A Problem Solving Approach to the teaching of Mathematics at the

Secondary level

Developing Habits of Mind

Judith SediDavion Leslie

Objectives

To explore the features of problems

To establish a framework for developing

problem solving skills in students.

To explore the benefits of adopting a

problem solving approach to teaching

math.

Have you ever met students who

can perform operations and algorithms but are unaware of

what they are doing?

slavishly follow algorithms regardless of what they are

doing?

cannot respond to context based questions – even though

they can perform the operations implied in the questions?

require an example before they can ‘solve a problem’

are not able to try different approaches in order to arrive at

a correct answer?

Thinking-based curriculum

Thinking based curriculum

What is the thinking based curriculum?

How can it address some of the problems

mentioned before?

What is the true purpose of teaching math in

school?

Does what exists now in schools qualify as the

thinking based curriculum?

What is a Problem ?

Any task or activity for which the

students have no prescribed or

memorized rules or methods, nor is

there a perception by students that

there is a specific ‘correct’ solution

method (Hiebert et al. , 1997)

What is a Problem ?We can safely say that most worded

“problems” are simply “dressed up

algorithms”, exercises that give prompts by

using specific words such as “altogether”,

“shared among” and students do them

without truly understanding what the

question is about.

“Problems that are truly problematic do not

have clear or single solution paths”

(Schoenfeld, 1985, p. 34).

“Computational exercises for which students

do not have a readily-accessible method or

approach can be truly problematic” (Yackel,

Cobb and Wood, 1988, p. 87).

Problem solving

Problems are not to be seen as traditional word

problems.

Traditional word problems “provide contexts for

using particular formulas or algorithms but do not

offer opportunities for true problem solving”

(NCTM, 1989, p. 76).

Problems are, therefore, NOT contextualised

algorithms

According to NCTM

Benefits of Problem solving It develops students’ higher order thinking skills

It allows the student the opportunity to express

their understanding of the concept

It reduces dependence on memory

It creates multi-dimensional classroom setting

It caters to the multiple intelligences of children

Developing problem solving skillsThe aim of mathematics teaching is not to make

students solve problems but to make them into

problem solvers.

Lessons should not only expose students to

problems but also develop the habits of mind that

enable them to become problem solvers.

Students must be made to see an activity as an

opportunity to think and not a question to be

unpacked with and through algorithms.

Developing habits of mindIntroductory problem

A tournament is being arranged among 22 teams.

The competition will be on a league basis, where

every team will play each other twice – once at

home and once away. The organizer wants to know

how many matches will be involved.

Habits of mindAt first the problem appears difficult

No known algorithms exist and students may not

have an example of a similar problem.

Students may not know how to approach the

problem.

How would you intervene at this stage?

What would you say?

Would you give an example?

Would you model an approach?

Would you do anything at all?

Habits of mindHow do we start?

How about simplifying the problem?

Suppose instead of 22 teams, there were only 4?

But still, how do we start?

Well figure out a system for recording how many

matches 4 teams will play.

Habits of mindAvoid this:

A v B

B v A

C V A

A v D

C v D

C v B

C v A

A v D

Be systematic and

organised like this:

A v B A v C A v D

B v A B v C B v D

C v A C v B C v D

D v A D v B D v

C

Randomly

listing

possible

matches

may cause

repetitions

and/or

omissions.

Habits of mindOr better yet, like this:

Habits of mindBy now you should realise that 4 teams will play

12 matches.

Does this mean that the number of matches will

be 3 times the number of teams?

Will 22 teams play 66 matches?

Perhaps we should try a few more cases to see.

Which cases would you try and why?

Habits of mindYou now know how many matches 3 – 6

teams will play.

It’s perhaps best to make a table to

capture your findings.

Teams 2 3 4 5

Matche

s

2 6 12 20

Habits of mindNow that you have a table, look for patterns.

Write down the patterns that your are observing

Look for horizontal (side to side) as well as vertical

(top down) patterns

Look for

Differences (1st and 2nd)

Relationships

Rules

Patterns such as symmetry, odd-even, number types, etc.

Habits of mindUse your patterns to solve the original

problems with 22 teams.

Which patterns/relationships are more

helpful – vertical or horizontal?

Can you find a general rule that tells you the

relationship between the number of teams

and the number of matches?

Can you make it into an algebraic

expression?

Habits of mindTry some simple cases

Find a helpful diagram/creating models

Organise systematically

Examine results (make a table, etc)

Spot patterns

Explore/use the patterns

Find a general rule

Some final thoughtsThe problem was stated with 22 teams; was

this number too small, too large or

challenging enough?

What information in the question would

require some explanations or background

information?

Is this best done as group work or as

individual work?

What could we change about the question to

create an extension/variation?

Habits of mindNow, in your groups, attempt problems 1 – 3

on the activity sheet.