Sunday, 10 May 2015

What's the Problem?

A Mathematical Problem is...

-when you can't get the answer immediately
-unclear about what strategy to use or how to work through a method

A problem should be....
motivates the need to solve it
context that interests them
uses their name if possible
at the appropriate level
provides a challenge but allows success

Big Ideas 
Work systematically
- organising systematically supports logical and systematic approach to math. Teachers need to model and remind students often about this.

Look at solved problems
Some problems can be solved in similar ways. Comparison of strategy use rather than context.

Solve a simpler Problem
use smaller numbers to demonstrate the strategy then scaffold to larger numbers

Find a pattern
- patterns in math can be easy or complicated. Looking for them is problem solving

Work Backwards
- justify your answer if you know it.

Strategies to teach

Guess and Check - simplest strategy however develops complexity when justifying the pattern
guess is a conjecture, checks are called proofs, a guess that has been checked becomes a theorem
- limited possibilities
- gain better understanding of problem
- might know the answer
- systematically try possible answers
- our choices have been narrowed down by other strategies
- no other obvious strategy to try

Guess and Improve- students use initial guess to improve their next one

Act It Out  - great for demonstrating to class
- problem is about doing things
- aids understanding

Use equipment - should be used by all students of all ages
-may be difficult to keep track of the solution.
-encourage students to keep a record of what they are doing
- supports logical thinking and sequence of events to solve a problem

Make a Table - vary in form
- efficient way of finding number patterns

Make an organised list
- should be in natural order
-categorised under headings

Draw a diagram - anything that is not a picture, more symbolic dots for animals etc
some frequently used diagrams have names - tree diagram 
- physical situation is involved
- geometric figures or measurements involved
- gain understanding
- need a visual representation

Draw a Picture - only drawing enough detail to explain thinking to solve problem

Questioning - scaffolds, creates ownership, directs thinking, empowers the learner, maintains momentum of the learning, encourage ideas, 

Getting Started
- What are the important things in this problem?
- What is the problem asking you to find out?
- What information has been given?
- Can you guess what the answer might be?
-Has anyone seen a problem like this before?
-What strategy might you use to get started?

While working on the Problem
-Tell me what you are doing.
- Why did you think of that?
-Why are you doing this?
-What will you do with the result when you've got it?
-Why is this idea better than this one?
-Can you think of another way to find the answer?
-Can you justify this step?

At the End
- Have you answered the question?
-Have you checked your answer?
- Does the answer look reasonable?
- Is there another answer?
-Can you explain your answer to the class?
-Is there another way to solve this problem?
-Can you make the problem harder /trickier?
-What if....

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