Conception of the good

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:

• Introducing the number family
• Manipulating the number family for complex worded problems
• Just when you thought the problems couldn’t get more difficult

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

Conclusion

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!

 

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 second write-up of a sequence of blog posts:

• Introducing the number family
• Manipulating the number family for complex worded problems
• Just when you thought the problems couldn’t get more difficult

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.

This post will go through how the set-up of a number family is utilised by pupils to answer complex worded problems. 

1. Problem Type #1: P and Not P questions

In Lesson 16, P and P’ problems were introduced. For example:

There were 8 new shows. The rest were old shoes. There were 46 shoes in all. How many shoes were old shoes?

The number family problem has been written up into words where pupils need to do the following:

1. Label the three groups; P, P’ and all. In this case, New, Old and All
2. Assign the number stated to the correct group
3. Decide whether to add or subtract to form the calculation

It is explained in the presentation book how to deliver the teaching of such a problem type. Then the following questions are given to follow throughout future lessons:2. Problem type #2: More than or Less than problems

In Lesson 24, a problem type is introduced which has a new layer of complexity, pupils need to:

a) Learn the terminology of difference words: more than, less than, greater than, smaller than

b) position labels and numbers on the number family arrow by understanding the difference words of more than, less than etc.

Here are some examples:

• If you know that one value is more than another value then that one value is the big number. The other number is the small number. E.g. 14 is more than 11. 14 is more. It’s the big number. 11 is one of the small numbers.

Second example:

• If you know that one value is less than another value then that one value is the small number. The other number is the big number. E.g. 24 is less than 30. 24 is less. It’s the small number. 30 is the big number.

Pupils are also taught to write the smaller value as the second small number. This is deliberately mentioned in the Teacher’s presentation book. Pupils are learning how to use the number family to interpret the question. This can be extended to similar questions where pupils are taught to structure the calculation before performing it:

The presentation book emphasises the importance of the key words: shorter, heavier, fewer, further etc. These words indicate which item will be the small number label and which item will be the big number label.

In regards to question a, the pole is shorter than the tree so the pole is the small number label and the tree is the big number label. The number 8 is referred to as the difference number, this is deliberate language used within the Teacher presentation book. The difference number is the first small number label. The number before the difference word (8 feet), in this case shorter than, is the difference number. 

The next three lessons pupils are taught how to label and structure the number family. They do not perform any calculations they just place each label and the difference number given. Lesson 28 they are taught to perform the calculation fully. For example,

The calf weighed 345 pounds less than the horse. The horse weighed 509 pounds. How many pounds did the calf weigh?

3. Problem Type #3: Asking to find the difference number

By lesson 43, there are number family problems that ask about the difference number. For example: A coat cost $134. A radio cost $23. What is the difference in the price of these objects?

Pupils are being asked to compare the cost of two items. Using the number family set up a pupil positions the big number and the small number. In this case the coat costs more than the radio. They also learn that the small number is the second small number too. The first small number is the difference number.

4. Differentiating between problem type

By Lesson 43, an exercise to differentiate between problem types is introduced:

Similarly, what is interesting to note in regards to the distribution in the teaching of these problem types and the spread of independent practice exercises is that what is mostly practised is the P and Not P problem types. I think this is because it is the first worded problem type that pupils use the number family to reach the answer.

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

Each number represents the Lesson no. E.g 16 = Lesson 16

In the next post, I will go into the problem types that became progressively more difficult yet were simple to answer because of the set-up of a number family.

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. The textbook series comes with:

• A pupil textbook
• A pupil workbook
• Prepared Tests (after every 10 lessons)
• Teacher Presentation book

Each lesson would have:

• 4/5 Teacher-led exercises (using  the presentation book)
• Independent pupil work

I know that some teachers would feel uncomfortable using a presentation book with prepared scripts but when I read through it I realised that Engelmann had written it better than anything else I had seen. The script is accurate, deliberate and is economical with text. This is something I will come to later on.

This blog post is one of a series to come. I will go through one specific skill that was taught and covered within the textbook and workbook for the pupils to really master. I will outline how this one specific skill was taught from the simplest problem type to the most difficult problem type. I will also show the same algorithmic set-up, as suggested by Engelmann in the Presentation book, which was used consistently when pupils worked through different problems which became increasingly more complex.

Here I will be looking at the relationship between addition and subtraction using something called a number family.

Here is the sequence of posts:

1. Introducing the number family
2. Manipulating the number family for complex worded problems
3. Just when you thought the problems couldn’t get more difficult

1. What is a number family?

A lot of the addition and subtraction problems are based on using a number family.

A number family is made up of three numbers that always go together to make an addition and subtraction fact.

Engelmann introduces it as a family made up of two small numbers and a big number. The big number is at the end of the arrow. Here are the two facts taught:

To find the big number, you add the small numbers.

       

To find a missing small number, you subtract. You start with the big number and subtract the small number that is shown.

         

Engelmann also shows the two visual possibilities of the same problem. This explicit instruction within the presentation book is allowing pupils to develop a schema around the set-up they will be using to complete different addition and subtraction calculations. Engelmann is also pre-empting misconceptions which are most common by stating that “you start with the big number and subtract the small number that is shown.” How many instances have you seen pupils writing it in the reverse manner?

 

2) Deciding whether to add or subtract?

Engelmann states in the presentation book that the first question to ask pupils before they perform the calculations of the number family is: “Do we add or subtract?”

This is really valuable for pupils because they are being asked to decide what operation to use between the two numbers stated. They have to decide whether they have two small numbers or one big number and one small number. Through the number family it is also visually really clear for pupils to determine what combinations of numbers they have. Thus, helping pupils to see what operation to use.

Also, note that there are two and three-digit numbers included. This is to show pupils that we apply the same decision making question to calculations with digits of all size.

3) Number family where subtraction calculation requires no regrouping (#1)

The first few lessons pupils were asked:

“Do we add or subtract?”

“Write out the calculation from the number family.”

The subtraction facts did not include numbers that would require regrouping. Why? This is because subtraction which requires regrouping was a separate skill which was going to be taught within the sequence of lessons. This is really important! It is essential when you are teaching a pupil one specific skill at a time and also that you aren’t making it difficult for the wrong reasons. You are testing whether pupils can decipher which operation they need to use with the values given, not their subtraction skills with regrouping.

4) Number family where subtraction calculation does require regrouping (#2)

Once the subtraction with regrouping had been taught as a different skill then such subtraction calculations were included in the number family.

5) Visual variations of the same problem type

Engelmann introduces number families where the arrow is vertical. He shows that the calculation works the same even if the arrow was horizontal. Visual variations of the same problem type show pupils that some changes are irrelevant. The set up with a vertical or horizontal arrow does not change:

• The question we ask whether to “add or subtract”
• Which is the big number or small number because the big number is always at the end of the arrow. Small numbers are always along the arrow.

To then include questions sets of the similar type:

This has now been taught explicitly by the teacher using the presentation book. From lesson 11 this skill is practised within the Independent task section of future lessons to ensure pupils don’t forget what they have learnt. Below, I have a table listing the amount of lessons where horizontal and vertical number families are present, in which lessons and in what capacity:

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

Each number represents the Lesson no. E.g  6 = Lesson 6

In the next post, I will go into the complex worded problems pupil were able to attempt and accurately complete by manipulating the number family set-up.

 

 

On Wednesday, I visited Great Yarmouth Charter Academy and I was truly blown away by how extraordinary the school has become in the space of 5 weeks. This blog post will summarise what I witnessed to be excellent practice:

Chanting Poetry: Invictus

Whilst pupils were entering the hall to go to assembly, I could hear lines from Invictus being chanted with such rhythm and passion. The drama teacher would recite a line in an animated fashion and he would project his voice to resonate to every corner of the room. Pupils would recite back the line or complete the sentence of the line. If it wasn’t good enough then the teacher would repeat the line and get the pupils to recite it again.

Peaceful Corridors

Pupils walked in silence, single-file, on the left hand side with their bag in their right hand. Pupils transitioned through the corridor smiling and greeting their teachers wishing them ‘Good Morning, Sir!’ ‘Good Morning, Miss’. There was no chaos. Pupils would arrive at their teacher’s door with a sense of urgency, purpose and desire to learn. I saw the transition between form time and period one and it was incredibly peaceful and pleasant. Teachers stood in the middle of the corridor reminding pupils of the corridor behaviour expected. One of the teachers remarked that this transition took one minute where in previous years pupils would be arriving 15-20 minutes late to class.

Uniform

There has clearly been a huge push on uniform because pupils looked incredibly professional. Pupils’ shirts are tucked in, skirts at the knee, ties are immaculate and also hiding their shirt’s top button. I didn’t see any pupils wearing hoodies, coats or trainers. Pupils with scrappy uniform would be dealt with straight away.

Silent lessons

The school’s headteacher,  Barry Smith, took me around to see lots of different lessons. This was the highlight of my visit. Every classroom I walked into I saw pupils SLANTing. They were in silence listening to the teacher talking or watching the teacher writing on the board. I didn’t see any low level disruption in all of the classroom I walked into. I didn’t hear or see pupils whispering to each other, passing notes or speaking rudely or being disrespectful to their teacher. More importantly, I didn’t see a difference between how the kids were SLANTing before Barry entered the room compared to after he had entered. Why? This is because they were SLANTing throughout the lesson.

I went to two English lessons, and in one lesson pupils were in silence writing an essay about different overarching themes of the play, Romeo and Juliet. In the second lesson, I saw a pupil detail the context behind the poem, London, using a wide range of sophisticated language. I was almost brought to tears hearing an eleven-year old pupil share accurate insights into the poem in such a confident and articulate manner. His teacher was so pleased and she showered him with praise.

I watched the Head of Maths teach and she was the respected authority in the classroom. Her pupils were listening intently, they were SLANTing and speedily working away to complete their worksheet of questions. I saw a full 40 minutes of teaching and learning. No disruptions. Every second mattered and every second was utilised for learning.

Pursuit of Happiness

Pupils and teachers are genuinely happy at Charter Academy. Pupils are learning because teachers are able to teach with a clear and consistent behaviour policy being implemented in the school. I ate lunch with the pupils and lots of them shared their joy of being able to learn. More importantly, teachers are beaming with pride over how well-behaved and keen their pupils are to learn. One pupil was very honest and admitted to being the pupil who would truant and hide in the toilets. She admitted how behind she is academically due to the hours she spent outside of the classroom and is now doing extra revision at home and attending after school catch-up classes.

One of the Prefects I found standing in the crowd during break shared with me “I am really loving school. I am learning more than I ever thought possible. My teachers are great. Mr Smith has turned around my school. I am proud to be Charter.”

It is very early days but the school is a wonderful place to be for pupils and teachers and I’m very excited to see what Barry and his team achieve. Thank you to all who had me in theirclassroom!