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?