Around the world, more and more educators are realising the significance of asking their students to prove what they know and why they think as they do. It is now an expectation in most curriculums – as a Proficiency Strand in the Australian Curriculum (1), as part of the Eight Mathematical Standards in the Common Core in the USA (2) and as a key aim in the UK National Mathematics Curriculum (3), to name just a few.
As a classroom teacher I know I was guilty of not doing this enough – often making assumptions based on test results, practice question sets or students’ recordings without requiring sufficient evidence of the reasoning process the students used to arrive at their answers.
While working with a Year 3 student the other day I learned a short but powerful lesson on the value of asking for proof and how this process can transform teaching and learning.
Jessica* and I were working on extended facts. I asked her what 2 + 3 was and then what other facts she knew from this. She confidently scrawled:
The temptation to use this as evidence that she possessed conceptual understanding as well as procedural fluency and to move on to a different question was strong. We had a lot to get through that day and it all looked pretty cut and dried. However, the simple act of asking for proof revealed a powerful new learning direction that I could easily have missed.
Using virtual HTO blocks Jessica created the first fact as seen in Figure 2 and then recorded the number sentence to match and explained her thinking when asked. So far, so good.
Her 2nd fact (Figure 3) however, revealed much about her level of conceptual understanding.
She quickly replicated the same representation as the original fact. When asked to record the number fact and explain her thinking, she recited the fact to me while circling the blocks.
When asked if the facts were the same she replied “No”, and when pressed, explained that the 2nd one was bigger. I then pointed to the blocks she’d used and asked her why she’d chosen them and that they looked the same to me as the ones used in the original fact. She struggled to answer, and then quickly erased what she’d done and created the fact in Figure 4.
“So that shows 20 + 30 = 50?” I asked. She nodded assuredly, “Yeah, I forgot to make them different before.”
This brief exchange showed me that while Jessica could reproduce the facts with relative ease, her conceptual understanding of what they actually represented was limited – the blocks appeared to be just blocks, disconnected from their intended representational purpose and the number facts were just that – facts to be recited. When prompted, she did make connections with her prior learning of building numbers with tens and ones blocks but again, this understanding was fragile and based on misconceptions. Place value would be the focus of our next session.
What I’d learned by asking for proof had been powerful – not for Jessica so much at this stage – but for me as her teacher. I knew more about what she knew, and where to go in the next session. This next session would in turn deepen my understanding of her understanding, and so it would go on – bringing about a richer teaching and learning process.
While this session was a one on one, I can elicit the same kind of information in a classroom setting – by asking questions as I circulate the room, by creating the expectation that as learners we reflect upon, explain and justify our ideas and by making sure class discussion times are valued and not a quick 5 minute session before the bell (sometimes guilty as charged, Your Honour).
It appears that the burden of proof rests with me.
Primary Mathematics Publisher
Dip T (ECE), B Ed (Primary)
*Name has been changed