numeracy forum 3 - 4 april, 2012 [email protected] reasoning – for learning mathematics and...
TRANSCRIPT
Numeracy Forum
3 - 4 April, 2012
Reasoning – for learning mathematics and long term benefits
Kaye StaceyUniversity of Melbourne
Australian Curriculum (2010 March)
Australian Curriculum Proficiency Strands
• Understanding
• Fluency
• Problem Solving– Students develop the ability to make choices, interpret, formulate,
model, and investigate problem situations, and communicate solutions effectively
• Reasoning– Students develop increasingly sophisticated capacity for logical
thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising
• Also “general capabilities” across subjects which include “thinking skills” and “creativity” (p. 5)
WhatHowUseWhy
Mathematics content and process – concerns
• Different nature of four proficiency strands means they need different treatment– Understanding and fluency – inherent part of learning content well– Problem solving and reasoning – more than this
• Important outcomes of learning, independent of content • Part of the fabric of any real mathematics lesson• Also contributing to learning content
• Dilemma of separation from content VS integration with content– in class and in curriculum specification– Need to identify relevant goals for broad age groups
Everywhere and nowhere
Reasoning and PS are not just “forms” of classroom interaction (e.g. discussion)
• Reasoning is not evident in the form of classroom interaction, but in the substance
• External evidence of reasoning– Classroom discussion
• Find reasons and arguments• Compare reasons and arguments• Analyse reasons and arguments
– Writing arguments• Consolidate reasoning• Check reasoning
• Reasoning happens when students work by themselves too!
Why an interest in reasoning?
• Strong data that it is missing
• Best data is getting old, but new information paints a consistent picture
ACER Research Conference 2010
TIMSS Video Study 1999: Methodology
• Video-taped 638 Year 8 lessons from the seven countries– Video-taping spread across school year– Lesson selected at random; little warning given– Random sample of schools and (volunteer) teachers
• Australia: 87 schools, 1950 pupils
• Comparative analysis of many characteristics
Hiebert, J., Gallimore, R., Garnier, H., Givvin, K.B., Hollingsworth, H., Jacobs, J., Chui, A.M.-Y., Wearne, D., et al (2003). Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study (NCES 2003-013). Washington DC: National Center for Education Statistics.
Hollingsworth, H., Lokan, J. & B. McCrae, B. (2003) Teaching Mathematics in Australia: Results from the TIMSS 1999 Video Study. Melbourne: ACER.
Australia in comparison
• Many strong features of Australian lessons– especially teachers (overwhelming strongest point from our survey)
• Shallow Teaching Syndrome – in comparison Australian Year 8 lessons exhibited:– A very high percentage of problems that were very close repetitions of previous
problems– A very high percentage of problems that were of low procedural complexity (e.g.
small number of steps, not bringing different aspects together)– General absence of mathematical reasoning– [Somewhat low on percentage of problems using real life contexts]
• Aspects of absence of reasoning– No lessons contained ‘proof’ (even informal)– Very few problems requiring students to ‘make mathematical connections’– When problems required connections, these were not emphasised in discussing
solutions• Often give the result only • or often focus on the procedures employed
ACER Research Conference 2010
ACER Research Conference 2010
Outline
• Examples relevant to primary school, but interesting to adults
• Use examples to illustrate aspects of reasoning that – Can be incorporated into teaching to improve students’
mathematical reasoning– Can be used to improve students‘ understanding of content topics
• Discuss ‘proper’ mathematical explanations (deductive) and also ‘didactic explanations’ which are appropriate to convince students
• Basic premise: all students can understand why mathematical results are true (at an appropriate level). Doing so helps them remember and apply what they learn.
Reasoning in Mathematics
• Reason like a scientist --- inductive reasoning– (experimental) – gather data and see patterns
• Reason like a lawyer -- deductive reasoning– from definitions and agreed laws (theorems)
• Reason like a detective – abductive reasoning– using clues to think about what might happen– used in finding arguments
• Reason like an artist – analogical reasoning– drawing analogies and underlying similarities
• Children use all these forms of reasoning from an early age, although there is much to learn.
Even and Odd Numbers
• An example of the different types of reasoning that can be present from the beginning of school.
• Especially contrasting – Inductive reasoning– deductive reasoning
even + even = eveneven + odd = odd
Inductive reasoning (experimental demonstration) – look at examples
1 + 1 = 2 2 + 4 = 6 3 + 3 = 6
2 + 6 = 8 13 + 5 = 18
12 + 4 = 16
14 + 5 = 19
even + even = eveneven + odd = odd
Inductive reasoning (experimental demonstration) – look at examples
1 + 1 = 2 2 + 4 = 6 3 + 3 = 6
2 + 6 = 8 13 + 5 = 18
12 + 4 = 16
14 + 5 = 19
Definition of even (and hence odd) numbers - Multiple of 2- Last digit is 2, 4, 6, 8, 0- Is made up from pairs
Even numbers make pairs
Odd numbers – pairs and one left over
even + even
even
even + odd
odd
odd + odd
even
http://www.fgworld.co.uk/Schooldays/Mary%20Hines/st%20mikes%20walking%20day%20kids.jpg
Explanation?
To prove that the sum of two even numbers is even
Let and be even numbers
Then
2 for
2 for
Therefore 2 2 2( )
Since ( ) , is an even number
a b
a m m
b n n
a b m n m n
m n a b
Definition of even (and hence odd) numbers - Multiple of 2- Last digit is 2, 4, 6, 8, 0- Is made up from pairs
Extending and generalising reasoning (and to look through child’s eyes)
• Even numbers end in 0, 2, 4, 6, 8
• oven + even = eveneven + odd = oddodd + odd = even
• Even/even fractions cancel
• Patterns in multiplication tables
• Any others?
• Investigate “threeven” and “throdd” numbers
• Examples 1, 2, 3, 4, 5, 6, 7, 8, ..
• Investigate “foureven” and “fourodd” numbers
• Examples 1, 2, 3, 4, 5, 6, 7, 8, ..
Observations
• As reasoning develops, argumentation becomes more precisee.g. even and 0, 2, 4, 6, 8 end digits (if and only if) foureven and 0, 2, 4, 6, 8 end digits (if, but not only if)
• Even very young children can make valid mathematical arguments using deductive reasoning
• Note that young children “see the general in the particular”, whereas with algebra the general is directly worked with.
• Note the importance of the right definition - teachers should choose the best they can.
What is a circle?
Australian Curriculum:
• “obvious features” (Yr 1)
• “key features” (Yr 2)
• symmetry (Yr 3)
• measurement fomulas Yr 8
What is a circle?
Australian Curriculum:
• “obvious features” (Yr 1)
• “key features” (Yr 2)
• symmetry (Yr 3)
• measurement fomulas Yr 8
This (round, curved) or this (has a centre)?
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/space/SP25001P.htm
Key features of a circle
• Round, curved line, all points same distance from centre, constant width, don’t pack well,....
• Which key features are relevant to these uses:– Lid of jam jar– Space to open a door– Car turning circle– Sports field “goal circle”– Street manhole cover– Proving circle theorems– Spinning around
Important to stress the mathematical definition of circle (as Taiwan examples) for further reasoning
Angle Sum of a triangle
• Multiple ways of convincing – students get a sense of their own mathematical power and a sense that they can think things out
• Different ways have different mathematical status
• Many ways have didactic ‘value’, even without formal mathematical ‘value’
Measuring Angles
Triangle Angle A
AngleB
AngleC
AngleSum
1 65o 67o 47o 179o
2 27o 64o 90o 181o
3 20o 35o 125o 180o
4 27o 128o 24o 179o
5 155o 8o 15o 178o
6 8o 90o 84o 182o
Evaluating inductive reasoning
• Provides concrete examples, so students understand what is being claimed
• Convincing to all
• Good place to start – can provide ownership through discovery
• Not a proof
• Leads to many isolated facts to remember
• Why is it so popular now?
Dynamic Geometry (or a long rope?)
• Open file dynamictriangle
• Evaluation of this method
Reasoning sometimes leads astray
• Angle sum might change if the triangle changes size
• Limits need extra care (e.g. false Pythagoras’ theorem)
Tearing off corners & putting together
From experimenting to reasoning
Order: C-A-B Order: C-B- A
Use these methods to find angle sum of quadrilaterals and pentagons
• Measuring – this will work – no need to try
• Dynamic geometry?
• Tearing off corners and reassembling
• Making a proof from tearing off corners method
Varying teachers’ opinions on nature of appropriate arguments for Year 8
• Role of (possibly informal) proof following a dynamic geometry investigation of sum of exterior angles of a pentagon– “I would be happy to students
just explore these results”– “there are good opportunities
for deductive reasoning here”– (proof) “not particularly
appropriate for many students”– “I would not do that”– “have students generalise the
result to other polygons”– “too obvious to prove”
Sum of exterior angles is 360 degrees
Observations about Angle sum reasoning
Area and perimeter – confronting misconceptions and common errors
• Reasoning about rich tasks helps to – clarify and distinguish concepts– confront errors, and eliminate them
• Examples– Confusion of area and perimeter (next problem)– Area = length x breadth for any shape– Bigger shapes have bigger area and bigger perimeter
Evaluating mathematical statements.
Is it true that if you cut a piece off a shape, then the area and perimeter are reduced?
And now understanding.....
“When I have eaten all my sandwich, I will only have perimeter left”
Rich tasks promote reasoning (Swan
• Draw a shape on squared paper and plot a point to show its perimeter and area.
• Which points on the grid represent squares (or rectangles, or…)?
• Draw a shape that may be represented by the point (12,4)
• Now generalise the above, allowing any shapes, not necessarily following grid lines
• Draw a shape that may be represented by the point (4,12)
• Which points on the graph are possible and which are impossible?
Decimals – the good and bad of reasoning by analogy
• Using models relies on students reasoning by analogy– What happens with the model will happen with the numbers
• Reasoning by analogy is a basic cause of persistent misconceptions
Research tool: Decimal Comparison Test
Lots of interviews and careful checking…….
Result: about 12 types of thinking
• Longer-is-larger (L)
whole number thinking string length
column overflow thinking zero makes small
• Shorter-is-larger (S)
denominator focussed
reciprocal negative
• Apparent Expert (A)
true understanding, rule followers, money thinking, expert except 0.6/0, place value number line,
Teaching & Learning about Decimals CD Caitlin, whole number thinking
• Profile of person & her thinking
• Annotated copy of Decimal Comparison test
• Four interview extracts – slide show format– movie format
• Self-test with answers
• Links to teaching strategies
Understanding children’s thinking
Courtney (Year 8) is a “reciprocal thinker”.
Interpreting decimals by analogy with fractions . (one of the S thinkers)
Another analogy – with negative numbers
• With the pair 2.516 and 2.8325, Anita explained:
• "I felt more comfortable selecting the number with the least digits [as the larger numerical value] as I thought the longer the number, the further it was down the number line in the negative direction."
Place value number line
-2 -1 0 12
thousands hundreds tens units tenths hundredths thousandths
Percentages of students able to compare size of decimals reliably (Victoria, 1996-1999, 3000 students)
Change of category over 6 months (N=5497)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
A U S L
Original category
Pe
rce
nta
ge
in c
ate
go
ry a
fte
r 6
mo
nth
s
A
U
S
L
Change of category over 3-4 years (N=64; 6 tests per person)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
A U L S
Original category
Pe
rce
nta
ge
in c
ate
go
ry a
fte
r 3-
4
year
s (6
co
nse
cuti
ve
te
sts)
A
U
S
L
Teaching Ideas
Even a little bit makes a difference!
OUR MOTTO The decimal point marks the ones column
• Avoids thinking there are two separate parts (c.f. separates the whole number part from fraction)
• Minimises problems with false symmetry across the decimal point
hundreds tens units tenths h’ths th’ths t’th’ths
4 5 6. 7 8 9 1
Endless Base Ten Chain
Stressing generality – not special facts
Number Expanders(Click here)
Linear Arithmetic Blocks
Assemble to represent numbers by lengthActions on the model guide actions on numbers
Linear arithmetic blocks can model a number, a number line and demonstrate number density
Linear model easier to understand than a volume model such as MAB.
Model number operations by combining.
Decimals and reasoning by analogy
False analogies
• Whole numbers
• Fractions
• Calculator junk
• Negative numbers
• Place value number line
Misremembered rules
• Zeros don’t matter
• Add a zero to multiply by ten’
Using the model
• Length of pieces represents the value of the number
• Compare numbers by comparing length etc
• Operations link to practical meaning (e.g. Multiply or divide by ten)
• Illustrate errors such as 5.13 x 10 = 5.1300.377 = 0.37
Reasoning in Mathematics
• Reason like a scientist --- inductive reasoning– (experimental) – gather data and see patterns
• Reason like a lawyer -- deductive reasoning– from definitions and agreed laws (theorems)
• Reason like a detective – abductive reasoning– using clues to think about what might happen– used in finding arguments
• Reason like an artist – analogical reasoning– drawing analogies and underlying similarities
• Children use all these forms of reasoning from an early age, although there is much to learn.
How does reasoning develop with age?
• All forms of reasoning present from beginning of school.
• Can reason about more sophisticated mathematical objects (fractions instead of just whole numbers)
• Can create and follow longer arguments (more steps).
• Can present arguments more clearly, integrating words and symbols better
• Become more aware of pitfalls of reasoning (e.g. All multiples of 6 are multiples of 3, not vice versa)
• Can better decide if an argument is mathematically convincing
To encourage reasoning:
• Use rich tasks that require exploration, discussion, argument, and then presenting reasoning
• Establish classroom norms of explaining your thinking
• Give as few ‘rules without reasons’ as possible – “didactic explanations” where mathematical explanations not possible.
• Begin discussion of types of evidence in maths and other subjects
• Set practice examples that involve reasoning (e.g. area not just of rectangles, but composite shapes such as L made from two rectangles) – we are falling down here now!
