# Unit Coordination

### What Every Student Needs to Know for Multiplication (Part 4)

***This is part 4 of a series. Click the links to go back and read part 1, part 2, and part 3***

In this series, we have been discussing a progression of tasks that give students the opportunity to construct meaning for working with two types of units, the towers and the cubes that make up the towers. Last week, Brook discussed an activity in which the students would establish the first set of towers and then their partner would pretend to add in more towers and tell the student the total number of towers after the pretend ones are added. In that activity, the student would have to imagine the additional towers and work with them and the cubes they are constructed from to answer questions.

So what would be the next level of a task? Do you already know the answer?

Remember, the ultimate goal of this task is for students to be able to switch their thinking between the towers (groups) and the cubes that make up the towers (number of items in each group). This is the big switch in thinking for students working with multiplication. With addition and subtraction they are working with the same unit type, but with multiplication they are moving between two different unit types.

The next change in the task promotes the ability to switch between towers and cubes. The sender tells the bringer to bring back a certain number of towers made up of a certain number of blocks (we let them build the towers at this point because it engages the students more) and then the towers are hidden. Next, the sender pretends to add cubes and tells the bringer how many cubes were added. The bringer then reflects on the following questions:

- How many towers were added?
- How many total towers are there?
- How many total cubes are there?
- How did you figure it out?

Notice how the bringer has to now switch back and forth between towers and cubes with the collective experiences of building towers and reflecting on them throughout the series of tasks.

The goal is to provide situations in which the student can construct the understanding of groups and members of a group in order to build toward reasoning multiplicatively. In the next part of this series we will be looking at the next level of complexity in this progression.

As you try these tasks in your own classroom or have questions or thoughts, please share by replying in the comments.

### What Every Student Needs to Know for Multiplication (Part 3)

***This is part 3 of a series. Click the links to go back and read part 1 and part 2.*** In last week’s post, David Pearce described a modification of the Towers Task in which the students are asked to build two sets of towers and combine them. For example, the student may be asked… Continue Reading

### What Every Student Needs to Know for Multiplication (Part 2)

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

### What Every Student Needs to Know for Multiplication (Part 1)

The topic of the latest AIMS Center colloquium was “What every student needs to know for multiplication” (Video Archive)(Resources). This presentation highlighted the work that we have been doing around understanding how students develop multiplicative reasoning. One of the things we are doing is implementing a task with second and third grade students called the… Continue Reading

### The Power of Imagination in Mathematics

I have been reading and thinking a lot about the power of imagination in learning — specifically, learning mathematics. In this and successive blog posts, I will discuss one role imagination plays in helping children form number sense. Merriam-Webster’s definition for imagine: “to form a mental image of (something not present),” is what I mean… Continue Reading

### Same Task, Different Situation

When I taught middle school, I always found it interesting that my students could do this task: Johnny rode 34 miles on Tuesday and on Wednesday he rode 27 miles. How far did he ride over the two days? Yet they often had no idea what to do when I gave them this task: Johnny… Continue Reading

### Permission to Fail

I wrote a blog post at the beginning of the school year talking about our plans for research this semester. I’ve been reflecting on our project and the progress we have made so far, and I thought I would share a few of those reflections with you. As I mentioned previously, we have been working… Continue Reading

### Christian (Part 3)

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

### Is Counting Really So Easy?

I have heard the claim “calculus is easy, algebra is difficult, and arithmetic is impossible,” but if that is true, then what does that make counting? We often hear little ones proudly singing the alphabet song or reciting a string of numbers from 1 to 20. Have you ever asked one of those who now… Continue Reading

### Not Another Ball!

I can’t believe it’s been a year since I embarked on this journey as a Research Associate at the AIMS Center for Math and Science Education. It has been an exciting transition for me, having the opportunity to pursue my passion for understanding how children develop their knowledge of mathematics. Over the last year, I… Continue Reading