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.

Maths problems using Christmas fun

While Christmas is a little while away, it will be here before we know it. So we'll share some of our favourite maths problems - combining maths and Christmas that will certainly get the students equally enthused about the two topics. These are for grades 1 and 2 - we'll share problems for higher grades throughout the month so stay tuned.

Whizzing Yoyos
Mrs White wants to give every one of the 20 children in her class a whizzing yoyo for the Christmas
yoyo trick competition but she doesn’t know how many packets of yoyos to buy.

There are 4 whizzing yoyos in a packet and each packet costs $5. How much will it cost her to buy
the yoyos?

Lucky Dip

Mrs White bought 6 bags of lucky dip prizes and found that there were 4 prizes in each bag. She

wants to share the toys fairly into three lucky dip barrels.

How many toys should she put into each barrel?

Secret Santa

I need to buy 12 Christmas cards for the secret Santa we are having in the staff room and I was wondering which will be cheaper, buying cards in packets of 2 for $5,or packets of 3 cards for $6.60. Oh I just noticed that the packets of 6 glitter cards for $18 are marked down $3.

Which cards should I buy?

Xmas Muffins

I found this decorated Christmas muffin recipe. It says that for 6 muffins you need 36 stars, 18

walnuts and 12 silver balls.

I want to make 24 muffins though. So how many stars, walnuts and silver balls will I need to decorate the tops with?

Santa’s Reindeers

Santa’s nine reindeers are ready to be harnessed to his sleigh.

The leading reindeer is on his own and after that they are harnessed in 2 lines of 4, making 4 rows of 2. But where should they be placed?

Reindeers with the same initial quarrel so badly that there has to at least one row of other reindeers between them. These reindeers are also arranged alphabetically, front to back.

In each row, the reindeers are arranged alphabetically left to right. For example, Prancer would be to the left of Rudolf if they were in the same row, which they are not, and Prancer would be to the right of Comet if they were in the same row, which they are not. The reindeers that come first and last alphabetically are such close friends that they have to be in a row together.