Summer School Insights: Week 3 Negative numbers

Craig Barton
August 24, 2020

Hello everyone,

This is the third post in our Eedi Summer School series, where I provide insights from the cohort of Year 6 students who are taking part in our seven week programme. My hope is that this may give you an indication of where our Year 7 students might struggle come September.

The subject of Week 3’s quiz is Negative numbers.

This proved a trickier proposition than the first two weeks, with our group of 10 and 11 year olds scoring an average of 76% across the 10 questions.

Here are the proportion of students getting each question correct in reverse order from the most poorly answered to the best answered:

We can see that Question 9 is the one that caused our group the most problems. What is Question 9, I hear you ask? It is this…

Take a moment to consider where you predict students might go wrong with this question.

Here are the results from our Summer school students:

It is perhaps no surprise that our students found this quiz as a whole more challenging than the previous two on place value and integer arithmetic. I suspect most teachers reading this email will have experienced their students making all kinds of mistakes with negative numbers. Likewise, if asked you predict the type of questions students are likely to struggle with this most, one involving not one, but two, negative signs may have been top of your list..

We can see that a significant proportion of our students struggle with this question, and it is answers A and C that are most alluring.

Let’s start with answer A. Why might students think -9 is correct? Well, here are the explanations two students gave whilst answering the question:

Because if you did the inverse (-4 add 5) the answer is -9

Because -9 - 5 is -4

Here we have two solid approaches - either attempting to do the inverse operation or solving this as a missing numbers style problem -  but still getting the wrong answer.

The students who believe the correct answer to be C, 9, have different reasons for their selection:

i think it is c because if you do 5+4 you would get 9 and if you do 9-5 you would get -4

Two minuses make a positive

the difference between 5 and -4 is 9

We see a combination of misapplied rules in the first two explanations, followed by an interesting attempt to think about this question in terms of differences - an approach that is sensible in a subtraction problem, but which is problematic with this question.

As with all these misconceptions, I find it useful to ask myself two questions:

  • Why are students going wrong?
  • What am I going to do about it?

Negative numbers - particularly operations involving negative numbers - is one of those areas of mathematics that teachers tend to have very strong views on how they approach it. I have tried most of them over the years - rules, number lines, temperature, analogies involving sandcastles, sins and debts. I have never been entirely happy with the results. And this is such a key concept to get right because negative numbers creeps its way into many different areas of mathematics, including solving equations, operations with fractions and standard form.

As I describe in my book, Reflect, Expect, Check, Explain, my approach to operations with negative numbers these days is to expose students to a carefully selected sequence of examples. By holding key aspects of examples constant, students attention can be drawn to what has varied and its impact on the outcome. Here is a personal favourite:

This sequence is available to download here.

An approach that can be used alongside this sequence - and one which can be used with most Diagnostic Questions - is to challenge students to change the original Diagnostic Question to make each of the wrong answers correct. So, how could you change the question so option A, -9, was indeed the right answer? This should help focus students’ attention on the impact changes make, and move them away from clutching to rules and approaches that can be misremembered or misapplied.

Let me show you the second and third worst answered questions from this quiz, together with the proportion of students opting for each answer:

Again, it might be useful to ask yourselves the questions:

  • Why are students going wrong?
  • What am I going to do about it?

You can access the whole quiz here, and the Insights page that will give you access to the percentages and student explanations here.

Next week we have Decimals, and I will be back with some more insights based on students’ responses.

Take care of yourselves, thanks for reading and stay safe
Craig and the Eedi Team

Written by
Craig Barton
Head of Education