# Unit Construction

### Procedures Built from Conceptual Understanding

A few weeks ago, I saw a post of some students dancing and singing to a set of procedures for solving a long division problem. The person who shared the video raved about how she had never seen students love math so much. Several of my friends responded by saying that they didn’t love math, they loved singing and dancing! There was another aspect of this video that was not mentioned that may have an even more damaging effect on the learning of mathematics by these students; they were learning a rote procedure. Nothing in the song or the explanations gave any indication that they understood why the steps they used worked.

The National Council of Teachers of Mathematics (NCTM) has published a small book, Principles to Action, that discusses eight teaching practices which promote mathematical development in students. These practices support the new state math standards in a narrow context ,and deep understanding and appreciation of mathematics in a broader context. One of the teaching practices is to build procedural fluency from conceptual understanding. When teachers teach a memorized set of steps without conceptual understanding the learning is disconnected to other math they have learned and does not lay a foundation for math they could be learning in the future. It teaches students that math does not make sense. We refer to procedures of this nature as *empty procedures*.

In our work with students at the AIMS Center, we came across a student who had learned one empty procedure. Over my next few blog posts, I would like to talk about her as a way of continuing to understand how students come to understand number. This story will serve as an example of the problems that can be created when we build procedures without the conceptual understanding that supports them. This is the story of Chloe.

We begin our story of Chloe as a 1st grader with an experience in May 2015. Chloe would use her fingers to solve tasks like combining 6 and 3, when not given counters to solve the problem. She would show six fingers and then make a three with the other hand and count all the fingers, arriving at the answer 8.

In December 2015 she had adapted her adding strategy to solve 6 + 3 by simultaneously lifting six fingers and then sequentially lifting three more fingers. She answered 9, but when asked, “Nine what?” She said 93.

When solving problems where she didn’t have enough fingers like 8 + 4, she attempted to use her finger pattern strategy, lifting simultaneous fingers for eight and then sequentially lifting four fingers and counting “1, 2, 3, 4”. She was left with seven fingers lifted and answered seven.

In January 2016, when given the problem 8 + 5, Chloe lifted five fingers simultaneously for the second addend and counted-on “9, 10, 11, 12, 13.” When asked about how she figured this out she described putting the first number in her head and counting the second number out.

Someone might think that this was a big improvement and for this problem she did get the correct answer, but in other tasks we gave her she was not able to use this strategy. For her the strategy was just a set of steps to get an answer. We knew this because of how she had recently solved problems and by the limitation of her strategy in other tasks.

Next time, we will further explore the implications for Chloe in other tasks and the connection to the other teaching practices from NCTM.

*(Special thanks to Beverly Ford for her assistance in writing this blog.)*

### Change Unknown

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

### Little Steps, Part 1

There you are, sitting in your classroom after all the students have left for the day and you’re pondering just how much you think your students have grown academically throughout the school year. I know this situation, I can remember being in it many times. Unfortunately, I focused a lot more on what my students… Continue Reading

### Mathematics of Grace: Limitations and Perturbation (Productive Struggle)

In my last blog I wrote about one of the first things I noticed about the mathematics of Grace. She used her fingers to solve addition situations like 7+4 by constructing more advanced finger patterns, where one finger could mean one or eleven and six fingers could mean six or sixteen. This allowed her to… Continue Reading

### An Update

Things are really hopping around the AIMS Center. Everyday becomes better than the last. I wake up and I’m challenged and excited by what I get to do during the day. As most people know, we really want to find a way to share what research tells us about children’s construction of number with classroom… Continue Reading

### Fingers as Math Tools – Part 2 “Let’s Get Tapping!”

In my last blog, I highlighted various ways you have probably observed children using their fingers when they are counting. In this blog I will continue that discussion and show you how observing the way children are using their fingers can help you understand where a child is in their construction of number. I pointed… Continue Reading

### Watch Your Student’s Steps

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

### Reflections on the 2017 California Kindergarten Association Conference

January 13-15, 2017 I was very fortunate to attend the 32nd Annual PK1 Conference, sponsored by the California Kindergarten Association. The conference was attended by many preschool, transitional kindergarten, kindergarten, and 1st grade teachers. This was a great place to gather with other teachers and early childhood leaders that were interested in early childhood education.… Continue Reading

### Mathematics of Grace: Using finger patterns

The mathematics of students is a powerful tool for a teacher. It allows a teacher to hypothesize what is happening in the mind of a child and plan a next step that will allow that child to construct more sophisticated understanding. Today I want to look at the mathematics of a student we call Grace… Continue Reading

### What is Perturbation and Why Didn’t My Cookies Turn Out?

We know that children do not learn simply because we have given them information that we find to be important. And I hope that we know that just because we list the objectives on the board, cover each one with diligence, and check it off doesn’t mean that we have taught the objective. So what… Continue Reading