Conception of the good

On February 15th, the world lost an educator who spent his life developing an approach to accelerate the learning of disadvantaged pupils.

Engelmann was a Marketing Director turned Professor Emeritus of Education at the University of Oregon. He co-authored the famous ‘Theory of Instruction’ with Douglas Carnine and co-developed the term ‘Direct Instruction’ while working with Carl Bereiter. Through grant funding, they set up the Bereiter-Engelmann Pre-school which demonstrated the extent to which disadvantaged pupils could accelerate their learning in comparison to the performance of middle-class pupils.

I have spent the last couple of years becoming familiar with Engelmann’s work, taking aspects of his Theory of Instruction and applying it in my resource creation.

So, what have I learnt from Engelmann?

Answer: That a learner’s inability to respond appropriately to a form of instruction may not be the fault of the child; instead, it can be a problem with what she’s being taught.

This means it’s possible to teach a syllabus in a way she can respond to appropriately without dumbing it down. Here are the four things I keep in mind when creating resources that allow the highest percentage of pupils to understand the course content on the first attempt.

Atomisation

When I used Engelmann’s Connecting Maths Concept textbook series with my Year 7 and Year 8 Intervention pupils, I saw that Engelmann had taken a concept and broken it down into several sub-tasks. A sub-task is a small aspect of a concept. For example, A sub-task of how to add fractions with denominators would be finding the lowest common denominator. For a pupil to develop a flexible understanding of a concept, she needs to be taught as many sub-tasks of the concept and then shown the connections between each of the sub-tasks.

Atomising does exactly this. When I plan a unit of work, I take a concept and break it down into its sub-tasks, and I explicitly teach even the most nuanced aspects of the concept. For example, before I teach pupils how to factorise an expression, I teach pupils how to divide an algebraic expression by an integer, or by an algebraic term. Before I even do this, I teach pupils whether we can also divide an algebraic expression by a number or an algebraic term. An example is shown below:

Here are some examples, of where we can simplify the algebraic fraction:

Here are some examples of where we CANNOT simplify the algebraic fractions because we cannot divide ALL the terms in the fraction:

The value of this exercise is two-fold:

1)      Pupils are taught the most nuanced aspects of a concept which are usually the most difficult parts of the unit being taught. If the most challenging part of the concept isn’t taught explicitly then how can we expect pupils to attempt the most complex applications of the concept? We need to be more thorough and comprehensive than you might think and teach the most complex elements of a concept as well as the most basic.

2)      Pupils develop a flexible understanding of the concept because they can see the big picture. If you plan an entire unit rather than isolated lessons parts, you are more likely to teach as many sub-tasks as possible and not miss anything that’s essential to a student’s understanding. Missing out sub-tasks inevitably means you have to re-teach. Engelmann set up his textbook series to avoid the need to re-teach. If re-teaching is required, Engelmann provides appropriate correction and reinforcement exercises for each unit of work.

Sequencing the learning in the most effective manner

Engelmann’s Connecting Maths Concept textbook structures the content of a unit of work in just that sequence where the learning can be delivered most effectively. Engelmann believed that all future learning is dependent on prior learning and that there is an optimal sequence for each concept. Provided the lessons are sequenced in the most effective way, pupils always have the knowledge required to access the topic they are about to learn. At United Learning, scheme of work is structured and resourced with the same philosophy in mind. The underlying idea is that how effectively the pupils learn depends on the sequence in which they learn about a particular concept.

Scripting the lessons – Pedagogy

Scripting how you communicate the concept is essential. Now, many teachers despise scripted lessons, and some with good reason, e.g. the script they’re expected to follow is sub-optimal. Another reason for their scepticism is the belief that there is more than one optimal way to teach pupils about a particular concept. However, Engelman persuasively argues that there is only one optimal way to teach a particular concept – and his scripted lessons were field tested with tens of thousands of pupils and constantly being refined in response to feedback. Consequently, he was confident that the scripted lessons he and his colleagues developed embodied the most optimal learning sequences.

When I created my resources at Great Yarmouth Charter Academy, I started scripting how I would communicate concepts, to ensure pupils received the most effective and efficient form of instruction.

Then, I would think carefully about what method to communicate.  For example, I didn’t want to teach pupils how to add fractions using a method which was limited to only a few problem types, and then create a different method for another set of problem types. Instead, I tried to create methods that could be applied consistently to as many problem types as possible. This allowed pupils, especially the weakest, to master each concept in all its myriad complexity; evidenced by ever increasing scores in weekly quizzes.

Lastly, my scripted lessons were designed to give pupils the grounding they needed to articulate their understanding. Here is a video showing how a pupil using this knowledge to subtract negative fractions:

Low-stake quizzing and providing appropriate corrections and reinforcements

Engelmann’s Connecting Maths Concept textbook has many opportunities for pupils’ understanding to be tested. The script includes hundreds of questions for teachers to ask. Pupils are given exercises to try with the teacher, as well as independent exercises. Similarly, after every ten lessons, there are also small quizzes recapping what pupils have learnt, not only in the last ten lessons but in the previous 20, even 30.

At Charter, one visitor tallied the number of questions I asked pupils in a single lesson, and they totted up 76 questions in about 25 minutes. I learnt from Engelmann’s teacher scripts how to ask pupils’ questions which test their ability to recall prior knowledge, articulate their knowledge of a concept, to explain a misconception, etc.

In summary, I believe that Engelmann is one of the most important educators of the 20th and 21st Century. I think his work will stand the test of time. By applying his teaching principles to resource creation, I have helped my pupils learn more, and remember it for years to come.  My experience confirms, for me, that teacher quality is a function of the resources they have access to.  Children are more likely to be successful with a teacher, who has access to exceptional resources, than a teacher who doesn’t, and never has.

Engelmann’s work has taught me more than any educator that I studied with during my PGCE and MA. My next post will look into the evidence for the effectiveness of Engelmann’s approach and the reasons why his work hasn’t been more influential.

After my podcast with Craig Barton, I have received many emails asking to share more booklets. I have attached the booklets that I made during my time at Charter. They aren’t perfect, and with my current workload, I am not in a position to refine them. However, I do think they are useful for teachers who want to start designing their own booklets. I used each booklet with all my classes. I hope they are helpful.

There will inevitably be mistakes in the booklets. I take full responsibility for any errors that you see.

https://drive.google.com/open?id=16UtxWsL3T5M2wz65YjwfRYyJ3BZSNZZp

 

 

In February 2018, I heard Andrew Blair speak at a CPD event about Inquiry Maths. Andrew is a Head of Maths and founder of his website Inquiry Maths where teachers can gain access to ideas about teaching Mathematics using the Inquiry Maths model.

Andrew and I tend to disagree over the best pedagogical approach when teaching Mathematics, but we agree in teaching many of the problems that he has shared on his website. When I create resources, I check out different sites for ideas, and I regularly check Andrew’s website. It’s fantastic! When I saw Andrew speak at this CPD event, he showed a mathematical inquiry task that I made a mental note to include in my resourcing for the following year. The problem is below:

40% of 70=____% of _____

I love this type of mathematical problem. There is so much mathematics that needs to be communicated, and I would recommend readers to check out Andrew’s page on this inquiry. At United Learning I’ve finished the Fraction, Decimals and Percentage booklet for the Y8 curriculum and I’ve made resources for this type of mathematical inquiry to be shared with the kids.

Here is an example of how I broke down the teaching sequence so all pupils can access a problem like this but also support all pupils to attempt more complex forms as well. 

The teaching sequence is designed to increase the probability that all pupils can be successful in learning the subject content. 

1) Showed a multiplication model for the following problem types

I have listed some of the potential decimal multiplication calculations and then demonstrated how to draw the model for the equivalent decimal multiplication calculation:

Pupils are only shown the first line because they are taught that the mathematical model of rewriting the decimal multiplication calculation.

A few points: I’ve deliberately kept the position of the mixed number or decimal and the integer in the same place within the calculation. In the second example, the first term of the calculation is a mixed number and the second is an integer, it is the same on the right-hand side of the equal sign. This is because I want pupils to focus on the digits changing position, and that only. Everything else is kept constant.

This is how I would communicate the model in the classroom to pupils:

I am multiplying a two decimal places decimal by an integer, so my equivalent calculation will have a two decimal places decimal and an integer.

I would get kids to draw the underlines to show me how their equivalent calculation would look. I have done this because learning the mathematical structure of the calculation is another thing a pupil needs to learn. This also reduces the likelihood of a future error caused by pupils by not knowing whether the calculation will have an integer and a decimal, or an integer or a decimal, and how many decimal places the decimal will have.

2) Introduce the steps to go from the problem to the equivalent calculation

a) Rewrite decimal place structure
b) Fill in the digits from the right side
c) CHECK: Multiply out both sides of the calculation to see that the result is the same

In my teaching, I write the generalised steps for the calculation pupils are about to do because they can see that even if I am rewriting a multiplication calculation with

• an integer and decimal or;
• two decimals or;
• decimals with the same number of decimal places or;
• decimals with a different number of decimal places

the generalised steps remain the same.

Generalising method stops is something that Siegfried Engelmann calls ‘Logically Faultless Communication’ which has been blogged about here.

A logistical thing, it also allows pupils to follow the steps better if they can flicker their eyes from the live worked example demonstrated by the teacher and the steps already written on the board.

3) Introduce the Example-Pupil attempt sequence

At this point, I then design the teacher-led worked examples and the pupil attempt questions. The problem type that the teacher goes through and the pupil goes through after seeing the teacher demonstrate the same problem type.

Here is the sequence:

Pupils aren’t evaluating the calculation. They are rewriting the calculation so their answers would look like this:

I have deliberately started with explicit examples meaning that the 2 decimal places decimal and two-digit integer have digits more than 0. Only from example 4, I have introduced a single figure of 5; this is because pupils must write the digit 5 as 0.05 where there is a zero between the decimal point and the 5. Starting with explicit features such as digits more than 0 in decimals and integers makes the change from one calculation to the equivalent one more clear.

Pupils then will complete a practice exercise where they will rewrite a decimal calculation but ending in the same result as the first decimal calculation.

They will then transition onto a filling in the blank exercise like shown below:

4) Equating Percentages: Introduce the Example-Pupil attempt sequence

I believe that I’ve now taught all the prior knowledge pupils need to complete this statement:

40% of 70=____% of _____

The prior knowledge is listed below:

• Writing a percentage in its equivalent decimal form
• Rewriting a decimal multiplication calculation in a visually different form but where the result is the same as the original calculation
• Decimal place value
• Evaluating decimal multiplication calculations

Given that the prior knowledge has been taught and committed to long term memory, there is only one new thing pupils need to learn to complete the statement. They need to write the percentage as a decimal and replace the ‘of’ with a multiplication sign. Then every other line of working is prior knowledge that has already been covered.

Here is the working out, and the new line of working is in bold:

40% of 70 = ____% of _____
0.40 × 70 = __.__ __×____
0.40 × 70 = 0.70 × 4

5) Comparing Percentage Increase/Decrease

What has been learnt here can also be used to complete statements like this below, and place the correct symbol between the statements <, > or =

6) Equating Percentage Increase/Decrease

The next level of difficulty that is accessible because of the prior knowledge taught is that pupils can evaluate the complete statement and equate the result to the incomplete statement.

Here the new knowledge being taught is dividing the result by the value in the incomplete percentage calculation.

However, what makes this more complex is that you must evaluate the complete calculation and equate the result to the incomplete calculation to then find the missing percentage. So pupils must:

1) Convert the decimal into a percentage
2) If the calculation is a percentage decrease, then subtract the percentage from 100%
3) If the calculation in a percentage increase, then subtract the percentage from 100%

Summary

I appreciate all the beautiful problems and mathematical tasks that Andrew provides on his website. They are incredibly rich in knowledge and can create a valuable mathematical conversation between teachers and pupils. There is a great deal of importance in teaching inquiry tasks because it is an opportunity to develop a flexible understanding of the subject. However, where I would disagree with the Inquiry Maths model would be how the mathematical task would be used and how the content would be taught to children. I think there is a great deal of prior knowledge that needs to be sequenced, so all future learning is supported by prior learning, and that only one new thing is being taught at each part of the sequence. This is to avoid cognitive overload.

I hesitate to have pupils taking responsibility for directing the lesson because each child will focus on a different aspect of the mathematical problem in front of them. More importantly, they will focus on various aspects depending on what knowledge they already have. It is inevitable that in any learning experience, each pupil will be starting at a different point based upon how much knowledge they have. However, when pupils direct the lesson that will result in more knowledgeable peers leading the lesson, and weaker pupils struggling to identify or learn aspects of mathematics that they need even to attempt the mathematical problem. By laying out the prior knowledge and sequencing it, you are allowing the more knowledgeable pupils to consolidate any existing knowledge they may already have, but you are letting the weaker pupils close any gaps they may have. Overlearning is an essential part of the learning process; it helps the more knowledgeable pupils to learn something to the point that they never make a mistake.

This attempt, I believe allows all pupils regardless of their knowledge gaps to have any missing gaps filled, but also allows all pupils to be able to access the mathematical problem.

 

 #Mathsconf18: Atomisation Pt 2

Atomisation: Breaking down your teaching as you have never seen before…

On Saturday 9th March I delivered a workshop at the La Salle Mathematics Conference in Bristol. This blog post is a summary of some of the points made in the session.

For this workshop, I chose to look at an uncontroversial topic such as Angles on parallel lines. I think it’s uncontroversial because teachers know that it is an undeniable part of geometry that is commonly assessed. Also, I think it’s a topic which is taught with a poor sequence of examples. Lots of angle problems on parallel lines feels like an angle chase – when you find one angle then how can you find the other angle. However, in the process, pupils can’t have the rich mathematical discussion between the relationships of different angle facts on parallel lines. Part of teaching this topic effectively is dependent on the sequence in which the examples are organised.

When I started making the worked examples for this topic, I thought about the simplest application of angles on parallel lines for each angle fact and the most complex application. What I realised is that I could have spent hours or even days making lots of different worked examples. To avoid this, I thought of how I could cover all the myriad of complexities for each angle fact within the fewest number of worked examples.

The first step was to write out all the sub-tasks that I planned to teach:

–   Vertically Opposite Angles are Equal

–   Alternate Angles are Equal

–   Co-interior Angles sum to 180o

–   Corresponding Angles are Equal

–   Basic Angle facts on parallel lines

–   Angles on parallel lines – Algebraic

o   Simplified expressions equal to 180o or 360o

o   Simplified expressions equal to form an equation with unknowns on both sides.

After I listed the sub-tasks, I realised that I wanted to try a different pedagogical approach from the status quo approach. Historically, pupils are told that one unknown and one known are equal, and they are to accept it, and then identify the unknown and known angle pair which are equal in a similar looking example. Instead, I showed a selection of worked examples where the angles were of equal size, and I would state that these two angles are equal. I used Geogebra which is an online graphing programme where I would have an interactive set up so if I moved one of the parallel lines or the traversing line, the equal-sized angles would change from what they were before, but they would still be equal.

So, using a sequence of worked examples, I would

1)   Show the relationship with the position of angles and the angle fact

2)   Find the missing angle

3)   Find the missing angle by using a basic angle fact

NOTE: Diagrams aren’t drawn to scale here.

Vertically Opposite Angles are equal

Here is an example sequence for Vertically Opposite angles being equal

Show the Relationship

At this point, I transitioned to asking pupils in a whole class discussion the size of the unknown angles because they had seen the relationship between the position of two vertically opposite angles.

I deliberately used more challenging examples for pupils to identify one missing angle which is vertically opposite to one known angle. This is because I knew it is these type of angle problems, they would struggle with the most so I went through it with them so they would be successful when they would attempt similar issues independently. I felt that showing the relationship was explicit enough for pupils to attempt the simplest applications of identifying vertically opposite angles being equal.

The third section is using basic angle facts from the list below:

1. Angles in a triangle sum to 180^o
2. Angles in a quadrilateral sum to 360^o
3. Angles on a straight-line sum to 180^o
4. Angles around a point sum to 360^o

To either use the angle fact to determine one of the two vertically opposite angles or find one of the vertically opposite angles to then find another angle using one of the basic angle facts.

Here are some worked examples of this with an explanation:

I made Example 4 deliberately to highlight that the small triangle FCB and the large triangle KCE have the same angles.

In Example 5 and 6 I have included 2 parallel lines and 1 parallel line segment to allow Example 5 to include an opportunity to use the ‘Angles around a point sum to 360^o’ fact. Similarly, in Example 6 I deliberately didn’t label the vertically opposite angle because I want pupils to start finding angles that aren’t labelled but are required to find another unknown angle.

In summary, I followed the sequence structure of:

1. Show the relationship
2. Find the missing angle using the angle fact given
3. Find the missing angle fact using basic angle facts to then determine angles using the new angle facts learnt.

Alternate Angles are equal

Here is a worked example sequence showing the relationship in positioning of Alternate Angles being equal

Now using the same geometric structure as the worked examples used to show the relationship of alternate angles, I’m asking pupils to find the size of the missing angle:

Here is a sequence of worked examples where basic angle facts have been interleaved:

Co – Interior Angles sum to 180^o

Here is a worked example sequence for Co-interior angles summing to 180^o

In Example 7 and 8, pupils can see that each triangle formed by both traversing lines and one of the parallel lines all have the same size angles.

Here are examples where pupils are asked to use the fact that co-interior angles sum to 180^o to find the missing angle:

 

In the last two examples we can explore so many interesting mathematical patterns between co-interior angles and parallel lines. In Example 6, A = I and G = C

Here is the next section of worked examples where basic angle facts are being interleaved:

Corresponding Angles are Equal

Here is a worked example sequence for Corresponding angles being equal

Here is a worked example sequence where pupils are using the angle fact that corresponding angles are equal to find the missing angle:

 

Here is a worked examples sequence where basic angle facts are being interleaved with the angle fact that corresponding angles are equal:

 

Atomisation: Breaking down your teaching as you have never seen before…

On Saturday 9^th March I delivered a workshop at the La Salle Mathematics Conference in Bristol. This blog post is a summary of some of the points made in the session.

This presentation was on the topic of atomisation. Recently, atomisation has created a lot of conversation on Twitter and at a couple of conferences. What is atomisation and is it the next fad?

It’s not the next fad.

Atomisation is the process of breaking down a topic into its sub-tasks.

Atomisation is a term coined by Bruno Reddy who taught me about it during my second school placement in 2014.

Atomisation is the starting point to every booklet I create in my role as Curriculum Advisor at United Learning. I sit down and list all the sub-tasks that I need to teach that is within the topic. An example of this for Perimeter is available here.

What are the benefits of Atomisation in respect to the big picture?

Atomisation is a process in which teachers can collaboratively identify the specific and detailed knowledge that pupils must know to be academically successful.

Identifying this specific knowledge means that pupils can learn as close to 100% of the domain of knowledge that they need to know. Exam boards and the national curriculum, truthfully, may provide high-level specificity of what needs to be taught, but not the finer details or detailed knowledge that goes into making pupils able to do a task.

For example, pupils are expected to know the following fact below:

Fraction x Fraction’s Reciprocal = 1

And upon reflection, this seems understandable in respect to multiplying fractions, index notation, ratio, proportion etc. but it is not necessarily explicitly stated in exam board specifications or Curricula.

The lack of detailed knowledge outlined by exam boards or textbooks results in teachers reducing the subject content being taught. And the source of information of what needs to be taught from children usually comes from exam tasks, and then the curriculum becomes an endless repetition of exam materials. Similar thoughts have been shared on Daisy Christodoulou’s blog.

Mathematics as a subject is vast. There is so much to teach a child in the space of their academic career. Here is a visual:

If the bubble represents 100% of the domain of knowledge that needs to be taught. It is the case that we teach small samples of that domain. We teach concepts and how they overlap with other concepts, and we also teach connections between different concepts. However, if we use exam boards and textbooks as our source of information of what needs to be taught in the curriculum, then we inevitably miss out teaching other parts of the domain.

The process of atomisation enables teachers to focus on the concept and identify the finer details that aren’t readily available. Atomisation allows the teacher to teach as close to 100% of the domain of knowledge.

What are the benefits of Atomisation in respect to the teaching process?

Identifying and sequencing all the sub-tasks and specific knowledge that needs to be taught for a concept increases the probability each child will be successful in the learning process.

This means that pupils will be able to appropriately and accurately respond to planned questions, or that they will remember what has been taught at that moment after a long period.

Teaching aspects of a concept that are usually overlooked undermine how successful each pupil will be in respect to their future learning.

For example, not teaching pupils that:

Fraction x Fraction’s reciprocal = 1

undermines a child’s ability to attempt questions like this:

Show that the ratio in the form 1:n can be written as 1: 1.5

If pupils get these questions wrong, then the incorrect inference is made. Instead, they must be retaught the topics of ratios from the beginning, but they only need to be taught this fact. And this is a fact has been mentioned several times in the Year 8 Curriculum so far. I’ve resourced a section on problem types like this for the most recent ratio booklet coming out.

Teaching all the sub-tasks of a concept and sequencing it in a logical sequence prevents pupils from being cognitively overloaded because each sub-task taught is being used or covered in future learning. In the teaching process, pupils will have committed prior knowledge in their long term memory so any future learning will occupy space in their limited working memory.

What are the costs of Atomisation?

The cost of atomisation is a valid one. The initial stages of teaching a concept take more time to cover the content. If we explicitly teach more sub-tasks than we would normally then we would have more time available to teach the rest of the curriculum. However, by not explicitly teaching all the sub-tasks of a concept it will result in the following consequences:

1. Reduce the probability of a child learning parts of a concept on the first attempt
2. Undermine a child’s ability to access future learning

Atomisation avoids these inadvertent consequences by:

1. Guaranteeing a greater likelihood a pupil will learn the sub-task on the first attempt
2. Increasing the probability of success when learning future content or more complex applications of the concept
3. Saving time in the future which would be inevitably spent re-teaching
4. Revealing sub-tasks that need to be taught that are usually overlooked in the curriculum.

In summary, Atomisation has improved my teaching and as a result my pupils learning more and remembering it a half term later, a year later etc. More importantly, it benefits the children that struggle with learning the most. Atomisation has allowed my weakest pupils learn more in less time, and in my small world, I believe that this is a path that I will continue down when creating resources.