See and understand
Developing conceptual understanding in mathematics
Alex Cull, director at Mangahigh, shares her experience of developing children’s conceptual understanding of mathematical skills before, during and after Covid.
Calculations without understanding
Like many of us, my memory of learning long multiplication using the calculation below, would involve me starting at the right-hand side of the sum, and multiplying the two values, eight and two making 16 and carrying over the ten value of 1. Then I’d move on to multiply the four with the eight and add the ‘1’ that I’d carried over to this total.
Why did I do it this way? well that’s what I’d been told to do. I didn’t understand that 2 x 8 was 16 which was one group of 10 and 6 units. I’d probably get the answer right, but I wouldn’t be able to explain why and nor would I be able to apply this skill to any other ‘real life’ problems. In other words, I had no conceptual understanding, or mental representation of what I was doing.
The importance of conceptual understanding
As we all know, developing each student’s conceptual understanding of each mathematical idea is the only way to ensure that they’ll be able to transfer this knowledge to other situations and apply it to new contexts.
Whether a classroom activity involves using drinking straws to explore vertical angles, Triangle Sum Theorem, exterior angles or transversals, or jugs of water to test the mathematical theorem of volume, these conceptual models are the necessary abstractions of physical things in the real world.
And then came Covid-19!
From physical to graphical representation
As routines were turned upside, schools and colleges had to shift towards online and remote learning. In Mathematics, one part of the challenge became finding effective ways to provide students with these visual representations to develop their in-depth conceptual understanding of each mathematical skill. Increasingly, teachers turned to high-quality online graphical rather than physical representations of mathematical concepts. Some concepts, of course, are easier for teachers to demonstrate than others. Topics, such as solving inequalities with a negative coefficient, can be more challenging to find the right way to ‘teach’ the concept through graphical representation.
And it’s not just about having a single graphical representation.
Developing a student’s conceptual understanding isn’t a ‘ping’ moment. It needs to be developed after repeated exposure to a particular mathematical idea in various contexts. This process takes time.
As with all teaching, the more ways a concept can be represented through activities, challenges and games, the more likely they are to meet a variety of students with a range of different learning needs. Several students with different levels of competence can also work on the same problems, and each of them will find certain aspects of the problem to be solvable. Students need to engage with each mathematical concept through graphs, graphics video representations, lists, and other mathematical notations. When developing conceptual understanding, it’s imperative to give students freedom of choice in how they might potentially respond.
New concept? Keep the numbers simple
With new concepts it’s important to start by keeping the numbers easy. If the numbers are too difficult at the outset, students can get overwhelmed and will often shut down. Easier numbers help give students an access point to the concept and once they have this in their mind it is easy to increase to more challenging numbers.
The impact of virtual learning tools
Using virtual tools that help contextualise a mathematical problem can help provide the variation and repetition needed for students to develop conceptual understanding. With educational games for example, the content should develop a student’s ability and curiosity to observe, hypothesise, test, evaluate, conclude and refine ideas in a way that is not always possible with traditional pedagogic teaching. Whether a game involves number, algebra, geometry, measurement, statistics or probability, students can practise (and dare we say enjoy??) tricky concepts that challenge their learning at home.
A range of representations
Narrowing experience to a single representation too often causes students to try to find the one ‘correct’ path towards a solution, rather than thinking expansively and for themselves. It is therefore crucial that mathematical games are continually developed by researchers and teachers using a range of pedagogical approaches, so that students remain effectively engaged in their studies. The more ways found to conceptualise a mathematical notion, the more effectively they will be able to apply the learning to their daily lives, too.
Engaging students in the visual representations of mathematics really is vital for their learning. By balancing conceptual understanding with procedural skills when, for example, multiplying decimals, students can learn to perform the operation fluently while understanding what they’re doing and developing number sense. In this way students develop the ability to construct viable arguments and critique the reasoning of others, developing their analytical skills.
Alex Cull is the Global Marketing Director at Blue Duck Education Ltd, the publishers of Mangahigh.com, organisers and sponsors of the 2021 Council of British International Schools (COBIS) Maths Challenge.
Support Images with kind permission from Blue Duck Education