Conception of the good

Deciding the first step is a type of question where pupils are only asked to decide explicitly which step to perform. A pupil reviews the problem type and identifies what steps he/she needs to take. After acquiring knowledge of a particular step then automatically the end of each step triggers the start of the next one. I will be looking at some questions which test whether pupils can decide on the correct step to perform, and compare them to more commonly asked questions which make the decisions for pupils.

Compare these three questions being asked by a teacher to a set of pupils

What is 4 times 2?

What two numbers do I multiply first?

What is the first step?

The correct answer to all the above questions are the same. However, the first question being asked makes the decision for the pupil because they are told that they are to multiply, and which numbers to multiply. The second question is testing whether the pupils know that we multiply the numerators, and then multiply the denominators, but the decision is being made for the pupils. It is possible that a pupil will give a wrong answer by stating two numbers that we do not multiply (4 x 11). The final question is specifically asking a question to test if a pupil knows what step to perform, compared to the first two question.

Compare these two question being asked by a teacher to a set of pupils

What do I cross-simplify first?

What is the first step?

Again, the first question makes the decision for the pupils compared to the second question. The second question is testing if a pupil can recall the first step in answering this question. This problem type is different from the one above because you can cross simplify. And I would consider cross-simplifying the first step to the procedure but simplifying the product of both fractions can happen after a pupil has multiplied the two fractions.

Compare these two questions being asked by a teacher to a set of pupils:

What is the first fraction as an improper fraction?

What is the first step?

Again, the first question makes the decision for the pupils. Furthermore, the first step of the previous two problem types cannot be applied here because there is a necessary step before both fractions can be multiplied.

The decisions pupils have to make to attempt the following calculations, vary:

The following question – “What is the first step?” not only helps pupils to learn explicitly which step to perform first but it also allows pupils to distinguish between different problem types. This is because a pupil is then attaching their knowledge of what decision to take depending on the make-up of the question. They are acquiring surface knowledge of the problem type – If I see mixed numbers when I am multiplying fractions then I must convert them to Improper fractions first.

This form of questioning allows pupils to develop mathematical reasoning around different procedural calculations. For example, a pupil recognises that to multiply mixed numbers we must convert them into improper fractions because we cannot multiply 1 with 1 and 4/5 with 2/11 because 1 and 4/5 is not 1 x 4/5 but 1 + 4/5 which equals to 9/5. This also consolidates pupils existing knowledge of mixed numbers and their equivalent forms as improper fractions.

Another example which is interesting is making decisions about the first step when comparing two negative fractions. Here are the different problem types and the first step to each one. The question posed for all the problems below is “Which is greater?”

Even though the problems may look the same to pupils, through teaching pupils explicit decision making around the problem types the pupils are doing two things. They are identifying the first step they need to take, and they are attaching their knowledge of the first step by identifying the features of each problem type.

For the most competent pupils their mathematical reasoning will make the features of each problem type and the first step they need to take seem obvious. For pupils who’s mathematical understanding isn’t as fluid they greatly benefit from being asked such questions, because they are identifying the features of each problem type which makes them different from each other, and thereby helping them know what is the first step they need to take.

Last week, I explored different number based circle theorem problems that can test (a) a pupil’s ability to identify the circle theorem being tested and (b) problem types where a pupil has to find multiple unknown angles using their circle theorem knowledge as well knowledge of basic angle facts.

In this blog, I’m displaying a few different problems within the topic of circle theorems where each angle is labelled as a variable or a term. I am interleaving lots of different knowledge:

• Forming and simplifying algebraic expressions
• Forming algebraic equations
• Equating an algebraic expression to the correct circle theorem angle fact
• Equating two algebraic expressions which represent equivalent angles and solving for the value of the unknown. Furthermore, using the value of the unknown to find the size of the angle represented by the algebraic expression.

I have interleaved fractional coefficients into a couple of questions to add some arithmetic complexity to the questions. Enjoy!

A triangle made by radii form an Isosceles triangle

The angle in a semi-circle is a right angle

The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)

Angles subtended by an arc in the same segment of a circle are equal

The questions for this circle theorem differ in nature from the problem types shown above. Here you are equating two algebraic expressions which represent equivalent angles. We are no longer forming a linear expression and equating it to an angle fact like 180^o.

The angle subtended at the centre of a circle is twice the angle subtended at the circumference

In these problems types the key mistake that a pupil may make is equating the angle subtended at the centre to the angle subtended at the circumference without doubling the angle at the circumference. This can be pre-empted by asking pupils a key question of “What is the first step?” The answer I would be looking for after going through a few worked examples would be “you need to double the angle subtended at the circumference.”

Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)

Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

I would be keen to hear any thoughts or feedback. Please don’t hesitate to email me on naveenfrizvi@hotmail.co.uk.

Last summer, I made as many different problem types for the topic of Circle Theorems. I looked through different textbooks and online resources (MEP, TES, past papers). I did this because when I last taught circle theorems at my previous school there weren’t enough questions for my pupils to get sufficient deliberate practice. This was a two fold issue. Firstly, I would find a practice set of questions which would not provide enough questions for a pupil to practise one particular problem type. Secondly, the sequencing of questions in terms of difficulty would escalate too quickly or not at all. Here I will outline the different problems types I created (using activeinspire) and then explain the thinking behind them. I have been very selective with the problems I have included here; I have made more questions where certain problems types are more complicated which I shall discuss at the end. I shall more in the following posts.

I made two different categories of problems for each circle theorem. The first type would explicitly test a pupil’s understanding of the theorem to see if they could identify the circle theorem being tested.

The second type would be testing two things. Firstly, such a problem type would be testing their ability to determine the circle theorem being applied in the question. The second aspect of the problem type would be testing related geometry knowledge interleaved which can be calculated as the secondary or primary procedure in the problem e.g. finding the exterior angle of the Isosceles triangle.

One common theme in these questions is that procedural knowledge applied is executed in a predetermined linear sequence. Hiebert and Lefevre wrote that “the only relational requirement for a procedure to run is that prescription n must know that it comes after prescription n-1.” Multi-step problems such as the ones that you will see show that procedures are hierarchically arranged so that the order of the sub procedures is relevant. Here are the different problem types for each circle theorem where I explain how many items of knowledge is being tested in each question, and what each item of knowledge is.Figure 1: A triangle made by radii form an Isosceles triangle

Figure 2: The angle in a semi-circle is a right angle

Figure 3: The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)Figure 4: Angles subtended by an arc in the same segment of a circle are equal

Figure 5: The angle subtended at the centre of a circle is twice the angle subtended at the circumferenceFigure 6: Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)Figure 7: Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

To conclude, there are many different problems types for the topic of circle theorems and the complexity of the problem can be addressed in many ways such as:

• arithmetic complexity
• Orientation of the problem
• Multiple representations of the same problem type
• multiple subprocedures to determine multiple missing angles
• Interleaving the application of multiple circle theorems.
• Interleaving the use of basic angles facts
  1. as a necessary step in the procedure to find other angles
  2. as an independent step in the procedure where finding one angle is not necessary to find another angle.

I would be keen to hear any thoughts. Please don’t hesitate to email me on naveenfrizvi@hotmail.co.uk.

In my last post I outlined how I developed pupils’ knowledge of rational numbers and simplifying surds in preparation for pupils to solve the following equations.

I asked them if we could have any number that could be squared to result in a negative number. This was the point in the session that I introduced the idea of imaginary numbers. I gave them a brief history referring to the work and discussions by Rene Descartes and Leonard Euler.

Now, “If I can square a number to get a negative result then it is an imaginary number because when we square a positive number we get a positive result, and when we square a negative number we get a positive number. This is how we display the square of an imaginary number”:

I told the kids that this is a fact that we are going to accept. Now let’s move on. “We are now going to square root both sides to see what  is equal to. We can’t square root a negative number but I am going to present it like this, and we can present it like this, again we are going to accept this for now and move on.”

These two knowledge facts are the foundation in solving the next few problems which I reiterated again and again to the kids. I then showed how we can evaluate the two calculations shown at the start where the square of an imaginary number will result in a negative integer answer. I explicitly outlined each step, one by one, and I kept each line of the algorithm between the equation and the solution consistent for each question I demonstrated.

“I am going to square root both sides, and I am going to include the positive and negative sign in front of the square root. We must include this.”

“I am going to separate √-25 as a calculation of √25 and √-1 because I can then square root positive 25.”

“I am going to evaluate √25.”

“How handy! I know that √-1 is equal to . We are using our knowledge facts which I showed you midway into our session,”

“This is my final answer.”

I demonstrated another example, and then asked pupils to attempt a selection of questions by themselves.

I then told the pupils, “We are now reaching the peak of our lesson where we are going to combine our knowledge of finding the solutions of a negative number, but this time the number that is going to be square rooted will not be a square number, this is where our knowledge of simplifying √24 and √500   will help us here.”

I modelled the next example with the following teacher instruction:

Here is my problem:

“I am going to square root both sides, include the positive and negative sign in front of the square root, -72 will be within the square root”

“I am going to separate -72 where it is shown as a product of -1 and 72, both values will be within a square root.”

“I am going to simplify the square root of 72 where I have an integer and a non rational number, because the square root of 72 is positive, no longer negative!”

“I am going to replace the square root of -1 with .

“I am going to rewrite this so i is next to the integer, you will do the same for all your answers too. We are finished.”

This is an example of a session that was delivered where the goal at the end was to have pupils being able to solve equations where they had to combine their understanding of rational numbers, and imaginary numbers. Pupils who attend this session are learning about different concepts in a succinct and limited manner to then apply their understanding to specific problem types selected by myself and the department. The questions the kids did to apply their understanding were carefully crafted by me to ensure that pupils were deliberately practising what I had modelled on the board. I think it worked quite well. If anything It made me realise that through a strong foundational understanding of number and times tables and thought out instruction, we can teach something as abstract as imaginary numbers. Furthermore, to get the delivery of the worked example to be as tight and accurate as possible I did script this lesson and rehearse it as well.

The idea of imaginary numbers is huge! There is so much that can be taught but I narrowed the focus to ensure that pupils could achieve the goal required. I have attached a few images of pupils’ work, and a video of a pupil dictating his understanding to me. I stop him because he did not simplify the surd where we had the greatest possible k integer. The pupil was correct in the method that he was using as it had two steps, but I wanted pupils to simplify the surd using the largest factor.

This post  explained how I taught pupils the procedure of solving equations where the result has an imaginary number. The video posted shows the outcome of the session where two pupils solve one equation. Enjoy the video available at this link with a selection of photos of the pupils’ work.

https://1drv.ms/f/s!AjSSrbwXqTsRhFjuw0ViPISNYBns

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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.
3. Add/subtract the numerators.

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.

Yesterday, I had the pleasure of delivering a session with Craig Jeavons at #Mathsconf7 titled ‘Implementing Academic challenge in KS3 using Nuanced Problem Types’.

Craig and I joined forces because we teach in different parts of the country, in two different schools, but also because we work in contexts expressing the same philosophy and mindset about mathematics education. We both believe that teaching knowledge in year 7 and year 8 can be made more academically challenging by creating nuanced problem types, strengthening pupils’ understanding of a concept through designing a problem set using intelligent procedural variation. We both believe that memorisation of facts and creative thinking are not mutually exclusive.

Craig and I chose a topic each, in which to create different problem types: I chose indices whereas Craig chose fractions. We chose these two topics because we wanted to select topics that some may consider mundane and standard to teach, proving to our audience that challenges can nevertheless still be presented in the teaching of both topics.

In this blog post, I shall outline the six different ideas used in implementing a challenge in teaching the topic of indices.

1. Basic factual recall

Indices, as a topic, relies on committing the knowledge of the first 15 square numbers and square roots, and the first 10 cube numbers, to memory. How can pupils commit this to memory? Ask them to try and recite it in the smallest amount of time possible? Give them a series of practice questions where they have to evaluate 9³ (figure 1).

Figure 1 – Factual Recall exercise evaluating the first 15 square numbers and first 10 cube numbers.

Figure 2 – Facutal Recall exercise evaluating the first 15 square roots and first 10 cube roots

Figure 3 – Factual Recall exercise evaluating a mixture of square roots and including cube roots from 11 cubed onwards.

Give them a similar practice set of questions using square roots and cube roots (figure 2), and why not include questions where pupils have to evaluate calculations with square roots and cube roots (figure 3). Give them more than two terms in the calculation, include all four operations (figure 4).  I like the second question, in particular, because pupils must distinguish the difference between √64 and ∛64  despite both looking visually similar (figure 5). This is deliberately designed to make them stop and think.

Figure 4 – Calculations with square and cube numbers      

Figure 5 – Calculations with square and cube roots

I wanted my pupils to learn how to evaluate 11³ because of the link between the powers of 11 and Pascal’s triangle (Figure 6). Each line of Pascal’s triangle relates to one of the powers of 11, so the first line of Pascal’s triangle is equivalent to 11⁰, and so forth. My kids loved this so much! I would ask the following:

Figure 6 – Linking the the powers of 11 to Pascal’s triangle

Me: “What is my favourite cube number?”

Pupil: “11³, Miss”

Me: Who can evaluate 11³?”

Pupil: “1331, Miss,”

Me: “What is the name of the special triangle I mentioned last week, the triangle is not a shape but the numbers form a triangular structure?”

Pupil: “Pascal’s triangle, Miss”

Me: “What is the first line of Pascal’s triangle…What is the third line of Pascal’s triangle?”

Pupil: “1…121, Miss.”

Me: “How can I express 121 using Index form?’”

Pupil: “11², Miss.”

Me: “Now, what is the 6^th line of Pascal’s triangle?”

Pupil: “15,101,051, Miss”

…so on and so forth. This type of questioning is reliant on pupils having memorised 11⁰, 11¹, 11², 11³ etc.

2.Pre-empting misconceptions

It is all well and good giving pupils the correct knowledge by using correct examples, but it is incredibly valuable giving pupils the non-examples too. Why? Pupils do incorrectly associate squaring with doubling; cubing with tripling; and square rooting with dividing, because 2² = 4 and √4 = 2. So, show them some non-examples too (figure 7 and 8):

Figure 7 – Non examples of square and cube numbers      

Figure 8 – Non examples of square roots

Have conversations around why visually similar problem types are not equivalent when evaluated (figure 9). Similarly, have conversations why visually similar problem types are equivalent when evaluated (figure 10). It does not matter the number of the root of 1, it will always equal 1.

Figure 9 and 10 – Different problems which are visually similar

3. Evaluating to the power of 0

This is an awesome piece of knowledge that the kids can learn and pick up. The best way for them to learn this is for them to identify that, regardless what the base value is, when evaluated to the power of 0 it shall equal 1. Include a base value where it is a one digit number, two digit and three digit example; make the base a decimal, variable and fraction. Craig later on mentioned that, when he was visiting schools in Shanghai, a teacher at one of the schools said “If you have taught it, then use it.” If pupils have learnt about decimals, fractions and variables, then show them that when you evaluate anything to the power of 0 the answer is 1. (figure 11)

Figure 11 – Evaluating to the power of 0 using any base value

This then allows you to make your initial practice set of questions incredibly challenging, by including complex examples such as evaluating to the power of 0, evaluating 1 to any index number, evaluating any base number to the power of 1 etc (figure 12) . You can then scatter these newly learnt problems with previously attempted problems; and you have varied the problem set where you expect pupils to still be thinking about each and every question (figure 13).

Figure 12 – Exceptional cases of Indices – because of their answers

I did add questions where the power was greater than 3, additionally, to add some more challenge to the practice set (figure 13).

Figure 13 – Practice set with intelligent varied problem types

4. Multiple representations of square rooting and cube rooting

This relates to my first point in respect of basic factual recall. I will now always teach pupils the fact that cube rooting a cube number is the same as evaluating a cube number to the power of 1/3. Show them a different example, the fifth root of 32 is equivalent to evaluating 32 to the power of 1/5. They are seeing the pattern between the number of the root and the denominator of the fractional power. (figure 14)

Figure 14 – Demonstrating the pattern between the number of the root and the denominator of the fractional power where the number being rooted is equivalent. Multiple representation of the same knowledge fact.

Why did I not start with the square root of a square number? The pattern is not obvious visually because the square root does not have the 2 visible like the third root of 27 or the fifth root of 32. Start with examples which state the pattern in an obvious and explicit manner, showing the exceptional examples at the end.

5.Complexity in structure rather than content

Now, if I return to the initial practice set of questions, which includes calculations with square roots and cube roots (figure 15), I can use essentially the same questions but only now using a different set of visual representations of the same problem (figure 16). I have just made the teaching of the concept more difficult by making the structure of the problem more complex. The content here is the same as the previous exercise when I showed calculations with square roots and cube roots. The content is also the same because it relies on the same factual knowledge I expected pupils to memorise in the initial stages of teaching the topic.

Figure 15 – Initial problems using square roots and cube roots in calculations

Figure 16 – Same problem types from figure 16 but using a different visual representation where square rooting of 49 is displayed as evaluating 49 to a power of 1/2.

Remember, in the previous image where I showed that any number to the root of 1 will result in an answer of 1. I can now present students with the last problem type where 1 has a power of 1/100 (last problem in figure 16). Each questions relies on the same factual knowledge but whereas this is displayed differently – visually –  and this is challenge I am referring to because it is still difficult for students to identify the structure of the problem and make the link that 343^1/3 is equal to 7 and that 1^1/100 is equal to 1. I love it!

6.Partial knowledge recall

We have asked pupils to attempt a full problem, such as ‘evaluate 5³’, and to then determine its answer. Now, give them an incomplete problem with the answer given. Can they fill in the blanks? This is extremely difficult because they can no longer rely on their factual knowledge to determine the answer; they must instead manipulate such factual knowledge to complete the problem using the answer. Can they identify the missing index number? Can they identify the missing base number? (figure 17)

Figure 17 – Problem types requesting partial knowledge factual recall

Figure 18 – Problem types where more than one part of the problem or answer is missing.

Figure 19 – Problem types requesting partial knowledge factual recall using square and cube roots. 

Can they identify the missing base number and index number in the same problem?  What about the fourth problem type? Even better, what about the last problem type? (last two problems in figure 18)

Pupils will be thinking, what power can I evaluate 31 with which gets me an answer smaller than the actual base number? 0! How do pupils know this? They know the fact that if you evaluate any base value to the power of 0 we always get 1. Let’s include examples with missing digits in the square root? Here the question marks represent the same digit. What about the last example – the only possibility is that both the question marks are 1. (figure 19)

Give them a practice set of questions where they have to attempt filling in the blanks in not only the problem but in the answer too. (figure 20)

Figure 20 – Partial knowledge recall exercise. 

And so, that was a whistle stop tour on how teaching the topic of indices can be made more challenging, and in particular in respect of year 7 and year 8 classes, as presented at #mathsconf7. These six different ideas allow children to consolidate their knowledge of different facts about square numbers, cube numbers, square roots and cube numbers, through the six different ideas discussed above.

I have attached a PDF of the powerpoint presented at La Salle via a link below.

https://www.dropbox.com/s/l73m3sc87nuos77/PresentationLaSalle%23mathsconf7%20Naveen%20and%20Craig%20.pdf?dl=0]

If there are any questions then please do not hesitate to get in contact with either Craig or myself via Twitter (@naveenfrizvi and @craigos87) or via email (naveenfrizvi@hotmail.co.uk). Enjoy!

Thank you to everybody who took the time out of their weekend to attend mine and Craig’s workshop; we really appreciate your time and enthusiasm. Thank you to La Salle who hosted another great conference – you guys are awesome!

Whilst teaching, I asked a question to my  class after delivering a worked example listing all the factors of a number. Specifically, I was demonstrating that the number of factors of a square number will be odd because we have a repeated root, and therefore that we don’t write it twice. For example, 16 has the following factors 1, 2, 4, 8 and 16, whilst 36 has the following factors 1, 2, 3, 4, 6, 9, 12, 18 and 36. Also mentioning a non-example such as 15 having an even number of factors, because it is not a square number with a repeated root, and so 15 has the following number of factors 1, 3, 5 and 15 etc.

After the worked examples were delivered and pupils had completed a selection of questions in the practice set, I posed the question to check if pupils had remembered the fact taught: “Why do 16, 36, 81 and 144 have an odd number of factors?”

Pupils were given 20 seconds thinking time, and I could see hands all going up in anticipation to answer the question. I selected a pupil and this was her response:

“When you square root a square number you get the root number twice, so the root number is repeated, and that is why a square number will always have an odd number of factors when you write a list of factors.”

I responded “Incorrect,” and all the pupils in the class looked at me stunned, not because I said Incorrect, but because they thought that what *Sally had said was correct. Now, I know what Sally meant but that is not what was said. I then corrected her: “Let’s clarify that when we find the square root of a square number we get one value which is the root number. Therefore, √16 = 4. What *Sally is trying to say is that when you list the factors of a square number, like 16, you get a repeated root because within that list of calculations we have 4 x 4, and because it is repeated we only state one 4 as one factor out of many.”

As written in Doug Lemov’s Teach Like a Champion:

Right is Right starts with a reflection that it’s our job to set a high standard for answers in our classrooms and that we should strive to only call ‘right’ or ‘correct’ that which is really and truly worth those terms.

Many readers may be thinking well actually that is pretty obvious. However, it has its challenges. If I said to Sally “Well, you are nearly there, or halfway” then I would be doing her a disservice. Why? Because what she said was incorrect. It was not mathematically accurate, and I know that if I let that misinterpretation hang around in the air then all pupils in the room will develop a misconception around it. As a teacher, children believe that everything we teach them is correct, and if I allow slightly incorrect answers to seem ok, even though I knew what Sally meant, then my standards of Sally and her peers would not be high enough. As Lemov states in his field notes, “teachers are not neutral observers of our own classrooms,”— it is simply the thought that I know what the pupil means when they say something inaccurately that resonates the most with me.

For example, whilst marking an assessment a pupil wrote the unit for a compound area question in cm² instead of m² where metres was the unit used in the question for each length: I marked her answer incorrect. There are many debates on this and I do understand that this is not a calculation error but maybe an error in reading the question, or in stating the correct units, but nonetheless she did not write the correct units for the question. It is not right. By holding pupils to account, you are striving to equate the term ‘right’ with ‘correct’.

As Lemov has stated in his field notes, there are many caveats posed in implementing the strategy of ‘Right is Right’ in the classroom. He mentions the problem of time. That to fix *Sally’s mistake I needed to spend more time than planned in my lesson to correct her, but this is an investment that will be appreciated later on when a question such as this arises in a high stake exam.

Secondly, pupils who are shy or timid may become discouraged in putting their hand ups ever again because they made a mistake. However, I think that comes down to the culture you have in your classroom. Lemov talks about having back pocket phrases for moments like this, and here is the one I use frequently: “Thank you Sally for letting us all learn from your contribution, because of you, you have learnt so much more and so has everybody else.” Then at Michaela, we would give an appreciation with two claps to follow for Sally.

Right is Right is a challenging strategy to implement in the classroom. However, I wholeheartedly believe that it enables pupils to raise the standards of what they can achieve, and, for teachers, it ensures that expectations of what pupils can achieve also remain high.

*The child who I am referring to has been referred to as Sally, and not by her real name.

I remember marking books; I didn’t find it very useful. However, I found marking exit tickets incredibly informative because I was able to act on the assessment of pupils’ learning the very next day. Compare this to marking books every two weeks on work that I would be giving feedback to pupils who completed this work two weeks ago…and I am giving them feedback too late. Does anybody else see the elephant in the room?

I do think that marking books is a time consuming task which requires a great deal of input from teachers for a nominal amount of benefit for pupils. For this reason, we have a no marking policy at Michaela. What we do is mark our weekly quizzes. For each subject, let’s say pupils are given their maths homework on Friday, they will have a quiz on content that has been taught in the past week, and that has been given as homework on the following Monday. This is a whole school policy. It works in science, as I have observed as a teacher. Similarly, it works with all our other subjects. It is incredibly effective. It gives me an idea of how much pupils have learnt, and how much knowledge they have retained. More importantly, I can act on the feedback within my teaching in a timely fashion, and not two weeks later.  

On several occasions, I have been asked how pupils are assessed at Michaela. This post will go into how we assess pupils using low stake weekly quizzes; we then have bi-annual assessments which measure how much pupils have learned and retained over a large period of time.

Every week, pupils are given a low-stake quiz testing how much of what has been taught in the week before it has been mastered and committed to memory. At Michaela, we believe that if it hasn’t been committed to memory then it hasn’t been learned. I cannot reiterate how much I believe in this, and indeed even more so after starting at Michaela from September 2015.

On Friday, pupils will be given their maths self quizzing homework. Joe Kirby goes into detail here about self quizzing in a previous post of his. Pupils will then have a quiz on Monday testing them on how much of their self-quizzing on maths definitions has been committed to memory, and whether the procedural calculations learned the week prior have been mastered.

Before I go into how the quiz is made it is really important to decide as a faculty what the purpose behind the assessment is. Is there any benefit in the quiz that I have made? What is the intention behind this low stake quiz? Our low-stake quizzes are testing whether pupils have committed the knowledge from their self quizzing to memory, and whether the procedural calculations have been mastered. Knowing such a purpose, this guides the structure and content of the quiz.

Each quiz has 8 – 12 questions testing pupils on procedures that have been taught in the previous week. We do not test pupils on content that has not been taught. I repeat, we do not test pupils on content that has not been taught. Why? This is because we are testing whether pupils have mastered the content that has been taught, and that means that the sample of knowledge that is being tested is small, rather than large.

Quiz 1:

Here is a year 7 quiz testing pupils on their week’s worth of teaching on short division. Each question selected is testing pupils on a specific skill:

1) Short division where there is no remainder – but with one digit where it is smaller than the divisor. For example, when dividing 8420 by 4, 2 is too small therefore the digit will be 0, and then carry the 2 as the remainder to the next digit to become 20.

2) Short division where there are no remainders, and each digit in the dividend is greater than the divisor.

3) and

4) The first digit in the dividend is smaller than the divisor. Also, stating the remainder as a fraction, where the remainder is the numerator and the divisor is the denominator. Also writing the answer where the remainder is a fraction and a decimal.

5) Short division where there are remainders that need to be carried to the next digit. The answer is perfectly divisible by the divisor.

6) Short division where there are remainders that need to be carried to the next digit. The answer has a remainder which needs to be displayed as a fraction as well as a decimal.

7) Short division of a decimal (less than 1) where there is no remainder

8) Short division of a decimal (integer and decimal) where there is no remainder

9) Short division of a decimal where pupils must write their answer in decimal format. They have to put additional zeros to continue the decimal to allow the remainder to be carried.

10) Short division of a decimal where pupils have to put several zeros to continue the decimal to complete the division.

11) Question where pupils have to identify the closest square number to the dividend and identify that the divisor and the integer as a result will be the same.

Quiz 2:

Here is a year 7 quiz testing pupils on their week’s worth of teaching on GEMS (Groups, Exponents, Multiplication (and division) and Subtraction (and addition). We teach GEMS as opposed to BIDMAS or BODMAS. Each question selected is testing pupils on a specific skill:

1) evaluate exponents before addition (GEMS)

2) multiplication comes before addition where addition is visually first in the question (GEMS)

3) two groups of multiplication and division where they come before addition. Must identify both and then add. (GEMS)

4) multiplication comes before addition where division is visually first in the question (GEMS)

5) Another group similar to Q3 – additional but not necessary

6) Multiplication comes before addition but can they identify that the division will result in a fraction which can be added to 21. (GEMS)

7) Practise left to right when we only have addition and subtraction operations because they are equal in GEMS.

8) More complex GEMS question because of a mixture of operations (GEMS)

9) Can pupils identify whether they can apply GEMS correctly where we have exponents, and then go left to right because of multiplication and division being equal in GEMS.

Quiz 3:

Here is a year 8 quiz testing pupils on their week’s worth of teaching on the topics: formulae, rearranging equations to change the subject of an equation.

Q1 + 2) Mastery on deciphering whether the length given is the radius/diameter, and identifying whether we must substitute into the formulae for area or circumference, and whether they can recall the correct formulae for either concept.

3) Calculate the area of a trapezium given the slant height and perpendicular height. Shape is also orientated. Can they distinguish which is which, and which length must be substituted into the formulae? Furthermore, can they recall the formulae memorised?

4) Mastery of understanding how to substitute into the formula for the volume of a cone, and recall the formula too. Deliberately radius is given to ensure that pupils are being tested on whether they can correctly substitute into the formula.

5) Substituting into the equation M = DxV but must identify that the volume is not given. Can they calculate the volume of a cube first, and then find the mass?

6) Substituting into the equation D = SxT but must identify that time is given in minutes and must be converted into hours. Also, that the speed is given in kilometres instead of miles.

7) Calculate the volume of half a sphere given the radius. Can pupils identify that they must half the result after using the formula or use the formula 4/6pi(r) ^3 or 2/3pi(r)3?

8) Rearrange the subject of the equation where we must expand the brackets. Implied in the question by stating “simplify your answer fully” hoping to see the result b² + 32b + 256.

9) Rearrange the subject where the unknown is the denominator.

10) Rearrange the subject for basic one step rearrangements besides the last one where the unknown is negative. Can pupils spot that the we must multiply or divide both sides by -1 to get m as the subject of the equation.

The quizzes are testing whether pupils have mastered the procedural questions. To make questions challenging we have given the diameter instead of the radius when calculating the area. We have orientated the trapezium, and given both the perpendicular and slant height. We have made it more challenging because we are testing mastery. The rigour in the assessment allows for the rigour in the teaching, and pupils do indeed perform. We spend a significant amount of time at Michaela talking about how the only score to celebrate is 100%. I write postcards for the pupils that get 100% in their quizzes and make a huge deal out of it. Pupils love the feeling of success, and appreciate the admiration from their teachers who recognise the success in scoring 100%.

Low stake quizzes are incredibly powerful because they inform your teaching and planning. If lots of pupils have made a mistake on the same question, this informs me that either they have all missed the point, that I have to reevaluate how I teach the concept in the first place, or both of these points. I hope you find this useful.

This post will be outlining the academic rigour I have seen in the teaching and in the assessments created by the Uncommon Schools Network. I shall explain how we can develop such rigour in our classrooms as individual teachers.

Each grade (year group) has an interim assessment exam which is determined centrally by a subject lead teacher; although collaboratively made, a subject lead teacher will finalise the exam produced. Once it is finalised, all teachers within the network will have access to view the assessment but will not have the access to make any amendments/alterations.

I was very impressed by the interim assessments made, because they are created in a way in which pupils’ mathematical knowledge is tested in a manner where they are forced to manipulate the knowledge they have: it tests and masters the fine balance between procedural fluency and conceptual understanding in each question. It also made me reconsider how I plan my teaching, how I now combine concepts in different problem types and how I try to make rigourous low-stake quizzes including questions like the ones you are about to see.

I am going to look at one question which I thought was brilliant, then explain how I would plan a selection of lessons, in a sequence, inducing pupils to become mathematically proficient and confident in order to attempt such a question.

Here I am assuming by convention from how I was trained and from what I have seen in my experience, previously I would begin by teaching the concept of square numbers and square roots, by getting children to identify the square numbers by multiplying the root number by itself, and then drawing the link: that once you square root a square number you will obtain the initial root number which, multiplied by itself, then results in that square number.

Then, I would have possibly explored completing calculations with square numbers and square roots. After that, in years to come, I would teach square numbers and square rooting in relation to fractional indices and with surds. When teaching surds, I would bring in the idea of rational and irrational numbers. Here is what I now suggest!

This is how I would map out teaching the topic of square numbers, square roots and holding a root number, to an index number greater than 2, and introducing rooting greater than a square root. Kids must have gotten to a stage where such calculations are memorised:

Why not bring in the idea of interleaving fractions, decimals etc?

Then, I would build upon the previous questions systematically, in a way in which students can create a fraction which can be simplified.

Also, I would introduce from an early stage the proposition that when we square root a square number we get positive and negative of the root number, which can be a really valuable discussion to have with children. Introduce calculations afterwards with square numbers and square rooting. This type of thinking is driven by another question I saw in a North Start Academy assessment:

The question is touching on key knowledge that when x is equal to a square root then we assume a positive value as an answer, however when x is held to an exponent the values of x will result in a positive and negative value.

Eventually I would introduce a base number to an exponent greater than 2 and interleave it into calculations:

I could get pupils to then go into the following questions:

Or, have pupils determine the value of unknowns through trial and error:

Now, I am going to draw a link between square numbers, square roots and rational numbers without linking it to fractional indices and surds. The purpose behind this is to (for now) restrict what I would teach at Year 8 or 9 but, at the same time, providing that conceptual scope. Simultaneously, it also provides the opportunity for multiple choice questions that have the academic rigour for our pupils.

I would define a rational number and then go into the necessary vocabulary such as ‘a rational number is a number which can be written as a fraction of two integers. A rational number cannot be written as a non-repeating or non-terminating decimal’, for example, in comparison to 0.45454545 which is a rational number. I would then provide a selection of example and non-examples.

I could test children’s understanding through mini whiteboards or hands up: 1 if it is a rational number and 2 if it is an irrational number; selecting the order of questions such as 2, 6, 100, 15, 1/2 , 5/6, 10/100, then bringing in the square numbers and asking pupils to show why it is a rational number or irrational number such as √16, √36 or √144

Pupils could type in the value of √12 or √3  in a calculator to see that you will get a non-repeating or non-terminating decimal that cannot be written as a fraction of two integers therefore categorising such non examples as irrational numbers.

Also then mentioning that square rooting a square number will result in a rational number, and cube rooting a cube number will result in a rational number etc.

You would then be able to explore the idea of rational numbers in which you are square rooting square numbers using your calculator:

Following this, discussion will be made of the notion that the following are rational numbers in regard to the definition of a rational number being a number that can be written as a fraction made of two integers:

Then, the idea of either the numerator or denominator being a whole number should be brought into my teaching:
After that, I would introduce problem types where you can simplify the fraction:

Discuss that the problem below results in an irrational number because the numerator is irrational and overall the solution cannot be written as a fraction of two integers:

Will the following number sentences produce rational numbers?

Children need to be comfortable in identifying that  is not a rational number because it cannot be displayed as a fraction of two integers. The definitions need to be consistent so that children are confident in differentiating between a rational and irrational number using square numbers and square roots, and then moving onto powers greater than 2. If we expose children to the following above we are providing the academic rigour which future exam boards are attempting to build upon.

All of this is just an idea which I am trying to explore in order to provide students with the academic rigour, in terms of their classwork and in the assessments, that they will be taking.

Everything that I have written, the problem types I have created and the interconnections between square numbers, square rooting, and rational numbers, is a result of looking at such questions above. These questions guided me to plan a selection of lessons which provide the balance between enabling students to become procedurally fluent in calculations and in developing conceptual understanding of this topic.

Through creating difficult assessments you can guide your teaching to induce that academic rigour. I have never taught maths like this, but I do plan to in the future. This was all inspired through looking at this question at North Star Academy in New Jersey. The academic rigour present in the interim assessment guides to create academic rigour in what is being taught and learnt within the classroom.