I would imagine that you are not alone. In the good old days, where teaching was more traditional this phrase would generally mean giving children some pre-written word problems that were linked to some of the work they were doing. These were very staged and mostly unrealistic. However, these days problem solving means something entirely different and requires and a very different way of looking at maths.
Over the past few years maths teaching has developed and changed and there is now a strong belief of teaching for understanding. Basically we really want your children to understand what they are learning and how it works, rather than just to know a formula and how to put the numbers through it. Part of this teaching for understanding is problem solving. These days we want to teach children through problems and discussion. By doing this your child has to think about what they know and apply this to the problem and can often extend their own learning and what they know by working through a problem. Initially, this can be quite a difficult thing to understand and get our heads around, as parents. But once you start to get into it this can be great fun and a learning curve for all. The great thing about problem solving is it can be a learning adventure for everyone at any level.
There are a wide range of methods for using problem solving to teach. Children can solve collaboratively or independently. New ideas can be taught or old ones consolidated. Problems can extends a child's knowledge of a particular concept or just give them the vision to understand what they thought they knew.
I thought that today I would introduce problem solving to you as a parent / tutor and start showing you how this can be used to help your children.
As I mentioned above, problem solving is very open ended and, I believe, the part that we, older people, struggle with the most is the fact there are no answers! We were brought up on the concept of maths being right and wrong. Problem solving is all about exploring and experimenting with numbers, a very different way of looking at things.
So, here is today's activity.
This is an incredibly open ended problem which makes it a fab problem as there is so much that can be done with it but also one where lots of encouragement could be needed as it's quite easy to be lazy on this problem.
When it comes to doing this problem with your child you may be lucky and have a child that is used to this kind of thing and will just get stuck in and play with those numbers. I would expect though, that most of you will have children that will look at you blankly and not know how to start. Below I have given you some question that will hopefully get the problem started and then just go from there. You will need paper and pens for all the jottings you might do.
When you introduce this problem keep it very simple and try not to give too much info and see what your child does. If they are not sure how to start, use some of the questions to help. The idea behind the questions is that they will spark other ideas and thoughts and hopefully you and your child, together, will start to explore the numbers without the aide of my questions. The questions are there purely to get you going enough to get the ideas and conversation flowing with these numbers. I have only given questions and not broken this down into levels this week as it is so open ended. Go with the flow, start with the very simple questions whatever level your child and just keep nudging the conversation deeper and deeper depending on what level your child is. I have added in some more difficult question to really push that thinking if some children are struggling to do this on their own.
What's the biggest number you can make using all of these digits?
What's the smallest number you can make using these digits?
Can you count with them?
Can you count forwards? backwards?
Can you count every other number? What are those numbers? What about the ones you missed out?
Can you join one of the numbers to each of the other numbers and make a sequence? What sequence did you make? Can you count further than you made?
Can you add any of the numbers together?
Can you add the first number to each of the other numbers? What happens to them? What about if you add the second number to each of the others? the third? etc.?
Can you add all the numbers together?
What happens if you add 1, 2 and 3 together? Now add 2, 3 and 4, what do you get? What about 3, 4 and 5? Continue.........look at all the answers.
Can you add any of these numbers together to make 10? Do any of the other numbers add together to make 10? How many different ways can you make 10? What about doing the same for 11, 12 etc.
Can you start at 9 and take each of the the numbers before it away? eg 9 - 8 - 7 - 6.......... (you may get 'no you can't' or you may get 'yes' and be able to start exploring the negative numbers they go into - particularly good with 9, 10 and 11 year olds)
Which numbers can you pair up so that you can take one away from the other and stay with positive numbers?
Can you take more than one of the numbers away from any of the numbers? What's the highest numbers of these numbers that you can take away from one of these numbers?
Can you multiply any of these numbers together?
What happens if you multiply the numbers all by the last number: zero?
Can you multiply more than 2 of these numbers together?
Can you multiply all these numbers together?
What happens if you multiply all the numbers by one of the numbers? Now do this again with another of the numbers?
Can you multiply the numbers together in groups of three? Are there any patterns? What about if you multiply them together in groups of 4?
Can you divide any of these numbers by one of the other numbers? Which ones? Did you develop a system to help you work this out methodically? If not, would this help?
Can you find a number that all of these numbers divide by? Can you find a number that none of these numbers divide by?
Can you put all the numbers together except one. The ones that are together should make a number that divide by the one not included (eg. 123467890 divided by 5)
Can you find another way to explore / play with these numbers?
Have fun and enjoy this weeks problem. I would love to hear about some of the discussions you have with this problem and how far into number exploration it took you.