The past few weeks I have been learning and researching a lot on multiplicative thinking. Chris Hirst’s article “The Multiplicative Situation” has been my go to, it’s an incredible article that makes it very clear on what we need to do to help develop multiplicative thinkers. This post was going to be about multiplicative thinking but something more interesting came up. I was working with a teacher while she was diagnosing her students before starting a learning sequence on multiplying/division. What came up was a truly “aha” moment for this teacher. This “aha” moment was seeing that two students who she thought were way ahead in their multiplicative thinking based on some problems they had solved were actually not.
The teacher had done some multiplying problems before and both these students were solving the problems and coming up with the correct answers while showing their thinking. We then decided to interview her students as the diagnostic using Alex Lawson’s continuum
and two of the prompts from her book to see/hear what the students shared with us. The two prompts were:
You have 5 boxes of chocolates with 4 chocolates in each box. How many chocolates do you have altogether?
You have 15 guppies and want to put 3 in each jar. How many jars do you need?
Here are two videos of the students answering these prompts:
As you can see in the videos both students are using a combination of strategies from the direct modelling and counting more efficiently stages of the continuum. Both modelled the groups and then skip counted. These students are still counters and not yet using multiplicative thinking when looking at multiplying and dividing problems. Making the move from being an additive thinker to a multiplicative thinker is one of the most important steps as we move from primary to junior. If they don’t make this leap it can stall much of their math learning from junior through to high school.
During these interviews she realized that two of her best math thinkers, who get the answer correct most times are actually still using counting strategies. We then discussed what specific instruction they need to help move them into using more advanced additive strategies and then on to multiplicative strategies for the transition to being a multiplicative thinker in the junior years. Most research I have read says they should make this transition between grade 3 and 4 and then extend their understanding into more complicated multiplicative thinking (rate, combinations) as they move into later junior/intermediate years. We realized there is no reason for this teacher to panic, these girls are in grade 3 but it was still eye opening for her to see where their thinking is located on the continuum even though they were consistently answering problems correctly.
Here is some pics of written product assessment that gives further evidence these two students are still counters:
Looking closely at the two work samples you can see they both counted all the squares. If they were more advanced additive thinkers we would have seen something more like this:. 5+5+5+5=20 or 10+10=20. You can see they even wrote the number in each box as they counted.
The math consultant for my school board and I chatted at a PLC last week and she tells me this conversation is starting to take place more around our board as teachers start to know their learners and the math more deeply. Correct answers on assessments can be deceiving especially when looking at bigger ideas like moving from counting to additive and then to multiplicative thinking. Precision is important too, but seeing and hearing their thinking, going deeper and learning what strategies they are using and where their conceptual understanding lies, to me is so much more important! Please leave some comments, I would love to hear other peoples thoughts on this topic.