When using bigger numbers e.g. 91, be careful with your word problem If you use 91 students, we see 91 individual people. If you use $91 dollars, then we imagine nine $10 notes and one $1. The students automatically imagine the groups.
When students are first learning something we need to give them structure (which is why using money is good) but once they have the learning then we can use other contexts e.g. 53 children are on the bus and 27 get off. How many are still on the bus. Once students are fluent in their learning, they can now generalise and impose the structure themselves.
Introductory questions need to link to the equipment/structure of the numbers e.g. packets of biscuits (10 in a pack) or a packet of lollies (10 in a pack)
The second step -is to push without the scaffolds e.g children on a bus question mentioned above.
For students struggling to order simple 3 digit numbers - go back to can you make 86?
Telling the teacher what to get to make a number (if they say an 8 and a 6 say no that’s 14, I need 80 and 6). Explore the 2 digit place value understanding and then they can transfer this to 3 digits quite easily.
For students struggling with 35+8: Use models that would work - 10s frames or packets of biscuits "There are 10 in each pack - how many have I got here? I have 8 more biscuits (counters). What should I do?" They will tell you to fill up the 10s frame/packet - 5. "How many left?" 3. So my answer is 43. When you are teaching something for the first time go back to the place value structure.
Practice models and teaching models:
10s frames as the first choice as they are visible.
Next use beads in packets of 10 (they are hidden). "This one is full, this one is full, this one is full this one only has 5 in it. I pull out 8 jellybeans what should I do?" Students can still see 5 more drops into the container. "How many are in my hand?"
Use believable examples e.g. iceblocks in a freezer.
I have 36 and I get 8 more - tell your buddy what happens. Self-directed at this point
Next they need mileage - keep practising it with them for days - they need to do them quickly and without help (this is called 'fluencey')
Write 38+ (students need to know straight away they are going to need 2 to make the next group).
If you get students to think there is a pattern there they will find it. "what happens if the number was 36 what would I need?
As scaffolding goes down - variation goes up turning :I can do it" to" I am fluent."
Once students know th system and are getting fast use the claim game e.g. using a deck of cards, each player flips over a card, every time the cards add up to a multiple of 5 say "Claim" and pick up the pile. Don't tell the students the rules - see if they can work it out by themselves as you play. The idea of the game is to add really quickly.
Level 3 students need to read large numbers. Use a sheet of A4 paper - simple works best. Fold a piece of paper in three pieces - write three numbers - unfold the paper and read it as one number
28,451,862 - in the gaps we say millions, thousands, etc.
Subtraction:
When they get to triple numbers use the algorithm - before this use the maths.
Then we can investigate different strategies: algorithms, reversibility, rounding.
For algorithms use a story like:
"I went to an ATM machine. It only had 100 bills and 10 bills - it was an AI one and it had attitude big time.
On Monday (because it’s a smart ATM, I didn’t have to use a card). It asked me what I needed (Level 1 place value), I said I wanted $420. It gave me:
4 hundreds
2 tens
On Tuesday - I went again and said I wanted $420. The ATM machine said to me, "If you can tell me another way I can give you $420, I'll give you $420. (Level 2 place value)
42 ten dollar notes
On Wednesday I went again and said I wanted $420 and it said, "Ok but I won’t give it to you like Monday or like Tuesday." What did it give me?
3 hundred
12 tens
Repeating this process - swap 100 to 10 10s.
You want them to get to the 12 tens and 3 hundred etc
$510 - 3 ways of making this number. Once all the students can verbalise this then we can do this subtraction problem.
I have $510 but I owe … $140 and she wants a 100 dollar note and 4 tens. What would I need from the ATM machine …
4 hundred 11 tens
4 11 0
-1 4 0
3 7 0
This is where we can show what the algorithm is showing us - this is where we can show the faster way of writing it.
Without this lesson they do not understand why we do this or why it works - they trust themselves first!
90/10 Rule - Teaching children top end of level 3 -
Everyone gets a 2 digit number:
e.g. 47 61 38 24 79
Tell us the number to get to 100 (90/10 Rule).
Students say the pattern to get to 100 - 90 and 10.
What goes with 626 to make 1000?
377 to get to 1000.
Place value is always in collections of 10 - "You are all accountants. The budget for the project is $1 million.
You have spent $329, 876. How much do you have left? You have ten seconds to answer. The boss is walking down the corridor!"
This is a fantastic bridge to decimal fractions
9.2m - 3.615m (take the decimal up to the next whole number).
3.615 to 4 is 0.385 then add the 5.2 onto it to get 5.585m remaining.
Use this rule again and again - it is faster than an algorithm - round it up works using the 90/10 Rule.
Staff Meeting - Numeracy and Pedagogy
Acceleration and Planning:
Acceleration
Prior meeting was on planning from a big picture view and the key progressions. The planning today is the planning within and between the lessons when you have already decided what the teaching is going to be.
Acceleration - Bruce’s idea is that we don’t accelerate kids what we are hoping to do is not need it! That we don’t have any kids not keeping up. Long term we don't want to have remedial programmes in maths. If you can name the children who are struggling in maths or groups or cohorts who are struggling then we do something about it. We are looking at saying that we are going to move the vast majority of kids on - we will always have strugglers but if you can name them then we are winning!! The numbers are small if you can name them.
The math teaching you do is for a deeper, longer lasting purpose - our aim is that they are independent from us!
It’s about throwing out the stuff we don’t need and doing the stuff we do need very very well.
Acceleration is the rate of learning against time - those students who need acceleration haven’t learnt very much or very quickly. They’re pretty much flat lining. What can we do to shift them??
First and most important is to identify, do the diagnostics with them. Find the starting point!
Own that starting point! Don’t ignore where the child is - you will do them a disservice. It’s not what you taught the child it is what they have learned.
It’s about finding out the last thing that the child actually understood, that is then what you use to take them to the next step.
So we identify the key progressions needed for that child’s learning and it may be a Y4 student going back to doing place value at the Y2 level. AND CELEBRATE that progress! Once the children start that process then we can get some acceleration.
Acceleration happens because once the children develop the key understanding then the work they have already been exposed to throughout the year can actually begin to make sense and they can, most times, learn the rest quite quickly.
To get acceleration you have to be prepared to take the step back in order to take the step forward. You have to keep trying to get the children to make the connections faster so the learning is accelerated.
Year 1’s we are teaching, those kids who are on track we keep on track and keep on pushing ensuring that we are covering the curriculum, those who are behind we FIND THEIR STARTING POINT
One of the things will kill acceleration is if what we are teaching is not what we are assessing! Assessment must mirror the learning that has occured. Assessment should also be grounded in what the curriculum identifies what the students should be learning.
Don’t undermine student confidence by not accessing the student’s learning!
Assessment should be an affirming process for our students not a discriminatory process.
Planning
Long term we want to apply the same principles wherever you are teaching. There are a number of simple repeatable things we do that work.
You already know your starting point.
Piaget’s disequilibrium model - Piaget described learning in a line. Learning involves stress but not to the point of distress. The teacher induces stress by increasing the demand. This is normal. If you know what the child knows then you can reaffirm with the child that they know how to do this. You make the situation more comfortable like maybe bringing out materials to aid them Now the activity becomes enjoyable.
Need to allow learners to get things right again and again and again!! Don’t make it harder, let them feel success and know that they can get it right until they get to the point of fluency. If it always gets harder and harder students lose incentive, motivation and belief in themselves as a learner. Let the kids get to the point where they just nail it! Practice is important - do it to an effective depth!! We do have time to get them to fluency! If we recognise and know what is necessary, we remove excess useless teaching and this frees up time.
We know what our students ZPD (Vygotsky) is then we know they will be able to learn! If the teacher is confident than that is transferred to the students - “Oh she thinks I can do this ergo I think I can do this”.
Didactical or teaching purpose - if your math lesson is light then your students will do better. Having a laugh is actually helpful, it de-stresses and aids rational thinking.
Barely sufficient scaffolding - is about supplying only just enough help and no more!!!
Teacher lust - the almost irrepressible urge to go and help someone! We don’t shut up and we don’t stop helping. We need to know and recognise when our help is needed and when it is not. We can’t develop independent learners if we have this!
The help we supply to children over time impacts their independent level of work. We transfer the ownership of the math problem from the teacher over to the student so that they are doing the maths and NOT YOU!! Give them more than one at a time - allow them the freedom to go ahead and get on with it. Put up multiple problems for them to do. Check by asking if they all have the same answer - you don’t have to mark it. You only come in when the answer is NOT the same and you can go in and help figure out how they solved it.
Children should be allowed to work together and figure it out together, talking and trying out their ideas. Teachers need to ensure that they provide enough time for the students to work together, wait before providing help - give them sufficient time. When they start becoming fluent let them just get on with it. Plan to differentiate within your lesson - let some with fluency just get on with it and teacher helps those who are still hesitant.
Robert Sieglar - real children don’t learn in stages, there are ups and downs, gentle rises and steep inclines. There are no clear cut demarcations on how children learn. Children start off with a particular strategy they use all the time as they know it. The teacher introduces a new way of doing it ( a new strategy). The children then enter a new zone where they can choose to do it the new one or to do it the old way. We only find that an issue if we have the mistaken belief of the children learning in stages. What we have done is allow them a choice which they use. Eventually they will change - the teacher's role is to help them make the choice. Effective, logical and efficient will be the three drivers of change. If the students recognise the new way of things only if these three factors are present! We deliberately must attend to these three drivers of change. Itf the students do not believe in what you're saying then they will not tune into the new learning - they will do it with you on the mat but independently will turn back to the prior strategies.
MATHS IS ABOUT MAKING THINGS EASIER FOR CHILDREN NOT HARDER. MATHS TURNS HARDER THINGS INTO EASIER ONES.
