Building on Seeing Double

For the past few weeks I have been blessed with the opportunity to work with a fourth grader and really enjoy the learning and relationships that have been fostered. The unfolding story in this entry is to mark the evolution of this journey, knowing there is much more to be written. Enjoy the journey, remark on the learning, and never forget to relish the relationships.

She sat there tapping her clenched fist as waited for her parent to finish the introductions, and then began to share what we had tried to do via Zoom the week prior. My new student had work from her teacher she wanted to share. She had her work from our first interaction she had questions about. Mostly, she had an inquisitive nature with a mix of both question statements and revelations she excitedly buzzed about.

As we moved toward the work we were supposed to be addressing, she shared a statement that centered our work and became the linch pin for our journey. On a worksheet of many subtraction problems, she pointed to a problem and said, “I got that one, because it is really easy…” The problem is shown below:


I asked why she thought it was easy and she said it was because they were the same. Since she could see doubles as easy, this gave me a thought, so I tried drawing the same representation with dots. I drew three filled in dots to represent 3 and three empty dots to represent -3. We discussed how these come together to form zero pairs, which she accepted as an easy fact. When I gave her the same problem with one small change as shown below:

Doubles Again

I asked her how she saw this change would affect the result, and how that relates to our picture. Wanting to extend our understanding, we tried making one more small modification, I added one to either the addend or the minuend. After we discussed this for some time, I drew a box around the two problems, and we considered how the previous understanding was being visualized. We also discussed how this helped us make sense of the problems, by seeing what is “new” in the problem and what we have already done in the doubles problems.

It was here where she taught me something new, the connections between the four operations and the Division Algorithm. Since she was unclear how the sum and difference were connected to multiplication and division (her areas of need), she was very excited to see that she’s been multiplying and dividing her whole life. So we talked about the Division Algorithm, is really just a way of viewing how the four operations work together. Noting that repeated addition is multiplication and repeated subtraction is division. In both cases, the remainder, r, is what is the same connective tissue. For example, 4 divided by 3, requires one subtraction of a group of 3, with a remainder of 1, and the product of two groups of 3 and 1 more is the same as the sum of two products of 2 (2×2=4) and adding one more group of 2 plus one more yields 3 groups of 2 plus 1 i.e. 7=4+3=2*2+(2+1)=(3*2)+1. Such a simple statement, but the use of the associative property, combined with groups of and flexibility of this kind of number sense thinking is profound.

Practice = The Games

To provide space for sense making and to make this experience fun, I was stoked to use two of my new tools from the incredible Marilyn Burns: 1) Circles and Stars and 2) Pathways.

Starting with Circles and Stars, we modified the original game, so that each pair rolled one dice, the first partner rolled for how many stars, the second for how many circles to draw. Each partner took turns, counting the total number of stars and recorded their score. The visual of having the circles as groups of was really powerful, especially when I asked if she could write her total as a sum, as a product, as a quotient, and as a difference. This pushed our thinking quite a bit and we had some great conversations about the various ways to combine numbers. The beauty of our conversations was especially poignant as a tracker of her learning, what was becoming firmer and what was still needing support.

Over time, we moved on to include Pathways, which we use the first example from Marilyn’s entry, and we haven’t gotten to the point of adding the additional complications to the game. The game play is to choose a starting side, from which the opposite marks the finish line. The winner of the game is the first one to connect a touching sequence of squares on the 5×4 grid, where connecting means the two squares share a side or a corner. The players are able to choose any square they wish, and they can claim the square if they can write a product that equals that value. For example, if the square I chose had the number 12, then if I wrote 3×4 or 6×2, I could claim the square. We agreed that we couldn’t claim the square by saying 12=12×1. On our last meeting, the victory here is my student ended up computing both my score and hers. On the pathways to doing so, she was choosing numbers that end if 5, since she could skip count to that number by counting her fingers. We discussed this and we were feeling good about it. Once she ran out of 5’s, she tried the same strategy with 2, and was excited to see anything ending in an even number was divisible by 2. When I chose the square 64, she checked it by counting by 2’s stating 64 takes 32 quantities of 2 to equal the same value. The value of the whole experience provided a deep connections she is seeing in recognizing repeated addition as multiplication.

So far what I’ve learned or had reinforced through this wonderful experience are: 1) Relationships are key, nothing goes forward without a solid foundation in treating the learner first as a person. 2) By listening to the student, you will find out so much more, including an entry point that builds on a student’s strengths rather than focusing on their weaknesses. 3) There are many deep connections in studying something as seemingly simple as arithmetic, including the underlying observations of the operations and their relationship. 4) The power of visuals to connect learning and being able to see mathematics, not just writing numbers. 5) The power of asking questions, regardless if the student is right or wrong, challenging them on both, and asking for justification, adds a new dimension of struggle and understanding. 6) As always, I end up learning way more from my students than I think I ever actually teach them. And, 7) Working with students in the best part of any work day.

Published by mathkaveli

I'm a math geek.

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