Stretch your Math Muscles

This week in class, we explored what makes a task a rich learning experience for our students. The criteria included accessibility for all learners, engaging material, various entry points, encourages creativity and collaboration, allows students to make connections, and broadens mathematical skills. All ideal points of interest that teachers should consider when planning any learning activity.

A post about rich mathematical tasks fittingly follows my previous post about adapting for gifted learners. Rich tasks allow for all learners to engage in a problem-solving situation at their own level of readiness without planning multiple lessons. I mentioned using open tasks to engage all learners, especially gifted students, last week. During class, we participated in an open task in which our teacher gave us the answer, but we were to give the question. 

The answer was 32 = ?

As a class, we brainstormed a variety of ways to represent 32, and I’m sure it was only the tip of the iceberg. It was an interesting process for me. Initially, I began with simple equations such as 32 = 30 + 2 or 32 = 16 x 2. But then I had an urge to challenge myself and I began to think a little deeper about what 32 holds. It has 3 tens and 2 ones. So I could represent 32 as 10 + 10 + 10 + 1 + 1. Or even employ some BEDMAS with (3 x 10) + (2 x 1). I could break down 32 a little differently. What about taking the 30 and dividing it in half to get 15? I could do (15 x 2) + 2. I could use exponents 25 or (2 x 2 x 2 x 2 x 2). I could use integers and fractions, I could show it on a graph, I could draw a picture…You get the point.

All this came from one simple idea that the answer was 32.

We discussed this same task in Brittany and Kursten’s webinar about differentiated instruction. As a group, we came to the consensus that this task is rich because it promotes creativity, which in turn engages our students, and it allows for various entry points so all students can contribute. I’d like to add that this task allows students to naturally engage their prior mathematical knowledge and make connections between numbers and various representations. It also encourages students to explore number flexibility.

It's like Yoga for Math

Number flexibility allows a person to look beyond the surface of a number and truly understand the components of the numbers. This flexibility will allow students to make connections between numbers and operations to assist them in problem-solving tasks. For example, I could just look at 32 as only the number stated. 32. But when I really think about what 32 is, 3 tens and 2 ones, or 6 fives and 2 ones, or 32 ones or two sets of 16 or two sets of 10 and 6... I am now able to manipulate what those numbers mean. So if I have 3 tens in 32, I also know that I have 4 tens in 42 or 9 tens in 92. I can connect those numbers just by unpacking 32 slightly deeper beyond “It’s 32.” Number flexibility is discovering the relationships between numbers.

http://brownbagteacher.com/number-talks-how-and-why

Some strategies to encourage number flexibility include using the above brainstorming task of finding the question. This presentation given by Sherry Parrish about how to host Number Talks to encourage students to explore relationships between numbers while focusing on the process rather than the answer. Math strings also invoke the same outcome. Exploring number flexibility can begin with allowing students to demonstrate various representations of numbers or simple equations.

Ready, SET, Go 

Subitizing

Another strategy we explored in one of our first classes also reinforces number flexibility and relationships. Our teacher very briefly showed us an image of random dots, and then we had to state how many dots were on the screen.

But what was interesting was the many different strategies everyone had employed to “connect the dots”. This is called subitizing, the ability to know an amount without counting using sets. Below is a fun video with many examples of sets to subitize with students.

To make subitizing rich, allow students to develop and share their process in determining the number in each set. You can even extend the activity by providing students with the opportunity to create their own series of subitizing examples to which the class must solve!

Not only does this task allow for number flexibility, but also practices number fluency, which is something I believe is important for students to make connections in higher level math. 

How do you encourage number flexibility for your students in a rich task? Let me know below!