemble unknowns. The link from n and t(n) to x- and y-coordinates on a graph might not be clear. If I were the author, I'd want to try out at least these two forms. If I were the teacher, I'd assign the work in class, to be done in groups. I'd quietly give half the members of each group a paper with [n, t(n)] and the other half, one with (x, y), and let them sort it out together. Incidentally, from the point of view of cognitive development, notation is the only thing standing in the way of introducing this work early in elementary school. Young children could build the shapes with plastic tiles, point to the interesting parts, and talk about them.

Students are likely to get the formula wrong initially, because the author decided that the starting figure should be numbered Figure 1. On the back side of the same worksheet is a question about y-intercepts, for which x (n, here on the front side of the paper) will have to be brought down to 0.

A teacher might teach the habit of drawing Figure 0 independently of what worksheets or textbooks expect, but students grumble about extra work, and the disjoint tiles in this particular shape would be a distraction. The fundamental problem is that the author didn't think and didn't care. Humans, even very young, can see 5 tiles in (what they are told is) the starting figure, and they can see the increase of 2 tiles in each new figure. Sense-making places a premium on what we can see with our own eyes!

The author sows confusion and wastes the limited time available for teaching and learning. The formula turns out to be t(n) ≟ 3 + 2n

Figure
Number,
n:        0    1    2     3      4
        ▌▌   ▌▌▌  ▌▌▌▌  ▌▌▌▌▌  ▌▌▌▌▌▌
               ▌    ▌▌    ▌▌▌    ▌▌▌▌
Tiles,    ▌    ▌    ▌     ▌      ▌
t(n):     3    5    7     9     11

An attentive author would have crafted something like:

Figure
Number:                   2      3       4
                       ▌▌▌▌  ▌▌▌▌▌  ▌▌▌▌▌▌
                       ▌ ▌▌  ▌ ▌▌▌  ▌ ▌▌▌▌



Figure
Number:                   2      3       4
                       ▌▌▌▌  ▌▌▌▌░  ▌▌▌▌▌░
                       ▌ ▌▌  ▌ ▌▌░  ▌ ▌▌▌░
Change:                         +2      +2
Tiles:                    7      9      11


Figure
Number,
n or x:       0     1     2      3       4
             ▌▌   ▌▌▌  ▌▌▌▌  ▌▌▌▌▌  ▌▌▌▌▌▌
Tiles,       ▌    ▌ ▌  ▌ ▌▌  ▌ ▌▌▌  ▌ ▌▌▌▌
t(n) or y:    3     5     7      9      11
Accessible description of the proposed alternative pattern

Shapes consisting of tiles on a grid…

Proposed alternative figures are made up of 2 equal-width rows. In each figure, all grid spaces have a tile, except the 2nd space on the bottom row. Each figure has the same total number of tiles as the original. From Figure 1 onward, the shape is a boxy lowercase "n", with the right-side vertical line getting wider.

Beyond avoiding disjoint tiles in Figure 0, this shape offers another advantage: an extra pattern that is readily visible and easy to express in words. Some students might start by describing the shape as: "2 equal rows, except the bottom row is missing 1 tile." From there, groups of students would surely arrive at:

t(n) = 2(n + 2) – 1 = 2n + 4 – 1 = 2n + 3

What a way for students to confirm that simplification isn't just about following rules and moving symbols around!

t(n) = (n + 2) + (n + 1) would be another good first step, without multiplication. Some students will realize that the location of the missing tile isn't important, and that laying the tiles out in two rows was a visual convenience.

If I were teaching, I wouldn't necessarily prescribe drawing Figures 1 and 0. I'd leave ample space and provide a grid, but I'd ask questions that would require students to use their own minds, and possibly to exchange ideas with one another.

  1. "What other figure or figures in the sequence are worth drawing, to help you find the formula for the number of tiles in Figure n?"
  2. "Some of you drew Figure 1 whereas others drew Figure 0, too. What are some advantages of counting from 1? Of counting from 0?"
  3. Later in the algebra course:
    "Would counting from 1 instead of from 0 [or vice versa] change the arithmetic and geometric sequence- and series-related formulas that you've learned?"
  4. Much later, in an accounting course:
    "What's the difference, mathematically speaking, between an 'annuity due' and an 'ordinary annuity'?"

As for Figure 8, students have to "wait for teacher" to find out whether their work is correct. Asking instead about Figure 6 and leaving space on the right for just two more drawings would make it practical for students to check the formula and the arithmetic. The usual verification process, substituting n = 8 and the t(n) result back into the formula, would not help in this case. Students would merely re-do the algebraic steps that they had just finished. Repetition is not a verification independent of the original work. Supporting students' use of multiple methods (including recognizing the advantages and disadvantages of particular methods) suits individuals and, in this case, also grants autonomy. In adult life, there will be no teacher to supply the answer.

Next time you write a report or presentation, or prepare your code for a pull request review, ask yourself whether you've done everything possible so that your work will make sense to your colleagues.

Goal Example in this article
Anticipate your conclusion and build up to it intentionally Leaving space to the left and right, for more figures in the sequence
Show instead of telling Choosing a coherent example: the starting point, the shape, the notation, the questions
Focus Choosing a shape with no disjoint tiles, and offering a familiar alternative to function notation
Engage your colleagues Providing an independent verification method, and one that can be used autonomously

The extra care might help you clarify your own thinking. It did for me. By taking another look, I progressed from making 0 the starting point to choosing a more interesting shape as well. At work, many times I've been able to simplify my code and explanations to facilitate a less perfunctory and more meaningful code review, with all the benefits that entails.

Thanks for reading! Comments by e-mail are welcome and I will acknowledge corrections publicly. This article has also been published on LinkedIn.

About me: I took some years out of my software engineering career to qualify and work as a teacher. I mostly taught math but also earned credentials in world languages, business, and elementary education._

Thank you to Priscilla Jo Elsner, PhD, the Silicon Valley Mathematics Initiative (SVMI) (of the current staff, I know Tracy Sola), the authors of the CPM curriculum, and my parents (my late father and my mother spent their entire careers as teachers), for teaching me how to teach math.