Conception of the good

Insights into our current education system

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

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!

 

Engelmann Connecting Maths Concept Textbook series: Number Families Part 2

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.

Engelmann Connecting Maths Concept Textbook series: Number Families Part 1

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.

 

 

Charter Academy: Teachers Teach, Kids Learn

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!

 

 

 

 

Teaching a concept from every angle

At Trafalgar College, we have a centralised planning system put in place where all Maths teachers within the department are teaching from the same textbook. This textbook has been designed and agreed by myself and Ian Burchett, Executive Principal.

The rationale behind our decision to use a textbook as our main teaching resource was to ensure that all pupils in the year group, across all ability spectrums, are being taught the same content. The only difference between each class is the amount of time the teacher spends on teaching a particular concept.

This week I taught all my year 8 classes how to write a fraction as an integer (given the numerator is divisible by the denominator) and in the planning process I was thinking of the different problem types that fall under the concept of writing a fraction as an integer, here they are:

  1. Simplifying a fraction where the numerator is divisible by the denominator
  • Using the first 12 multiples of the first 12 times tables

  • Large numerator that requires short division

 2. Finding the missing value which can either be:

  • Numerator
  • Denominator
  • Integer

3. Filling in the missing blanks where the integer is equal to a string of fractions

4. Writing a fraction from a sentence and simplifying it to an integer

5. Deciding if the following equations are true or false

6. Deciding whether the equation will be a mixed number or integer, where the answer would be ‘integer’ or ‘mixed number’

7. Deciding which fraction is the odd one out given a list of fractions which simplify to an integer. Including fractions that do not simplify to an integer

8. Given a number line, state the equivalent fraction for each integer given the denominator of that fraction

What I found was that pupils were genuinely thinking about the concept that was being taught simply because they were trying a variety of different problem types. Essentially, they were doing the same thinking again and again but applying their knowledge in different instances. What I enjoyed was watching the kids ponder, stop and think whilst they were attempting each question on their mini-whiteboards. I saw the pupils correcting their mistakes before I even needed to highlight their mistakes whilst circulating the room. This is simply because I thought about the different ways of assessing a pupil’s ability to write a fraction as an integer.

Rolling Numbers

This September, I started at Trafalgar College in Great Yarmouth and my biggest push for the first week was to ensure that all Year 7s would be able to recite their times tables.

I did this through rolling numbers. This is something I learnt from my visit at KSA, and saw the successful implementation of whilst being at Michaela Community School.

Rolling numbers is a call-and-response chanting of times table facts while pupils are counting off on their fingers. The chants are catchy and funny. There are specific hand motions to each times table chant which allows pupil to distinguish between different times tables – also resulting in many pupils desperately wanting to roll certain times table chants over others.

What worked well?

Kids at Trafalgar College have been successful in learning their times tables. The most effective aspect of it all is the counting off on their fingers. Kids can now associate that 21 is the third multiple of 7. This goes above and beyond having kids just listing out all twelve multiples. This enables pupils to answer certain calculations easily. How did I know this? I could see an improvement in the time it took for pupils to answer times table questions when I would test them before or after my lesson.

The lyrical and rhythmic aspect of the chants really motivates pupils to get involved because it makes reciting times tables fun and exciting. They find the opening line the teacher chants “Team! Team! Good as Gold! Let me see your fingers roll…the threes” motivating because the team element gets the reluctant kids to take part. When I say “roll” the kids start rolling their arms. From the front it looks really impressive to see 32 kids rolling their arms and smiling. Why? Simply because they are having a great time!

What were the struggles?

There are some pupils who will really want to take part more so than other pupils. I did have a small population of pupils who were reluctant, and thought that the whole process was cringeworthy and unnecessary. However, I didn’t let those pupils opt-out. The process of learning your times tables is imperative. My Headteacher put it nicely that learning the times tables is equivalent to learning the alphabet before you learn how to write. I would motivate these reluctant pupils in a positive manner by reminding them that we are a team and that they don’t want to be the person who lets everybody down. If you take part then you will enjoy it. If you take part then you will be able to list off your times tables effortlessly. What really helped was picking the most enthusiastic pupil to stand at the front with me and chant the teacher part. This made reluctant pupils see how successful other pupils have been who took part and motivated them to get involved too.

Why did I push rolling numbers in my first week?

It helps kids learn their times tables, successfully. Kids go through school learning their times tables in a very touch and go fashion and by repeatedly rolling numbers on a daily basis it pushed all pupils across the ability spectrum to commit these facts to their long term memory. The chants are catchy. The kids love the physical movements because they are loud and dramatic. Most importantly, it has made learning something potentially mundane (as children may think) incredibly enjoyable.

Routines in the classroom

This blog post is a summary of a workshop delivered at Teach First’s Summer Institute with Nick Hutton.

Why are Behaviour routines effective?

Routines are behaviour tools to help teachers train pupils to behave or complete tasks in a certain way. Routines can serve many different purposes, they can:

Save time

There are many tasks that pupils take a significant amount of time to complete but those tasks are necessary, for example: passing out or collecting in exercise books, entering the classroom, exiting the classroom.  If they are completed in an inefficient manner then huge amounts of time accumulated is wasted. Routines can save precious seconds that total to hours.

Improves Behaviour

With explicit narration of how you want pupils to behave or what you want pupils to be doing at different stages of the lesson can improve behaviour. Narrate to pupils what you want them to be doing and what you don’t want them to be doing.

Creating a positive and purposeful

Part of implementing a routine is narrating to pupils the purpose behind the routine. When they see that there is an important purpose behind a routine pupils can understand why they are praised for complying and accept demerits when they aren’t. An understanding of routines create that purposeful environment and positive environment.

Here are a few routines that can be implemented in the classroom. I have included the instructions I would use.

Handing out books.

I have my classroom organised in rows and the pupils’ books are placed on the windowsill. Pupils were taught how to pass their books down in the most time effective manner. Here are the instructions that were given to do this:

“The person sat at the windowsill, raise your hand please. You are the most important person in handing out the books! The faster you pass the books the faster your peers will have their books to start the task. You want to make sure that your row is the fastest one… 

Your book will be on the top of the pile. You will take the pile and take your book. You will then pass the pile of books to the pupil next to you.

The next pupil, your book will be next in the pile. Take your book and pass it on.

So, the pupil at the window will have their book at the top of the pile. The pupil next to you will have their book next in the pile.

The pupil at the end of the row will have their book at the bottom of the pile.

Once you have your book, open it to your last page of working out and then SLANT to let me know that you are ready. We aren’t looking out of the window. We want to be ready!”

“When we pass our books back. The pupil at the end of the row. Your book will be at the bottom of the pile. When the next pupil gets the pile, you ALWAYS put your book on top. Not below. ALWAYS on top. Where do we put our book in the pile, Hamza?”

“We then put our pile of books on the windowsill. Once you have passed on your book you will slant.”

My instructions to the pupils are crystal clear. I have emphasised the role of the pupil who needs to take the pile of books. I have explained:

  • where they can find their books
  • what they have to do with the pile
  • who they pass the pile on to
  • what to do once they get their book
  • what to do once their book is open to show me they are ready

I have also added a bit of flavour to my instructions in the sense that I have hyped up the kids to try and make sure they are the fastest row in passing out their books.

To get the pupils to get faster at handing out their books it would be a good idea to use a timer. I would then record the times for each day, and motivate the kids at the start of each lesson by saying

Yesterday we passed the books in 42 seconds. Today we want to break that record. Let’s see if we can pass the books in 41 or 40 seconds? Ready…(scan the room to build suspense)…Go!”

Or you would set a countdown. This gives each pupil a sense of urgency that we don’t want to waste any time. It gets the pupils excited to beat a record.

The instructions for every lesson would be:

“When I say go, and not before. I want you to pass your green books and textbooks. Once you have your green book open it to the last page of working out and slant. That way I know that you are ready. You have 30 seconds. Let’s see whose row is the fastest. Ready…(watch class and pause to create suspense)…GO!

25-24-23…Lauren is ready…18…17…16…Hannah is slanting already…9…8…7…6….Emily’s row is ready…4…3…2…1…and slant! WOW, we broke our record – 29 seconds.”

A query that participants mentioned was “What if I haven’t got a windowsill? What if I haven’t got my tables in rows but in groups of 4 instead?” If I had my tables in rows and I had no windowsill or had my tables in groups of four then I would layout each rows’ pile of books just before the lesson.

This routine can be adjusted depending on the layout of your classroom. This routine simply saves time on a job that needs to be done. If I didn’t do this routine in a certain way then time would be wasted with books going up and down the row because a pupil could have another pupil’s book. The first pupil will waste time sifting through the pile of books to get their book. The time wasted accumulates to become hours. Pupils get frustrated filtering through books and I get annoyed watching this happening.

In summary:

  • Provide pupils with explicit instructions on how to do the routine before they hand out the books
  • If the pupils are doing the routine poorly then send the books back to the windowsill and get pupils to do the routine again. Keep doing the routine on a daily basis.
  • When pupils are passing out books you need to be vigilant. Watch the pupils. Praise the ones who are ready early. Nudge the pupils who are taking their time.

Here is a clip –

https://drive.google.com/file/d/0B3tfMNA84AZoakZPNkJnMkdtYXc/view?usp=sharing

How to use Mini-whiteboards (MWB).

Using mini-whiteboards in the classroom can be stressful unless you have taught pupils how you want them to use them. Here are a few questions to think about beforehand:

Where do you want the pen to be when you are talking? In the pupil’s hand? On the table?

Where do you want the pen lid to be when pupils are using their pen?

When pupils are writing on the mini whiteboard do you want it to be on show to their peers or hidden away?

When pupils have completed their work on the MWB do you want the board up on s

how or hidden?

These are all questions that I didn’t even think of when I started using MWBs in my first year of teaching. Here are a few routines to put in place to help pupils know what to do:

Using their MWB pen

“When I say go and not before, you will take your MWB, take off the pen lid and place it at the end of the pen. This is because we don’t want our pen lids to go missing. I am going to give you 5 seconds to do that and show me in 5. 5-4-3-2-1-and show!”

This tells pupils what to do when they are about to start the mini whiteboard activity. This also doesn’t cause 30 MWB pen lids to go missing.

Closing their MWB pen

“When I say go and not before, you will close your MWB pen by taking the pen lid and placing it onto the pen. We want to click the pen and pen lid. I am going to give you 5 seconds to do that and I want to here the click in 5-4-3-2-1-and click!”

This prevents 30 pens drying out over time. Pupils now know what to do when they are finished with their pens. Also, they know how to check that their pen lid is actually on because they will hear a click.

Using their MWB

“When we use our MWB we don’t want to show our working out to anybody else because we don’t want to encourage any cheating. I trust that nobody would want to cheat because you are not really letting yourself learn. It is ok to not know the answer because it is my job to help you. So when you do your working out you have to hide your MWB. You want to keep it a secret. In three I want you to pretend to be writing on your MWB so I can see who is the best secret keeper?

This instruction is explicit in the sense that the pupils know how to do their working out. I have also narrated the purpose behind why I want pupils to be hiding their working out. I have also tried to discourage any cheating. There is also an opportunity for pupils to demonstrate doing their working in the way that I want and the competitive edge gets kids keen to do their best version of the routine. Offer some praise when the kids attempt the routine:

“I think Emily is the best secret keeper because she is using her arm to cover her MWB.”

What do you want pupils to do when they have completed their working out?

“When you have completed your working out you still want to hide your work. You will hold your MWB so the working out is facing your desk. That way nobody can see it. You will hover your MWB above your desk and track me. That way I can see that you are ready. We do not put our MWB up in the air because that means somebody else can see your work. I’ll give you a warning and if you do that again then you’ll have a demerit.”

I would then give pupils an attempt to try hovering their MWB.

“When I say go and not before, I will give you a question to complete on your MWB. Once you are finished hover your MWB. When I countdown from 3, you will put your MWB up in the air when I say ‘show’ and not before.”

If pupils aren’t putting up their MWB altogether then I would get pupils to do it again

“We aren’t showing our MWB as a team so do it again. 3-2-1-and show! Better, I have 100%”

These are just a couple of routines that can be applied in your classroom. The main take away is that your instructions have to be explicit for pupils to understand what they need to be doing at each and every stage of the routine. Explain to pupils what the routine should look like. Explain to pupils what the routine does NOT look like. Narrate the purpose. Add a bit of fun factor to the routine. Time the routine so pupils know that they have to act speedily. Pupils like to be told what they need to be doing. You are happy and so are the pupils.

 

Gladwell: The English number-naming system is highly irregular

I’m currently reading Outliers: The story of Success by Malcolm Gladwell and in one chapter there is a brief extract from Stanislas Dehaene’s book, The Number Sense. This chapter discusses why pupils in China, Japan and Singapore experience less confusion when learning elementary maths. Gladwell states that “a part of the disenchantment is due to the fact that math doesn’t seem to make sense; its linguistic structure is clumsy, its basic rules seem arbitrary and complicated.”

Here are a few key differences  between number-naming systems in Western and Asian languages:

1) Chinese numbers are brief

For example, 4 is ‘si’ and 7 is ‘qi’ which takes a quarter of a second to say compared to pronouncing the same numbers in English which takes a third of a second. Dehaene believes that “the memory gap between English and Chinese apparently is entirely due to this difference in lengths”. The Cantonese dialect of Chinese allows a memory span of about 10 digits.

Here is an experiment conducted in Dehaene’s book. Attempt to memorise this list of numbers: 4, 8, 5, 3, 9, 7, and 6. English speakers will have a 50% chance of remembering this list perfectly. Chinese speakers will list the numbers correctly every time. This is simply because “as humans we store digits in a memory loop that runs for about two seconds. We most easily memorise whatever we can say or read within the two-second span.” Chinese speakers are able to fit all those numbers into two seconds

2) The English number-naming system is highly irregular

The numbers above twenty such as twenty-two, thirty-five etc all follow the structure where the ‘decade’ comes first and then the unit number second. Compare this to the teens (18,16) which is the other way around where the unit number comes first fifteen, eighteen. China, Japan and Korea have a logical counting system, examples:

Eleven: ten-one

Twelve: ten-two

Thirty-five: three-tens-five

3) Asian children can perform basic functions (e.g. addition) far more easily because of the regularity of their number system

English speaking children need to convert the calculation of thirty-five plus twenty-three from words into numbers to then complete the addition. Compare this to adding three-tens-five and two-tens-three makes it easy to do the calculation since the order of the digits is within the sentence.

These are just a few facts that I was not aware of before reading Gladwell’s book. I highly recommend it for many more reasons as well as this one section about mathematics.

References:

Gladwell, M. (2008). Outliers: The story of success.

Dehaene, S. (1997). The Number Sense: How the Mind creates Mathematics

 

#Mathsconf10: Worksheet-Making Extravaganza

On Saturday, I spoke at #mathconf10 in Dagenham. The workshop was titled “Worksheet-making Extravaganza!” Indeed it was that.

When I started teaching I regularly went onto TES to download resources for my pupils. Usually, I would be annoyed that I couldn’t find a good enough worksheet because the worksheet I wanted would never be available. I then realised that I needed to start creating my own. However, in the process I realised that I found making problem types for different topics to be difficult because my subject knowledge wasn’t up to scratch. Now, I am very good at maths but that doesn’t mean that I am good at making a worksheet of questions for pupils to do and learn from. Then after spending a summer between my first and second year of teaching where I made different resources combined with my own experience of working at Michaela Community School I finally realised how to make my own worksheet. Here are the two things that make a good worksheet:

  • High quality content
  • High quality structure (deliberate practice)

What do I mean by high quality content? I am referring to the questions that are made to test a pupils’ understanding of the concept or procedure that has been taught. What is the starting point of gathering this content? If you are planning a lesson on how to simplify fractions then list out all the problem types from the easiest to the most difficult one. More problem types can be found by looking at different textbooks, asking colleagues and reading blogs etc. Let’s look at the problem types that can be included when planning a lesson on simplifying fractions, simplify a:

  • proper fraction
  • (an) improper fractionImage 1
  • fraction that simplifies to a unit fraction
  • fraction that simplifies to an integer
  • mixed number
  • fraction with large numbers (still divisible by a common factor)
  • fraction that simplifies to 1

It is all well and good gathering a bunch of problem types but I have deliberately chosen the ones I have listed. This is because these problem types present four features that make the content of worksheet high quality:

  • Arithmetic complexity
  • Visual complexity
  • Multiple steps
  • Decoding

Arithmetic complexity basically means including more difficult numbers or large numbers in your questions. Can a pupil simplify one hundred and eleven-thirds? Also, it means including questions which have decimals and fractions, and creating questions where the answer is a decimal or fraction as well.

Image 1aVisual complexity refers to creating questions which look a bit scary. Again, this can entail including large numbers to make a pupil think. For example, in terms of calculating the area of a square you can write a question where the length is a large two digit number or a decimal.Image 2Another example is to only label one length of a square, more than two lengths, four length etc. A question like this is testing whether a pupil can calculate the area of a square from squaring one length. Another very simple approach to make this type of question visually complex is to have the image at a certain orientation.Image 3Multiple steps is a feature in a question where you have increased the number of steps between the question and the answer. This can include having calculations in the numerator and denominator before a pupil is asked to simplify a fraction.

Image 5Decoding is a feature of a question where its set up is testing whether a pupil can rearrange or manipulate their new found information of the concept or procedure that has been taught. For example, providing pupils with questions which have incomplete answers. Or asking if the following equations are true or false?

Image 4 Image 6An exercise on a worksheet can look something like this. Question (g) is testing a pupil’s misconception where dividing a number by itself is not 0 but 1. Similarly, question (k) is testing a system 1 mistake where a pupil is not fully thinking about the question, they simply see that the half of 18 is 9 therefore the answer is 9.

Image 7Can I apply these four features of high quality content when planning a series of lessons where I want pupils to learn how to add and subtract fractions with like denominators? Again, start with listing out the potential problem types. Adding or subtracting:

  • Two proper fractions where the result is less than 1
  • A proper and an improper fraction
  • Two proper fractions where the result is greater than 1
  • Two mixed numbers where the sum of the proper fractions is less than 1
  • Two mixed numbers where the sum of the proper fractions is greater than 1
  • Two mixed numbers where the sum of the proper fractions equals to 1

Image 8

I can make these type of question arithmetically complex by having large numbers in the numerator and the denominator. I can include a question when I subtract two proper fractions and the result is negative.

Image 9Questions can be visually complex by including more than two terms, or a string of fractions including fractions, mixed numbers and different operations.

Image 11Similarly, making questions where pupils are asked to add or subtract a proper fraction with a mixed number can be visually complex too.

Image 10Questions where pupils have to rearrange their information of adding and subtracting fractions with like denominators can include true or false question. The first question below is conflating adding a numerator and multiplying the denominator.

Image 7bSimilarly, it is a really powerful form of testing pupils’ knowledge by including questions where the answer is incomplete. Can the pupils identify what the numerator is? Can pupils identify that 1 can be written as 10/10 and that 2 can be written as 20/10.

Image 12The following questions demonstrate the feature of questions having multiple steps between the question and the answer. Yes, you can easily write the answer, but if you wanted a pupil to write their answer as an improper fraction, you would want them to write the integer as a fraction.

Image 13Lastly, can we apply the following features to creating high quality content for the topic of powers and roots with fractions. Here are the problem types:

Image 14

Image 16

Image 17The mixed number examples are tailored so pupils are able to create an improper fraction where the numerator is still less than 15. Why? Simply because the kids at Michaela have committed the first 15 square numbers to memory as well as the first 10 cube numbers, but if I made the fraction more difficult where the numerator was greater than 21 then I would be making the question difficult for the wrong reasons. If the fraction was four and one-fifth then the numerator would be 21, and squaring 21 isn’t valuable. What is valuable is spotting the first step of writing the mixed number as a fraction, and then squaring the fraction. The mixed number examples demonstrate questions being visually complex as well as arithmetically complex.

Image 15The following question is demonstrating questions which have multiple steps to the answer, where calculations in the numerator and the denominator need to be simplified before squaring.  Again, calculations were designed where the numerator and denominator values would be less than 15.

Image 18The same form of thinking in applying the three features (arithmetic complexity, visual complexity and multiple steps) can be applied when fractions are being square rooted or cube rooted. However, ensuring that the numerator and denominator values are numbers that can be square or cube rooted.

What is new is including negative fractions when they are being rooted by an odd number. Furthermore, visual complexity as well as multiple steps to be performed to get the answer make these questions challenging and rigorous.

In summary, the four features which make a worksheet consist of high quality content is creating all problem types where they are arithmetically complex, visually complex, require decoding of current information from instruction and multiple steps to be performed to result in the answer.

I am still trying to wrap my head around creating a worksheet with high quality structure which will be my next blog post.

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