making sense of the mathematical literacy caps document
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AMESA Congress 2011
MAKING SENSE OF THE MATHEMATICAL
LITERACY CAPS DOCUMENT
Marc North
Workshop outline:
1. Discussion of the key components of the ML CAPS
2. Developing activities based on the philosophy/approach outlined in the CAPS
Key components of the ML CAPS
Guiding question behind the development of the CAPS:
What does it mean to be mathematically literate?
Sowetan, 10 August 2010
Background: Government offering 7,5% increase in salary + R800 housing allowance Unions demanding 8,6% increase in salary + R1 000 housing allowance
Mathematical model:Consider a teacher who earns R10 000 per month and does not qualify for a housing allowance:
1. “No work, no pay”:Strike for 1 week: salary = ¾ of R10 000
= R7 500,00Loss in pay = R2
500,00
2. How much will the teacher earn at the Government and Union rates?
Government:Government offer = 7,5%→ Increase = 7,5% × R10 000
= R750
New salary = R10 000 + R750
= R10 750
UnionsUnion demand = 8,6%→ Increase = 8,6% × R10 000
= R860
New salary = R10 000 + R860
= R10 860
Extra money every month = R110.
3. How long will it take for the teacher to recoup the money lost as a result of the “no work, no pay” principle if they strike for 1 week?
Extra money every month = R110Loss of pay from the strike = R2 500
No. of month needed to recoup the lost pay = R2 500 ÷ R110/month= 22,727…≈ 23 full months.
Current pay R10 000 Strike Pay Pay LossNew Salary offered by Government (@ 7,5%) R 10 750 Pay after 1 week of strike R 7 500 R2 500
New salary demanded by unions (@ 8,6%) R 10 860 Pay after 2 weeks of strike R 5 000 R 5 000
Salary difference R 110 Pay after 3 weeks of strike R 2 500 R 7 500
Months worked
after strikeTotal pay recovered
Months worked after strike
Total pay recovered
Months worked after strike
Total pay recovered
Months worked
after strikeTotal pay recovered
1 R 110 13 R 1 430 25 R 2 750 37 R 4 0702 R 220 14 R 1 540 26 R 2 860 38 R 4 1803 R 330 15 R 1 650 27 R 2 970 39 R 4 2904 R 440 16 R 1 760 28 R 3 080 40 R 4 4005 R 550 17 R 1 870 29 R 3 190 41 R 4 5106 R 660 18 R 1 980 30 R 3 300 42 R 4 6207 R 770 19 R 2 090 31 R 3 410 43 R 4 7308 R 880 20 R 2 200 32 R 3 520 44 R 4 8409 R 990 21 R 2 310 33 R 3 630 45 R 4 950
10 R 1 100 22 R 2 420 34 R 3 740 46 R 5 06011 R 1 210 23 R 2 530 35 R 3 850 … …12 R 1 320 24 R 2 640 36 R 3 960 69 R7 590
Does this mean that teachers shouldn’t strike?
The point?
To be mathematically literate implies:
having the capacity to use mathematics and other techniques and considerations to make sense of authentic real-world problems (IF YOU WANT TO)
having an awareness that daily life is (more often than not) not structured around mathematical principles:o this means having an understanding that although we
can use mathematics to make sense of a situation, there are often non-mathematical considerations that affect our decisions and actions;
o this means having an understanding that mathematical models and mathematical solutions have limitations, and do not always present the most appropriate solution;
o this means recognising the role of informal or less formal techniques used for solving problems.
4,5 kg
R27,99
10 kg
R56,99
Which bag is better value for money?
Classroom
÷ 4,5 ÷ 4,5
× 10 × 10
4,5 kg
R27,99
10 kg
R56,99
4,5 kg : R27,99
1 kg :
10 kg :
R27,99 ÷ 4,5
R62,20
4,5 kg ≈ R28,005 kg ≈ R30,0010 kg ≈ R60,00
4,5 kg
R27,99
10 kg
R56,99
10 kg ≈ R57,001 kg ≈ R5,70½ kg ≈ R2,804,5 kg ≈ 1 kg + 1 kg + 1 kg + 1kg + ½ kg ≈ R25,00
4,5 kg
R27,99
10 kg
R56,99
Summary:
Currently: assessment of mathematical competency
CAPS document: exploring authentic real-life contexts in detail using a variety of both mathematical and non-mathematical techniques and considerations.
Philosophy behind the CAPS
Authentic real life problems
Using maths to make sense of
the world
Recognising that mathematical
models & solutions are
limited
Awareness of informal
techniques
Awareness that real-life can be structured in non-mathematical
ways
Maths Lit #1
Everyday practices
Maths Lit #4
Critical Citizenship
Maths Lit #3
Modelling
Maths Lit #2
Numeracy
Components of
Mathematical Literacy
Frustrations with the NCS
Vague assessment standards: → uncertainly over what is teachable content
→ variation between textbooks
Limited direction on suitable contexts
Uncertainly over progression
Contradiction between emphasis on mathematics and real-world application
Issues for consideration in
the CAPS document
1. Content: This aspect is very loosely defined in the curriculum statement Mathematical skills / calculations (e.g. percentages; ratios; etc) Daily life applications (e.g. constructing a budget)
2. Contexts: Authentic & realistic Appropriate?
3. Integration of content and skills
4. Progression: Content Contexts Problem solving skills (i.e. ability to make sense of contexts and problems without guidance)
5. Assessment Number and types of assessments Structure of examinations Taxonomy levels
Approach adopted
Mathematical content & skills & other considerations
Problem / Context
Problem / Context
“IN ORDER TO”
Problem / Context
In other words:
Content Context+ → Application
CONTENT
CONTEXT
APPLICATION
Activity:1. Identify the content, context and application
2. How are the requirements for this section different for Grades 10, 11 and 12?
Overview of the CAPS Document
CAPS Document Structure
Approach / Philosophy Curriculum Assessment
Content
Context
Application
Prog
ress
ion
Generic intro
Curriculum Component of
CAPS
Maps, plans and other representations of the
physical world
Finance
Measurement
Data Handling
Inte
rpre
ting
and
com
mun
icat
ing
answ
ers
Num
bers
and
op
erat
ions
with
nu
mbe
rs
Pat
tern
s,
rela
tions
hips
and
re
pres
enta
tions
Probability
Bas
ic S
kills
To
pic
s
Application Topics
Maps, plans and other representations of the
physical world
Finance
Measurement
Data Handling
Inte
rpre
ting
and
com
mun
icat
ing
answ
ers
Num
bers
and
op
erat
ions
with
nu
mbe
rs
Pat
tern
s,
rela
tions
hips
and
re
pres
enta
tions
Probability
Bas
ic S
kills
To
pic
s
Application Topics
All Grades
G10
G10 & 11
Grades 10, 11 & 12
Why not stick to the existing 4 LO’s?
Why “Basic Skills” and “Applications”?
Basic Skills Topics
Topic Section
Numbers and calculations with numbers
Number formats and conventionsOperations on numbers and calculator skillsRoundingRatiosProportionRatesPercentages
Topic Section
Patterns, relationships and representations
Making sense of graphs that tell a storyPatterns and relationshipsRepresentations of relationships in tables, equations and graphsWorking with two or more relationships and/or representations (Grade 11 & 12)
Grade 10: Constant (i.e. horizontal / vertical) Linear and direct proportion Inverse proportion
Grade 11 & 12: Compound growth graphs (e.g. graph showing the amount outstanding on a loan over time) Combination of the above (e.g. step function; cell phone scenario with free minutes) Graphs that arise out of a problem / context for which no pattern is available or obvious.
Topic Section Grade 10 Grade 11 Grade 12
Finance
Financial documents
Contexts are limited to those that deal with personal and/or household finance.
Contexts are limited to those that deal with personal, household, workplace and/or business finance.
Contexts are limited to those that deal with personal, household, workplace, business, national and global finance, and more complex financial scenarios.
Tariff systemsIncome, expenditure, profit/loss, income-and-expenditure statements and budgetsCost price and selling price ----
Break-even analysis ----Interest Contexts are limited
to those that deal with personal and/or household banking.
Banking, loans and investments
Inflation ----
Taxation Contexts are limited to VAT.
Exchange rates ----
Topic Section Grade 10 Grade 11 Grade 12
Measurement
Conversions
Simple tasks in the familiar context of the household.
Larger projects in familiar contexts of the household and school and/or wider community.
Complex projects in familiar and unfamiliar contexts.
Measuring lengthMeasuring weightMeasuring volumeMeasuring TemperaturePerimeter, area and volume
Time
Work with time formats and calculations to plan and complete daily activities in the familiar context of the household.
Work with time formats and calculations to plan and complete daily activities in the household, school and wider community.
Work with time formats and calculations to plan and complete daily activities and trips in both familiar and unfamiliar contexts.
Topic Section Grade 10 Grade 11 Grade 12
Maps, plans and other representations of the physical world
Scale Maps and plans of familiar contexts and/or simple structures (e.g. school).
Maps and plans of less familiar contexts and/or structures (e.g. office space) and models of packaging containers.
Maps and plans of unfamiliar contexts and/or complex structures (e.g. RDP house) and models of packaging containers and buildings.
Maps Maps and plans of less familiar contexts and/or structures.
Maps and plans of possibly unfamiliar contexts and/or complex structures.Plans
ModelsWork with actual tins and boxes to explore packaging arrangements
Build 3-D scale models of packaging containers to investigate packaging arrangements.Draw 2-D scale pictures of 3-D packaging containers.
Build 3-D scale models of packaging containers and buildings to explore what the final product will look like.Draw 2-D scale pictures of 3-D buildings and packaging containers.
Topic Section Grade 10 Grade 11 Grade 12
Data handling
Developing questions
Data is limited to contexts relating to the personal lives of learners.Learners are expected to work with only one set of data.
Data is limited to contexts relating to the personal lives of learners and wider social issues.Learners are expected to work with two sets of data and comparisons thereof.
Data is limited to contexts related to the personal lives of learners, wider social issues and national and/or global issues. Learners are expected to work with multiple sets of data and comparisons thereof.
Collecting data
Classifying and organise data
Summarising data
Representing data
Interpreting and analysing data
Topic Section
Grade 10 Grade 11 Grade 12
Likelihood
Expressions of likelihood
Explore likelihood in scenarios involving: games using coins
and dice; weather
predictions.
Explore likelihood in scenarios involving: games using coins
and dice; weather
predictions; tests where there is
the chance of inaccurate results;
cosmetic and other products making statements regarding likelihood.
Explore likelihood in scenarios involving: games using coins
and dice; weather
predictions; tests where there is
the chance of inaccurate results;
cosmetic and other products making statements regarding likelihood;
lottery and other gambling games;
risk assessments; newspaper articles containing references to likelihood.
Prediction
Evaluating expressions involving likelihood --- ---
CAPS vs. NCS
NCS CAPSLO 1: Numbers and Operations Basic Skills
Topics
Numbers and calculations with numbers
LO 2: Functional Relationships
Patterns, relationships and representations
LO 3: Space, shape & measurement
Application Topics
Finance
LO 4: Data handling Measurement
Maps, plans and other representations of the physical world
Data handling Probability
LO’s vs. Topics
Content from NCS not included in CAPS
Scientific notation
Financial indices
Solving equations simultaneously using
algebraic methods
Quadratic formula
Angles
Transformation geometry
Cumulative frequencies and Ogive curves
Standard deviation and variance
There is not much “new” content.
Rather, the document is much more specific (compared to the NCS) about precisely what must be taught in each topic and section.
“New” content
NCS (2003)
CAPS
Advantages: There is much more description given as to what must be taught / learned. There will be far more continuity between textbooks.
Disadvantages:
Refer to document on CD titled:
“NCS vs. CAPS”
Other differences
Assessment Component of
CAPS
Weighting of topics
Inte
rpre
ting
and
com
mun
icat
ing
answ
ers
Maps, plans and other representations of the
physical world
Finance
Num
bers
and
op
erat
ions
with
nu
mbe
rs
Pat
tern
s,
rela
tions
hips
an
d re
pres
enta
tions
Measurement
Data Handling
Probability
Bas
ic S
kill
s To
pic
s
Application Topics
35% (+-5%)
Min 5%
25% (+-5%)
15% (+-5%)
20% (+-5%)
Examinations (Grades 10, 11 & 12)
GRADE 10 GRADE 11 GRADE 12
TERM 1 Control Test Control Test Control Test
TERM 2Paper 11 hour
(50 marks)
Paper 2: 1 hour
(50 marks)
Paper 1: 1½ hours
(75 marks)
Paper 2: 1½ hours
(75 marks)
Paper 1: 2 hours
(100 marks)
Paper 2: 2 hours
(100 marks)
TERM 3 Control Test Control Test
Control Test Control Test
Paper 1:3 hours
(150 marks)
Paper 2:3 hours
(150 marks)
TERM 4Paper 1
1½ hours(75 marks)
Paper 21½ hours
(75 marks)
Paper 1: 2 hours
(100 marks)
Paper 2: 2 hours
(100 marks)
Nationally set
Paper 1:3 hours
(150 marks)
Paper 2:3 hours
(150 marks)
Examination Paper Structure
5 questions
Finance Measure Maps … Data Integrated
Likelihood
Numbers + Patterns Interpreting+
PAPER 1 – “Skills Paper in familiar contexts”
Scope of contexts for Paper 1: limited to those listed in the CAPS document (i.e. “familiar” contexts)
Intention of the paper: assess proficiency of concepts, content and/or skills
Distribution of marks according to the taxonomy levels for Paper 1:
Grades 11 and 12
Paper 1
Level 1: Knowing 60% 5%
Level 2: Applying routine procedures in familiar contexts 35% 5%
Level 3: Applying multi-step procedures in a variety of contexts 5%
Level 4: Reasoning and reflecting
4 or 5 questions
Integrated Integrated Integrated Integrated Integrated
Numbers + Patterns Interpreting+
PAPER 2 – “Applications paper working in familiar & unfamiliar contexts”
Scope of contexts for Paper 2: not limited to those listed in the CAPS document
Intention of the paper: assess ability to use both mathematical and non-mathematical techniques/considerations to explore familiar and unfamiliar contexts
Distribution of marks according to the taxonomy levels for Paper 2:
Grades 11 and 12
Paper 2
Level 1: Knowing
Level 2: Applying routine procedures in familiar contexts 25% 5%
Level 3: Applying multi-step procedures in a variety of contexts 35% 5%
Level 4: Reasoning and reflecting 40% 5%
Grades 11 and 12
Paper 1 Paper 2 Overall allocation
Level 1: Knowing 50% 5% 10% 5% 30% 5%
Level 2: Applying routine procedures in familiar contexts
40% 5% 20% 5% 30% 5%
Level 3: Applying multi-step procedures in a variety of contexts
5% 35% 5% 20% 5%
Level 4: Reasoning and reflecting 5% 35% 5% 20% 5%
Total distribution of marks according to the taxonomy levels for both papers:
Possible challenges
Philosophy of the document vs. mathematical basis of the taxonomy
How is the weighting of topics going to play out in exams?
APPENDIX 3FURTHER INTERPRETATION OF THE DIFFERENT TAXONOMY LEVELS ACCORDING TO TOPICS
Designing activities based
on the philosophy of the CAPS
Time for work!
OR
231.18
Call Time Call Cost
Any time R1,50
TNS Research Surveys – www.tnsresearchsurveys.co.za. Sourced 14 Feb 2011
Marc North
mnorth@telkomsa.net
083 627 8188
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