Developing Personal Strategies
Jan 19, 2016
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