Thinking back to my schooling experience Primary School through to High School I was always “good at maths” as I was told by peers, teachers and family. I was quick with number facts, could recall any times table up to my 15x facts (division included) and would rarely struggle with number races and punching numbers into formulas. I tell you this not to brag but to help set the story.
I now look back and wonder “was I just quick?”, “was fluency my strength?”. I remember being able to problem solve well but probably wasn’t that strong with understanding or reasoning of concepts.
We all learnt that the area of a triangle is “1/2 b x h” or some variation of that. Give me a triangle with dimensions of the base and height written on there and I could punch the numbers in and work the answer quickly (see below)
I was good mechanically with it, but would struggle with a triangle that didn’t didn’t look this straight forward.
When I began as a teacher, I taught students “the formula is 1/2 b x h”… so “here’s some problems”.. and wonder why kids would struggle. There were a cohort that just got it and would punch in the numbers and fly through 20 questions but fall down when given questions requiring some reasoning and problem solving.
What happened next was the question that changed the way I taught maths….
Student: “Mr Millar why is the area 1/2 base times height?”
Me: “Great question, let me check” (in my head I’m thinking “Bugger how do I answer this question”)
I actually had no explanation as to why this was the case, as I didn’t have a deep understanding of the concept.
I began thinking… “How can I provide kids with an opportunity develop conceptual understanding before drilling formulas and pages of problems into them?”
The next lesson I took a completely different approach by providing kids will rectangular pieces of paper and creating a triangle from that (see example below). I then posed the question “What is the connection between the original rectangle and the triangle you created from it?” Something interesting happened…we weren’t practicing problems, formulas weren’t being used, numbers didn’t matter. We did go through half a ream of coloured paper though!
Kids were trying all different types of triangles and combinations. There was a buzz in the room not normally present in a maths lesson. They we coming up with their own formulas and variations of the formula… “Times the base and height together and half that answer” was one I recalled. They were all able to explain why the formula was what it was. They had a better conceptual understanding.
Fast forward to 2011, when I started as a Numeracy Coach and I came across Peter Sullivan’s “6 Principles for Effective Teaching of Mathematics” (More can information can be found at this link in Section 5). This reinforced my thinking…
Principle 6: Promoting Fluency and Transfer. Sullivan discusses “With mechanical practice, students have limited capacity to adapt the learnt skill to other situations. With automatic practice, built on understanding, students can be procedurally fluent while at the same time having conceptual understanding”.
I have also heard this referred to as “Experience before Instruction”, giving kids experiences to develop conceptual understanding before getting into the mechanics.
This has now filtered to my current practice and planning. I’ll be teaching multiplication facts to my 4/5’s over the coming weeks. Rather than drilling the mechanical practice, I will be starting with creating arrays using manipulatives and making connections between these and the facts to give kids a deeper understanding.
Are we providing kids with the opportunities to develop conceptual understanding and reasoning skills before moving to fluency?