# Author Archives: Brook Lewis

### 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 to build five towers of four, and then three more towers of four. The student is then asked to find the number of towers, number of cubes in each tower, and the total number of cubes in all towers. It is important to note that students are usually still counting by ones (1,2,3…4,5,6… etc) as they also keep track of how many times they have made three counts. They are still developing a meaningful skip count.

The modification to the task we will discuss this week has a different configuration of towers, as well as a different set of questions for the student. We don’t present the task in this format until students are consistently successful combining two sets of towers of three (or four) cubes each.

Since the students are now proficient at keeping track of how many times they have made three (or four) individual counts, it is time to increase the sophistication of the arrangement of towers. They will still be working on developing a meaningful skip count because this modification will require them to find a total number of cubes, but many students will already be skip counting meaningfully up to a certain point. So here are the steps:

1. Student is asked to build a set of towers, and the towers are covered (Example: Please build five towers of four cubes each).
2. Student’s partner pretends to add more towers behind the cover, and tells the student the total number of towers that are now hidden (Example: Now that I have added more towers, there are a total of 9 towers hidden).
3. The student is asked the following questions:
1. How many towers were added?
2. How many cubes were added?
3. How many cubes altogether?

Here’s what this situation looked like in one of our classroom interactions:

This problem is similar to a missing addend problem, but the addends are towers. In this modification of the task, students work on their ability to switch between units. They have to think about the towers as wholes, and switch their thinking to the cubes, while maintaining a separation in their mind between the towers they built and the towers that were added.

Notice this is the first modification of the task where one of the sets of towers is not built at all. It is a progression towards imagining towers without ever seeing them. Initially, students can struggle to make sense of this problem, and it can be helpful for them to act out the entire task. Their partner can actually build the second set. After a couple of times, students should be ready for the pretend scenario.

There are still further modifications to challenge students that will be revealed in next week’s blog. Let us know if you try any of these tower tasks and how your students responded.

### 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

### Where Are You?

As an instructional coach, I would travel from school to school working with different teachers every week. While I would visit the same sites repeatedly, I would use my navigation system to find them initially. After a couple of months though, I noticed that I still needed directions to get to some of the same… Continue Reading

### Taking it to the Classroom

ZAC or Zone of Actual Construction: what the student will be able to accomplish or solve unassisted. ZPC or Zone of Potential Construction: the range determined by the modifications of a concept a student might make in, or as a result of, interactive communication in a mathematical environment. This year we are excited to be… Continue Reading

### What Did I Learn?

Now that the school year has ended, our research team has been gathering our data from time spent working with students and analyzing it to answer the question: “what have you learned this year?” More importantly, I wanted to figure out what I have learned that will actually enable us to help kids. After completing… Continue Reading

### Mean Math

Two experiences this month have opened my eyes to the value of the research we are learning about at the AIMS Center. One involves my 5th grade daughter Neva, and the other was a conversation with a second grade student I have been working with. Neva says she doesn’t like math, but she has been… Continue Reading

### Recognizing Meaningful Skip Counting

A few posts ago (December 13, 2016 and February 21, 2017), I discussed the importance of skip counting with meaning. If you are teaching particular grade levels, you may have students who skip count to solve problems already. So, how can we tell if skip counting is meaningful to them? Often times, a student will… Continue Reading

### Composite Units and Fractions

In my previous blog, I talked about a composite unit, what it is and how it plays an important role in many different aspects of students’ construction of mathematics. One of these areas is fractions. So how does the student’s ability to take a number as something that is countable affect their understanding of fractions?… Continue Reading

### Composite Units

Counting-on is one of the things I have come across in Les Steffe’s research that is crucial, but not necessarily an obvious goal to have for students. It would seem that if a student could count-on (ex: given the problem 6+5, would start at six and count-on five more rather than starting from 1 and… Continue Reading

### Perturbation Instigating Reflection in Students

Hopefully last week you read the blog by Tiffany Friesen in which she discussed perturbation.  In it, she gave a couple of examples from her own experience. Both of the situations that she mentioned were familiar enough to her that she had the capacity to resolve her confusion. She had familiarity with making the cookies… Continue Reading