Thinking Mistake: Multiplicative Comparison

When I work with students, I love to celebrate mistakes and promote a growth mindset (Dweck, 2016). I encourage all students to see that through mistakes comes learning. I like to label misconceptions as ‘thinking mistakes’. ‘Thinking mistakes’ are in direct contrast to ‘silly mistakes’- ones which I want my students to consciously avoid- these are made when our brain is not ‘switched on’. In stark contrast, ‘thinking mistakes’ are made when our brain is working hard, struggling to make thoughtful and logical conclusions. It is during these times that real brain growth occurs.

This series of blog posts presents six common ‘thinking mistakes’ which have repeatedly surfaced in my research into Year 3-6 students whole number place value understanding. Along with each thinking mistake I also present a piece of teaching advice to help you address these issues.

In this blog I am looking at the ‘compare/order’ aspect of place value. In my research I defined this aspect in the following way: Compare numbers to determine which is larger or smaller. Compare numbers in a multiplicative manner, for example ten times larger than 54 is 540. Place numbers in descending or ascending order and locate numbers on empty, partially marked or complete number lines.

Thinking Mistake:

Comparing numbers is an important concept in early maths. We want students to understand  ‘more’, ‘less’, ‘larger’ and ‘smaller’. However, in Years 3-6 we also want to students to be able to compare multiplicatively.

Multiplicative comparison requires a sophisticated form of understanding related to the relationship between two quantities. It relies on the ability to think multiplicatively, something we know takes many years to develop in students. Students find multiplicative comparison difficult as we rarely provide them opportunities to think in this way. They often think in an additive way when they hear the words ‘smaller’ and ‘larger’ and do not understand the meaning of these words in the context of the word ‘times’. The image below shows an example of a student thinking in an additive manner to solve this problem. They have simply read ‘ten’ and ‘smaller’, and subtracted 10. An understanding of the x 10 relationship between each column in our place value system, is a critical link for students to make when completing multiplicative comparison involving factors of 10. For example, 10 times, 100 times larger/smaller. Making direct connections with our place value system is critical to deepen our students number sense.

Teaching Tip: A fabulous book and activity to reinforce this idea is to read “How Tall was a T-Rex” by Alison Limentani. In this book students are prompted to consider multiplicative comparison scenarios. For example, a T-Rex is as long as 6 lions. This is the perfect context to talk about a lion being 6 times shorter than a T-Rex or a T-Rex being 6 times longer.

Students can begin their own multiplicative comparison investigation. In the above example, the student compared a western long beaked echidna with a T-Rex and discovered the echidna is 15 times shorter than a T-Rex. Alison Limentani has several beautiful books that have excellent links to maths. I encourage you to check them out!

This week I would love you to provide your students some multiplicative comparison experiences and let me know how you go!

Ange🎲🎓

Want to learn more about teaching place value?

Check out my 'Assessing Place Value in Years 3-6' mini course. Click here to find out more.