Author Archives: David Pearce
In the latest series of blog posts by the Coordinating Units team, we explored our recent work in the classroom developing multiplicative reasoning with students. One area that we focused on was creating different contexts to present the task. If students can generalize the mathematics and transfer it to different contexts, then they can use it in problem solving instead of memorized procedures.
To begin the school year we used cubes from which the students built towers.
In October, we changed the context of the task, sending them out “trick or treating,” by having them place a predetermined number of candies (small blocks) in bags.
In November, we changed the context yet again, having the students retrieve turkey cut-outs, for their Thanksgiving dinner, with a specified number of feathers on each turkey.
In all the variations, the essence of the task was preserved as the students were asked to make equal groups of items and then asked a series of questions. For example, they were asked to make towers with three cubes, bags of candies with three candies, and turkeys with three feathers, and then asked to determine the number of groups (towers, bags, turkeys), the number of items in each group (three), and the total number of items (cubes, candies, feathers).
Many teachers have experienced students struggling when the context of a concept is changed.
One example of this is:
14 x 10 = 140
So, when multiplying by 10 just add a zero to the end of the number.
Except, an example like this:
1.4 x 10 ≠ 1.40
doesn’t follow this rule even though the problems look the same.
One of the reasons that students struggle is that they are taught to focus in on keywords or formats and then to use a set of prescribed procedures that go along with them. Problems arise when the keywords are not found or the problem is formatted differently. The students see the situation as different and a new set of procedures is needed to solve it. The result is the impression by some students that mathematics is complex with lots of rules and procedures that must be memorized in order to be successful. Teachers want their students to generalize concepts and extrapolate them to situations that have a common underlying mathematical structure even when they are not familiar with the problem. Changing the context gives the student the opportunity to solve problems by understanding the underlying concept instead of memorizing a set of procedures.
In our last teaching session before the Thanksgiving break, one of the students asked us how the task would be changed when we returned. This prompted the other students to start making suggestions such as placing ornaments on trees, presents under trees, bells on a string, and many others. What remained the same in the ideas was that they all could be made into equal groups with a constant number of items in each group. The students have generalized the tasks and realize that the situations contained a common underlying structure of distributing a set of items over a set of groups. They can now produce their own situations using this structure, knowing that the context could change while the math remains constant.
This post continues the Constructing Units team’s discussion about developing composite units with the goal of building children’s multiplicative reasoning. You can read part one here. In the Towers Task, the teacher uses the child’s understanding of composites as a starting point, and then provides modifications to the original task which encourage opportunities for the… Continue Reading
In the August and September installments of my blog, I’ve been telling the story of Christian and our mathematical interactions with him. Christian is a second grader who came to us with mathematical skills that had been taught through his first years of schooling. He was bright, eager to work with us, and considered, by… Continue Reading
In my previous blog I introduced Christian. He had a prescribed method for solving addition tasks, but many times his answers were not accurate. In our second session with Christian our primary goal was for him to use a counting strategy when adding two numbers. We began by presenting cards to him with the numerals… Continue Reading
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years.—Common Core State Standards for Mathematics, p. 8 CSS.MP1 Make sense of problems and persevere… Continue Reading
In the Common Core State Standards for Math, counting-on is considered “a strategy for finding the number of objects in a group without having to count every member of the group.” Counting-on is an efficient way to add and we want children to count-on. Yet, many young children begin by counting-all. For example: Teacher [placing… Continue Reading
My 4-year-old grandson and I have been spending a lot of time together. Without him realizing, we have been doing and talking more mathematics around the house. The other day I was rebuilding a gate and while I was measuring a board to cut, he picked up the tape measure wanting to help out. So… Continue Reading
I recently attended the Annual Conference of the National Council of Teachers of Mathematics (NCTM) in San Antonio and came away invigorated and hopeful about our children’s future in math education. The creativity and passion on exhibit within the many sessions and workshops was impressive. I had numerous conversations with awesome teachers that eagerly shared… Continue Reading
Recently, while working with students, we offered up a situation where nineteen counters were placed under a cloth. Seven of the counters were pulled out and the students were asked how many remained under the cloth. One child extended ten fingers, pulled them back, and then re-extended nine. He pulled back seven fingers, one by… Continue Reading
We at the AIMS Center have been digging into research on how children come to know number. What I realize is that knowing the journey from perceptual to conceptual which children must take is important for teachers to understand, in order to effectively help children on their path to understanding number. We have discussed the… Continue Reading