Recently I began tutoring a university-level student in a subject called Scientific Practice. The course involved all of the basic mathematical concepts and processes necessary for the successful completion of science subjects like chemistry, biology, and even physics. So topics like fractions, percentages, and ratios, all the way up to logarithms and algebraic thinking. It was an intense course designed to fit four years of mathematics learning into eight months of teaching, which sounds difficult, but in many of the concepts covered, students would have had extensive practise...or at least they should have.
When the student approached me they were having a lot of difficulties remembering all of the processes they had learned in their high school maths classes. As well as struggling to remember their multiplication facts the topic that they needed help with the most...you guessed it, fractions!
Someone recently commented that I seem to have a large focus on fractions in my work, but this has not been intentional, just born from necessity. This is because every time I get approached to work with either schools or students, fractions will inevitably be a part of the conversations.
Here are just some of the reasons:
1. Fractions are not as intuitive as whole numbers.
2. The processes learned for working with whole numbers do not always work for fractions.
3. We move students through the learning of fractions quickly and unilaterally...usually, using the area or 2D representation.
4. We limit conceptual models, especially as the students approach higher year levels, and focus solely on the procedural methods of working with fractions.
Let us go back to my student. This beautiful young lady was determined but very stressed. She desperately wanted to succeed in the STEM course she had chosen, but knew that her limitations in maths would hold her back. She had recently received a diagnosis of ADHD which was not the reason, but a major contributing factor to her lack of understanding and retention in process-based maths concepts. You see, people with ADHD, have issues with their executive function, meaning that process-based learning will not be received well. Aside from the fact that drawn-out instructions, will not hold their attention for long, people with ADHD need to engage with their learning in order to be able to understand it. They need to see the patterns and understand the why.
I'm not a cognitive psychologist, but here's the thing. Many studies that I have read, point to the fact that what works for neurodivergent students and students with a learning disability, can work for EVERYONE...and work well! So why do we continue to beat that dead maths process horse? With the increasing number of neurodivergent students in our classrooms, shouldn't we be designing courses that we know will engage and build understanding for all?
Armed with this knowledge, and my trade tools, i.e. paper strips, grid paper, whiteboard, and calculator, I set out re-teaching her everything fractions, decimals, and percentages. Surprisingly, this didn't take as long as you would think.
I began by making the connection between fractions and percentages, in a similar way to this diagram
We talked about how percentages relate to decimals as decimal fractions, all the while backing up any information with visuals, paper to fold, and grids to colour. We then used the same processes to move into simplifying fractions and operating with fractions and percentages. I made sure that everything I did had a conceptual element to back up the process, otherwise, I knew that they would walk away and potentially forget everything taught in that lesson. For example for the problem:
I made her fold paper similar to this:
and then show me what that meant. Here's a mockup of what she did:
I then got her to connect this to the mathematical process that she had previously learned in fraction division and suddenly the penny dropped! I could see her face light up like a curtain had suddenly opened. She finally got it!
We did the same with adding, subtracting, and multiplying fractions, and she continued to use the representations that she had formed in her mind to help her remember what methods to use when attempting to solve the problem. In fact, solving problem after problem, and very confidently. When she got stuck, she glanced back at the representation that she had drawn for a previous question and that's all it took.
Now, I know that I haven't touched on the problematic multiplication facts here...I'll have to save that for next time...but I will proudly say that after discussing a method of recalling the 9 facts, this student, who previously could not convert 5.3% to a decimal, was able to immediately determine that the denominator in the fraction 104/72 was divisible by 9 because 7 + 2 = 9, and 9 x 8 = 72 because the digit 7 was one less than 8. Furthermore, 104 was divisible by eight but not 9 so the highest common factor was 8 making the simplified fraction 13/9.
It was a proud teacher moment! Not because she got the answer correct, but because she was able to confidently reason through and justify her thinking. She actually solved it faster than I did, which for those who know me is not always hard, but made me supremely happy nonetheless!
Better still, she achieved a grade of 93% for her end-of-unit test, making only minor attention errors, but zero process-based errors! this was after only 2.5 hours of conceptual-based practise with multiplication, fractions, decimals, and percentages.
I know for those of us who have learned maths procedurally, and have been successful at it, conceptual based mathematics can be difficult to grasp, but we owe it to our students to give them every opportunity to learn. If that means a procedure, then great, but if not, then what are you going to do to help make a difference to those students?