The transition from arithmetic thinking to algebraic thinking

1
The transition from arithmetic thinking to
algebraic thinking

• Kaye Stacey
• University of Melbourne
• K.stacey_at_unimelb.edu.au

2
Outline

• Example from Singapore students that reveals some
  students intuitive thinking about equations
• Example from Australian students that highlight
  differences between arithmetic and algebraic ways
  of solving problems
• Showing some ways to prepare students better for
  algebra. Teachers need algebra eyes and ears.

3
Algebra is . . .

• gateway to higher mathematics?
• OR
• wall blocking path for most students?

4
Relationship between arithmetic and algebra?

• Historically, algebra grew out of arithmetic,
  and ought so to grow afresh for each individual
  (UK Maths Assoc, 1945)
• Significant differences between the approaches of
  algebra and arithmetic more than teachers
  commonly realise.
• Arithmetic leads to algebra, but the path is not
  straightforward!

5
Monks and Buns

• Ming dynasty (500 years ago) probably older
• Mathematician Ching Taai Wai
• Book Suen Fatt Tung Chung
• Calculation algorithm generalise group
  solutions

6
Data from Singapore

• There are 100 buns to be shared by 100 monks.
• The senior monks get 3 buns each.
• Every 3 junior monks have to share 1 bun.
• How many senior and junior monks are there?
• Data from Wong Khoon Yoong, MERGA 31, June 2008
• 124 Grade 6 8 students from 2 secondary schools
• Singapore often is worlds best in international
  tests

7
Averages across 12 similar items

• Approximately 10 of solutions by equations
• Approximately 50 success rate by equations
• All other methods have higher success rates
  (around 75)
• systematic listing,
• guess and check,
• logical argument.

8
Systematic listing not a good method! (about
70 success rate)
9
Guess check - improve solution (gt80 success
rate)
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
ensuring total of 100 monks
zooming in on the answer
10
Solution by algebra
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
11
Some algebra solutions look like proper equations
but are not!
v
12
Some algebra solutions look like proper equations
but are not!
Correct equations
13
Some algebra solutions look like proper equations
but are not!
Abbreviated sentences x senior monk y junior
monk 1 senior monk and 3 junior monks eat 4
buns 25 senior monks and 75 junior monks eat
100 buns
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Characteristics of students equations

• Students often see algebra as a shorthand (e.g.
  for writing abbreviated mathematical sentences)
• IN CONTRAST algebra is a formal system which only
  works if the rules are strictly followed
• Students often use algebraic letters to stand for
  objects (and they are sometimes taught to do
  this!)
• IN CONTRAST in school algebra, letters always
  stand for numbers
• MAIN THEME FOR THIS TALK Learning to use algebra
  requires a very new way of thinking

15
Australian textbooks sometimes teach that letters
stand for objects (EASY but WRONG!)
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Using algebra to solve problems

• Difficulties arise from the transition from
  arithmetic to algebraic thinking in
• idea of the unknown (transient vs fixed)
• usefulness of equations
• purpose of only describing the situation
• knowing what an equation can say
• solving methods
• Algebra provides a very different method, not
  just a new language for a familiar method.

17
Solve this problem (correctly) in several ways

• MARK AND JAN
• Mark and Jan share 47, but Mark gets 5 more
  than Jan. How much do they each get?

18
Equations
Equations
First, give Mark his extra 5. This leaves
42. Share equally, giving each 21. Jan gets
21. Mark gets 26.
Mark Jan Total 20 15 35 too small 25 20 45 too
small 26 21 47 correct
Logical reasoning
Guess-check-improve
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MARK AND JAN Student solutionsStudents asked to
use algebra. Mark and Jan share 47, but Mark
gets 5 more than Jan. How much do they each
get?

• Sara Guess-check-improve (Year 9 - very common)
• 15 32 47, difference 17 too big
• 16 31 47, difference 15 too big,
• . . .
• 21 26 47, difference 5 - this is the
  solution
• William (mid-Year 11) - wrote and solved equation
  
• x (x 5 ) 47
• 2 . x 5 47
• 2 . x 42
• x 21

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MARK AND JAN Student solutionsStudents were
asked to use algebra!

• Brenda (Year 9) - logical arithmetic reasoning
• 47 ? 2 23.5 - 2.5 x
• 47 ? 2 23.5 2.5 y
• Wylie (end of Year 10) logical arithmetic
  reasoning, writing answer as a formula
• y (47 - 5) ? 2 5 42/2 5 26
• x (47-5) ? 2 42/2 21
• (Some students write general formulas like y
  (T - D) / 2 D , x (T-D) / 2 )

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Characteristics of students solutions

• More successful without algebra than with.
• Dont understand algebra is helpful - often do
  the problem first by logical reasoning, and then
  sprinkle letters on the solution (Brenda, Wylie)
• The next solutions show
• Problems with meaning of letters
• Students algebra may look like teachers
  algebra, but the meaning is not the same!

22
Joel writes x for Jans money and x5 for
Marks money, then x 5 47

• Interviewer
• Points to x 5 47. What does this say?
• Joel
• (its) the amount they both get. The amount
  that Jan gets. I just like to keep the three of
  them, 47 dollars, x and 5 dollars and make
  something out of them.
• x is the amount they both get (42) and also x
  is Jans amount

Joel has 2 different meanings for x in one
solution
23
Les begins by writing 5 x 47

• Les x is what is left out of 47 if you take 5
  off it.
• Interviewer What might the x be?
• Les Say she gets 22 and he gets 27. They are
  sharing two xs.
• Int What are the two xs?
• Les Unknownsthey are two different numbers, 22
  and 27.
• Int So what is this x? (pointing to 5 x
  47)
• Les That was what was left over from 47, so
  its 42.

Les uses x for any unknown quantity, in this case
Jans amount, Marks amount and the amount
left after taking 5 off 47.
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Tim Writes x 5 for Marks amount and then
writes x5 x, saying the x after the equals
sign is Jans x

• Tim (pointing to first x in x5 x) Thats
  Marks x.
• Int And why do we add 5 to it?
• Tim Because Mark has 5 more dollars than Jan.
  No, thats not right, it should be Jans x plus 5
  equals Marks x.
• Int Could you write an equation to say that
  Mark and Jan have 47 in total ? You dont have
  to work out the answer first.
• Tim x divided by a half equals x (writes x
  ?1/2 x)

Tim uses x as a general label for all unknown
quantities
25
Leonie writes (x 5) y 47, and cannot
progress nothing interviewer say helps her.

• Leonie explains to the interviewer that
• y is the money that Jan has
• (x5) is the money that Mark has and that this
  says Marks money is 5 more than Jans money
• together they have 47
• Leonie believes that the equation (x5) x 47
  is wrong because y is not the same as x.
• Why?

26
Leonie writes (x 5) y 47, and cannot
progress nothing interviewer say helps her.

• Leonie explains to the interviewer that
• y is the money that Jan has
• (x5) is the money that Mark has and that this
  says Marks money is 5 more than Jans money
• together they have 47
• Leonie believes that the equation (x5) x 47
  is wrong because y is not the same as x.

27
Australian textbooks sometimes teach that letters
stand for objects (EASY but WRONG!)
28
Arithmetic to Algebraic thinking
thinking

• Work from knowns to unknowns
• Unknowns change through problem
• Equation as formula to produce an answer
• Chains of successive calculations
• Guess and check equation solving
• Intermediate results can be interpreted in
  problem situation
• Undoing operations one by one

• Working with and on unknowns throughout
• Unknown fixed
• Equation as description of relationship
• Chains of successive equalities
• Do the same to both sides equation solving
• Intermediate results are not interpreted in
  problem situation
• Undoing operations one by one

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• Chains of successive calculations
• Intermediate results can be interpreted in terms
  of the problem situation
• Find amount to share equally
• 47 5 42 Find each persons share
• 42 ? 2 21
• Find each persons amount
• 21, (21 5)

• Logical chains of equalities
• Intermediate results are not interpreted in terms
  of the problem situation
• x (x 5 ) 47
• ? 2 . x 5 47
• ? 2 . x 42
• ? x 21

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Building algebra on students arithmetic reasoning

• Build understanding of new algebraic concepts
  using arithmetic problem solving methods
• Tables of values leading to idea of variable and
  function in context
• Solving equations from tables of values of two
  functions in context
• Graphing tables of values and solving equations
  from graphs
• Guess-check-improve problem solving for solving
  equations
• Move to algebraic problem solving methods
  requiring working on the unknown when
  fundamental concepts of algebra are in place

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Systematic listing use table of values to build
understanding of functions
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
32
Systematic listing use table of values to build
understanding of functions but here two
separate tables of values
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
33
Guess check - improve solution
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
ensuring total of 100 monks
zooming in on the answer
34
CLOSER TO ALGEBRAIC THINKINGstudent sees one
independent variabletable uses one condition
(equation) (100 monks)checks the other condition
(equation) (100 buns)
100 monks 100 buns Senior 3 buns each 3
Juniors share one bun
ensuring total of 100 monks
zooming in on the answer
35
Building algebraic concepts on students
arithmetic reasoning

• Build understanding of new algebraic concepts
  using arithmetic problem solving methods
• Tables of values concepts of variable and
  function
• Good arithmetic solutions use the conditions
  (equations) in the problem better
• Move to algebraic problem solving methods
  requiring working on the unknown when fundamental
  concepts of algebra are in place

36
For teaching

• Give problems with algebraic potential
• Use students strengths. Teachers highlight the
  algebraic ideas in students numerical solutions
  (eg idea of variables and idea of function and
  relations not simply systematic listing)
• Through discussion, teachers help students
  develop more sophisticated numerical solutions.
  These use algebraic conditions more strongly and
  lead to equations.
• Later, use symbols and formal algebra solving
  methods
• Teachers need algebra eyes and ears to see the
  many good examples that occur for algebra before
  symbols are introduced

37
Making the arithmetic to algebra transition more
smoothly

• International research has focussed on
• early algebra how to teach arithmetic in a
  way that prepares students for algebra by
  focussing on
• operations and their properties and their
  meanings
• generalisations and structures
• contexts which make new mathematical objects
  meaningful
• variables to express generality
• numerical and graphical approaches, often with
  technology e.g. spreadsheet

Examples in the written paper
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Algebra is . . .

• gateway to higher mathematics?
• OR
• wall blocking path for most students?

39
Algebra is . . .

• gateway to higher mathematics?
• OR
• wall blocking path for most students?

Highlighting differences between arithmetic and
algebraic thinking Teaching arithmetic with
structure and generalisation in view Introducing
algebra concepts numerically and graphically
40
Thank you

• Kaye Stacey
• University of Melbourne, Australia
• k.stacey_at_unimelb.edu.au