Conception of the good

Insights into our current education system

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Engelmann Insights: Structuring Teaching for the Weakest Pupils (Part 2)

This blog post is the second in a selection of posts outlining the teaching and sequencing of various fractions skills taught in the Connecting Maths Concept Textbook series. Specifically, Level D. The first blog post can be found here. The following content was shown at La Salle’s National Mathematics Conference in Kettering.

Breakdown of each teaching point

5. Simplifying a fraction to an integer

This is commonly taught. What made it so effective in Engelmann’s textbooks was the regular times table exercises (multiplication and division) which were included in every lesson.

Around Lesson 59, pupils were asked to simplify a fraction where short division would be required. Again, this was introduced only after pupils had learnt how to divide using short division.

Similar exercises were given where pupils were only asked to write the division problem before working out the problem. This was to ensure pupils were avoiding the common misconception of writing a division calculation were the divisor is written before the dividend.

6. Stating whether a fraction is an integer or not an integer

In this exercise pupils are asked to state whether a fraction will simplify to an integer, or will it not simplify to an integer. Note that pupils are stating a term for the positive case, and the ‘not’ term for the negative case. The language of mixed number is not introduced. This is because pupils will have to learn two separate terms for two positive cases. When you learn what something is and what it isn’t then you are learning where one concept is true in one instance and when it is false, and the case for when it is not true is the ‘not’ case. It is much easier for pupils to grasp than introducing two different terms for two positive cases.

7. Adding or subtracting fractions with like denominators

Pupils are taught that when the denominators of a fraction are the same then you can add the fractions in they are written. It is explained further that each fraction has each unit divided into the same number of parts, and this is why we can add fractions when they are written this way.

You can’t add these fractions the way they are written because the denominators are not the same.

The most important part of the wording is ‘the way they are written’. Pupils are taught to visually spot when it is possible to add fractions by spotting the denominators being the same. Pupils are then given exercises to decide whether you can add these fractions the way they are written.

Engelmann does go into the reasoning behind why fractions with different denominators cannot be added or subtracted in the way they are written.



8. Listing multiple equivalent fractions for a specific integer

Since pupils practice their times tables in an exercise in each or every other lesson within the textbook series, pupils can quickly learn how to write multiple equivalent fractions for an integer. Pupils learn the multiplicative relationship between the integer and the denominator of an equation. Including ‘1’ as the denominator is important because I think it is sometimes overlooked.

9. Writing fractions from a sentence

Engelmann creates questions which force pupils to deliberately think. Here is a perfect example: pupils are told to write a fraction which meets certain conditions, such as the fraction being more than or less than one. Pupils are also using prior learning in this instance.

Here are a few examples

10. Introducing Mixed numbers 

Mixed numbers are introduced as a sum between an integer and a (proper) fraction.

It is visually presented on a number line, you go to the marker for the whole number. Then you count parts for the fraction.

The skill is revisited where pupils write the addition sum between an integer and a fraction as a mixed number, without the number line. Pupils are then asked to do this again with three or four digit numbers.


Here is an example of a pupil’s work.


In my next blog post, I will  outline the remaining fraction skills that are taught in Engelmann’s sequence of lessons.

Engelmann Insights: Structuring Teaching for the Weakest Pupils (Part 1)

Engelmann’s Connecting Maths Concept Textbook series teaches concepts that all future mathematical study relies upon. For example: the four operations, the relationship between addition and subtraction as well as the relationship between multiplication and division, fractions to simply name a few. In this blog post I will be looking at how the topic of fractions is taught within the first 100 lessons of Engelmann’s CMC textbook series (Level D). I will go through a number of knowledge facts or skills that Engelmann outlines.

Connecting Maths Concept Textbook Series

The CMC textbook series have multiple levels to select from to best suit the children who will be learning from the CMC textbook. I taught the Level D series. The textbooks served the purpose of acting as a remedial programme where pupils received 3-5 one-hour lessons per week on top of their mainstream maths lessons. There is a scripted teacher presentation book, a pupil workbook, pupil textbook and an answer key.

One note to mention on the scripted teacher presentation book. Many teachers do not like being given a teacher script because it seems unusual to teach using somebody else’s words. I felt this way initially. I then completely changed my mindset on this because the script was incredibly accurate, deliberate and economical with text. I realised that Engelmann has written it better than anything else I had seen. I found it to be a really humbling process because it made me realise that I had been very verbose with my teacher talk. I could say much more with fewer words. This is still a controversial topic in the Education world. However, I found that the script helped the children to articulate their understanding because they were able to repeat a precise explanation back to me, that was simply provided through the teacher script. This was something that @MrBlachford commented on too, on Twitter:

A few overarching points

The key theme that Engelmann demonstrates throughout his textbook series is using:

teaching methods of future learning which never contradicts the teaching methods used for prior learning. 

There is total consistency throughout the teaching methods used and applied when learning a specific concept. This will be shown by outlining the teaching of 20 skills on the topic of fractions over a 100 lesson period.  There is also a smooth track of gradual complexity in the problem types that are used to test a pupil’s understanding. The content that is taught looks deceivingly easy but pupils are taught teaching methods that can be applied in the most simplest of cases as well as the most complex of cases.

Engelmann’s lesson spread looks like this. The red sections are the teacher-led parts and the green sections are the parts where pupils work independently. Each lesson covers a range of 5-6 different topics such as adding, subtracting, fractions, times tables, ratio etc.

Breakdown of each teaching point

  1. Stating a fraction from a diagram 

Engelmann introduces how to write a fraction from a diagram by using images where there was more than one unit, as shown below:

The wording to write a fraction was:

The top number is the total number of shaded pieces.

The bottom number is the total number of pieces in one unit. 

The wording was precise, and fool-proof. I did alter the working to ‘the bottom number is the total number of pieces in each unit’ simply because my pupils responded to that with more accuracy. The wording applied to cases where pupils were writing proper fractions, improper fractions as well as whole numbers. The questions that were selected which tested this also:

The wording also avoided the common misconception that pupils have in writing the denominator as the total number of pieces that they can see between the number of shapes shown. This is because the wording say ‘in one unit’.

2. Stating a fraction from a number line

The same wording was introduced. I did explain to pupils that one unit was the gap between each whole number. 

There were exercises where pupils were practising both skill #1 and #2 side by side.

3. Stating whole numbers from a diagram or a number line

If you see a diagram or a number line where a certain number of units are shaded then the diagram or number line represents a whole number. The point is also mentioned that if there are no leftover parts are shaded then we definitely have a whole number. The opposite case is presented that if we do have leftover parts that are shaded then we don’t have a whole number.

The scripted nature of it states: “3 units are shaded. So the fraction for this picture equals 3.”

Similarly, there is a negative example of this where the script words what the teacher says as: “There are more than 3 units shaded, but less than four.”

The testing stage phrased the question as: which picture shows a whole number?

4. Fractions as integers from a diagram

For it to be the case that we are showing an integer visually then there is only one part in each unit, meaning that each unit is one whole shape. The number of shaded units is the top number. The number of parts in each unit is the bottom number. This method is consistent with previous problem types of stating a fraction from a diagram or number line. What I like the most about the sequence of questions is that the nuanced example is taught after the explicit examples. #4 is the nuanced example of stating a fraction from a diagram because its not as straightforward. Engelmann’s presentation book still uses the same wording to state a fraction from a diagram, which shows pupil that they are equipped to answer even the trickest of questions.

The wording to teach pupils how to state a fraction as an integer from a diagram goes as follows:

There is one part in each unit. 4 parts are shaded. The fraction for the picture is 4 over 1. Here is the equation: 4/1 = 4.

There is one part in each unit. 6 parts are shaded. The fraction of the picture is 6 over 1. Here is the equation: 6/1 = 6

I remember one of the boys in my intervention class, looking at the question and attempting it before I started going through it. He did say the correct answer, and he was really impressed with himself. I then asked him how he did it. His response was ‘using the way that you taught me’. He was able to apply the method taught for previous problem types in this case too with no teacher guidance.

Pupils were then asked to state a fraction as an integer from a number line as well.

Pupils completed testing exercises where they had to state a fraction which could simplify to an integer where each unit had more than one part. These exercises also tested a pupil’s prior learning alongside new learning content.
The most difficult problem testing skill #4 was asking pupils to write the fraction for each image that shows a whole number. Pupils are now applying collection of separate component skills where pupils are to: read a number line, write a fraction from the number line, and state whether that fraction is a whole number. 

The next degree of complexity introduced where now pupils need to state the fraction for each whole number on a number line.

Here is the wording used in the textbook:

Here is a number line:

There are three parts in each unit. So the bottom number of each fraction is 3. The bottom number for each fraction is the same:

The top numbers are the numbers for counting:

There are exercises further on in the series of lesson where pupils are given the same task but the number line doesn’t state the number of parts in each unit, instead the denominator is provided.

In my next blog post, I will continue to outline the remaining fraction skills that are taught in Engelmann’s sequence of lessons.

Resourcing: Applying Variation Theory

Teachers spend hours creating their own resources for their pupils to use after a concept has been taught. Over the years, I have made lots of resources which I change every year, simply because I think this exercise won’t help this set of pupils grasp the concept being taught. I’m now at a point where I can make an exercise of questions which all pupils can use to learn the intended concept. What has changed for me? I have developed a better understanding and can now apply principles of variation theory when creating resources.  This has allowed me to use the same practice exercise with all my classes.

Variation Theory

Variation Theory is posited on the view that “when certain aspects of a phenomenon vary which its other aspects are kept constant, those aspects vary are discerned.” (Lo, Chik & Pang 2006:3) It has been previously pointed out that “to see or experience a concept in a certain way requires the learner to be aware of certain features which are critical to be the intended way of seeing this concept.” (Lo and Marton 2012) Here is an example, I am attempting to teach pupils how to factorise a quadratic expression. The critical aspect I want pupils to focus on is determining the factors of ‘c’ that sum to give the ‘b’ coefficient. I have kept the ‘a’ coefficient the same, and I have kept the ‘c’ constant the same too. I want pupils to work on the aspect that the ‘b’ coefficient is always the sum of two factors of the constant. Here is a similar example:

Here is another example, but here I am focusing pupils’ attention on the critical aspect of determining the factors that multiply to give the ‘c’ constant. The factors will always sum to 10 which is the ‘b’ coefficient.

I have deliberately included the last example. Where pupils are taught that two numbers that add to give 10, but the product will be 0. I have included it at the end because it is a nuanced example of the set of expressions above. It is nuanced because its explicit features are not visible for pupils. It does not follow the structure of (a + __)(a + __) because the bracket of (a + 0) = a

What about introducing a practice exercise where you want pupils to focus on their negative mental arithmetic? The critical aspect for pupils to focus on is determining the factors of -24 that sum to give the ‘b’ coefficient stated.

What I really like about the above exercises is that pupils are experiencing different forms of variation from the previous set of expressions. However, in this practice exercise, I have changed only one aspect of each expression, which is the focal point of the intended concept being taught. If I change both the ‘c’ and ‘b’ coefficients of a quadratic expression then I am asking pupils to make two decisions, essentially to focus on two different things, two numbers that sum to ‘b’ and the same two numbers that multiply to ‘c’. This above exercises constrains a pupil’s thinking deliberately for two expressions at a time so the intended process of factorising a quadratic expression becomes clear. They can also see how the negative arithmetic changes the value of the ‘b’ coefficient. 

I can now bring a practice exercise where pupils are now able to put their understanding of how to factorise a quadratic expression to the test:

A few key points about the design of this exercise set:

  1. I have only changed one aspect of each expression from the previous expression.
  2. I have focused on changing the surface of each expression. The procedural process is the same.
  3. Between (b) and (c) I have deliberately changed the focus from finding the sum of the constant’s factors to then finding the product of two factors of 12 that also give a sum of the ‘b’ coefficient.
  4. I have structured the calculation for the coefficients of ‘b’ and ‘c’ because I want pupils to explicitly see how they are determining the numbers that go into the factorised form. I wanted pupils to see the depth underneath the surface of each quadratic expression. I also want pupils to see the procedural relationship between the factorised form and its quadratic expression. 

To summarise, this is simply an attempt to widely implement variation theory when resourcing. I have seen that pupils are able to draw the intended mathematical understanding being taught sooner. A greater proportion of the class are able to apply their understanding of the intended concept correctly. The design of this exercise also aids pupils to deliberately think and apply their understanding at each expression they had to factorise. The above exercise is an improvement from previous exercises I have made in the past, and can be improved further after seeing how pupils respond to it.

Paper used:

TLAC Show call: Using Mini-whiteboards in the Maths Classroom

Show call: Using Mini-whiteboards in the Maths Classroom

The use of mini-whiteboards in the classroom can go either way. It can be a really great resource to check pupil understanding, identify and close knowledge gaps etc. Or it can also be a nightmare unless pupils are taught how to use mini-whiteboards effectively in the classroom. This has been blogged about previously, which can be found here.

I use mini-whiteboards because it is the most effective way of using a teaching technique known as Show Call¸ from Doug Lemov’s, Teach Like a Champion 2.0 (TLAC).

**Craig Barton’s Podcast with Doug Lemov goes into further detail about Show Call. Highly recommend it.**

What is Show Call?

From TLAC Show Call is defined as a teaching technique which:

“Create(s) a strong incentive to complete writing with quality and thoughtfulness by publicly showcasing and revising student writing – regardless of who volunteers to share.”

Here is the context: I pose a question which is projected on the screen using a visualiser which I want pupils to complete. I have provided an exemplary worked example, stating all the steps to go from the question to the answer. I have emphasised the areas of the working out which pupils will struggle with or most likely make a mistake at a specific point. I have demonstrated a piece of quality written work that I want the pupils to aspire to. I then give a similar question for pupils to try on their mini-whiteboards.

At this point, pupils are scribbling away on their mini-whiteboard, I am circulating the room or watching pupils writing on their boards from the front of the classroom. I am looking for potential mistakes pupils are making that I may not have pre-empted, or spotting pupils who are greatly struggling or who are finishing of an exemplary answer, like the one projected on the screen.

I start the count-down. 10 seconds left. Start counting down from 5…4…3…2…1…and show. Pupils simultaneously show their whiteboards.

What does Show Call achieve in this instance?

Pupils are given an opportunity to put their initial thoughts and understanding of the concept being taught on a mini-whiteboard for me to then suggest improvements. The public showcasing incentivises pupils to produce quality written work.

I show off top-quality work which other pupils can see and aspire to. This creates an environment of success and positivity balanced with an expectation that pupils need to develop and improve their work if what they have produced isn’t parallel to what is being shown. The act of finding exemplary work is fully intentional.

Similarly, it gives me the teacher an opportunity to take a pupil’s piece of work which may present a mistake, or a common mistake that many pupils have made, or an area of improvement and start a conversation giving pupils precise and actionable feedback. The ability to give an entire class specific edits or points of revision empowers pupils to be constantly learning and improving their written work. At this point, you see pupils thinking, their eyes following the calculation on the board, and you see their eyes light up with a smile of satisfaction that they now know how they can go from “good to great”. Pupils are coming closer to making their work like the exemplary piece of work that has been celebrated. The process is efficient and productive.

What type of revisions and edits is Show Call trying to achieve?

When I started to use this technique in my second year of teaching, I didn’t use it correctly because I was focused on getting pupils to correct one-time mistakes rather than revising the structure of their work, or the content of their work and/or focusing pupils to revise a replicable skill.

The one-time mistakes can be pre-empted by using very good worked examples where a teacher goes through them and points out possible mistakes that pupils may make. These are mistakes that pupils need to avoid when attempting similar questions on their mini-whiteboards. The revisions that I needed to get pupils to get in the habit of could all be categorised under the umbrella of ‘applying a replicable skill to multiple concepts’. E.g. when using the four operations when working with fractions, to always place a 1 in the denominator of a whole number. Or when expanding and simplifying an expression like the following. Being able to underline each expression with the number and sign which will multiply the entire expression in the bracket. Then drawing each table where we write -3 rather than 3. This is because pupils need to multiply each term of the expression by -3. (Image 1)

OR drawing a ‘boat’ around the two single brackets, including the sign on the left. This is not the case in the first image, but then achieved in the second image with a different example.  (Image 2)


Image 1 and 2

A constant push on making such revisions motivates a pupil to actively listen to the teacher and implement the feedback provided. Lemov calls the act of reviewing work, as seen by champion teachers, as using the language of “check or change”. Time does need to be provided for such revisions to be made, but with a sense of urgency so pupils are going at a productive pace. I have seen instances where pupils are taking ages because they lack that sense of urgency. It usually is the case that with a reminder that you have a minute left, along with a countdown then pupils pick up the pace.

The Take and Reveal

Lemov goes into detail about how the deliberate act of taking a pupil’s work and the reveal of the pupil’s work can increase the impact of Show call.

For example, when circulating the room, I’ll say “I’m looking forward to picking the best piece of board work to show to the class so we can learn from it.” Pupils then feel the desire for that piece of work to be theirs. Once I say, 3…2…1…and show! I will look across the room, and ask a pupil if I can show their piece of work so we can all learn from it. Sometimes, I’ll take a pupil’s board work without saying anything, in TLAC this is referred to as an “unnarrated take”. Giving pupils a sense that this is a normal thing to take the best piece of work to show to the class. Usually, I’m grinning from ear to ear over the best piece of work, and so is the pupil whose board I pick.

If I see a mini-whiteboard that demonstrates a common mistake, I may take the board without mentioning the pupil’s name, or I may ask the pupil “Do you mind if I can take your board to share a common mistake that some others have made, so we can learn as a class to make everybody’s work better?” I could hold the board up and say “Can anybody tell me how this piece of work can be improved?”

Image 3 – In this piece of work, the pupil’s working out is correct. The final expression is incorrect. They have identified that they need to sum the values of 4 and 2, but they have written +6c rather than -6c.

Image 4 – Another pupil in the class made a similar mistake as demonstrated in Image 3. They were able to make the required revision in another question.

I then asked how the following answer could be written in a more elegant way, implying that we do not need to show the coefficient of b, as 1.

At this point, it is important to mention that sometimes this act can be pointless because the mistake may be too difficult to spot or to even understand, that it may make sense for you to go through it. Sometimes, I think it is powerful because it gets pupils to deliberately spot a mistake. The act of spotting a mistake and changing it is a form of learning, an effective one. At this point, I’ll see a couple of hands fly up in the air and then slowly the number of hands in the air will increase.

What are the long-term benefits of Show call?


The main benefit I have found is that I can see pupils checking and/or changing their written work to the point of complete accuracy. This then results in pupils completing written work in their books to be thoughtful and well-constructed work. I am not drowning in marking work with multiple mistakes that could have be corrected through the use of mini-whiteboards. The quality of pupils’ work in their books, then in their weekly quizzes, as well as their working out in the exam has drastically improved. The pupils are more successful and capable in the subject.

The content of this blog is heavily borrowed from Doug Lemov’s Teach Like A Champiom 2.0. I would highly recommend reading this book to gain a greater understanding of Show Call, as well as other teaching techniques. 

Engelmann’s Faultless Communication in the classroom: Algebra

Engelmann’s, Theory of Instruction, introduces the concept of Faultless Communication. It is a form of communication which conveys only one interpretation of the concept at a time. The result of this is having the learner either respond by learning the intended concept, or the learner fails to do so. Faultless communication is designed in a manner where the learner’s performance is framed as the dependent variable. Faultless Communication also

“rules out the possibility that the learner’s ability to respond appropriately to the presentation, or to generalise in the predicted way, is caused by a flawed communication rather than by learner characteristics.”

For this to be the case, some assumptions have to be made about the learner:

Assumption 1:The capacity to learn any quality is exemplified through examples

Assumption 2:The capacity to generalise to new examples is on the basis of sameness of quality (and only on the basis of sameness)

Structuring Examples

The examples that are chosen demonstrate a quality of sameness of a concept, and such communication, as outlined in Engelmann’s Theory of Instruction, must meet these structural conditions:

  1. The examples present only one identifiable sameness in quality – not more than one
  2. The communication requires a signal for the quality of sameness that each example demonstrates. A second signal is also required to identify the examples that do not share the quality of sameness.
  3. The examples chosen must demonstrate the range of variation that typifies the concept. Each positive example will be slightly different from each other but the examples will all share the quality that is to be generalised.
  4. Negative examples, as well as positive examples, must be shown to show the limits of variation in quality that is permissible for a given concept.
  5. The communication must provide a test to check whether the learner has received the information provided by the sequence of examples.

Here are a few examples of teaching sequences that I have used in the last week to communicate which terms are like terms:

In regards to the conditions stated above, this sequence meets the conditions because:

  1. It is communicating one quality of sameness – exactly the same unknown in each term
  2. There are two signals, one for the positive examples and one for the negative examples
  3. There is a wide variation that typifies the concept, negative terms, integer powers, fractional powers, negative powers etc. I could potentially include more examples with decimal powers etc.
  4. Negative examples have been included and positioned deliberately within the sequence to contrast with the example presented before.
  5. A test was given to follow.

How did the lesson proceed?

After getting the pupils to do a mini-quiz on whiteboards to identify whether the following group of terms within the expression were like terms or not like terms. I went through a series of teaching examples of how to collect terms to write one expression in its simplified form. In this lesson, pupils learnt how to:

  1. Differentiate between like terms and unlike terms within an expression
  2. Simplify an expression by collecting the like terms.

Here is the teaching sequence of examples used to teach the second point:

Pupils attempted to collect like terms on their mini-whiteboards using the following set of examples:

A few key points to mention here: I deliberately gave pupils examples where they had to collect negative terms. I taught pupils that the sign on the left side of the term was part of that term. For example, in regards to example 2, a pupil would say:

Pupil: The two like terms are a^2 and 4a

Pupil: The unlike term is -5a

I wanted pupils to practise adding negative terms because that is what pupils struggle with the most when learning how to collect like terms. My examples gradually escalated in difficulty but that difficulty is important for pupils to practice with the teacher.

In regards to example 6, I wanted pupils to see that when two terms give a result of 0, we don’t write 0a^2b and we don’t need to write + 0 to the simplified expression.

This was the second set of teaching examples that were used to collect like terms where a term had more than one variable:

There are some topics in Mathematics where it is easier to create an sequence of teaching examples in line with the structural conditions of Faultless Communication.

I attempted to apply the a similar structure when teaching Year 11 how to change the subject of a two step equation where the desired subject either has a power or root. Here is my attempt:

Make ‘f’ the subject:

  1. The quality of sameness I attempted to demonstrate was that when you list out the order of operations of what happens to ‘f’ we reverse that order. For example, I have ‘f’, I square it, then add 2. Now, I reverse the order of operations. Therefore, I first subtract 2 on both sides of the equal sign, and then square root both sides of the equal sign. The sameness was in the procedure that pupils used to change the subject of the equation.
  2. There is no signal that I used here because It wasn’t applicable for what I was intending on teaching – how to change the subject of an equation.
  3. The examples demonstrate that the if I add 2, subtract 2, multiply by 2, divide by 2 I will always reverse the order of operations of what happens to the desired subject.
  4. I didn’t have a set of negative examples because it wasn’t applicable for what I was intending on teaching.
  5. There was a test of similar examples to attempt

I do think it isn’t always possible to create a sequence of examples which meet all the structural conditions of Faultless Communication. The example where pupils learn how to rearrange the subject of an equation is a perfect example of this. However, what I have done is take some of the structural conditions which are applicable and applied that to create a sequence of examples. It is possible to design a better set than the one above where all conditions are met.

Final Reflection

Faultless communication has allowed me to design a sequence of teaching examples and questions for pupils to attempt on mini-whiteboards which allow pupils to infer the correct generalisation with the highest percentage of pupils understanding what has been taught on the first attempt. The intended concept has been taught and pupils have been able to take what they have learnt and apply it to an independent practice exercise. I am finding this to be successful with my classes of all abilities. This is simply an attempt of applying Engelmann’s work from his Theory of Instruction within the classroom. From what I have seen in terms of pupil work, it has been working. I hope this to be something I continue to use in my planning.

Typical Lesson (Substituting Fractions)

I saw a couple of tweets asking teachers how they plan their lessons, what is a typical lesson look like, what happens behind the scenes? I was keen to read people’s responses and then to be able to compare it to how I have been planning my lessons over the years. This blog post will outline the typical planning process and I will use an example: substituting fractions.

Before I begin, I thought it would be useful to know that during the planning process I outline a progression model. I pick a topic that I want to teach, e.g. substitution, and I break down the topic into complex skills which I want to teach and form exercises which pupils will complete. Breaking down each skill into small component skills prevents pupils overloading their working memory during the initial stages of teaching new information.


The start of every lesson I give pupils a selection of questions to recap what has been taught over the last few days. Pupils do this on their mini-whiteboards. This is a quick exercise taking no more than 5-10 minutes. The questions have been prepared in advance because I want to ask questions which really test a pupil’s understanding. There is no point giving pupils a question to do on their mini-whiteboards which they will all more definitely get right. You want to tease out any confused understanding or misconceptions that pupils may have. This is because if a pupil proceeds to learn new content with misconceptions in mind then more misconceptions will form, or become consolidated.

Recap: Required pre-requisite knowledge

Recapping prior knowledge is essential. If pupils don’t have prior knowledge that future learning depends on then no successful learning will take place, and furthermore pupils will be cognitively overloaded from learning multiple skills. The core knowledge that needs to be recapped will be used again and again throughout the lesson when learning new content. Before I taught pupils how to substitute fractions into different expressions I recapped how to :

  1. Add and subtract fractions with common denominators
  2. Multiply fractions
  3. Multiply fractions and integers
  4. Divide fractions
  5. Add and subtract fractions with different denominators

Initial Teacher Instruction: Faultless Communication

I will think about the worked examples that I want to use to teach pupils how to substitute fractions in a manner which works for all problem types. Engelmann refers to this form of instruction as Faultless communication which I shall blog about soon. I try and design my teacher instruction so nothing that I say contradicts what I have taught in the past.

I want to teach pupils how to substitute fractions into different expressions. I plan my worked examples and I think about how I can design them so pupils can see that the process of how to substitute a fraction into an expression is always the same. The process is the same when I am substituting (adding and subtracting fractions with common denominators):

  • two different fractions
  • three fractions
  • if the result after substitution gives me a fraction which I can simplify to an integer
  • if the result after substitution gives me a negative result
  • if the result after substitution gives me zero
  • if the result after substitution gives me a mixed number

I will also pre-empt misconceptions:

  • Question A: “When we add or subtract fractions with the common denominators. The denominators are the same, the denominator remains the same, add or subtract the numerators.”
  • Question O: “I have ¼ – 6/4 they are going in different directions, I find the difference which is 5/4. The larger fraction is negative so my result is also negative
  • Question G: “Here I have three terms that is ok. I can have questions like this…
  • Question I: “I have eleven-tenths minus one-tenths and the result is ten-tenths which simplifies to 1. That is correct. That is possible.”
  • Question P: “I am adding an integer to a fraction so my result is a mixed number. That is correct. That is possible.”

Not showing these examples means that when pupils try the following questions then they will be freaked out by getting answers which are negative or zero or mixed numbers. The process of substituting fractions where you have to add or subtract fractions with common denominators is the same regardless of the question set up or the answer you get.

Show Call

After going through my worked examples, I will give pupils a very similar set of questions to attempt on their mini-whiteboards. Here pupils are showing their written work to maximise the likelihood of pupils creating perfect answers and working out during independent work. I put a question on the board; give pupils 30 seconds to complete the work on their mini whiteboards. They hold their mini-whiteboards with the answer hidden so other pupils can’t copy each other. I then ask them to show me all at the same time – 3, 2, 1 and show. This lets me see all 32 pupils’ working out. I will give whole class feedback over mistakes that I see. I will show the class a pupil’s work if it is excellent. I will show a sub-optimal answer too and stay ‘Jonny, would have gotten a merit if he had done this…’ or ‘Jonny can improve this. Who can help him improve this? Thank you for letting us all learn from your work Jonny.’ Doug Lemov goes into more detail about the effectiveness of Show Call in his podcast with Craig Barton.

The purpose of this exercise is for me to fix any mistakes and close those gaps. Also, to identify the pupils who are still struggling and support them during the independent practice part of the lesson.

Independent Work

I will have designed a practice exercise of questions for pupils to attempt independently. I will circulate the room. Help pupils who are struggling. Glance at pupil work as I go around and fix any mistakes that I spot. At the end of this timed exercise I will read out the answers and get pupils to tell me how they did:

“Raise your hand if you got 10 or more questions correct. 15 or more. 20 or more. All correct so far.”

“If you made a mistake but you now understand that mistake raise your hand please.”

“If you made a mistake that you don’t’ understand and want me to go through it on the board then raise your hand. We can all learn from you.”

No Plenary

I don’t usually have plenaries in my lessons. I plan my lessons so pupils can complete all the exercises, even more so the hard staff at the end that I want pupils to have the time to complete. I don’t use Exit Tickets because I do a weekly quiz which informs me of the same information that an Exit Ticket does.


I think this approach is similar to many teachers. Nothing here is extraordinary but ordinary. However, what is most important to understand is that the typical lesson above includes features and details that allow pupils to learn, and more importantly it allows pupils to remember what they have learnt. Everything that I have outlined is a combination of what I saw as good teaching from my own education but also from what I have witnessed and learnt from many exemplary teachers that have kindly let me watch them teach.

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.

Engelmann: Underestimating the Number Family

In my last series of blog posts I outlined how Siegfried Engelmann teaches the relationship between addition and subtraction using the number family.

This blog post was inspired by an epiphany whilst teaching a very weak pupil basic addition. I had taught her the following in a sequence of lessons:

  1. Adding integers with no carrying
  2. Adding integers with carrying
  3. Adding decimals with same number of decimal places (carrying)
  4. Adding decimals but with varying number of decimal places (carrying)

In the last half of a lesson I was teaching pupils how to attempt the following set of problems.


I explained using the last problem that when I am adding 2 in the units column to the 9 I am not really getting 1, instead I am getting 11, and the tens is being carried to the column on the left. All pupils besides two understood this. I then demonstrated using a number family that:

  1.  2 is the small number
  2.  I am adding 9 which is the second small number
  3. To get a big number. This big number must be bigger than the two small numbers.

I also showed a non-example of the number family:


  1. 2 is the small number
  2. I am adding 9 which is the second  small number
  3. I will have a big number at the end of the arrow. It can’t be 1 because the big number cannot be smaller than the two small numbers.

At this point, she got it. This is my weakest pupil in year 7. I definitely underestimated how powerful the number family structure can be!

Engelmann Connecting Maths Concept Textbook series: Number Families Part 3

During my second year at Michaela Community School, the department used Siegfried Engelmann’s Connecting Maths Concept Textbook series as the main teaching tool for Intervention.

This is the third write-up of a sequence of blog posts:

In my last post, I outlined how Engelmann teaches pupils within his textbook series of the relationship between addition and subtraction using the number family. Furthermore, how the set-up of a number family is utilised by pupils to answer complex worded problems. 

This blog post will go through the most difficult problem types using the number family set up that were taught.

  1. Problem Type #4: Number families with fractions and integers

By Lesson 67, fractions were included in the number family calculation. For example:

If a number family shows a fraction and a whole number, you have to change the whole number into a fraction.

The bottom number of the fraction you change the whole number into is the same as the bottom number of the fraction in the family.

The above sentences in bold are directly lifted from the teacher presentation book

Here’s a number family with a fraction and a whole number:

This is taught after pupils have learnt and practised the following:

  • Writing an integer as a fraction with a specific denominator
  • Simplifying fractions into integers
  • Differentiating between fractions that do and do not simplify to an integer
  • Subtracting fractions which have common denominators

The teaching of the above then allows such a problem type to be taught.

To have further similar questions included in Test 7

Similarly, including problems in Lesson 72:

2. Problem Type #5: End up, Out and In 

The next number family problem that is introduced tells pupils about getting more and then getting less of an amount. The number family has three fixed categories: End up, Out and In.

Here is an example before I explain the set up of the problem type:

End up is the first small number. The big number is the number that goes in. The next number small number is shown by the number for out. Whatever is left is referred to the number that you end up with. It is the difference between the number for in and the number for out.

Then the problem type became more complicated because more terms were added to the number family, For example:

Fran had $36. Then her mother paid her $12 for working around the house. She spent some money and ended up with $14. How much did she spend.

Here you have two values for in that you must find the total of: 36 + 12 = 48. The total is the big number which is labelled under in. We know how much Fran ends up with so this is what is left.

The sequencing of this problem type and the transition between the first example to the second example with multiple terms is gradual for pupils to learn without experiencing cognitive overload.

The first example of the problem type is only taught in the teacher-led part of the lesson, it is not set as independent work for pupils to do. I think this is simply because it is very difficult for the weakest pupils to complete by themselves that to ensure success it is taught with teacher assistance. Also, pupils did find this hard to learn! There is a lot of reading, a lot of information to organise but by the end of the year pupils were able to do this will little to none teacher assistance. It may be covered in the next set of Engelmann books that follow after this set.

We can see the distribution of how the problem type is covered over different lessons.

Blue = Teacher-led exercise using prepared script + teacher-guided qus

Green = Independent task where pupils are being tested

White = not present in the teacher-led exercise or independent task

End up, Out and In

End up, Out and In with multiple terms


Here we can see the progression in difficulty from taking the most simple addition and subtraction calculations and introducing more complex problem types were addition and subtraction are being tested in various ways. Furthermore, Engelmann looks into including visual variations of the problem type, including large integers and fractions, and multiple terms. Yet, each and every time the algorithmic set-up of structuring the information using a number family allows pupils to draw the correct inference about what they need to do next – Do I add or subtract?

Next, I shall be looking at the teaching of fractions!


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