I recently read this quote that really stuck with me

“In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). Additionally, they found that training students through visual representations improved students’ math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.” (see article by Jo Boaler for more: https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/

So this sounds amazing – we all want to optimize our students’ learning…how do we do this?!

I’m going to share with you a few examples that I’ve been working with in the classes I’m currently teaching.

"The regular classroom teacher told me today that the difference in his students’ understanding of multiplication from when he taught this traditionally (procedurally) to how we taught it visually was night and day – he was blown away by how much they improved AND the added bonus of learning about shape and measurement at the same time, not to mention the struggling learners were finally understanding and succeeding! "

Grade 6: We are working on developing strategies for mental fluency in multiplication. We have been working on creating strategies that would allow students to mentally compute a fact that they either have forgotten or don’t know readily. An example that some of the students came up with is 6 x 7.

We asked our students to show us their strategies visually and here is an example of a way they created:

3 groups of 7 plus another 3 groups of 7. You can see that this is a blend of a pictorial model with symbols but what it allows students to do is actually visualize the strategy of finding 3 groups and then doubling it.

Another group decided that it would be easiest for them to do this:

 2 x 7= 14and 3 groups of 14 = 6 groups of 7 = 42 Although these ways are similar, the idea is to allow the students to explore working with partitioning and understanding how numbers can be broken down to simplify mental calculations. This is also an important pre-algebra skill as this relates directly to this: (which they will learn about when they get to high school).

Other teams decided 5 groups of 7 plus another 7 was the best strategy, which they showed visually as well:

In a grade 6/7 class we are reviewing multi-digit multiplication and area using area models. So students were actually creating area of models like:

Then we moved onto sketching them as well as integrating the meaning with procedures and how multiplication is connected to area. Here is a sample of how we did this:

Students are now integrating their understanding of dimensions (length and width) with area and how this is directly related to multiplication of two numbers. Furthermore, they are now better able to visualize the areas of these large rectangles and so their estimates have also been getting more accurate. The students struggled most with understanding the length and width in terms of ‘rows of’. For example: 27 x 45: the dimensions are 27 by 45 and what that really means is 27 rows of 45.

In both of these examples the students could have improved their models by: Showing 48 rows of 74 rather than 48 columns of 74 and by drawing their sketches more to scale. Students were given feedback on their sheets and we looked at several students’ responses of excellent answers and then they had more opportunities to improve on their understanding and representing it clearly.

The regular classroom teacher told me today that the difference in his students’ understanding of multiplication from when he taught this traditionally (procedurally) to how we taught it visually was night and day – he was blown away by how much they improved AND the added bonus of learning about shape and measurement at the same time, not to mention the struggling learners were finally understanding and succeeding!

My last example is from grade 8, where we were exploring square numbers and square roots (including estimating). Before we moved into this lesson, we were reviewing factors (among other things) by using square tiles to build rectangles with different areas and exploring how many rectangles (including squares) we could make for each number. For example: using 12 tiles to build a rectangle (filled in – so the area is 12). This way we could discuss how each number has a certain number of factors (prime and composite numbers are easily recognizable as having either one rectangle only for prime numbers or many rectangles for composite numbers).

 If the area is 12 then…. The factors are: 1,12,2,6,3,4 (all of the dimensions of the rectangles we can make using exactly 12 tiles) We also know 12 is a composite number because we can make more than 1 type of rectangles
 If the area is 7 then…. The factors are: 1,7 We also know 7 is a prime number because we can only make 1 type of rectangle