# Conception of the good

### Insights into our current education system

#### Month: March 2018

This blog post is the third out of four 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 second post can be found here.  The following content was shown at La Salle’s National Mathematics Conference in Kettering.

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.

Pupils also complete exercises where they:

1. write an improper fraction as a sum between an integer and a proper fraction
2. show the sum of an integer and fraction as a sum of two fractions

11. Multiplicative relationship between an integer and a fraction’s denominator

Pupils have been taught previously how to list a string of equivalent fractions equal to an integer, listed as skill 8. This was done through the use of their times table facts. Similarly, pupils are now taught how to find the missing numerator of an incomplete fraction equal to an integer. 12. Equivalent fractions from a diagram

Pupils are taught that:

If fractions are equal, pictures of each fraction will have the same shaded area.

If the fractions are not equal, then the pictures do not have the same shaded area.

A teaching point to mention, I got the pupils to use a ruler to check to see if the shaded areas matched to make them see the connection that equivalent fractions are equal in size, the equivalent fractions presented vary visually. This helped pupils to attempt multiple choice questions like the following:

This was also tested using equivalent improper fractions. In the image titled Part 9, I particularly like (c) and (e) because the parts are split horizontally and vertically. Engelmann here has varied the non-relevant aspects of each image: the number of units for each image, the shape being cut horizontally and vertically etc. What matter is the whether the area between each image is the same, if so then those fractions are equivalent. At this stage, pupils are not being asked to demonstrate equivalence using times tables between two fractions.

13. Multiplying fractions to demonstrate equivalence Pupils are taught that if you multiply an integer by a fraction which is equal to 1, then the result is the integer in the question. Images are provided to show that you are taking two whole units, and splitting the shape into more parts in each unit.

Pupils are also learning to spot visually that when you multiply an integer by a fraction with the same numerator and denominator that it is equivalent to multiplying the integer by one.

Pupils are taught how to structure their working when multiplying fractions where they write the integer with a denominator of one, then multiplying the numerators, and multiplying the denominators. 14. Placing fractions on a number line  (work on)

This was an exercise that pupils did find difficult. Pupils had to place an improper fraction on the number line. What pupils found the most difficult was picking an integer equivalent fraction which two-fifths came after. For example: I want to place two-fifths on the number line. I will go to the integer with a fraction which is just before two-fifths. I will go to zero-fifths, and count two parts to find two-fifths.

I want to place nineteen-fifths on the number line. I will go to the integer with a fraction which is just before nineteen-fifths. I will go to fifteen-fifths, and count four parts to find nineteen-fifths.

I want to place thirteen-fifths on the number line, I will go to the integer with a fraction which is  just before thirteen-fifths. I will go to ten-fifths, and count three parts to find thirteen-fifths.

Exercises were included where pupils were asked to also label fractions onto a number line where the integers, or the integers’ equivalent fractions aren’t labelled.

15. Equivalent Fractions – Multiplying a fraction by 1

In this exercises pupils are demonstrating equivalence between two fractions by writing the fraction to multiply the first fraction by to get the second fraction.

Pupils were using times tables to identify what the first fraction’s numerator and denominator are multiplied by. This fraction has the same value in the top and bottom of the fraction which then simplifies to one. Equivalent fractions are multiplied by a fraction which simplifies to one. This is also visually demonstrated.

Below is an example of an exercise, where pupils are asked to state the fraction for each image, and demonstrate equivalence by multiplying fractions. A teaching point to mention: I did have to structure the working out for the pupils. I would say:

1. I will write the first image’s fraction
2. I will write a times sign
3. I will draw a box for my missing fraction to show equivalence
4. I will write an equal sign
5. I will write the second image’s fraction
6. What number goes in the top of the fraction?
7. What number goes in the bottom of the fraction?
8. Check that this fraction simplifies to one?
9. Are these fractions equivalent? Yes or No? It was important to include this because I wanted pupils to know exactly how to demonstrate their working out. The pupils in this group struggled to structure their understanding on paper in way which made sense to somebody reading their work. By sticking to the following, essentially sticking to a ‘script’ then pupils were able to demonstrate equivalence consistently in this type of exercise throughout the book.

Pupils were then given exercises where one fraction was equal to an incomplete fraction where they filled in the blank and the fraction that they were multiplying by. 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.

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.

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’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:

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.