Thursday, 25 February 2016

Posing "Challenging" Problems

Posing problems is a appropriate title to this post as they tend to pose me problems trying to write them.

My school was lucky enough to recently have Mignon Weckert talk during a maths job a like. She is passionate about how important posing challenging maths problems are.

So off I went...

The process I went through is based on readings and advice from people who are leaders in inquiry maths and with more knowledge than myself, most notably Mignon.

First, I started with a Central Idea:

Number operations can be used to solve problems in a variety of ways
and with the idea that...
Multiplication and division are effective and efficient ways to solve problems.

From my formative assessment (see previous post) I found that...
  • Some students are skip counting
  • Some students are using repeated addition
  • Some students are using multiplication facts
  • Some students are using materials
  • Some students are answering it ease!
  • Some students have no idea (they told me so!)
  • Some students counted all. 
Great 7 different "groups"! 
NOTE: I don't group students any more. The benefits of students working in mixed groups are just too great. I am just aware of where each student is at in their learning.

Now, my first thought was, "How do I pose questions that cater to all of these groups?" 

The advice I was given was..."pose problems that challenge them all!"

Great, but easier said than done!

So, I looked at my "groups" and focused on the central idea. I wanted my student to realise that multiplication and division are efficient and effective methods of solving problems. 

Problem Number 1:

So more by chance than good planning. My class were going on a field trip. I was going to divide them into groups, but I thought... well why don't you let them do it. Now with 21 students the problem immediately presented a challenge as we needed to get into groups of 5. It was also difficult as it dealt with division even before we explore multiplication. I decided to give it a go and see how it went. 

  • It was challenging for most, due to the 21 
  • To extend those who needed it.  I said each group should have even boys and girls (knowing we have more boys than girls).
  • I had rich conversations with my strugglers. We used materials, talked about skip counting or using basic facts.  One student even said that multiplication makes it easier BINGO!
  • Some kids drew pictures others used materials.
  • I challenged the higher, some even said that it was tricky.
  • Some said 4 groups of 5, that didn't work. So we then talked about 5 groups of 4, with 1 more remaining. Commutative properties! 

 Problem Number 2:

I gave students 4 different sets of arrays. Instead of using easy amounts, I made arrays that went over groups of 10, for example 15 groups of 2 or 2 groups of 15. The reason for this was to challenge students to experiment with breaking groups ups (division), combining the use of repeated addition or using skip counting to help solve the problem. The problem was simply to find the totals of each array and show me how.

  • Posing harder problems opened up a range of strategies.
  • Students worked in groups and learnt from listening to other strategies.
  • Students had the flexibility to solve problems in different ways using different operations.
  • Students made connections between all 4 operations. 
  • It didn't really challenge those higher working students. This is something that I still need to work on. 

My big take away so far is that... that when you make questions challenging, you bring out questions, challenge understandings and encourage discussions because you are making the students think! 

Problem Number 3.

I went to hang out my washing I had 19 pieces of clothing to hang up and 35 pegs. If each piece of clothing needs two pegs do I have enough?

  • We discovered the link between addition, subtraction, multiplication and division. i.e. split 19 into 10 and 9 or round up to 20 to help us solve 19 x 2.
  • Kids who tried 35 divided by 19 were stumped. Then one said lets use multiplication 19 x 2 as it is easier to solve.
  • When working with larger numbers such as 19, it allows you to challenge those who need it and to also support the others. I could work with students on 9 x 2 by creating arrays, but still allowing them to solve the problem. 
Some more questions I have come up...

I have between 15 and 23 apples. How many different ways could I divide them into equal bags?

50 / 5 + 10 = 20 write a story to explain what happens here?

3 X 5 = 30/2 - true or false why?

Use 23 linking cubes to show your understanding the 2, 3, 5 times tables.

Create your own stories using the 2,3,5 times tables.

9 x 0 = 0 Why?

I will continue to add to this post as I move through the unit. If you have any questions that have worked in your class I would love to hear them. 

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