Cognitive levels and types of mathematical activities

Prof Margot Berger

Lynn Bowie

Prof Lovemore Nyaumwe

Was the 2008 matric maths

exam a good one?

SAGM

Knowledge Routine Procedures

Complex Procedures

Solving Problems

Recall, Use of formulae or algorithms

Well-known procedure

Mainly unfamiliarNo direct route to solutions

Non-routineUse higher level cognitive skills and reasoningBreaking down problem into constituent parts

Porter

Memorize facts/ definitions/ formulae

Perform Procedures

Demonstrate an understanding of mathematical ideas

Solve non-routine problems/ make connections

Conjecture/ generalize/ prove

Recall basic factsRecall basic formulae or computational procedures

Follow proceduresSolve equationsRead or produce graphs

Communicate mathematical ideasUse representation to model ideasDevelop relationships between concepts

Apply and adapt strategies to non-routine problemsApply maths in contexts outside maths

Determine the truth of a mathematical pattern or proposition Write formal or informal proofs

Stein et alMemorization Procedures

without connections

Procedures with connections

Doing Mathematics

Reproduction of previous learnt fact, formula, rule, definition

AlgorithmicUnambiguousRequire no explanation

Engage with underlying conceptual ideasMake connections among multiple representations

Complex, non-algorithmic thinkingDemand self-monitoring of cognitive processesAnalyse task

NAEP

Low complexity Moderate Complexity High Complexity

RecallItem specifies what a student is to do

Student must decide what to do – bring together concepts and processesRequire more flexibility of thinking and choice among alternatives than do those in low complexity category

Use reasoning , planning, analysis, judgment, creative thoughtMay be expected to justify/construct an argument

1%

62%

25%

12%

SAGM

KnowledgeRoutine ProcComplex ProcProblem Solving

1%

51%48%

Stein

Memoriza-tion

Procs with connections

doing maths

Procs without connections

Routine, but with connections

Having drawn the graph of f(x) = 2(x-1)2 – 8 :

The graph of f is shifted 2 units to the left. Write down the equation of the new graph

Solving problems/non-routine

QUESTION A

Given

1) Write down x2 in terms of x

2) Hence determine the value of x

QUESTION B

Given x = 999 999 999 999, determine the exact value of

6 6 6 6 ....x

2 4

2

x

x

Proof

Proof

• NCS: In SAG and Exam guidelines:– Content– Cognitive levels

• OTHER: Weighting in terms of objectives“Read, interpret and solve a given problem using appropriate mathematical terms”“Formulate a mathematical argument and communicate it clearly”“Organize and present information and date in tabluar, graphical and/or diagrammatic forms”

NCS

• Mathematics is important for critical democratic citizenship

• Mathematics is relevant and practical. Mathematical modelling is an important tool

• Mathematics is created and organised using particular processes

• Mathematics is a body of knowledge

NCS Maths

• Purpose: “An important purpose of Mathematics in the FET band is the establishment of proper connections between Maths as a discipline and the application of mathematics in real-world contexts. Mathematical modelling provides learners with the means to analyse and describe their world mathematically...”

Modelling

• “Modelling as a process should be included in all papers, thus contextual questions can be asked in any topic”

• Word problems vs standard applications vs modelling problems

NCS MATHS

• Purpose:“Competence in mathematical process skills such as investigating, generalising and proving is more important than the acquisition of content knowledge for its own sake”

Process skills

• In number patterns:– NCS: investigate, make conjectures, generalise,

explain, justify, prove– EG clarification: investigate, identify, extend, explain,

determine, calculate • Proofs of the sum of arithmetic and geometric series are

examinable• Prove properties of polygons using analytic methods• Proofs of trig formulae excluded• Proofs in transformation geom? Proofs in algebra?

What possibilities?

Low complexity

Moderate complexity

High Complexity

Solving real-world problems; Mathematical modelling

Mathematical practices, such as conjecturing, generalising, justifying providing

Mathematical knowledge (facts, procedures, forms of representation, notation)