Teaching Tricks Can Be Helpful, If Done The Right Way!

It is human nature to want to get things done as quickly and easily as possible. Heck, our brains are actually wired that way according to Chip and Dan Heath as they share in their best selling book, Switch and articulated a little differently by Daniel Khanemann’s book, Thinking: Fast and Slow. Our fast thinking, seemingly “lazy” part of our brain is trying as hard as possible to make decisions quickly and easily in an attempt to conserve energy. When we are forced to think deeply, we are asking our slow thinking, more intentional side of the brain to kick into gear which requires energy. The reason our brain works this way comes down to survival; saving as much energy “just in case” is the goal despite the fact that we live at a time where saving energy for survival is not really a necessity any longer. 

Interestingly enough, these two parts of the brain are actually helpful when used appropriately to become more efficient in our everyday lives. For example, it is helpful for us to leverage our fast thinking part of the brain to do daily routine activities so we can save our slow thinking side of the brain for more important tasks like writing a report, analyzing data or engaging in planning meetings. 

In order to fine-tune both our fast and slow thinking systems in our brains, it is helpful to get them working together. In the context of learning mathematics, it is often helpful to first engage our slow thinking system in the brain to observe, analyze, conjecture and make conclusions based on the behaviours of mathematics which can then lead us to build in more efficient (and energy saving) routines that will allow the fast thinking system to take over. 

When we look at the mathematics education space, often you find two extremes: those who believe math should be explored conceptually through inquiry, investigation and discovery (constructivist) as well as another extreme where math is viewed as rules, steps and procedures to memorize (behaviourist). By sitting solely in one camp, arguments quickly arise and ultimately the result is an overemphasis on one system of the brain instead of leveraging both systems to maximize the learning outcome. 

One such example is when the topic of “tricks” in math class are discussed. Those leaning towards a constructivist-only approach might argue that tricks shouldn’t be taught in math class, while those leaning towards a behaviourist-only side of things might argue that tricks are efficient and allow students to move on to more complex ideas in less time. 

The Make Math Moments camp can actually see both sides of the argument and believe that both sides are right. Teaching tricks to be memorized is certainly not a great way to approach mathematics, but limiting mathematics learning to exploring, investigating and inquiring without extending to building automaticity and efficiency paints an incomplete picture of what learning mathematics is all about. In a balanced mathematics classroom, we are leveraging the constructivist views to observe, analyze, conjecture and make conclusions about mathematical behaviours through strategies and models as a means to build automaticity for efficiency, fluency and flexibility. As long as we don’t stop at investigating, exploring and inquiring but rather continue to deepen that conceptual understanding as we construct procedural fluency – that flexibility and fluency we’re after – we are able to also meet the goals of the behaviourist-perspective as there is lots of repetition and efficiency being generated. 

When we approach mathematics through a balanced lens like I’ve described here, we also have the opportunity to explore what some might call “tricks”, but are really just a summary of the behaviours of the mathematics we are engaging with. 

So rather than simply giving students a list of rules, steps and procedures to memorize which we often see taking place when one shares a math “trick”, we can allow students to build off of their conceptual understanding to co-construct the “rules of the game” or the “trick” that we believe might be helpful for them to be more mathematically efficient the future. 

So how can we do this with say a multiplication trick?

Well, let’s look at a common trick you see shared all over the internet known as the “9s times table hand trick” or “the multiples of 9 finger trick”. 

If you have heard of the trick or maybe you even know the 9 times table finger trick, let’s pretend for a second that you didn’t know the trick. How might we help students to emerge this idea?

Well, we can begin as we often do here at Make Math Moments with a low-hanging fruit. What I mean by this is starting with something that I know students in the room are comfortable and capable of given their current level of mathematical readiness. 

We could leverage a choral counting approach (Number Sense Routines) or an around the circle approach (Minilessons) to get students skip-counting by 10s first. 

Let’s have a look…

As you go “around the circle” with your students or “choral count” as a group, the facilitator can write down the string of numbers:

00

10

20

30

40

50

60

70

80

90

100

Great! 

Now, we simply ask students to think about how helpful (and seemingly effortless) it might be to count by 10s starting at 0. Maybe, your group is even really proficient with counting by 10s starting at any number – awesome!

Imagine if we could leverage what we know (skip counting by 10s) to skip count by 9s?

As you ask students to skip count by 9s starting at 0, we’re going to record this next to the column of skip counting by 10s:

00

09

18

27

36

45

54

63

72

81

90

Note that it might take students longer to do this and you might even have some students attempting to count by 1s, 9 times to determine the next number. If we see this, we’re going to ask students to pause and ask them explicitly:

Wait a second. A moment ago, we were able to easily skip count by 10s.

How many “less” is 9 than 10?

Most students will quickly exclaim “1” and this is where you encourage them to skip count by 9s by simply skip counting by 10 and remove 1. It is a major bonus if some students have already landed on this idea and are already making this move to skip count more efficiently. 

We can then ask students to look at the list of numbers and to turn and talk to share what they notice about the behaviours of skip counting by 10s and skip counting by 9s. The goal here is for students to recognize that the 10s column is going up by 1 (since it is 1 group of 10) and the 1s column is going down by 1. 

Once students are comfortable with this idea and you’ve had an opportunity (potentially over a number of days) to practice this strategy for skip counting by 9s, we might help students see an easy way to represent this with our hands. 

Since we have 10 fingers, we can actually show this behaviour of skip counting by 9s (or the 9s times tables) by putting a finger “down” to show how many groups of 10 there are in the tens column and how many 1s are in the ones column (see video for demonstration). 

Through this work, we are able to not only address the aspects of the constructivists perspective for teaching mathematics, but we are also building in the aspects that the behaviourist perspective feels is important for learning mathematics. As you probably have noted here, both of these perspectives are necessary to work together to craft a balanced mathematics program that helps students construct a deep conceptual understanding of the mathematics they are learning as well as the procedural fluency that is necessary to avoid wasting our limited working memory on ideas that are no longer the important aspect of the new mathematical concept at hand. 

What were your thoughts on math tricks in math class? 

Has your perspective grown since viewing this video? 

Share your thoughts below!

Resources Related To The Video

• Scan the article above and access useful links related to these topics or search our blog
• Grab the Make Math Moments 3-Part Framework to help you run an effective mathematics program leveraging problem based math units with number routines built in
• Take the Grow Your Math Program Assessment to receive a customized report to help you determine what areas to focus your professional learning on next. Both Classroom Teacher and School/District Math Leader versions are now available!
• Learn more about our Make Math Moments School/District Improvement Program
• Classroom teachers can book a Math Mentoring Moment Conversation to be aired on our podcast 
• School/District Mathematics Leaders can book a time to chat about your math professional development plan with our team

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