First, let’s look at a brief history of mathematics instruction. When I was learning math, computers were giant machines that filled entire rooms (yes, I realize I am dating myself). They were not accessible to the ordinary citizen or business person. It was important to learn how to do calculations, because there was no alternative. But now, I carry a phone on my person at all times that has greater computing capacity than those old room-sized computers had. So why are we still emphasizing the memorization of rote procedures? And what skills will our students need to compete in the global marketplace? The graphic below lists the skills industry leaders valued in 2015, as well as the skills they will be looking for in 2020.

So how do we help students make sense of math? Here are a few suggestions:

### 1. Provide students with plenty of concrete and pictorial experiences

You simply can’t rush math understanding. Children need to *touch* the math they are doing. While 5 – 1 = 4 makes no sense to a 5 year old, show them 5 jellybeans and then eat one, and I guarantee that they will begin to understand the concept of subtraction. Now, a bright 5 year old could certainly memorize 5 – 1 = 4, but that would be the equivalent of memorizing a spelling word, but not knowing what it means or how to use the word in a sentence.

A fraction, such as 3/4, is about as abstract as you can get. I have had many students tell me that 3/4 would be somewhere between 3 and 4 on a number line. That tells me that they haven’t experienced 3/4 and are not able to visualize the meaning of 3/4. Concrete and pictorial experiences aren’t related to age–that is, they are not just for “the little kids”. Those experiences are necessary whenever students are learning a new concept, regardless of age.

### 2. Connect learning to real life situations

You can make math less abstract by connecting it to real life. Take, for example, the order of operations. We often teach this concept as a set of rules you apply to equations. What meaning does that have to students? Students need to investigate the meaning of the order of operations through the lens of real life situations. Take, for example, this problem and the discussion you might have:

Margo made treat bags for a bake sale. She put 2 chocolate chip cookies and 3 peanut butter cookies in each bag. She made a total of 12 bags. How many cookies were in all the bags?

How would you find the total number of cookies *(add 2 + 3 and then multiply by 12)*

What happens if you multiply 3 x 12 first and then add 2? *(you’d get 38, which is not the correct amount of cookies)*

How do we indicate what order to perform the operations? *(by using parentheses)*

Once students get the hang of it and truly understand why operations must be performed in a certain order, give *them* the equations and challenge them to write stories to match.

### 3. Encourage natural curiosity, but don’t force it

A colleague recently told me a story about her 4 year old child and an elevator that really made me think. They had been staying in a hotel for a couple of weeks, so they had been riding the elevator to the 4th floor several times each day. On one trip up, they were on the 2nd floor and her daughter said, *We’re on 2, we have two more floors until we get to 4. *My friend was kind of stunned, and she said she didn’t really know how to react, other than agreeing with her daughter. I think that was a good thing, because sometimes we get carried away trying to capitalize on teachable moments. The math that child did made total sense to her. She probably just used the buttons on the elevator like a number line and knew it was two jumps from the 2 button to the 4. I doubt she did subtraction or addition in her head. We should certainly encourage a child to think about numbers, but that doesn’t mean teaching full blown lessons. Table Talk Math is a great resource for parents looking to engage their children in authentic mathematical conversations.

So there you have it. Our challenge is to constantly reflect on how we are helping our students make sense of math. I’d love to hear other suggestions in the comments!

Randy Hockey says

Good post. However, your graphic is incorrect for the sum and difference of cubes .

Factoring a Sum of Cubes:

a3 + b3 = (a + b)(a2 – ab + b2)

Factoring a Difference of Cubes:

a3 – b3 = (a – b)(a2 + ab + b2)

Donna Boucher says

Ha! Good eye. That graphic totally came from a stock photo service. 🙂

Tara says

Love the elevator example! Excellent tips for all ages!