# Focus on Math

## Helping children become mathematicians!

### Step Out of Your Comfort ZoneJanuary 21, 2013

Most of us tend to teach mathematics in the same manner as it was taught to us. I think of that as our “default setting”. We are comfortable with it.; it “feels right” to us. Unfortunately, it is often not the best way to teach math (which is why most of the North American population does not understand mathematics!).

To teach otherwise, to use strategies and approaches that we did not experience in or school years, requires real effort to change. It makes us uncomfortable; it does not “feel right”.

I believe that when we teach mathematics meaningfully, we need to have students doing more than just following our instructions. When we show them how to do a particular computation (e.g., 27 x 46), demonstrating each step of the computation that leads to the answer, their subsequent work (i.e., the 50 problems to do on the page) only shows to us whether or not the students could follow all of the necessary sub-steps in order to arrive at the final answer. Such work does not show any understanding of multiplication, nor does it show that the students understand why the sub-steps produced the answer.

I contend (again!) that “understanding” lives in mathematical processes. The National Council of Teachers of Mathematics (NCTM) lists 5 math processes, namely these:
• Communication
• Connections
• Problem solving
• Reasoning and Proof
• Representation

If we regularly incorporate these processes into our mathematics teaching, students cannot help but build mathematical understanding!!

I will add one caveat: you cannot add the processes for a week, examine the results, and say, “this doesn’t work!” The truth is, we must help students build skills in these areas. If they have not been talking and/or writing about their math thinking already, such communication will take time to build. If students have not been problem solving (in the truest sense of the word) then they will need to learn some strategies and approaches to help them solve problems. A similar case can be made for making connections, reasoning and proving, and representing.

But building competency in the processes is worth the time that it takes! When students are doing the hard thinking in math (and not just following rules that are meaningless to them) you will find you and your class enter a new place of teaching and learning!

Will you step out of your comfort zone so the students can go “where the magic happens”?

Mathematically yours,
Carollee

### Math Bulletin Board: Creating Patterns with a Green Triangle as the 7th ElementJanuary 10, 2013

Often when we do patterns in primary classrooms, we have students extend them, name them, and even create them. But if we ask students to work with patterns in the context of a problem-solving task, the thinking can be even richer.

I recently had both of my grade two math classes create patterns with pattern block pieces. (I was using die cut paper versions of the blocks as I intended to have the students glue down their final product.) I started the lesson by having them create a pattern with the green triangle as the 6th element of the pattern. As I had expected, every child was quickly successful with that. The fact that 6 has factors of both 2 and 3 meant that students creating common patterns such as AB, AAB, ABB, or ABC ending with a green triangle would easily end up repeating the triangle as the 6th element.

The second part of the challenge was significantly more difficult: I asked students to create a pattern with the green triangle as the 7th element of the pattern. Most students were initially stumped as to how to do this. Some just created longer versions of their initial pattern and had to be encouraged to carefully check the 7th element of their pattern. Although the students started out working in a ‘trial and error’ manner, most moved eventually to the point of being able to predict whether or not a green triangle would “land” in the correct spot.

In the end, all but one student was successful in the time frame I had for the lessons (approximately 35 minutes for both parts). I was happy with the thinking and talking that went on in the class as students worked to create and then glue down their patterns.

Pictured here is the bulletin board made from the students’ work.

I would encourage you to do a problem-solving task with your primary students. Of course, feel free to borrow mine! I’d love to hear how it goes with your students.

Mathematically yours,
Carollee

### Factoring: A Visual Representation of NumbersJanuary 8, 2013

Are you interested in factoring, prime numbers, and composite numbers? If so, this is the link for you!

Not long ago someone posted this link on the BCAMT list serve. When I first accessed the link I was fascinated as I watched the progression of dots on my screen, each representing the next natural number. The configuration of each number of dots revealed information about the make-up of that particular number.

It made me wish that I had had access to such a visual when I was teaching about factoring, prime numbers, and composite numbers. I thought I would pass the link on to you folks as I know some of you are, indeed, teaching these concepts associated with number theory.

Many of you will be familiar with exploring these particular concepts through the process of creating rectangles from square tiles. In this method, for example, seven can be shown to be prime because seven square tiles can be made into only one rectangle: 7 x 1. Eight, however, can be shown to be composite because eight square tiles can be made into more than one rectangle: 8 x 1 and 4 x 2.

The visual presented here offers another way for students to literally see whether or not a number is prime, and, for those which are not prime, to be able to deduce some or all of the factors from the grouping of the dots.

I hope you will use the link and the accompanying picture here to explore primes, composites, and factors.

Here is the link to the animated factorization diagrams.

Mathematically yours,
Carollee

PS: Thanks Kelli Holden for commenting on the picture and sending along a link to Malke Rosenfeld’s blog where the picture has been turned into a game! Check out this link.