We’ve just started an addition and subtraction unit. The students are pretty comfortable with written and mental methods so I’ve made it a focus to develop the students’ mathematical language – both their use and interpretation of it.
A real bugbear of mine is ‘word problems’, even worse when the phrase is preceded by ‘real-life’. I still so often see these ‘problems’ given to students as an extension activity once they have mastered a concept or skill. These are simply stories with numbers that require no higher order thinking i.e. not a problem at all! (See Marian Small’s article discussing a definition of mathematical problems.) Present students with a page of questions with ‘Addition Problems’ at the top and the words could be in any language.
Browsing #elemmathchat I came across Numberless Word Problems. Whilst the final question absolutely typifies the sort of problem I discussed above, the rich conversation, reasoning, estimation etc generated by this one question was so valuable. Below is the transcript (as much as I could catch of a very animated discussion amongst eight 10 year olds!) of the lesson where I introduced this routine to the group. I borrowed the wording of the question directly from here.
Some girls entered a school art competition. Fewer boys than girls entered the competition.
- That is not gender equal
- It’s less boys
- Boys and girls like different things
- More girls like art – boys like football
- How can we work out the answer without numbers?
- We can’t work out an answer because there is no question!
- It would be more mathematical if we knew how many girls and boys there were
- And also how many students are in the school – that way we could decrease the number
- It would help if we knew what kind of school it is – we could be more precise with our estimate
135 girls entered a school art competition. Fewer boys than girls entered the competition.
- We know that boys is definitely less than 135
- It could even be boys are 134 – that’s still less
- Could it be zero boys?
- No – fewer is not none
- Fewer means just a little bit less
- Maybe 128-134 boys
- I think 100-134 boys
- It couldn’t be 134 as fewer is more than 2
- 133 is the maximum number of boys
- The conversation continued with a range of estimates being thrown around and discussed
135 girls entered a school art competition. Fifteen fewer boys than girls entered the competition.
- Fifteen fewer means 15 less than 135
- That’s 120 boys
135 girls entered a school art competition. Fifteen fewer boys than girls entered the competition. What questions could you ask?
- How many boys entered?
- How many boys and girls entered altogether?
- Did everyone in the school enter?
- How big is the school?
- What percentage of the school entered?
- Why are there fewer boys than girls?
- We need more information
- How many students are there in the whole school?
- You could do a survey to find out what things the students like to do and why.
Notice how there is no discourse or debate or questioning or estimating as soon as both numbers are revealed. Proof enough for me of the value of this routine.
The anticipation and excitement from the students as each level was revealed was high. I can’t wait to have a go at some more of these. I’ll probably use a ‘What do you notice, what do you wonder?’ routine to help focus the thinking/discussion when working with the whole class.
My aim in introducing this routine to the students is not to make them better at solving word problems. It is my hope that this kind of critical thinking and processing of language becomes habit across the curriculum, whether it be reading a newspaper article or working on a maths problem that really is a problem.