Developing Personal Strategies

Where are we in our thinking?

• Is mathematics a set a rules and procedures that we must acquire through memorization?

• Is being good at math remembering what rule to apply?

• Has genetics blessed some students to be able to do mathematics?

Or Are We Thinking?

• The focus of mathematics is problem solving.

• Children learn by constructing ideas.

• Everyone can learn mathematics.

Question on a provincial gr 9 assessement

• 1 + 1 = ? 2 3 A) 1

6B) 2 5C) 3 2D) 5 6

What are personal strategies?

• Approaches to mental math and estimation.

• Arithmetic operations

• Algebraic operations

• Drawing Algorithms

Various types that may defined by the tools we use.

• Mental mathematics

• Paper and pencil

• Technology

• It is important for teachers to realize that no matter what strategy they may teach, students will process it in many different ways.

Dangers that may appear!

• Blind acceptance of a strategy

• Overzealous application

• Belief that algorithms train the mind

• Notion that one can be helpless without technology being available.

A need to teach mental strategies constructively

• Most teachers admit teaching paper and pencil algorithms about 90% of the time.

• Formal written algorithms do have the advantage of working for all numbers.

• The disadvantage is that they are not flexible and discourage student thinking.

• Mental mathematics requires a choice of strategy and understanding.

Now let us talk about strategies for mental math.

• What is 36 + 79?

• How about 25 +83?

• At the very heart of personal strategies is how the student can relate a difficult calculation into an easy one.

• For example 28 + 27 is difficult

But 30 +25 is easy.

• 9x17 is hard

But 10X17 – 17 is easier.

Should we teach methods of mental math?

• It may be unhelpful as students will use the method without understanding.

• For example:

A strategy of removing zeros when adding 70 + 20 becomes confusing when we consider

70 x 20 or even more prone to error when we attempt . 150

What direction should we take?

• We need to stop the 10min of mental math that emphasize accuracy and speed.

• We should not concentrate on the right answer but instead look at the various ways the calculation is done.

• Have students present a strategy and allow time for the students to practice it.

• Mental math provides opportunities for students think for themselves.

Suggestions from the Australians.

• Instead of asking the students for a calculation do the reverse.

• Provide plenty of time to work out a calculation asking questions like: Who can share how you did it? Do you understand that way? Who did it a different way?

• Given a calculation and after finding the answer have students suggest a context for the calculation.

So now we have students inventing strategies…

• Teachers need to consider:

Is it efficient enough to be used regularly?Is it mathematically valid?Is it generalizable?

Where’s the research?

• Much of the research is based on the work of Piagets’ constructivism and Kamii’s work.

• They found that when children are allowed to think for themselves, they “universally proceed from left to right.”

• Kamii worked in schools from 1989-91 There were four grades involved:Grade 1 – none of the four teachers taught

algorithmsGrade 2 – One of the three taught algorithms, one

did not teach algorithms but parents did, and one taught no algorithms

Grade 3 – two of the three teachers taught algorithms

Grade 4 – all four taught algorithms

Answers from grade 2 students for 7 + 52 + 186 Algorithms n=17 Some algorithms n=19 No algorithms n=20

9308

1000

989

986

938

906

838

295

989

938

810

356 617

245 (12%) 245 (26%) 255

246

245 (45%)

243

236

235

200

198

30

29

213

213

199

133

125

114

138

Answers from grade 3 students for 6+53 +185

Algorithms n=19 Algorithms n=20 No algorithms n=10

838

768

533

800+38

800

444

344 284

246

244 (32%)

235

234

244(20%)

243

239

238

234

245

244 (50%)

243

238

213

194

194

74

29

204

202

190

187

144

139

221

Answers from grade 4 students for 6 + 53 + 185

Algorithms n=20 Algorithms n=21 Algorithms n=21 Algorithms n=18

1300

814

744

715

713+8

1215 848

844 783

783 718

713

445

791

738

721

10 099

838

835

745

274

244 (30%)

243

245

244 (24%)

234

224

244 (19%) 244 (17%)

234

234

234

194 177

144 143

134

194

127

144

138

134

225

Two other interesting findings in the grade 4 data were:

While about 8% of grade 3’s did not attempt an answer, this number jumped to about 25% of grade 4’s

A new way of writing answers such as “8,3,7” emerged.

• Kamii’s research led her to conclude that teaching algorithms can be harmful for two reasons.

1. It encourages children to give up their own thinking

2. It “unteaches” place value and prevents the development of number sense.

What is the argument for teaching strategies or algorithms?

o Many teachers believe that teaching algorithms is the most efficient method.

o Many students, in particular, those that struggle, need a method for getting answers.

How Teachers Undergo Change

For teachers who make the transition, they usually follow a pattern

1) They teach arithmetic by teaching algorithms

2) Teach algorithms after laying the “groundwork for understanding”.

3) Teach no algorithms at all.

Let us look at how students develop paper and pencil personal strategies

Teaching Multidigit multiplication

• Students will begin by using

DIRECT MODELING.

• For example if asked:

There are 6 trays with 24 eggs in each tray, how many eggs are there altogether?

Children may model using counters, base-ten materials, tally marks or other drawings.

• Students then might move to a complete number strategy such as repeated addition or doubling.

For example in solving the egg problem

A student might add 24 on six times to obtain 144.

Or

A student might add two 24’s three times to get three 48’s and then add these sums to get 144.

• Many students then move to a partitioning number strategy.

• For example If we have 12 boxes with 177 books in each box, how many books do we have altogether?

Students may calculate using an invented method such as

12 x 177 = (4 x 3) x 177 = 4 x (3 x 177)

Partitioning using decade numbers

• Example

In a building there are 43 floors with 61 offices on each floor. How many offices are in the building?

A student might find the sum of ten sets of 61 to be 610 and then add four sets of 610 to obtain 2440 and now add on three sets of 61 to obtain the solution of 2623.

Another example of partitioning – Can you understand the students’ thinking? • There are 17 containers with 177 books in each

container. How many books are there?Alberto wrote: 177 x 17

7 x 10 = 7070 x 10 = 700100 x 10 = 10001000 + 700 + 70 = 1770

= 8851770 + 885 = 2655177 + 177 = 3542655 + 354 = 3009

1770 2

Compensation Strategy

• How might we use compensation to solve these questions?

If I have five bags of jellybeans with 250 jellybeans in each bag, how many jellybeans do I have altogether?

There are 17 jars with 70 ladybugs in each jar. How many ladybugs are there altogether?

A pattern appears to evolve for students inventing mutiplication

strategies

• “When teachers understand students’ invented strategies and their developmental paths they can help students move towards more sophisticated strategies” (Baek, 1998)

• The learner should never be told directly how to perform any operation in arithmetic… Nothing gives scholars so much confidence in their own powers and stimulates them so much to use their own efforts as to allow them to pursue their own methods and to encourage them in them.” (Colburn, 1970)

Two important considerations.

• Lappen (1995) states that there is no decision that teachers make that has a greater impact on students’ opportunity to learn …then the selection of the tasks with which the teacher engages the students in studying mathematics.

• Kieren (1988) recommends that instruction should build on students’ understanding of fraction and use objects or contexts that have students acting on something or making sense of something rather instead of just manipulating symbols.

And finally

• Marilyn Burns (1994) “Imposing the standard arithmetic algorithms on

children is pedagogically risky. It interferes with their learning, and it can give students the idea that mathematics is a collection of mysterious and magical rules and procedures that need to be memorized and practiced. Teaching children sequences of prescribed steps for computing focuses their attention on following the steps, rather than on making sense of numerical situations.”

Some recommended resources.Fosnot, Catherine, and Maarten Dolk.

Young Mathematicians at Work:Constructing Number Sense, Addition and Subtraction. Portsmouth, NH, 2001

Fosnot, Catherine, and Maarten Dolk. Young Mathematicians at Work:Constructing Multiplication and

Division. Portsmouth, NH, 2001 Kamii, Constance. Young Children Reinvent Arithmetic. New York.

Teachers College Press

National Council of Teachers of Mathematics. The Teaching and Learning of Algorithms in School Mathematics. Reston, VA. NCTM. 1998