Embed Size (px)
Citation preview
2
Think of a number• Multiply your number by 4• Add 12 • Divide the number you now have by 2• Add four• Halve the number• Subtract the number you first thought of• Find the letter in the alphabet that occupies that
position• Think of an animal that starts with that letter
3
Was your animal
Grey?
(or pink?)
4
I’m thinking of a number. Four times the number plus three is the same as three times the number plus nine. What number am I thinking of?
Murray Britt - NZ
5
Change in pedagogy
• Students must take control of their own learning (teachers must be willing to give up control)
• Recognition of the importance of discussion in the classroom both in groups and whole class
• The use of language• Problem solving and investigations• Formative assessment – feedback which supports
learning• Evidence based approaches
6
• Impact of technology - CAS
• Algebra from early years on
Major movements
7
Impact of technology
• When ordinary calculators arrived teachers ignored them or actively argued against them.
• Calculators can be used as “answer getting” machines or as tools, particularly for learning
• Now they are accepted by many as useful learning tools
• Change in emphasis – move towards building number sense
• The “thinking” curriculum
8
New calculator technology
• Introduction of graphic calculators more planned• Need for professional development to use
creatively as learning tool rather than just answer getting tool
• We need to build “algebra sense” just as we build “number sense”
• CAS changes the range of problems and applications possible and opens up investigations and problem solving as well as providing a tool for learning
9
CAS
• Focus on understanding and making connections rather than routine skills
• Applications to “real” problems and to investigations opens up
• Classroom changes – students more in control – more group work
• Students still develop skills
10
Algebra in the early years
• New Zealand has acknowledged this for longer than most other countries
• Structure of systems
• Connections between arithmetic and algebra
• Patterns
• Generalisations
11
How many legs are there? 2 lions, their 4 cubs and 4 storks
12
How many legs are there? 2 lions, their 4 cubs and 4 storks
2 4 = 8
13
How many legs are there? 2 lions, their 4 cubs and 4 storks
2 4 = 8 + 4 4
14
How many legs are there? 2 lions, their 4 cubs and 4 storks
2 4 = 8 + 4 4
= 8 + 16 = 24
15
How many legs are there? 2 lions, their 4 cubs and 4 storks
2 4 = 8 + 4 4
= 8 + 16 = 24 + 4 2
16
How many legs are there? 2 lions, their 4 cubs and 4 storks
2 4 = 8 + 4 4
= 8 + 16 = 24 + 4 2
= 24 + 8 = 32
17
Standing 40 m away from a flagpole on level ground a man used a theodolite to find the angle of
elevation of the top of the flagpole as 60o. Find the height of top of pole
from ground if the angle was sighted from 2 metres above the ground.
18
h = 40.tan60o
= 40 1.732
= 69.28 60o
40m
h
2m
19
h = 40.tan60o
= 40 1.732
= 69.28 + 2
= 71.28 60o
40m
h
2m
20
h = 40.tan60o
= 40 1.732
= 69.28 + 2
= 71.28 m.60o
40m
h
2m
21
• Cos A = 0.5 = 60o
• 2x – 5 = 9
= 2x = 14
= x = 7
22
Understandings of “=”
• 7 + 8 + 9 = ‘makes’ or ‘now work it out’ also ‘now do the next step’ hence misuses shown earlier
• x = 3 assigning a value
23
Understandings of “=”
• 27 = 5 × 6 – 3 ‘wrong way around – the single number is always on the right’
(reinforced by classroom rules such as ‘x terms on the left and numbers on the right’ when solving equations)
• 20 + 4 = 6 × 6 – 3 × 4 ‘if you work out each side you get the same answer’ (quantitative sameness)
24
Understandings of “=”• 3x + 2 = 5 ‘both sides are the same only when x
is 1’
• 3 + 2 = 5 identical
• 3x + 2x = 5x ‘identical, equivalent – true for all values of x (identity)
and in fact also true for x as any object thus fruit salad algebra
‘both sides are the same when x is 1’
• 2x + 4 = 6(3x – 2) ‘the = means the two sides balance’
25
• Understanding of equals and the language used from early schooling on.
• Concentration on calculation outcomes
• Restricted understanding of the arithmetic operations – seeing them as combining only rather than also in terms of change (or relational).
(Elizabeth Warren)
26
27
2 + 4 = 5 + 1
282 + 3 ? 5 + 1
29
Relational thinking
78 + 34 = 112
78 + 35 = ?
69 + 57 = + 58
367 + = 562 + 364
30
If I know that 78 + 34 = 122 what else do I know?
If I know that 23 16 = 368 what else do I know?
31
Some reasons for difficulties in algebra
• obstructions caused by different understandings of the symbols between children’s arithmetic understanding and algebra.
• inappropriate generalisations and interpretations• alternative approaches to semantics deduced from
the “concrete” situation
32
Understanding of operation symbols
• + a sign meaning to combine two numbers and accompanied by action
• 5 + 7 is 12• Once it is 12 the parts are no longer visible• In a + 7 the + sign does not mean actively
combine the two parts as it did in 5 + 7• While a + 7 can be seen as a single object,
the components maintain their identity
33
Understanding of operation symbols
Seeing the operations as combining leads to the incorrect
4x + 3 = 7x
A critical part of algebraic development is “acceptance of lack of closure” (Collis, 1975)
34
Understanding of operation symbols
Another way of seeing 5 + 7 is in a relational way as 5 more than 7.
a + 7 then becomes 7 more than a number a
The use of this type of language rather than translating it into words as a plus 7 or a and 7 is one that seems to be of great assistance in making sense of algebra.
35
Inappropriate generalisations• x is any number• Guess and check being reinforced in spite
of teachers’ approaches• Backtracking leading to inappropriate
recording and limiting development
Equation to solve for x: (x – 8)/2 = 3Student response: x = 39
36
41 students, three schools, six weeks after completed a unit on algebra including equation solving
If c = 5b + 2,
and c = 27,
what is b?
37
41 students, three schools, six weeks after completed a unit on algebra including equation solving
If c = 5b + 2, and c = 27,what is b?
38 gave correct answerbut
38
41 students, three schools, six weeks after completed a unit on algebra including equation solving
If c = 5b + 2, and c = 27,what is b?
38 gave correct answerBut 3 said thought at first question must be
wrong – should have been c = b5 + 2 then b = 2
39
Those who were correct were asked
If g = 4f + 3, and g = 12, what is f ?
40
Those who were correct were asked
If g = 4f + 3, and g = 12, what is f ?
Initially only two were correct.
Only one used a teacher taught method.
A third changed answer when asked to explain
41
Those who were correct were asked
If g = 4f + 3, and g = 12, what is f ?
Initially only two were correct.
Only one used a teacher taught method.
A third changed answer when asked to explain
The rest said it was impossible.
42
Inappropriate generalisations• x is any number• Guess and check being reinforced in spite of
teachers’ approaches• Backtracking leading to inappropriate recording
and limiting development
Left to their own devices students are unlikely to develop the semantics of algebra as we know them because the experiences they have are limited and often lead to alternative representations which are situation specific.
43
• The sequential patterns focus the children’s attention on an aspect which actually limits their understanding of function.
Developing rules from patterns
44
x y
1 4
2 7
3 10
4 13
5 …
10 …
100 …
45
x y 3 7
9 1
6 4
5
8
20Ryan & Williams
46
Mary has the following problem to solve“Find the value(s) for x in the following
expression: x + x + x = 12 ”She answered in the following manner
A. 2, 5, 5B. 10, 1, 1C. 4, 4, 4
Which of her answer(s) is (are) correct? Circle the letter(s) for each correct answer.
47
Would your answer have changed if the question was
+ + = 12
A. 2, 5, 5
B. 10, 1, 1
C. 4, 4, 4
Which of her answer(s) is (are) correct? Circle the letter(s) for each correct answer.
48
Jon has the following problem to solve“Find the value(s) for x and y in the following
expression: x + y = 16 ”He answered in the following manner
A. 6, 10B. 9, 7C. 8, 8
Which of his answer(s) is (are) correct? Circle the letter(s) for each correct answer.
49
+ = 16
He answered in the following manner
A. 6, 10
B. 9, 7
C. 8, 8
Which of his answer(s) is (are) correct? Circle the letter(s) for each correct answer. ?
Again would you have answered differently if it had been
50
One question that arises in looking at early algebra is the use of symbols
When should letters be introduced?
51
Project: University of Hawaii
Began kindergarten with no number work. First semester was all about measuring and comparing using comparative language and symbols to represent what they found.
Barbara Dougherty
52
Big picture idea: Generalisation
53
Problem solving task
The problem was about a group of children and adults coming down to the river to cross. There was only one boat and the boat would only hold two people. There were the same number of adults as there were children. How many river crossings to get everyone across?
(The problem is from the Mathematics Task Centre. The work from Babro Anselmson – Malmö)
54
Anna, aged 6, is in her first year of school and has not yet learned about the symbols +, and =
55
56
A challenge
Why is it that we hide mathematics that we think may be too difficult for children from them?
We do not hide words from them. Imagine if we said children in the first year of school could not possibly cope with words longer than 4 letters.
57
A challenge
If symbols are introduced naturally as a way of generalising when the need arises why not use the correct approach rather than having to unteach incorrect ideas later – for example 2 – 6.
Why not use n for any number? Or some other such symbol?
58
• build algebra sense, making sure the concept of = and all of the operations are well developed along with the structure such as identity and commutativity
• Build relational thinking rather than concentrating just on calculating
• Be willing to explore symbols and numbers beyond the syllabus outcomes
• Recognise that concepts need to be introduced and explored a long time ahead of when we expect them to be well established and connected in a child’s mental framework of mathematics.
To meet the needs of the future we need to
59
• Support students to take control of their own learning
• Create classroom cultures where discussion and debate is an integral part of learning
• Use assessment formatively and to inform • Raise expectations of what is possible and not hide
things from students because we think they are too hard for them
• Include a variety of problem solving and investigations
• Utilise appropriate tools to support learning
To meet the needs of the future we need to
60
Sunrise
or Sunset ?
61
Sunrise
or Sunset ?Definitely sunrise.
