Last week we saw Hilary perfectly demonstrate the practice of reflection and adjustment when she went back to the problem of “four more dogs than cats” and clarified with her students how to interpret this as addition rather than multiplication.
The example gives us the opportunity to look at a subtle but important point about interpreting word problems with your students. Let’s take this problem:
There were some pizzas and some pies at the party. There were four more pies than pizzas.
What are some numbers of snacks there can be? What are some numbers of snacks there cannot be?
A tempting practice for teachers to engage in is the ‘word search’ approach to interpret the problem. So a teacher might direct the students to notice the word ‘more’ and infer from this that it is an addition problem. Although it may work out in this case, it is a problematic practice that can lead students to make many mistakes, and stop them from thinking more deeply about a problem. A better way to guide students with these type of problems is to look at the whole context to understand the meaning. Unfortunately, there’s no shortcut for this – we just need to expose them to lots of problems. Let’s explore what difficulties the ‘word search’ approach can create.
Firstly, this problem can be a subtraction problem – I could choose how many pies, say 7, and then subtract 4 to determine that I have 3 pizzas. Telling the students that ‘more’ = addition might stop them from thinking about the problem more flexibly and making deeper connections like the inverse relationship between subtraction and addition. The student who says this is a subtraction problem and interprets it as I’ve just mentioned is potentially demonstrating a more sophisticated understanding.
The general meaning of 4 more is not about either addition or subtraction exclusively, but about the relationship between pizzas and pies. I.e. they are 4 apart on a number line, and pies are to the right of pizzas.
Deep understanding of this problem is knowing that the (numerical) distance and direction between pizzas and pies is the only fixed thing. Everything else can be changed. BUT, because there is a relationship, if I set the number of pizzas, I consequently set the number of pies by the relationship (must be 4 more) and if I set the number of pies, I consequently set the number of pizzas (must be 4 less). I can even set the total number of pizzas and pies, which would automatically determine the amount of each, e.g. 24 snacks means it must be 10 pizzas and 14 pies. This is also an example of an extending prompt we could use for students handling the original problem easily – ask them how many pizzas and pies if the total number of snacks is 24.
This understanding of underlying general structures in numbers is what allows students to transfer their knowledge to different problems and be flexible problem solvers. It is also what sets them up for a proper understanding of algebra (This problem models the general relationship y = x + 4).
This relationship can be addressed using either addition or subtraction (because they are inverse operations). So, hopefully, you can see we don’t want to shut down this kind of thinking by saying something like, “the word ‘more’ means it is an addition problem.”
So, instead, if we are asking students to make sense of this problem in a group discussion and a student says “It’s a subtraction problem”, you might say, “How would you use subtraction to solve this problem? Give me an example.” Then they might explain something like what I’ve outlined above (in which case you could segue into a discussion about the number line and the inverse relationship between + and – and hence why they both work when used in the right direction). OR they might show you that they’ve misinterpreted the problem (in which case you could work to overcome the misconception).
Also, in some problems that use the word ‘more’, subtraction might be the appropriate operation to use, for example:
There are 4 more boys than girls in our class. We just counted and found out there are 12 boys. How many girls are in the class?
The most efficient solution here is 12 – 4 = 8.
When exploring how to interpret a word problem with our students, we should avoid focussing on individual words to decode the problem and think about how to make sense of the whole thing by making a picture (e.g. number lines, arrays, groups) or using manipulatives to model the problem (counters, Cuisenaire rods, MAB blocks etc.). We should aim to highlight the relationship between the different parts of the problem, rather than seeking out the operation to use.
I hope these reflections have given you something to consider and can support you in developing your own maths teaching practice.
Keep thinking and enjoy the struggle!
Find out more about the Alcoa Maths Enrichment Program