…Granny is coming for dinner! Now I’m not just saying this is a problem because she is my mother in law (seriously, I love my MIL!). I’m saying this because this is an example of an appropriate math problem for Oliver! We normally have 4 people at the dinner table, but we just found out, Gran is coming! Can he set the table for one more? Turns out he can!
What is a problem?
“A problem is any task or activity for which children have no prescribed or memorized rules or methods, and for which they do not have a perception that there is a specific “correct” solution method” (Hiebert et al., 1997)
“There must be some block preventing immediate resolution. If there isn’t a block, then the situation is not a problem for that student.” (Burns, 2015)
If you are a classroom teacher, this is a critical concept and it is the reason why differentiated instruction is so important. Gran coming for dinner was a challenging problem for Oliver. He struggled with it and could not solve it correctly until we put down the visual cues of 5 placemats. This was not an example of a problem for Rory though. Rory can count easily and adding one more to the table wouldn’t phase him; there would be no block to challenge him. In fact, he kept trying to confuse Oliver by saying he wouldn’t be home for dinner tonight (“he had sports”!) and that Gran was sick and couldn’t come! Trouble-maker!
If this had been a classroom setting, I would have needed to provide extension activities to allow Rory to experience a productive struggle; while also ensuring I had supports available to ensure other students are not engaged in an unproductive one.
The importance of problem solving
Problem solving has always been an important part of mathematics, but in the last few years, it has become even more so. Why? Because our society is changing…fast….and in order to be successful in it, you need to be able to reason, think critically and creatively, use logic, and apply your knowledge to new situations where you may be missing some or all of the needed information… in other words: PROBLEM SOLVE!
Now you may be thinking, at this stage of development, shouldn’t I concentrate on teaching my child the arithmetic first, like how to add or subtract? The answer is an emphatic no! Research shows that students who are given problems to work on before they are shown methods to solve it, actually perform at higher levels (Boaler, 2015). But there is a caveat, the type of problem you choose is important.
Choosing Rich Tasks
There are good problems, and bad problems, even at the pre-school age. Below are the important criteria that should be applied when choosing rich tasks for any age. I’ve also shown how my two problems measure up.
|RICH TASK CRITERIA FOR PROBLEM SOLVING||Oliver’s Task||Rory’s Task|
|A familiar context||X||X|
|The outcomes should matter to them||X||X|
|Involves math they are confident with||X|
|Low floor (lots of ways to enter problem)||X||X|
|High ceiling (can extend problem)||X||X|
|Have appropriate materials to solve it with||X||X|
|A perplexing problem that the child understands||X||X|
|Have more than one solution||X|
|Be interesting for them so they want to solve it||X||X|
|Challenging but accessible, provoke productive struggle||X||X|
|Encourage open thinking||X||X|
|Allows for connections to be made||X||X|
I chose a rich task for Rory, based on a problem from Van de Walle (2014), that has all the characteristics of a good problem. Rory will be turning 5 in a few months, so I’ve asked him to figure out how many different ways he could put 5 yellow or blue candles on his cake. This problem is great for introducing the following concepts: decomposition of 5, exploring 0, the commutative property, exploring part-to-whole relationships, counting, and cardinality…just to name a few! Let’s see how he does:
Well, Rory surprised me by taking this problem on a totally different tangent, and I let him! It’s important to allow children to try different strategies, develop their own solutions, and make mistakes and Rory did all three! Notice I did not provide him with answers or lead him to the ‘correct’ way to solve it. If I was in a class, I would get other students to share their strategies and have a number talk about the decomposition of 5, or have them work collaboratively to find different combinations. In this case, Rory worked independently (with a little help from his brother!). I haven’t told Rory the final answer; instead, we’ll re-visit the problem later. Perhaps we’ll see how he does arranging 3 candles on Oliver’s cake…stay tuned!
Your role in all this?
BEFORE the task, you should activate prior knowledge and be sure that the problem is understood by the child.
DURING the task, you need to step back and allow the struggle to ensue. Resist the temptation to lead them to the answer! Ask appropriate questions so the child experiences a productive struggle and not an unproductive one.
AFTER the task: talk with them about alternate methods and solutions so that they see there is more than one way to solve a problem.
Need help coming up with problems? No problem (ha!)!