# Conception of the good

### Insights into our current education system

#### Month: July 2016

At Michaela Community School we run a selection of extra curricular activities after school which complement pupils’ mainstream learning. In the Maths Department, we have a club called Mathletes. This club entails an additional hour of learning taking place once a week after school and it is specifically for one class that I teach (Top Set Y8).

This week I taught a session on imaginary numbers whereby pupils were able to solve the following equations independently: This blog is split into two parts. The first post will explain how I developed the pre-requisite knowledge pupils need to solve the equations above, with much focus placed on rational numbers because of the second equation. The second post will outline how I introduced a limited understanding of imaginary numbers, enabling pupils to solve both equations.

During my planning, I think of the pre-requisite knowledge pupils need in order to access the new topic that I will teach in the upcoming session. Before I started introducing the idea of imaginary numbers, I introduced the idea of rational numbers in a very limited sense. This was done deliberately – something which I will explain later on in this post.

A rational number is the square root of a square number which results in an integer answer. I introduced the concept like this.

This is not a rational number √2

This is not a rational number √3

This is a rational number √4

This is not a rational number √5

This is not a rational number √6

This is not a rational number √7

This is not a rational number √8

This is a rational number √9

This is not a rational number √10

This is not a rational number √11

This is not a rational number √12

This is not a rational number √13

This is not a rational number √14

This is not a rational number √15

This is a rational number √16

The above methodology was inspired by Kris Boulton’s talk on “The genius of Siegfried Engelmann”, delivered at The Maths and Science ResearchED conference in Oxford, and the National Mathematics Conference 7 in Leeds.

I asked pupils to raise their hands when they noticed the pattern. This was my introduction, because I wanted pupils to recognise that we can categorise the square root of a square number where the result is an integer as a rational number. I told them, for now, that this is a fact that you are going to accept and adopt, and that this is ok.

I am the classroom teacher of the pupils that attend the session and so I know how quick they are in recalling their square numbers and square roots. I also know they have memorised their times table facts from doing Bruno’s Times Table Rockstar programme, as rolled out in year 7 successfully by Bodil Isaksen, Head of Maths.

My next step was to pose the following question: “Now, we can spot which numbers are rational numbers, and which are not rational numbers. We are going to look at how we can rewrite the following  in the form k where k is the largest possible integer, and √a  is not a rational number.”

This is how it was laid out: “We are going to look at the number inside the square root. We are going to split it into two numbers each square rooted, where I have one square number, and one number which is not a square number.”

√72 = √36 x √2

“I am going to identify my square number which I can evaluate to get an integer.”

√72 = 6 x √2

“And I am going to rewrite it like this because we can”:

√72 = 6 x √2 = 6√2

Here k is 6 and  √a is √2.

I also reiterated that I deliberately chose the largest square number (HCF) that can divide 72 because it is more efficient. I left it there, because I didn’t want to go into more discussion which would deviate pupils’ attention away from what I had shown on the board.

I then asked the pupils the following questions, before I asked them to simplify a selection of surds. I didn’t use the word surd because my focus for the session was not “what is a surd” or “how do I simplify surds” but because the point of this task was to develop sufficient pre-requisite knowledge in order to understand how we were planning to solve problems such as x2 = -72. This is why I previously mentioned that I introduced the idea of rational numbers in a limited sense.

Before pupils committed pen to paper to simplify the following surds I asked these questions which pupils answered by raising their hands.

“How can I rewrite √98  as a calculation where I am multiplying a rational number and a non-rational number, the rational number is a square number being square rooted?”

√98 = √49 x √2

“How can I rewrite √242  as a calculation where I am multiplying a rational number and a non-rational number, the rational number is a square number being square rooted?”

√242 = √121 x √2

I demonstrated a complete worked example of evaluating a surd and then I rephrased my questions:

“How can I rewrite  √48 as a calculation where I have an integer and a non rational number? Say Step 1 then Step 2”

Step 1:   √48 = √16 x √3

Step 2: √48 = 4√3

“Why is the integer 4?”

One pupil responded, “because the square root of 16 is 4, Miss.”

Twenty questions later and we were moving on. These were the questions pupils were asked to complete to check they were able to simplify the following surds into the form k√a where k is an integer. After the exercise, I mentioned that we had previously learnt that when we square root a positive number we get a positive  solution, for example √9 is equal to 3. We have also learnt that when finding the solutions of  x2 = 9 then we have two solutions -3 and 3. I then posed the question: “Is there a number which I can square where I get a negative answer? If you think so, then on your mini whiteboard have an attempt, and if you think it is not possible then write it down your thoughts in your exercise book.”

Some pupils were determined to find a number that they could square to get a negative result. Some pupils were already light years ahead with an explanation.

In my next post, I shall explain how I introduced the idea of imaginary numbers and structured my teaching in order to enable pupils to solve the two equations outlined at the start using the pre-requisite knowledge showed here. There will be a video and some pictures of pupils’ work too.

When are teachers to generalise a procedural calculation to apply to all problem types? When are teachers to display all the different problem types for a particular generalised procedure? I always jump between generalising and displaying different problem types, including not stating that one technique is better than the other but stating that in some instances one technique can be prioritised over the other.

In Maths there are many generalised procedural calculations which are true for all problem types of a concept. For example, listing a step by step process for pupils to follow which exists for all problem types regardless as to whether the problems include integers, decimals, percentages, numbers in index notation and square roots etc. in the problem type. We can try to create all the possible visual forms of a problem type, where generalising a procedure is a great teaching technique because it allows pupils to demonstrate efficiently their understanding by outlining the algorithm between the problem and the solution – step by step.

Here is an example of generalising the step by step method in order to add fractions with unlike denominators:

1. Find the lowest common denominator.
2. Form the equivalent fractions.

This generalised algorithm is also applicable in instances of adding or subtracting like fractions; but pupils already have the lowest common denominator, and the fractions being added or subtracted are in their equivalent form, and so only step 3 is applicable. However, adding like and unlike fractions are seen as two distinct procedures, when a question with like fractions is a similar problem type to where step 1 and 2 are already completed.

A generalised procedure to adding and subtracting fractions is applicable for the following problem types where we have improper fractions, mixed numbers and a mixture of the two, or with calculations using more than three terms. Even through each example looks visually different, distinct and more or less challenging than the other, the generalised procedure is applicable.

Generalised procedures are great for pupils to develop their procedural understanding. Further, by discussing, in addition, the different problem types of adding fractions it can empower pupils further. The reason why I say this is because this generalised form of adding fractions neglects the different problem types that exist for this topic and many others. Here are the three different problem types I am referring to: 1. Fractions where the dominator are alike (LCD already present)
2. Fractions where the denominator are co-prime (share a factor of 1)
1.  Where the LCD is the product of the denominators
3. Fractions where the denominator are not co-prime (share a factor greater than 1).
1.  Where the LCD is provided as one of the denominators
2.  Where the LCD is not the product of the denominators.

Exploring different problem types leads to a greater development of knowledge and understanding which can be applicable in instances such as the problem types below: ‘A’ builds on the knowledge of the problem type where the denominators are not co-prime and where the LCD is provided as one of the denominators.

‘B’ builds on the knowledge that the denominators are co-prime and therefore the LCD is a product of all three denominators.

‘C’ builds on the knowledge where the denominators are not co-prime but the LCD is not the product of the denominators.

Can the teaching of problem types alongside the generalised procedure be beneficial in the other realms of mathematics such as algebra, geometry or data? Possibly, here is an example of expanding double brackets in a generalised form as well as exploring the different problem types.

The generalised form can be summarised into the following steps:

1. Draw a 2 by 2 grid.
2. Write each term of the expression in each section of the grid.
3. Multiply all the different terms.
4. Collect and simplify.

Furthermore, this generalised form allows even the most visually complex problem types to be solvable for pupils, as per the below:

1. (2x + 3)(3 + 2x)
2. (ab + cd)(2ab – fg)

The intention of generalising is not to make model algorithms for pupils to replicate. The point is to empower pupils to be able to look at the features of a problem, identify the problem and identify the route between the problem and the solution for every possible problem, regardless of how complex or simple the problem is.

The generalised procedure is applicable in all four different examples of multiplying two binomials to form a quadratic expression, where the expressions multiplied are in the following forms (a and b represent different values):

1. (x + a) (x + b)
2. (x – b) (x – a)
3. (x – a) (x + b)
4. (x – a) (x + a)

Discussions can then take place where pupils can begin spotting patterns, that:

1. the product of two binomials where the constants are positive, the second term will have a coefficient which is the sum of the constants. (x + 2)(x + 1) = x2 + 3x + 2.
2. the product of two binomials where the constants are both negative will have a constant which will be positive because the product of two negative numbers results in a positive number (x – 2)(x – 3) = x2 – 5x + 6.
3. the product of two binomials where one constant is negative and the other is positive then the ‘c’ term will be negative.
4. a difference of two squares will result in no ‘b’ term because the terms will result in 0 and the ‘c’ term will always be the square number of the constant in the binomial (x – 2)(x + 2) = x2 – 4.

To conclude, generalising a procedural calculation for all problem types, and exploring different categories of problem types for a concept, can be incredibly valuable because pupils start spotting patterns which can increase their confidence when learning. Primarily, enables pupils to be able to identify the problem and the correct generalised procedural calculation required to find the solution. Now, I believe that pupils will learn when to apply the generalised procedural calculation to a problem type if they are shown the different problem types that can exist. Yes, for some topics there are hundreds of problem types that can be explored, but then generalising the procedural calculation is even more important; for a select few topics the practice questions can be categorised into different problem types.

I am still getting my head around it all, but would love to hear people’s thoughts.