Bringing my learning from 2013 into 2014

Yes life is a little quieter at the moment and it is time to reflect on 2013. What worked, what didn’t and why? It is also time for me to be planning for the massive year ahead.

A large part of my professional development work next year will revolve around pedagogical moves in the maths class. During 2013, I tried out a few ‘moves’ one of which I call:

‘Do next to nothing’.

That sounds lazy but let me tell you it is hard work and it takes bravery and careful planning. So what do I mean?

In October, I wrote about the ratio and a pizza problem I gave to a Year 7 class at Mawson Lakes Primary School in Adelaide. The class had never done ratio and the fraction part was quite tough too. The students were asked to work with a partner. They were to listen to each other try to make sense of the problem and then explore ways of solving it. The main rule was that both partners must clarify their thinking and be able to explain the work that they did. They were also told that there would be no teacher assistance (even though there were 12 of us in the room).

After 10 minutes of agony, all sorts of incredible thinking and trial and error and a wide range of strategies were being trialled and the results showed that students could actually talk their way into an understanding of ratio and fractions.

Another example of ‘do next to nothing’ was a strategy lesson with Year 4-5 class at John Hartley Primary School. The Australian Curriculum Mathematics suggests the use of the area model for multiplication. I decided to put a worked example up on the board and again use partners with the rules outlined above.

I put the following worked example on the board and told the students that they had as much time as they needed to work together to find out how the model I had drawn worked and why. This was their first introduction to the area model. Initially there was stunned silence and then some ‘light bulb’ moments.

This was when the next pedagogical move kicked in:

‘Get the students to do the teaching and explaining’

Partners were asked to volunteer to come up to the board and explain one step at a time. The remaining students gave thumbs up, thumbs on the side or thumbs down to le t the volunteers know if they had understood the step.

A second partially complete example was put up and the process repeated. A few students who had struggled with the first example were now feeling secure enough to volunteer to explain a step.
As we listened to the students working together we gained many windows into student thinking, which included some place value issues. – 26 was easily partitioned into 20 + 6 but 15 was partitioned into 1 + 5 by a few students. This was soon rectified by listening to and questioning the volunteers.



The above examples were put on the board and partners were told that they could select the one that they wanted to try. Many students moved onto the two harder ones immediately and had no difficulty realising that there boxes would be needed for three digits not 2 as before.

The lesson went too fast and was too much fun. When I asked the students why they thought I was mean enough to put the multiplication on the board without explaining it to them, comments such as the following were made:

“Because you wanted to see what we could do without teacher help.”
“Because you wanted us to feel proud and successful.”
“Because you know we learn better when we explain for ourselves.”
“Because you wanted us to have fun working with a partner.”

I asked the class to give a thumbs up if they thought they learnt better without a teacher demonstration than they would have done if I had demonstrated and given them lots of examples to practice on. Their response was a resounding thumbs up.

Last year I was thinking a lot about Carole Dweck’s work around fixed mindset and growth mindset. It is my belief that because:

• The tasks that I set students are challenging 
• I expect them to work together 
• There are no winners or losers and struggle 
• Sharing is the norm.

I am seeing less and less fixed mindset even in those students who used to perceive themselves as good at maths and stick with the safe methods.

Next year I intend to push this even further and try to say even less than I did this year. I am also coming to the conclusion that a lot of teacher questioning is really just telling and playing ‘guess what’s in teachers’ minds, and, what is more, it interferes totally with the students’ thinking. Student questions are so much more interesting to them and to me and they usually lead to learning that they are ready for.

Making maths fun with Pop Beads

I was out shopping when I came across these pop beads.

Pop went my brain too – bringing maths and ‘The Brain likes Colour’ to mind.

You see, I am working on a Place Value to 100 and Beyond package at the moment in which one of the central themes is purposeful counting. Purposeful counting of large collections is a sure fire way for students to begin to see why we need speed counting (skip counting, if you prefer) as well as why organising in 10s makes sense. These little pop beads have 170 in a packet, plenty to share and to count and sort.

Make 50

My first game idea for two players was a collaborative one (which you can differentiate by changing the target number to suit the players). On their turn players dip into a paper bag and take a handful of pop beads. They set them out to see how many they have taken.

The next player then decides whether to take a small handful or a large handful because the aim of the game is to have 50 pop beads by the end of exactly six rounds. Players confer, and may use a 100 square to help them plan their next ‘take’. It might well be that for the last three goes players take only one pop bead but that is fine. Over time and through discussions about counting on from any starting number by 10 the ‘takes’ should become more evenly distributed. In fact you might invent a rule that says you may not take a single pop bead or even as few as two.

Of course you could do this activity with other materials but I am looking for an excuse to go shopping for appealing, unusual and very colourful maths resources that don’t break the classroom budget. If you find exciting cheap colourful and safe items, please add to the list. Let’s get stocked up for next year!

The brain likes colour

Back from England, refreshed and ready to go. I’m now wondering what supplies and ideas I can play with next year.

In an earlier post, I talked about how maths can sometimes be presented in a dull and grey way. We know the brain likes colour so why not include more colour in the maths lesson? One thing led to another and I began to think about a supplies list for maths next year.

Top of the list: coloured paper and mark making equipment. Checkout the piece below and you will begin to see that hole punches and glue sticks might be fun too.

Also on the list, brilliant picture books, even with older students. The butterfly was triggered by, 'Lots of Spots' by Lois Ehlert. I have written to her to ask if I can create and publish some activities based on this book.

As I looked at the book, the combination of fun, colour and  maths  jumped out at me (yes I know this is not a maths book!). I was seeing symmetry, subitising, counting, comparing, estimating and fractions with mental computation strategies all in just a flick through.

Then of course I had to play. I folded a piece of paper in half and drew a butterfly. Then I punched some holes. Before opening it I used a doubles fact to work out how many holes I would see. I also visualised roughly what the butterfly would look like when I opened it up. Then the moment I opened it and looked at it, I thought it was a bit boring so punched out some holes in a different colour and placed them carefully so that the line of symmetry was not disturbed.

As I looked, I couldn’t help but wonder what fraction of the spots were yellow. But there was still something missing...

We know we need to make connections between mathematical ideas and to consider the ways in which different aspects of maths are related. This little exercise crossed many normally atomised areas of the maths curriculum into one fun but rigorous activity. It links visual imagery, symmetry, fractions, halving and doubling for number facts and estimation as well as subitisation into one easy to differentiate activity.

Let me know what you think and if you try it with your class. We’d love to see pictures, which you can send to Sarah.



'Lots of Spots' by Lois Ehlert

Teaching problem solving strategies - achieving great outcomes

Having a sound understanding of maths is important from childhood through to adulthood. However, a lot of adults frown with confusion when presented with a problem solving situation. Memories of their school days trigger feelings of failure, lack of resilience ‘this is too hard’ and lack of executive planning function in terms of being able to find a way through and finding the steps required to solve the problem. 

We can do better than that for our students. 

We can use the STAR model explicitly, to help them sort out what the problem is about, what info they need to use, what the problem is asking them to find out and what the answer of the problem will actually tell them. We can then make it explicit to the students the thought and planning processes required to be able to get started on the problem. We ask “have we done a problem like this before?”, “what kind of problem is it”, “do I need to draw a table or work backwards, spot a pattern?” – these are questions we can encourage students to ask during this incubation time. 

If the ‘S’ and ‘T’ are well done, than action will follow fairly smoothly. Reflection is the crucial step where we can ask students what worked and what didn't work and what they can use if a similar problem was to come up again. Students can’t tackle the ‘think about it stage’ particularly well if they hadn't had earlier problem solving experiences where they have learnt how to apply strategies such as draw a picture, act it out, spot a pattern, select an operation, and so on. 

Structuring problem solving experiences with all of these steps in mind will ensure students have a positive disposition to problem solving as well as a well-equipped tool box to choose from. 

Contextualised maths problems: Santa's cooking gift sets

When it comes to teaching, there's nothing  more satisfying than seeing students get excited about maths as it finally 'clicks' for them. We're all to aware that each student has a different understanding and appreciation of maths. Therefore, it's important we teachers provide different entry points that allow each student to find their own way into a problem and use their own strategies. We need to promote that's it's not a race to the answer, rather they should take the time to think about what they have done before that may help them (our STAR model is handy here) and take risks to develop strategies that suit them.

Using problems that have context that is meaningful to them (so they can to relate to it) is a great way to further engage the students. So here it is, a problem with a Christmas theme:

Father Christmas was testing his favourite recipe while his elves were busy packing childrens' gift sets that Santa promises will have every child cooking up a storm in no time! Each set has 3 knives, 2 wooden spoons, a cutting board, a mini mixer, an apron and of course a Santa cook book, 'Santa's recipes made easy'. 

“Make me 15 sets right away and be quick.” Santa told the elves between mouthfuls of tender, tasty turkey.

The elves groaned: “That is a lot of sorting and packing!”

How many items do the elves have to pack into each set?

How many items will the elves have to collect from the shelves?

We'd love for you to share the results on Facebook: Natural Maths Facebook page

make connection between mathematical ideas and maths and how it’s used in the real world.

Reindeer Numbers

We delivered Christmas maths problems for earlier grades in a previous blog post recently, and now here's one for upper primary. 

This one will really get the studnets things:

One of Santa's best kept secrets is that there is a special link between a reindeer and its number. Indeed, a reindeer number has the property that it is equal to the product of two of the numbers (known as antlers) that can be made with its digits. For example, 1827 is a reindeer number with antlers equal to 21 and 87, because 1827 = 21 × 87. Sometimes one of the antlers gets broken (i.e. is 1 out) like
3456 which is the number of a reindeer with a broken antler:

3456 = 64 × 54 and just one digit is 1 out.

On the night before Xmas, the reindeers with these reindeer numbers reported for duty:

1827 2187 1435 3456
1932 2496 6880 8190
1530 3864 1395 7189

"It's wonderful to see you all," Santa said... "but I can only use reindeers whose antlers are not broken. It's a long journey, and you will need all the antler power that you can get if we are to deliver all the presents."

Ask students to help Santa sort out which of the reindeers he can use to pull his sleigh.

"Now, where is my friend Rudolf?" Santa said. "I can't go without him!" Rudolf has the smallest reindeer number and a complete set of antlers, but he hasn't reported for duty yet. What reindeer number should Santa go looking for if he is to find Rudolf?

There will be loads more Christmas themed problems to come that we'll share via our blog in the lead up to Christmas so stay tuned. 

Helping students gain a positive appreciation for maths

I'm often asked which key topics teachers can pay special attention to when it comes to maths. Students need to develop a strong numbers sense from an early age built on understanding of quantity.

They need to be able to pull numbers apart in many different ways so they can engage in value based not digit based mental computation strategies. From year four, students really need to have a deep understanding of fractions based on multiplicative strategies not additive strategies.

From the early years, notions of proportional reasoning need to be informally developed so that by year six students have an understanding of proportional reasoning and multiplicative reasoning.

You've heard us say it before, but we'll say it again: you can help students gain a positive appreciation for maths by giving them time to really think about a problem.

If we build on students’ earlier intuitive thinking, we can ensure all students like maths and are successful. Avoid 'rescuing' students and treating maths like it’s a race - trying to cover the curriculum rather than developing a deep understanding. Later maths is built on earlier mathematically building blocks. Enable the students to develop a good foundation and the rest will follow.