By Tierney Kennedy

I firmly believe that the most important and most effective thing that we can do with our classes is problem-solving. And yet far and away the biggest complaint that I hear teachers express on Professional Development sessions is their lack of time. So here is my proposition to you…

*What if I could magically make an extra 40-60 minutes of maths teaching time in your week, every week?… Would you then promise to take time out of your existing maths schedule to do problem-solving with kids?*

If you can spend the five minutes that it takes to get to the end of this article then I promise to do exactly that. To get to this, we need to start by thinking a little “outside the square” and enter the realm of philosophy.

A philosopher entered into his class and placed a large glass jar on the desk. He placed several large rocks into the jar until there was no room for any more. He posed the question to his class, is the jar full? They all agreed that it was.

The philosopher then reached into his bag and removed some pebbles which he proceeded to tumble into the jar, surrounding the large rocks. Repeating his original question the philosopher asked if the jar was now full. The students agreed that it was indeed now full… until the philosopher removed a sachet of sand and began to pour the sand into the jar, filling up all of the space between the pebbles. When the students posited that surely now the jar must be full, the philosopher produced a glass of water and added this to the jar as well.

His point was simple and fairly well-made: to have a full life we must firstly add the important things. These alone make for a full life, but once these are in place we can also add in extra things. If we start by focusing on the small issues – the sand of life – then we will never be able to fit in the large rocks.

This same thinking can be applied to a maths classroom to great effect.

Throughout our year we have certain large rocks – the most important things that need to be done well and have large amounts of time devoted to them. These underpin everything else, and if they are missing our students will suffer.

We also have pebbles. These are the annoying, irritating things that tend to eat up large amounts of our time and that are unavoidable. NAPLAN, report cards, student behaviour, implementing various policies and reteaching previous concepts might be examples of some of these. I’m sure you can think of numerous examples. If we get caught up with these then we risk not getting the most important things done. Think about how many schools you know that abandon all good teaching in the month prior to NAPLAN. We simply don’t have the time to allow a pebble to steal an entire month of our teaching from the big rocks!

*The magical time-saving that I am talking about though comes not from the pebbles, but from the sand… *

Try adding up the amount of time each week that you spend on the following:

- Lining up (e.g. waiting for the music teacher to finish so that your class can go into the music room)
- Waiting to start a lesson – making sure that everyone has arrived rather than having the stragglers miss out or having to begin all over again.
- Waiting for the 3pm bell – having packed up, sitting with the kids and filling in time until they are allowed to be dismissed.

For the average primary-school teacher I calculate that this time is often in excess of an hour each week. For the average high-school teacher, I estimate that this time is at least 25mins for each class.

*So here’s my proposal: If we could put this “sand time” to good use, then we would magically create an extra hour of teaching in our weeks, every week!*

**Making use of sand time:**

My premise is that Fluency is actually best taught in regular, small packets of time rather than in dedicated hour-long lessons. Spending long periods of time practicing or memorising is boring. Spending two minutes on a game is much more fun, and spending five two-minute segments of time on a fluency game during the course of a day is often far more effective than one ten-minute segment.

Here are some of my favourite two-minute fluency tasks that can make great use of your hour of “sand time” freeing up larger rocks in your schedule.

**When outside your classroom****:**

- Place two or three dice in a clear plastic container to keep with you. Quickly roll the dice. Have the kids use the numbers that come up to get as close as possible to a particular number using any operations that you define and some or all of the dice e.g. having rolled a 3, 4 and 5 try to get 20…
- 3 x 4 + 5 = 17
- 3 x 5 + 4 = 19

- Place two dice in a clear plastic container to keep with you. Roll the dice to get two numbers. The students pair up, and each partner takes a turn coming up with a third number that would go with the first two e.g. having rolled a 3 and a 5…
- 15, because 3 x 5 = 15
- 2, because 5 – 3 = 2
- 1 and 2/3 because 5/3 is 1 and 2/3

- Have students work with a partner/s to make a particular number with their fingers. They need to use some or all of their hands depending on how tricky you want to make it e.g. make the number 5 using…
- All four hands (2 + 1 + 1 + 1)
- Three hands (3 + 1 + 1 or 2 + 2 + 1)
- Two hands (1 + 4 or 2 + 3)

- Having rolled two dice, think of at least three numbers that would have both of the dice numbers as factors e.g. having rolled a 2 and a 5
- 10, 20, 30

**When inside your classroom****:**

- Play “snakes and ladders” with two dice. Kids can choose to move forwards by both, backwards by both or forwards by one and backwards by the other. This helps them to strategize how to get around the board, by avoiding the snakes and aiming for the ladders. You can even have a magnetic version of this on your wall/board and have students each lesson take turns moving a particular piece. The game is therefore ongoing and does not need to finish within a certain time period.
- Play “what happened?” Choose a starting number and an ending number and students have to work out what operation happened to get from the start to the end e.g. Starting number is 4 ending number is 20:
- I could have multiplied by 5
- I could have added 16

- Play “biggest, smallest and in between”. Roll three dice and have students create at least three different numbers using those digits. They then create a quick sketch of a number line, with the smallest number at one end, the largest at the other end and the third number spaced using relative size. You can play this game in teams by having a team come up with the largest and smallest number possible using those digits, then having each player come up with a different possible number and come and place it on the line between the largest and smallest in its correct position. Variation: multiply two of the three digits to come up with a number to place on a number line.
- Play a fractions version of “biggest, smallest and in between”. Roll three dice. Use two of the digits to make the smallest possible fraction. Repeat, making the largest possible fraction (you can decide if you want to include fractions bigger than one or not). Draw a blank number line with these two fractions at either end, and then make as many other fractions as you can and place them in their correct relative position on the line. To make this easier, just use a number line between 0 and 1.

Another advantage of using this “sand time” to do Fluency is that students tend to become much more efficient in their normal calculations. This means that when students are practicing a skill, their practice questions take far less time – magically creating you even more time!

*… So now that we have all this extra time, how should we best use it?*

Problem-based teaching is an incredibly effective tool for improving student results rapidly and permanently. The idea is fairly simple – when introducing new content, start by posing a problem to solve rather than by providing an explanation to memorise. Use this process to encourage exploration and experimentation by the students, and in the process uncover their deeply-held misconceptions.

Deep understanding requires connections to be made, principles to be uncovered and misconceptions to be confronted. It is not something that can be told to students – they need to experience it for themselves. To be truly effective teachers we need to stop doing all the thinking for kids – the more they think, the better they will understand.

Simply put, if we want kids to learn differently… we will have to start teaching differently.