Arithmetic to Algebra Chris Linsell

1
Arithmetic to Algebra Chris Linsell

• Long-standing difficulties
• School algebra
• Conceptual difficulties
• Results
• Summary

2
Poincare 1908

• One. ..fact must astonish us, or rather would
  astonish us if we were not too much accustomed to
  it. How does it happen that there are people who
  do not understand mathematics? If the science
  invokes only the rules of logic, those accepted
  by all well-formed minds. ..how does it happen
  that there are so many people who are entirely
  impervious to it?

3
Freudenthal 1973

• With the utmost patience teachers tried to
  engrave in their pupils' minds that letters in
  algebra mean something, that no formula is
  meaningful unless the meaning of its components
  is told, and that algebra is not a meaningless
  game with 26 letters.
• It was to no avail.

4
Major Reports

• The Cockroft Report in the UK (DES, 1982)
  highlighted the fact that algebra is a source of
  considerable confusion and negative attitudes
  among pupils
• School algebra Primarily manipulations of empty
  symbols on a piece of paper? (Brekke, 2001)

5
Hewitt 2001

• A girl in the class was looking at the following
• Example l x39 x6 Example 2 x710 x3
• I asked her whether she understood what was
  written on the board and she replied I don't
  understand, I just copy it down.
• I then asked What is the 'ex' on the board?
• To which she replied What 'ex'? That's a times.

6
What is algebra?

• recognising patterns and relationships in
  mathematics and the real world,
• generalised arithmetic,
• abstract thinking,
• using symbols, notations and graphs,
• unknowns,
• variables,
• a language to represent and communicate,
• a calculus for solving certain classes of
  problems,
• the activity of operating on expressions and
  equations,
• generalisation,
• proof and validation

7
Arithmetic and algebra

• arithmetic reasoning may be used to solve
  problems when there is an easily established
  connection between the known quantities
• algebraic reasoning is required when at the
  outset no connection can be established directly
  between the known quantities

8
A useful definition?

• Algebra is much more than the manipulation of
  symbols and may be viewed as the symbolising of
  general numerical relationships and mathematical
  structures and operating on those structures
  (Kieran, 1992)

9
Process / object duality and procepts

• Counting?natural?subtracting?negative?roots?imagin
  ary and complex
• Adding?sums
• Evaluating expressions?equations
• Procepts, pivotal symbolism

10
Cognitive development in algebra

• operational procepts of arithmetic
• evaluation processes in generalized arithmetic
• manipulation of procepts in manipulation algebra
• defined concepts in axiomatic algebra

11
How could we introduce algebra to Year 9
students?

• generalisation
• problem solving
• functions
• modelling
• algebra in schools is often reduced to rules for
  transforming and solving equations

12
The sign

• 35
• compute now
• is equivalent to
• Process/object again
• 358210
• lack of appreciation of the structure of
  mathematical statements
• 3x154475 and 3x475154

13
Abstraction

• mathematical constructs are totally inaccessible
  to our senses
• a symbol is merely one representation of an
  abstract concept
• it is a hallmark of mathematics that symbols are
  used as if they were the objects themselves
• Pirie-Kieren model

14
Numeracy

• x37 can be solved by advanced counters by guess
  and check, but can be solved much more easily by
  part/whole thinkers able to visualise 7 as 34
• 2x311 requires an understanding of numbers
  beyond simple additive part/whole or
  multiplicative part/whole thinking
• 2(x3)2x22 requires an understanding of the use
  of brackets, the hierarchy of operations, and an
  understanding of the commutative, associative
  and distributive laws

15
Other conceptual difficulties

• Lack of closure
• Binary / unary operations
• Operating on unknowns
• Unknown / generalised number / variable /
  parameter
• Conventions of symbolisation
• Patterns and relationships

16
Methodology

• Qualitative
• 4 Year 9 students studied for 27 lessons
• Activity based approach
• Data videotapes of lessons, SRIs, students
  work, field notes
• Transcription and analysis

17
Unknowns and variables

• Letter symbols used to represent unknowns,
  variables and generalised numbers
• Students often did not write relationships
• Sometimes students did not even use letter
  symbols
• Learning generalised number 3/4
• variable 1/4
• unknown ?

18
Arithmetic and Algebraic Reasoning

• Guess and check / formal solving
• Difficulties with guess and check
• Calculators for guess and check
• Short-cutting guess and check
• Algebraic structure clues from mistakes
• Unknowns on both sides
• Learning algebraic reasoning 2/4 perhaps

19
Process/object

• Models of Dubinsky and Sfard
• Action / interiorisation
• Process / condensation
• Object / reification
• Learning action / interiorisation 3/4
• process 2/4
• object 0/4

20
Process and procedure

• Superficial similarity
• Analogy of two-digit subtraction
• Getting the right answer
• Clues from errors
• Learning procedure 3/4
• process 2/4

21
Notation and Convention

• Algebraic notation
• Division and fractions
• Vertical arithmetic
• Series of horizontal arithmetic steps
• Appropriation of implied sign
• Appropriation of x
• Equations as formal conventions

22
Language

• Solving
• Equations

23
Prerequisite Numeracy

• Numeracy as a predictor of learning
• Arithmetic structure
• Inverse operations
• Operating on integers
• Binary/ unary view

24
Strategic Thinking

• Multiplicative part / whole
• Expressions as objects
• Value of guess and check
• Constraints on choice of parts
• Unknown parts
• Algebraic part / whole thinking
• Equations as objects
• Time spent on stages

25
Summary of Literature

• Algebra is much more than the manipulation of
  symbols and may be viewed as the symbolising of
  general numerical relationships and mathematical
  structures and operating on those structures
• Letter symbols are used as specific unknowns,
  generalised numbers, parameters and variables and
  children need to recognise and understand these
  different uses.
• Algebra is intimately connected to arithmetic and
  children need to have strong numeracy skills to
  develop algebraic thinking
• The process-object model of Sfard seems
  particularly useful for describing the
  development of algebraic understanding. Many
  children do not move on from a procedural view of
  equations to a structural perspective.

26
Summary of Study 1

• The level of numeracy displayed by each of the
  four students was a strong predictor of the
  amount of algebra they would learn during the
  topic. Only the students who displayed an
  understanding of arithmetic structure were able
  to carry out the process of unravelling
  expressions, rather than just following
  procedures. Lack of understanding of inverse
  operations was also an impediment to solving
  equations.

27
Summary of Study 2

• When attempting to solve equations by formal
  methods, the students either followed procedures,
  or carried out processes, rather than treating
  equations as objects to act on. Conventional
  approaches to the teaching of solving equations
  focus on algebraic statements and their
  transformations, and are therefore based on an
  object view of equations. It is suggested that
  this stage is difficult to achieve.

28
Summary of Study 3

• It was apparent that the part/whole thinking
  required for algebra posed conceptual
  difficulties beyond those inherent in arithmetic.
  Multiplicative part/whole thinking is argued to
  be a prerequisite for solving equations by formal
  methods if the student is to have an
  understanding of the process beyond merely
  following a procedure.

29
Summary of Study 4

• If operating on unknowns is regarded as one of
  the defining features of algebraic thinking, then
  the students remained largely at the stage of
  arithmetic thinking throughout the topic.
• Algebraic part/whole thinking is postulated as a
  strategy stage. This is seen as being more
  advanced than multiplicative part/whole thinking
  and a precursor to treating equations as objects.

30
Summary of Study 5

• The progress of the students in learning to solve
  equations was very slow. If algebraic thinking
  is part of a continuum of strategic thinking that
  starts with early arithmetic, and if it requires
  the movement from a process to object view of
  equations, then one might expect the majority of
  students to remain at their starting strategy
  stage for a year or more.