Problem Solving in Mathematics Colleen Young Mathematics, Learning & Technology Problem Solving

1 2 3 Definitions

Just what is problem solving? Ask the students Teaching Ideas

4 5 6 Questions

Making it stick The importance of recall Further Resources Problem Solving

Whats a problem anyway? Higher

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DfE GCSE Maths Assessment Objectives Higher

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Found 25% Whats a problem anyway? Polya (1945 & 1962) described

mathematical problem solving as finding a way around a difficulty and finding a solution to a problem that is unknown. Whats a problem anyway? Working out what to do, when you

dont know what to do Hiebert et al: A mathematical problem solving task must be problematic for a student to be viewed as legitimate mathematical problem solving.

Whats a problem anyway? A Level Mathematics Working Group Tasks have little or no scaffolding: there is little guidance given to the candidate beyond a start point and a finish point. Questions do not explicitly state the mathematical

process(es) required for the solution. Tasks provide for multiple representations, such as the use of a sketch or a diagram as well as calculations. The information is not given in mathematical form or in mathematical language; or there is a need for the results to be interpreted or methods evaluated, for example, in a realworld context.

A Level Mathematics Working Group Report Whats a problem anyway? A Level Mathematics Working Group Tasks have a variety of techniques that could be used.

The solution requires understanding of the processes involved rather than just application of the techniques. The task requires two or more mathematical processes or may require different parts of mathematics to be brought together to reach a solution

A Level Mathematics - Overarching themes Ask the students

Students on good teachers Good problem solvers need good teachers.

Students on good teachers. Good at explanations and lecturing. Someone who can explain in different ways. Someone who wont just tell you how to do something, but will explain how and why it works.

Provokes your mind to think beyond the syllabus Students on good teachers A teacher who provides the student with the opportunity to see what they need to revise. Regular tests and quizzes do this.

Doesnt mind repeating things. Pushes you to work on harder questions to extend your abilities. Students on good teachers Patient.

Good teachers ... Understanding. Approachable. Firm but kind. Someone you can feel comfortable with. Recognises achievements.

Genuinely caring about the students. Someone who knows who you are. Someone who you know wont judge you. Teaching Ideas

For problem solving in the GCSE classroom

Developing a problem solving classroom Teacher / Student relationships? What sort of questions do you ask? Open, closed? Are students comfortable to ask & answer questions? How do you respond to student answers? Acknowledgement?

Are problems a regular part of your lessons? From KS3? Written into schemes of work? Do students have thinking time? Time to play and experiment with problems? Is it OK to be stuck? Do your students persevere? Are your students confident? Determined?

Do you model problem solving techniques? Problem Solving Understand the problem. Do your students understand all the words used in stating the problem? Can they restate the problem in their own words?

Do something! Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. Draw a diagram Would a picture or diagram help to understand the problem?

To Try Guess and check. Look for a pattern. Make an orderly list. Solve a simpler problem. Consider special cases. Work backwards. Be ingenious. Being Determined

What do you think the question might be? Or less formally .

This is from a Year 7 student. Do we sometimes get too formal too quickly? Try diagrams to model problems

Heres the diagram.. ... and ask whats the question Try, heres the diagram whats the questi

on? Or try Algebra Snippets What could you work out? What else could we work out if we had

more information? Help students with vocabulary Good mathematicians can go backwards! From Nrich...

Working backwards at KS2 ; the ideas here could also be used at KS3. See this Nrich

article from Liz Woodham on Developing Problemsolving skills which includes the link above

Arithmagons Perfect for any topic for thinking backwards. See the ideas and resources here

everything from simple arithmetic to Calculus! Arithmagons from Flash Maths

Problems See for example the AQA problems incl uded here .

This type of backwards proble m really helps students think deeply.

Questions For problem solving in

the GCSE classroom Which numbers less t han 100 have exactly three factors?

The answer is 7, what is the question? Give me a pair of equations whose solutions differ by

Can you construct three. a triangle with sides 3, 4 and 9? Is a square a rectangle?

Can you give me a quadratic equation whose roots differ by 3? And another ...

Making it Stick The importance of recall.

Apply all the basics. Mini-Tests Usetheveryimportantideaofmaking

things stick andgivestudentsa mini-test . Reading MakeitStick (TheScienceofSuccessful Learning)whichdiscussestheuseoftestingasa learningtoolconvincesmeevenmorethat mini-tests areagoodidea!Studentsneedtorecall informationandtheevidencesuggeststhattesting

isabetterwayofdoingthisthansimplyrereading material,amethodoftenfavouredbystudents. StudentslikethealternativenameSelf-checks whichIhopehelpsthemrealisetheyareasthe learning tool, not

something to be stressed by. authorssuggesta exercisein repeatedly recallingathing

strengthensthe memory. Aristotle Making it stick

Mini-Tests the students I got it right in the test because I got it wrong in a mini-test On good teachers: A teacher who provides the student with the opportunity to see what they need to revise. Regular tests and quizzes do

this. Resources Further resources

for problem solving in the GCSE classroom GCSE Resources

Standards Unit Thetask(S4)istomatchthebarchartswiththestatistics(mean, median,modeandrangeareallgiven).Thishasworkedreallywell withmystudentsandIfeelleadstoadeeperunderstandingthan justsimplycalculatingthesestatistics.StandardsUnit

For each problem: Suggested questions to ask students How to get into the pr oblem

Complete solution KS4 Extension and Enrichment AQA Mock Exam Analyser

available throu gh All About Math s Axel and Lethna are driving along a motorway.

They see a road sign. The road sign shows the distance to Junction 8 It also shows the average time drivers take to get to Junction 8. The speed limit on the motorway is 70 mph. Lethna says

We will have to drive faster than the speed limit to drive 30 miles in 26 minutes. Is Lethna right? You must show how you get your answer. The CEO of GE explains why his math degree is more useful than his MBA December 2015

Immelt says his most invaluable qualification is not the MBA. It is his undergraduate degree in math. "I use my math major every day I don't use the MBA quite as much. My intellectual curiosity goes more toward problem solving than spreadsheets.

Running a company, to him, is really about problem solving. And that's something he learned about in his undergraduate studies, due to "the inherent intellectual curiosity around math and physics. That has stayed with Immelt throughout his career. "I'm just curious about everything," he said. "I can view every

situation as a problem to be solved.