By Michelle Chu, teacher and BC Association of Mathematics Teachers elementary representative, Burnaby
For much of my life, I couldn’t understand how people could just calculate math problems in their heads. I must have been quite a sight trying to answer 99 + 37 or 22 x 55 without writing tools. I would stand there, finger frantically air-writing the algorithms as I mouthed the numbers in failed attempts to remember both the question and the results from all the carry-overs and borrowing.
It wasn’t until later in my teaching career, as I focused my professional development on upper intermediate mathematics education, that I understood why. I was quite competent at using algorithms. I had a good memory and even found comfort in following a prescribed set of steps. What I was lacking was number sense. I didn’t think about what the numbers meant—I just calculated.
As I participated in sessions centred around number sense, I was amazed at the various strategies people used when approaching questions. I started working on developing my flexibility in strategy selection. Rather than lining up the numbers 99 and 37 to follow the standard algorithm, I used my understanding of constant difference to transform the problem to 100 + 36, which are friendlier numbers to use. The more I learned, the more excited I became. There were so many relationships between numbers that I had never considered.
I wanted my students to share in this excitement and flexibility as well. Many of my Grade 7 students solved questions much like I did—using the algorithm. The algorithm is a beautiful and useful tool; however, having the ability to select the most efficient one for the problem being solved is also important.
One way I try to help students build their number sense is through math chats. These are short activities that can last between 5 to 20 minutes. They can be connected to the day’s lesson but are often stand-alone activities with a major focus on active student discussion.
My math chats at the beginning of the school year are often chosen to emphasize the fact that there are different ways to view and approach a problem. Which One Doesn’t Belong and Subitizing Dots are both fantastic for this.
For example, in this Which One Doesn’t Belong (Figure 1), students might argue that 81 doesn’t belong, as it’s the only odd number or because it’s the only number that doesn’t end in 4. Some students might point to 144 not belonging, as the only three-digit number. What reasons can you find for 54 or 64 not belonging?
In Subitizing Dots, by Steve Wyborney, students are briefly shown several dots and asked to share the number they saw. This is followed by a class discussion about how students grouped dots together in their head to help them hold onto that number. After a few of these, students are presented with a challenge: How many ways can you find the total number these dots represent? Figure 2 shows examples of how dots can be viewed in different groupings.
Through these and other activities, I can establish math chat routines. These routines provide opportunities to practise multiple approaches, flexibility in thinking, using mathematical vocabulary in discussion, and providing peers time to think by putting a thumb up against one’s chest, rather than announcing the answer. Because of the short and approachable nature of these chats, students who view themselves as “not math people” often stay engaged and build their math confidence.
When students are ready, often within a few months, I introduce Math Strat Chat, by Pam Harris. We begin Math Strat Chat, by having students share their strategies aloud for a given question (Figure 3). As they describe them, I do my best to represent their thinking on the white board.
Other times, I might show students strategies that other people have used to approach the problems (Figure 4), asking them to explain the thought processes that are being demonstrated. Part of the process is to also share three or four strategies for a particular problem and ask students to share which strategy they find the most efficient. This helps remind students that sometimes our favourite strategies just don’t work with certain problems.
These activities allow me, as the teacher, to review and teach strategies without having to dedicate an entire unit to a particular concept. Regularly returning to these ideas over the course of a school year means that students have more exposure, helping build their retention of concepts and toolkit of strategies.
