## Summer School Insights: Week 2 Integer arithmetic Craig Barton
August 24, 2020

Hello everyone,

This is the second 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 2’s quiz is Integer arithmetic.

Once again, this quiz was very well answered by our group of 10 and 11 year olds (at the time of writing, 400+ students have completed the first quiz). On average, students scored 83% 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 10 is the one that caused our group the most problems. What is Question 10, 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:

I like this question. It brings an element of reasoning into integer arithmetic by presenting students with two strategies and challenging them to assess the validity of each one. Deeper thought than usual is required.

We can see that more than half of our students struggle with this question, and it is answers A and D that are most alluring.

Let’s start with answer D. Why might students think neither strategy is correct? Well, here are the explanations two students gave whilst answering the question:

Neither of them are correct but if you used Tom's method to multiply 9 by 2 it would equal 19 instead of 18. If you used Katie's method once it would be correct but if you double it again it would be 32 and not 24.

if you were to take away one from 20 you would get 29 and that is what tom says is 2x9 but 2x9 is 18 so tom is wrong. katies is wrong because if you were to double you would get 2 then 4 then 8 then 16 and then 32 which isn't 8x3 8x3 is 24

This fascinates me. It is so pleasing to see students as young as 10 creating their own examples to assess the validity of the strategies. And with Tom’s, it seems to work. However, when we turn to Katie’s method, there seems to be some confusion. 8 x 3 is a really nice example to use, but as we can see from the second explanation, something goes wrong. Multiplying by 3 and doubling three times are seemingly confused.

We see something similar in the explanations given by students who select answer A - only Tom is correct:

Katie is wrong because if you double from 8 then you would end up with a way bigger number than any question from doing 8 x tables. Where as with Tom his makes sense because 10-1 is still 9 so it would still be viable in the 9x tables.

i know that tom is right but i didn't understand kaities strategie so i think that only tom is right

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?

Will a number of examples be enough to convince students that Katie’s method works? If we use a wider variety of examples - including decimals, negatives and fractions - will it convince them more? Can we represent the strategies visually? Is there an opportunity to generalise?

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.

Next week we have Negative numbers (what could possibly go wrong????), 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