A bi-annual magazine for the Hong Kong Academy community.
Getting the Concept Right: The Importance of Conceptual Understanding in Mathematics Education by Shane McKinney and Maria Tullberg What do we mean by conceptual understanding and why is it important? When asked what they think about mathematics, some students might say, “It’s complicated. There are all these rules and facts to remember.” Many of us might remember the drilling of multiplication tables; we might even have a stereotypical image of children chanting in unison, “One times two is two. Two times two is four …” More recent rote methods include rhymes, songs, visuals and movements in an attempt to learn the ‘tricky’ multiplication facts. Now fast forward to 2018 and peek into the classrooms of Hong Kong Academy, where conceptual based learning and problem solving are part of the daily routine in Primary as well as Secondary School. So, what is conceptual understanding? Simply put, it is understanding mathematics in context and knowing more than isolated facts and arithmetic. Successful mathematicians do not only rely on memory to solve a multiplication fact, but also use number sense — they know how to be flexible with numbers. A successful mathematician is actively building new knowledge from “Working with manipulatives makes you better at maths because you can solve problems mentally and try different ways. The balance tiles are challenging because it really makes me think.” Brenton, Grade 3 mathematician, on using Marcy Cook tiles to solve algebra puzzles. 6
experience and prior knowledge and has the ability to transfer this knowledge into new situations. For students at HKA, this means learning mathematics by developing an understanding of how to approach a problem. They inquire into what skills they need to solve the problem and apply their knowledge accordingly. Interestingly, and perhaps surprisingly to many of us, this method of teaching mathematics is similar from Kindergarten all the way through Grade 8 and beyond. What does building conceptual understanding look like at different grade levels? In Kindergarten, children explore mathematics concepts by considering concrete situations they might themselves encounter. For instance, students might pretend to jump on a bus, sitting down on chairs lined up in five rows of two. The students observe that the bus can only take ten people, leaving some of their classmates without a seat. “What can we do?” the teacher asks, looking somewhat worried. “Let’s get on another bus!!” the children concur excitedly. This experience prepares the students for next steps, such as building ten frames. These frames are a key component to understanding our number system, and students with a solid grounding in this concept are prepared to develop the skills of addition and multiplication. Throughout the early Primary Years Programme, students continue to learn about ‘rows’ and ‘groups of’. In Grade 3, this groundwork turns first into an array and then later the area model. The area model is a cornerstone, and for some even considered a little bit of magic, to help us understand multiplication by using our knowledge in geometry. When learning a new concept, the children need to explore various manipulatives, such as base ten blocks or anything they can touch and move around, and build their own understandings at their own pace. The reward is that ‘aha’ moment when the student lights up and says, “Ahhhh! Now I get it!” Once that moment occurs, “It helps me to understand how subtraction works, especially when working with big numbers. It’s an easier way to look at it and understand it.” Rishi, Grade 3 mathematician, on subtracting using base ten blocks. it is difficult, perhaps even impossible, to unlearn that knowledge. The key is not to move on to the next step before they are ready; we teach the students to “build it, draw it, write it”, and this takes time. When the students are ready to understand and explain the concept, they will then find a more efficient algorithm such as ‘long division’. If that is not until the end of Grade 5, that’s ok. There is no need to rush. What using manipulatives and models looks like in Grades 6-8 For effective learning, students must create their own meaning. In Grades 6-8, this implies using manipulatives and models as they develop their conceptual understanding. This is the same process as in the Primary School, and by Grade 6, most students have a large toolbox of procedures and strategies to employ. In Grades 6-8, the greater opportunity for learning centres around exploring new concepts that challenge students to use what they know and build upon it. What better way to build on their understanding than using manipulatives and models? “Working with base ten blocks helps me because I don’t have to use my fingers, especially when working with bigger numbers. When subtracting we just feed the ‘minus monster’ the number of blocks we’re taking away.” Lucy, Grade 3 mathematician, on working on conceptual understanding of regrouping when subtracting. 7