Challenge Level

This task will help you unpick children's understanding of the equals sign. It may reveal misconceptions and help you to address them.

*This problem and the possible approach below have been inspired by the book 'Thinking Mathematically' written by Carpenter, Franke and Levi (2003). The full reference is included at the foot of this page.*

*This problem featured in an NRICH Primary webinar in January 2021.*

*Carpenter et al.'s research found that many teachers were surprised by their class' responses to these kind of calculations, sequenced in the way described below. So, before having a go at this task with your class, it might be useful to consider how you think your children might respond.*

To introduce this task, write 19+15=34 on the board. Invite learners to decide whether it is true or false, and why, firstly by giving some individual thinking time and then by asking everyone to share their ideas with a partner. Invite a pair to explain their conclusions to the whole group. You could then ask whether other pairs used a different approach to decide, and share a few alternative ways.

Next, write 34=19+15 on the board (leave the first calculation visible too), and again ask whether it is true or false, and why. As pairs are discussing their thoughts, listen out for their assertions and accompanying reasoning. You may wish to invite each pair to indicate their conclusion to you in some way e.g. by discretely putting a thumbs up or a thumbs down. If there is disagreement, choose a pair to explain why they think 34=19+15 is false. This may reveal some assumptions about the equals sign. Some learners believe that there can only be a single number after the equals sign and/or that an equals sign means 'the answer comes next'. Encourage other members of the class to comment on, or ask questions about, any reasoning that is verbalised.

Depending on the responses to the second example, you may like to offer a third example on the board for everyone to consider, or you may prefer to give pairs these cards, which include all the calculations in the first set. If the latter, allow time for all pairs to decide whether each number sentence is true or false, and why.

Draw the whole group together to share their thoughts. Again, try to facilitate a whole class discussion rather than you being the one to comment on or validate children's contributions. Encourage learners to see the equals sign as representing a *relation* between two equal numbers, in other words meaning 'the same number as'. Choosing words that express the relationship clearly
can really help learners. For example, "34 is the same amount as 19 plus 15".

At this point you could offer children the second set of number sentences to further consolidate and develop their understanding. This second set encourages learners to see the relationship between the left and right sides of the equals sign by including number sentences in which the left and right hand sides can be compared with each other, without the need to do any calculation. For example, listen out for learners who know that 346+289=349+286 is true by reasoning that 349 is three more than 346 but 286 is three less than 289, so the amount on each side of the equals sign is the same.

Throughout the lesson, you may choose to use the interactivity as a way of visually checking children's solutions, or even challenging their answers.

How would you 'read out' this number sentence? What does that mean?

Why do you think this number sentence is true/false?

It may be useful to sit alongside some learners and listen in to their conversation, and gently offer questions and prompts to probe further. You can then deliberately offer a particular number sentence next, which might help them grapple with any misunderstanding. For example, if a pair is adamant that there can only be one number after an equals sign, try offering them the number sentence 19+15=34+0 next.

You could invite learners to create their own set of true or false number sentences for a partner to try. You might specifically ask them to include examples where they think that calculation is unnecessary. This is a great way to reveal their understanding of relations.

*Reference:*

Carpenter, T. P., Franke, M. L. and Levi, L. (2003) *Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School*. Portsmouth, NH: Heinemann.