Mathematical Literacy

CAPS Document

Shifts and Rationales

Extensive ML CAPS document authoring team

Author Critical reader

Key question:

What does it mean to be

mathematically literate?

Mathematical Literacy Grade 12 Paper 2 Examinations (2009)

(i.e. “Applications” paper)

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 go on 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.

Abstract content

Basic numeracy

Real-world

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

Artificial constructs of the LO’s (i.e. lack of integration)

Issues for consideration in

the CAPS document

Brief:

“Unpack the existing curriculum document to make it more obvious precisely what must be taught.”

“Create one document to encompass all existing documents (NCS; SAG; Teachers Guide’ Examination Guidelines)”.

“Please give a workscheme, suggested timings (per topic) and resources needed (e.g. bills, maps, etc)”

“Oh, one final thing: please keep the document to around 40 pages.”

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

Possible approaches

Mathematical Content

Context

Context

Content

Content

Real Life Contexts

Mathematical content & skills & other considerations

Problem / Context

Problem / Context

“IN ORDER TO”Problem / Context

Example 1:

Example 2:

Example 3:

Overview of the CAPS Document

Chapter 1: Department blurb

Chapter 2: Mathematical Literacy What is Mathematical Literacy Progression in Mathematical Literacy Overview and weighting of topics Suggested teaching plan

Chapter 3: Content and scope of context per topic Overview Detailed outline of content / skills to be taught

per grade• Basic skills topics• Application topics

Chapter 4: Assessment

GRADES 10, 11 & 12

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

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ions

with

nu

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rs

Pat

tern

s,

rela

tions

hips

and

re

pres

enta

tions

Probability

Bas

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kills

To

pic

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Application Topics

Why not stick to the existing 4 LO’s?

Why “Basic Skills” and “Applications”?

Don’t be stupid Marc!

Topic SectionInterpreting and communicating answers and calculations

Interpreting answersCommunication

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 ----

A = P(1 + i)n

Months Opening Balance InterestBalance with

InterestMonthly

RepaymentClosing Balance

0 R 72 750.00 R 927.56 R 73 677.56 R 1 440.40 R 72 237.16

1 R 72 237.16 R 921.02 R 73 158.19 R 1 440.40 R 71 717.79

2 R 71 717.79 R 914.40 R 72 632.19 R 1 440.40 R 71 191.79

3 R 71 191.79 R 907.70 R 72 099.48 R 1 440.40 R 70 659.08

4 R 70 659.08 R 900.90 R 71 559.99 R 1 440.40 R 70 119.59

5 R 70 119.59 R 894.02 R 71 013.61 R 1 440.40 R 69 573.21

6 R 69 573.21 R 887.06 R 70 460.27 R 1 440.40 R 69 019.87

7 R 69 019.87 R 880.00 R 69 899.87 R 1 440.40 R 68 459.47

8 R 68 459.47 R 872.86 R 69 332.33 R 1 440.40 R 67 891.93

9 R 67 891.93 R 865.62 R 68 757.55 R 1 440.40 R 67 317.15

10 R 67 317.15 R 858.29 R 68 175.45 R 1 440.40 R 66 735.05

11 R 66 735.05 R 850.87 R 67 585.92 R 1 440.40 R 66 145.52

12 R 66 145.52 R 843.36 R 66 988.87 R 1 440.40 R 65 548.47

13 R 65 548.47 R 835.74 R 66 384.22 R 1 440.40 R 64 943.82

14 R 64 943.82 R 828.03 R 65 771.85 R 1 440.40 R 64 331.45

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 --- ---

“New” Content

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.

NCS (2003)

Advantages: There is much more description given as to what must be taught / learned. There will be far more continuity between textbooks.

Disadvantages:

Changes to Assessment

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

30% (+-5%)

10% (+-5%)

20% (+-5%)

20% (+-5%)

20% (+-5%)

Grade 10 examinations

5 questions

Finance Measure Maps … Data Integrated

Likelihood

Numbers + Patterns Interpreting+

Grade 10

Level 1: Knowing 30% 5%

Level 2: Applying routine procedures in familiar contexts 30% 5%

Level 3: Applying multi-step procedures in a variety of contexts 20% 5%

Level 4: Reasoning and reflecting 20% 5%

Distribution of marks according to the taxonomy levels:

Grade 11 & 12 examinations

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 50% 5%

Level 2: Applying routine procedures in familiar contexts 40% 5%

Level 3: Applying multi-step procedures in a variety of contexts 5%

Level 4: Reasoning and reflecting 5%

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 10% 5%

Level 2: Applying routine procedures in familiar contexts 20% 5%

Level 3: Applying multi-step procedures in a variety of contexts 35% 5%

Level 4: Reasoning and reflecting 35% 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