Mike Chartres F to 7 Mathematics Education Consultant

The role of challenge, questions and problem solving to engage learners with mathematics

Torrens Valley Partnership 23rd October 2018

Mike Chartres F to 7 Mathematics Education Consultant

Overview of the workshopThe workshop aims to to encourage teachers to explore:

what do we actually mean by problem solving in mathematics?; the purposes and intent of teacher questions to present mathematics as challenges and support learners develop their capability to think mathematically when problem solving; how presenting mathematics through engaging and interesting challenges supports learners to think mathematically, and the notion of teaching through problem solving rather the teaching about or teaching with problem solving.

Mike Chartres F to 7 Mathematics Education Consultant

Your first problem

Which domino is the middle domino and how could you show this?

What makes this a problem to be solved? What strategies did you use to solve this problem?

https://www.youtube.com/watch?v=ytVneQUA5-c

Mike Chartres F to 7 Mathematics Education Consultant

What are some of possible purposes behind teachers’ questions and which are the preferred purposes?

Possible purposes include: elicit an answer; suggest an error has been made; elicit student understanding; invite engagement; invite explanations and / or descriptions; invite comparisons; invite elaborations; invite justifications; invite to seek alternatives; invite making connections and / or transfer ideas and strategies; invite reflection and / or critique, and invite a choice or change a direction, making decisions.

Mike Chartres F to 7 Mathematics Education Consultant

Overaching theme 1

Several mathematicians who write about mathematics for the general public including Marcus du Sautoy, Simon Singh, Adam Spencer, Marianne Freiberger, Rachel Thomas, Ian Stewart, Arthur Benjamin, non mathematicians who write about a new found interest in mathematics such as Alex Bellos and scientists such as Leonard Mlodinow and Jim al-Khalili

describe mathematics as the science of pattern and

describe the role of mathematicians as pattern seekers and puzzle solvers.

So What Is Mathematics? What Actually Are We Teaching About?

Mike Chartres F to 7 Mathematics Education Consultant

Overaching theme 2 For Me, Two Panels For “Visitors To Note” From The Osher Rainforest Sum Up Learning Mathematics … And Science.

Mike Chartres F to 7 Mathematics Education Consultant

So what is problem solving? What do some curriculum documents say?

The Australian Curriculum: mathematicsdescribes problem solving in its key ideas / proficiencies as “the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable” (ACARA, https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/key-ideas/ )

Mike Chartres F to 7 Mathematics Education Consultant

So what is problem solving? What do some curriculum documents say?

The National Council of Teachers of Mathematics describes problem solving in its Standards as “engaging in a task for which the solution is not known in advance.” And goes on to describe the characteristics of good problem solvers having a "mathematical disposition" where they analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see.” (NCTM, 2013)

Mike Chartres F to 7 Mathematics Education Consultant

A critique

Posamentier and Krulik offer a more open ended approach to problem solving in school mathematics. “A problem is a situation that confronts the learner that requires resolution, and for which the path to the answer is not immediately known.” (2009, page 2)

They comment about some approaches to teaching problem solving in school mathematics. “It is this very definition of a problem (left) that reduces many of the “word problems” teachers do from “problems” to mere “exercises.” Teachers often group problems by types and demonstrate to the class how to approach them. Usually, students are shown how to do one of these problems and told that the others are very similar and should be done the same way, albeit with different numbers!

These we shall refer to as “exercises” rather than problems, because recognition of the type of problem immediately provides the learner with the path (or method) for arriving at the correct answer.” (Posamentier and Krulik, 2009 page 2)

Mike Chartres F to 7 Mathematics Education Consultant

Returning to Problem Solving:The impact of George Pólya

Hungarian mathematic ian George Pólya is of ten acknowledged either in word or in form for giving structure and understanding to mathematical problem solving in the curriculum. He suggests there is much more to problem solving in mathematics than word problems.

Pólya (1973, page v) argues that, “A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.”

Pólya goes on to say, “A teacher of mathematics has a great opportunity. If (s)he fills in allotted time with drilling students in routine operations (s)he kills their interest, hampers their intellectual development, and misuses his / her opportunity.”

Mike Chartres F to 7 Mathematics Education Consultant

The impact of George Pólya

Pólya suggests a four step process for solving problems. Namely:

Understand the problem. Devise a plan to solve the problem. Implement the plan. Reflect on the problem.

Pólya also suggested a variety of strategies which have been added to and “institutionalised” in many teacher resources to develop “problem solving programs.” Strategies include:

use materials and equipment; act out the situation; choose operation or operations; guess and check; draw a picture; search for patterns; write an equation find all the possibilities; use a table; eliminate possibilities; work backwards; logical reasoning, and solve a simpler problem

Mike Chartres F to 7 Mathematics Education Consultant

Some key elements of problem solving

Problem solving involves the ability to: search for pattern use mathematics to represent unfamiliar, meaningful authentic and situations; pose / formulate problems based on seen situations; make choices; interpret, formulate, model and investigate problem situations; analyse situations carefully in mathematical terms; think mathematically; design investigations and plan approaches; apply existing strategies to seek solutions; work individually and a collaboratively; be creative; appreciate and explore that a particular problem can be solved using a number of different strategies; verify that answers are reasonable; justify solutions and strategies employed; communicate solutions effectively; reflect on the solutions and strategies employed, and persist.

reflected in

Mike Chartres F to 7 Mathematics Education Consultant

The proficiencies and thinking mathematically

The Australian Curriculum: Mathematics includes problem solving as one of four intertwined proficiency strands:

Understanding Fluency Problem solving Reasoning

(ACARA, 2014 , page 6)

Interestingly the original document that these were taken from and adapted describes five intertwined proficiency strands Adding + It Up helping children learn mathematics:

Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition

(Kilpatrick, Swafford, and Findell eds, 2001. page 116)

Mike Chartres F to 7 Mathematics Education Consultant

Thinking Mathematically

Communicate and reflect

Critique Challenge

Engage with mathematics and science

Describe Compare Relate Identify Sort Explore Observe (notice)

Making sense of the mathematics and science

Search for pattern Predict Visualise Synthesise Classify Simulate Generalise Abstract Understand Connect Pose questions

Represent understanding and thinking

Construct Draw Sketch Graph Symbolise

Going further Investigate Hypothesise Model Apply Infer Justify Validate Test Prove Transfer

Mike Chartres F to 7 Mathematics Education Consultant

Questions - Dan Finkel and his five principles of extraordinary mathematics teaching

https://www.youtube.com/watch?v=ytVneQUA5-c

Mike Chartres F to 7 Mathematics Education Consultant

Playing with Number

relationshipsWhat do notice?

How are you connecting “what do you notice” to what you already

know?

What questions do you have?

Mike Chartres F to 7 Mathematics Education Consultant

A note about language that reflects doing mathematics

The language of doing and practicing mathematics includes:

completing exercises; simplifying; doing activities, and doing tasks.

The language of doing and learning mathematics includes undertaking:

a challenge; a puzzle; an exploration; an investigation, and resolving a situation.

Mike Chartres F to 7 Mathematics Education Consultant

Three approaches to engaging learners with problem solving

Van de Walle, Karp and Bay-Williams draw on Pólya’s construction of problem solving both from the view of the four steps to problem solving and many of the problem solving strategies described earlier. They offer teachers with some useful frameworks for thinking about problem solving in classroom programs including:

A problem is defined as any task or activity for which students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific “correct” solution of method. A problem may or may not have words. A problem has three key features;

it must begin where the students are, the problematic or engaging aspect to the problem must be due to the mathematics the students are to learn, and it must require justifications and explanations for answers and methods.

Mike Chartres F to 7 Mathematics Education Consultant

Three approaches to engaging learners with problem solving

Van de Walle, Karp and Bay-Williams explore three broad classroom strategies to engage learners with problem solving, with a clear preference for the third strategy.

teaching for problem solving - teaching a skill so it can be applied later teaching about problem solving - teaching specific strategies that can be applied eg, Pólya’s four step process and / or specific strategies such as draw a picture, construct a table teaching through problem solving - where students learn mathematics through real contexts, problems, situations and models and is often described as upside down of the teaching for problem solving approach.

(2013, page 32)

Mike Chartres F to 7 Mathematics Education Consultant

Adapted from NRICH http://nrich.maths.org/1016

A Challenge Using Dice

Mike Chartres F to 7 Mathematics Education Consultant

Australian Curriculum: mathematics - problem solving work samples …. what is the view of problem solving?

https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/key-ideas/ https://www.australiancurriculum.edu.au/resources/mathematics-proficiencies/samples/number-and-algebra-my-thinkboard-ws2/ https://www.australiancurriculum.edu.au/resources/mathematics-proficiencies/samples/number-and-algebra-fractions-and-decimals-and-percentages-ws4/ https://www.australiancurriculum.edu.au/resources/mathematics-proficiencies/samples/measurement-and-geometry-the-combined-area-of-spaces-ws3/

Mike Chartres F to 7 Mathematics Education Consultant

Ready for a challenge?

What is the same of all the numbers on a multiplications table?

Let’s start with a 4 x 4 table

Mike Chartres F to 7 Mathematics Education Consultant

Adapted from NRICH https://nrich.maths.org/1

Squares And Tangrams

Mike Chartres F to 7 Mathematics Education Consultant

Border Challenge

Adapted from NRICH

Firstly, how many tiles are required to border a lawn 2 squares long and 2 squares wide?

Mike Chartres F to 7 Mathematics Education Consultant

Adapted from NRICH https://nrich.maths.org/1816/note

Lady Birds On The Garden

Mike Chartres F to 7 Mathematics Education Consultant

Bedtime math - Laura Overdeck

Overdeck L and Paillot J. 2013 Bedtime Math an excuse to stay up late

Dogs in Charge

Big and small monkeysFire house horsesDogs in charge

Burst of bunniesIce cream truck

Mike Chartres F to 7 Mathematics Education Consultant

Some useful web-sites where learners are encouraged to be pattern seekers and puzzle solvers

YouCubed https://www.youcubed.org NRICH http://nrich.maths.org/frontpage NCTM Illuminations http://illuminations.nctm.org/Lessons-Activities.aspx Math4Love http://mathforlove.com Bedtime Maths http://bedtimemath.org Greg Tang Maths http://gregtangmath.com Dave TV Dara O’Briain’s School of Hard Sums http://dave.uktv.co.uk/dara-o-briain-school-hard-sums/item/dara-o-briains-homework-puzzles/ CSRIO Maths and Stats (maths by E-mail connected to the Double Helix club) https://blogs.csiro.au/helix/category/maths/