Tuesday, January 18, 2011

The Confusion Beneath Confusion:
Brief Glimpses of Number and Quantity

For the past few months, I’ve been spending a couple hours each week teaching remedial math to four fifth-graders in inner-city Brooklyn. My assignment: help my students to develop a conceptual grasp of numbers and elementary mathematics.

Over the course of my first week or two with my charges, I gauged my students’ knowledge and understanding—or so I thought. In fact, I had discerned only the superficial: their addition tables were shaky, but they could perform two-digit column-addition; they could carry but were prone to mistakes; three of them could borrow, but again with errors; they could name the place values of two- and in some cases three-digit numbers; they could quickly add multiples of ten in their heads.

It took me several weeks to realize what should have been obvious: that that was a procedural assessment, not a conceptual one. Beneath those weak basic skills were fundamental gaps in my students’ understanding of number and quantity.

Exhibit A: Counting Down in the Double-Digits

Several weeks in, I asked one of my students to subtract 11 from 58. She didn’t know the answer, so I suggested she count down. From 58 to 50 she did fine, but she didn’t know what number came below 50. Over the course of a minute or two of leading questions, she never even guessed in the 40s.

A couple weeks later, I asked the strongest of my four students to build the numbers from 1 to 19 out of the colored wooden blocks that I use to make numbers concrete and tangible. She built the whole set, using blue 10-blocks and differently-colored 1-blocks—not without a bit of effort. Afterwards, I asked her to count down through the numbers she had made, beginning with 19. I pointed to each row of blocks as we went, and she counted it off:

SHE: 19, 18, 17, 16… [pause. She counts the little 1-blocks above the ten.]... 15, 14, 13, 12.. [pause. There is just one little cube sitting above the ten, but she’s not sure what it means]… 14?
I: [shaking my head] Uh-uh. [pause] How many ones are there?
SHE: [pause] Eleven.
I: Right.
SHE: Eleven, ten… [she stops, uncertain; counts all the blocks in the next row.]… nine, eight, seven, six, five, four, three, two, one.

Exhibit B: Counting in the Hundreds

Inspired by these incidents, I had my students make “number scrolls,” an activity I remembered loving back in preschool. I taped together strips of loose-leaf to form the scroll, and told my students to write the numbers in order, starting with one at the beginning of the scroll and adding more strips of loose-leaf as they went.

They had no problem up through 100, but after 100, two of the three (the one who struggles the most was absent) got confused. One tried to go right on to 200, 300, 400, etc.; with a little prompting, she realized that 101 ought to come after 100, but wasn’t sure how to write it; when she got to 199, she didn’t know what came next; her first guess was 1100. After 200, though, she got the pattern and continued on with great enthusiasm until about 370, when we ran out of time.

The second did ok until 109, but after that she went straight to 200, 201, 202, up to 209, and then on to 300, and so on. With the help of some guiding questions (“What’s the number after nine?”) she realized that 110 ought to follow 109 but wasn’t sure how to write it. After 110, she put 120, then 130,and so on. She made no further mistakes, but remained hesitant about both the order of numbers and how to write them, up through the mid 120s, after which she too continued without stumbling for the remainder of the period. (The third student, incidentally, had no such confusion; he explained that they had had a number-writing race in his 4th grade math class, which he’d won.)

The conceptual gaps unearthed by this exercise seem to me even more fundamental than those which had inspired it. The inability to count down shows a weakness but not an absence of understanding, but the inability to count up from 100 indicates that these students literally did not have any conception of the order and size of numbers beyond 100. Numbers such as 200 and 300 were literally meaningless and arbitrary symbols to them.

Exhibit C: Comparing Quantities

Staircase made of blocks 1 through 10
The strangest deficit of all I actually noticed early on, but I didn’t appreciate its importance until later. The wooden blocks mentioned above include differently-colored blocks for each number below ten—the 1-block is a little cube; the 2-block consists of two cubes joined together, with a groove between them; the 3-block three cubes, and so on. Now, several of the exercises I do with the kids begin with putting these blocks in order, from 1 to 10, to create a staircase of even, equilateral steps. When I first started working with them, the kids had an odd way of finding the next block: they would pick up a block that looked about the right length and count the little cubes of which it was composed; if there were the right number of cubes, they knew they had the right block, and would place it next to the preceding one; if not, they’d pick up another and count that one. At first, I tried to point out that there’s a quicker way to do it, but they seemed determined to count, and I wanted to let them explore the blocks at their own pace.

Over dozens of experiences putting the blocks in order, they kept on counting. Then, a couple weeks ago, something wonderful happened. I had started a student on one such exercise, by putting the ten in its correct spot. I watched him pick up the nine, clearly unsure whether he had the right block; he placed it alongside the ten and, seeing that it was just one cube shorter, said to himself, “yeah,” and started looking for the eight. Dear readers, I was elated.


The four students whose math knowledge is on display in these three exhibits are at the far end of the spectrum, but I believe that the lessons they teach us about how and why math failure occurs are widely applicable, not only within grade-schools, but across the entire population. There is a lot to be said about this, and I’m trying to keep these posts short, so tune in next week for a close look at what can be learned from the above evidence.


  1. I used to tutor two grade school brothers when I was in high school. One was good at math and I would just have to monitor his steps, the other had a very tepid grasp of the finite nature of addition and multiplication. often times when he would get a problem wrong I would ask him what lead to his answer; "it felt right" he would say.

    This reminds me (indirectly) of dealing with them. I look forward to hearing more.

  2. Fascinating stuff, Max. Maybe "teaching to the test" wouldn't be so bad if the tests started with really basic, fundamental concepts and worked up from there.

  3. Thanks, Owen. You're absolutely right that teaching to the test is only a problem if the tests are bad. The tricky part is coming up with a test that really does assess conceptual knowledge; that's no easy task.

  4. This is a great post Max. I'm so impressed with how much you've developed your teaching skills since we worked together (I'm not implying in any way you were a bad teacher then...hope this isn't taken as a backhanded compliment). What great questions to push their understanding and fascinating conceptual holes that you unearthed. If only every child were able to sit down in a one of four situation and spend some time (as much time as needed) on these fundamental ideas with such a thoughtful adult.

    And then you took the words right out of my mouth in terms of Owen's comment. What Max did in these tutoring sessions is hard! It's even harder with the constraints of a format that is not interactive, statistically reliable, valid, replicable and cheap.

    Oh, hi Crispin. :)