Conception of the good

Insights into our current education system

Month: December 2017

Weekly Quizzes: Substitution

In my last blog post I discussed how curriculum sequencing underpins successful teaching of concepts. For examples, ensuring that pupils are able to add, subtract, multiply, divide directed numbers as well as raise a directed number to a power allows successful teaching of how to substitute directed numbers into expressions.

The second aspect of good teaching is ensuring pupils are regularly given low-stake quizzes which test whether a pupil has retained the small body of knowledge that has been taught explicitly over a short period of time. This is possible through weekly quizzes. I learnt this good practice during my time at Michaela Community School, and it has been the second most important aspect of my teaching practice that has allowed my pupils to be successful in the classroom. The first being the introduction of in-house resources in the form of a textbook.

Why are weekly quizzes powerful?

  • Testing is a form of learning which helps pupils retain new information

I didn’t think of testing in this way when I started teaching. I thought testing was a form of dead time which took up lesson time. There is some grain of truth to this because you don’t want to occupy a large proportion of your lesson time to testing because you want to teach pupils new content.

Where I was wrong in my understanding of testing is that when pupils are taught new content the only way pupils retain new information is through regular testing, this is the case by:

  1. Activating the information in regular, spaced intervals. The recall process needs to be an active one where a pupil responds to a question rather than re-reading or highlighting. Every act of recall and review of newly taught information the stronger the memory traces become. The stronger the memory the longer you are able to recall the piece of information.
  2. Testing a pupil’s understanding of a concept by creating a variety of test items. The greater the variety the more that pupil learns to connect new information with what they already know. This improves a pupil’s ability to retain information.

Equally, this is important because there is nothing more frustrating for a teacher, and a pupil, when a pupil has been taught a lot of information and then they forget it. This demoralises even the most competent of pupils to learn further.

  • Weekly quizzes are a good measurement of a pupil’s change in performance

This is dependent on a few factors:

1.The content of the test items has all been taught to pupils. Or pupils have been taught sufficient information to answer the question. For example, I didn’t explicitly teach this problem type below to my Year 10s or Year 9s because I wanted to see if they could do it themselves given the information they already had. In both classes 95% of pupils were able to do both questions correctly.

2. The way that I have organised my weekly quizzes is similar to what was done at Michaela. I give pupils a practice quiz before I give them the real quiz. The real quiz’s score I will total and collate into a spreadsheet. The practice quiz is a similar version of the real quiz but with different numbers, the question types are the same. Pupils do this practice quiz in my lesson in exam conditions. I then mark the quiz with the class, giving whole class feedback. I ask pupils how many marks they got per question. For example:

This lets me know which pupils have made the most mistakes, which pupils have been the most successful, if I need to reteach a particular concept that the whole class has misunderstood or that I haven’t taught well.

They are then able to take the practice quiz home, and they are also given a spare copy of the practice quiz. So on one piece of paper, double sided:

  1. class practice quiz with teacher feedback in green pen
  2. black copy of practice quiz to reattempt at home

The pupils have teacher feedback to act on so when they are revising for the quiz the following day they know what they need to revise over. They understand their mistakes and they know how to avoid making those mistakes again. The following day pupils are given the same quiz but with different numbers. They complete this in exam conditions where I invigilate to ensure pupils don’t speak or cheat.

This works because the test items haven’t changed. I have changed the numbers that the pupils will substitute but the problem types are the same in the real quiz and practice quiz. The test has stayed constant so I can infer a judgement on pupil performance. You can’t compare test results when the quiz continues to change. Keep as many test variables the same, so the only variable that is measured is a pupil’s performance.

From the data set of the most recent weekly quiz my Year 9 Set 2 did I can then make the right judgement about each pupil’s performance.

What makes this practice difficult?

It depends on how many lessons you have with that class because you do need time to teach the curriculum and do both a practice quiz with feedback to give, and time for another quiz. I am very lucky to have about 5 lessons per week with my classes, and some double lessons too.

If you teach five year groups then it can be hard as a single teacher organising these quizzes. If this became a whole department policy then it would be best to have one teacher designing the weekly quiz for one year group. This is manageable. If one teacher makes the quiz catering for all ends then each individual teacher can edit to make it suitable for their classes.

I have only been at GYCA for a half term, but the quizzes are informing me of how I can improve my teaching, how clearly pupils understand what I have taught them, how flexible their knowledge is and how well they are retaining what they have been taught. The kids love seeing percentage scores in the 80 – 100% range. Pupils love seeing the success of their hard work. It motivates pupils who want to learn more to be able to do so because they have been told (a) what to learn and (b) how to do it too. It also gives pupils who are lazy or find maths difficult guidance on how to improve.  I am looking forward to seeing the kids’ faces on Monday morning when they get their quiz scores back. Lots of them will be beaming with pride.



Effectiveness of Curriculum Sequencing – Substitution

After October Half term, I joined Great Yarmouth Charter Academy. I was given the task of teaching Year 10 Set 5 out of 6. The first topic I was assigned to teach them was substitution to then go onto teaching expanding brackets. I planned two booklets to teach both topics. I then realised due to a lack of prior knowledge, what I had prepared was out of their reach. I went back to the drawing board to teach the topic of directed numbers. This blog post is a reflection on what I saw to be effective teaching of substitution purely because of how the prior knowledge required was sequenced before teaching substitution.

Teaching Directed Numbers

Directed Numbers is a high-leverage topic. This is because if the content of the topic is mastered then pupils are able to successfully learn other concepts with great ease. Pupils are able to substitute negative numbers into expressions if they are able to add/subtract/multiply/divide negative numbers without having to process what they need to do in their working memory. For example, when I planned my lessons on substitution, I broke my lessons down into this order:

  1. Substitution of positive numbers
  2. Substitution of negative numbers
  3. Substitution of fractions
  4. Substitution of decimals

Before I wanted pupils to be able to substitute negative numbers into different expressions I had to teach the following:

  1. Adding and subtracting negative numbers (including large values)
    1. -4 + 5 –> -41 + 51 (crossing over 0)
    2. -4 + 3 (and 3 – 4) –> -41 + 32 (not crossing over 0)
    3. -4 + 4 –> -329 – 329 (meeting at 0)
    4. -5 + 6 – 9 (multiple terms)
  2. Adding and subtracting negative numbers (double signs)
    1. -4 – + 5
    2. -4 – – 5
    3. 4 – – 5
    4. 4 + – 3
    5. -4 – – 4
    6. 4 + – 5
  3. Multiplying and dividing negative numbers
    1. If I multiply an even number of negative numbers then my result will be positive
    2. If I multiply an odd number of negative numbers then my result will be negative
  4. Raising negative numbers to different powers
  5. Cube rooting negative numbers (including odd roots)

What worked?

Repetitive practice exercises

For the first lesson on teaching how to add and subtract negative numbers with no double signs, I had pupils completing exercises of one problem type. This was to help pupils recognise a pattern which I later shared. Pupils did lots of questions where they practised moving up and down the number line before hand.


This was also the case when teaching pupils how to multiply and divide negative numbers:

Similarly, this also applied when asking pupils if the result when a negative number is held to an even or odd power would result in a positive or negative number.

Daily Recap and Questioning

After the starter of my lessons, the next 5-6 minutes I would ask them the same questions on a daily basis, change the order of the questions which was important to ensure that pupils were made to think. Similarly, I used the same problem types each day but would change the numbers.

My questions also became less scaffolded over the weeks to ensure that I was really testing the kids. I also specified the answer to the question. For example:

If the numbers are going in the same direction then what do I do?

Answer: Sum the values

This was because I wanted pupils to have absolute clarity over what they needed to do, and with a variation different phrases for the answer and being verbose would have confused them. Also, the repetition helped pupils see specific patterns e.g when we have the following problem types -5 – 6 we add the values of 5 and 6.

These are the questions that I asked them on a daily basis:

Questions for adding and subtracting negative numbers – no double signs:

  • Are the numbers going in the same direction or different direction? ANS: Same/Different
  • If they are going in the same direction, what do I do next? ANS: Sum the values
  • If they are going in different directions, what do I do next? ANS: Find the difference
  • When I have found the difference, will my answer be positive or negative? ANS: The larger value is negative so my answer is negative.
  • What are the two things do I ask myself when I see this question?
    1. Same direction? Sum the values
    2. Different direction? Find the difference
  • What is the first step?

Questions for multiplying and dividing negative numbers:

  • How many negative numbers do I have? ANS: 2/3
    1. Follow up: Is that an odd or even number? ANS: Even/Odd
    2. If I have an even number of negatives then will my result be positive or negative? ANS: positive
    3. If I have an odd number of negatives then will my result be positive or negative? ANS: negative
  • Are there are an odd number or even number of negatives? ANS: negative
  • If there is an even number of negatives then will my result be positive or negative? ANS: positive
  • If there is an odd number of negatives then will my result be positive or negatives? ANS: negative
  • Will my result be positive or negative? ANS: Positive/negative
  • Is my power an odd number or even number? ANS: even/odd
  • Will my answer be positive or negative? ANS: positive/negative

Questions for adding and subtracting negative numbers – with double signs:

  • When I see two signs next to each other, what do I do? ANS: Circle the signs, and replace
  • What is my first step? ANS: Circle the signs, and replace

Why was the re-teaching of directed number crucial?

Pupils were to attempt the following exercises.

Substituting negative numbers with calculations requiring knowledge of how to add and subtract negative numbers

Prior knowledge they needed that was taught:

a = -3

a + 11 = -3 + 11 = 8

1.Given that a = -3, b = -4, c = -5 and d = 4, evaluate each expression

Substituting negative numbers with calculations (double signs)

Prior knowledge they needed that was taught:

a = -3

11 – a =

11 – – 3 =

11 + 3 = 14

2. Given that a = -3, b = -4, c = -5, evaluate each expression

Substituting negative numbers where pupils will be multiplying and dividing negative numbers

Prior knowledge they needed that was taught:

a = -3

2a = 2 x -3 = -6

-a = -(-3) = +3

3. Given that a = -10, b = -4 and c = -6, evaluate each expression

Prior knowledge they needed that was taught:


4. Given that d = -12, e = -3, f = 16, g = -8 and h = -20

It is very early days but they are very comfortably learning how to substitute negative numbers. We will be learning how to multiply a constant and an expression, next week. The teaching of directed numbers will support the pupils particularly when attempting the following problem types.