ADDENDUM A

SPECIFIC COMMENTS REGARDING THE PROPOSED MATHEMATICAL LITERACY UNIT STANDARDS

General

__Rationale__:

In light of Mathematical Literacy being an altogether new programme, the document would benefit from an introduction that sketches the rationale of the programme for teachers as well as sketching an image of the learner on completion of the programme.

__Language and clarity
__One of the greatest weaknesses of the document in its current form is that the language needs a lot of tightening up and improvement. Firstly it needs to be grammatically improved and secondly there is a need for an increased level of accuracy in use of language. Examples include:

*Identify, describe, compare, classify, calculate shape and motion . . . *(Unit Standard 2005)

*Describe, apply, analyse and calculate shape and motion . . . *(Unit Standard 3004)*.
Construct, analyse and calculate shape and motion in . . . *(Unit Standard 4004)

While one can "analyse shape and motion" (and even this it can mean a host of different things to different people), and "describe shape__s__ and motion__s__" and "compare (different) shape__s__ and motion__s__." One certainly cannot "calculate shape and motion" or "apply shape and motion" or "construct shape and motion."

*Mathematical justifications are provided the control of costs and maximising of profits in relation to data *(Unit standard 2002, assessment criteria 3.3)

This statement simply makes no sense

There are many such examples throughout the document and it is not within the scope of this submission to identify each one.

__Level of specificity
__There are too many instances where the descriptions are too general rather than being specific. Examples include:

*Number systems* (used in the titles of unit standards 2001 and 2002)

What does the "number systems" refer to here? There seems to be no mention of number systems in the specific outcomes and other design criteria. Furthermore does the use of "number systems" imply the inclusion of binary systems and complex number systems? —it certainly could and that would be beyond the scope of Mathematical Literacy!

The unit standards that deal with "geometry" speak of 2-D and 3-D shapes (technically this should read 2-D shapes and 3-D objects) do not list the shapes and objects to be considered i.e. are non-rectilinear shapes included and what about cardioids and ellipsoids not to mention volumes and surface areas of non-regular 3-D objects. Unless these aspects are specified, there exists the danger that different teachers will interpret the document very differently.

Our experience, as a nation, with Curriculum 2005 was that teachers could not implement the early version due to a lack of specificity and care should be taken to include an appropriate level of specificity throughout the document.

__Coherence between and within unit standards
__It is important that greater care should be taken to ensure greater coherence between the unit standards in the next iteration of the document. At this stage the document gives the feeling that it was written by a number of different authors who did not co-ordinate their efforts. Examples include:

Unit Standard 2002 Specific Outcome2 includes compound interest which is a specific case of the function A(1+r)^{n}, yet Unit Standard 2003 only lists linear and quadratic functions in the range statement, and no mention is made of exponential functions. In a coherent and integrated set of standards one would expect all functions included in "applied" units to be also included in more purely mathematical units. Also note that the hyperbola, i.e. functions of the form appear quite naturally in simple budgetary and other financial contexts and it would make sense for this function to also be included.

Specific Unit by unit comments

NOTE these comments are by no means comprehensive and AMESA cannot claim that they are consistent either. The time frame for comment has simply not enabled us to do a better job of this. It is, however hoped that they will provide the author team with points to consider. In addition to the comments below, an addendum is provided—in this addendum one AMESA member (Moses Mogambery) gives a detailed critique of Math 2001 and in so doing highlights some of the difficulties that teachers may have with the document.

Math 2001

- Some of the Assessment Criteria appear to be too academic in nature as opposed to them giving an emphasis to how one might deal with the issue in practice. For example Specific Outcome 1, Assessment Criteria 1.5 might be better expressed as follows: The result of the algorithm is checked against the context in which it was used.
- While the US is deals with tricky issues, and an understanding of them to the level indicated as well as guidelines for dealing with them in practice are important as the underlying implications for teaching and learning embedded in the Specific Outcomes and Assessment Criteria, a case could be made for dropping unit standards that deal with numbers altogether and developing the concepts within contexts and other unit standards when appropriate.
- What is meant by:
- Assessment Criteria 1.1 for Specific Outcome 1: "Rational numbers are used to develop different sets of odd, even and prime numbers"?
- Assessment Criteria 1.5 for Specific Outcome 1: "The viability of selected algorithms is verified and justified"?

More specifically:

- What is meant by the "viability" of an algorithm? Does it mean that a particular algorithm
- is workable/executable/practicable at all
- is workable/executable/practicable in a specific situation/context
- is a good choice of tool (selection) in a specific situation with reference to how much work/time/effort it will require in the specific situation
- produces the correct answer or not (i.e. whether it is "correct’ or "faulty")
- What is meant by "selected"? Does it mean that the provider of learning selects some algorithms for learners to assess the viability of the algorithms? Or does it refer to a situation where the learner has selected a specific algorithm to perform a task, and is now required to assess the viability of the particular (selected) algorithm for the particular task? Or does the "selected" mean something else still?
- What is meant by "verified and justified"? Is there a suggestion that there is a difference between "verify" and "justify"? If so, what is the difference between the two actions?

Math 2002

- An important Unit Standard for Mathematical Literacy purposes.
- Specific Outcome 2, Assessment Criteria 2.2 needs to emphasise the practical implications of the differences rather than the mathematical although the mathematical are important as generalised ways of dealing with them.

Math 2003

- Bullets 4 and 5 under the Range Statement for the Unit Standard appear to be out of sync with the SAQA Unit Standards for Mathematics. They are also too mathematically deep for Mathematical Literacy purposes.
- Since Math2002 deals with compound increase and decrease, should exponential functions not feature amongst the range of functions to be dealt with?
- Specific Outcome 2 Assessment Criteria 2.3: The features need to be considered in relation to the contexts in which the functions are acting as models and not considered solely in terms of the properties
- Specific Outcome 3 and its Assessment Criteria are central and the Assessment Criteria well expressed in relation to Mathematical Literacy ethos.

Math 2004

- This Unit Standard relates the Specific Outcome 3 of M2003 and is central to Mathematical Literacy. That said, we should be cautious about assuming that real life contexts are easily translated into mathematical models. Furthermore much emphasis needs to be placed on developing the skills of simplifying real life situations so that they can be mathematically modeled.
- The third bullet of the Range Statement mentions matrices. We do not feel that matrices are appropriate to a Mathematical Literacy course (furthermore they are not mentioned anywhere else).

Math 2005

- The title must be reworded—see earlier comments.
- Although trigonometry is mentioned in the broad Range Statement and again in the embedded knowledge, it does not feature specifically in the Specific Outcome s and Assessment Criteria. An additional Specific Outcome could cover this, though care should be taken in specifying the range of trigonometrical functions e.g. there is no place for secant in a Mathematical Literacy course.

Math 2006

- Specific Outcome 2 might better be expressed as follows:
- Use theoretical (based on equally likely events) and experimental probability to develop models, make predictions and study problems

Math 3001

- Specific Outcome 2's Range Statement could include a reference to the types of measuring instruments to use, e.g., micrometer screws, calipers with verniers, tellurometer.
- Units related to measurement including such prefixes as nano(seconds) could be included.
- What is meant by "substitution calculation"?
- The use of "number system" in Specific Outcome 3 could confuse. The specific stipulation of "Binary" is limiting. What of:
- Applications of numbers expressed in bases other than 10 and conversions to denary numbers.
- Embedded knowledge:
- Rational and irrational numbers
- Number bases
- Accuracy and error in measurement

Math 3002

- An additional Assessment Criteria for Specific Outcome 2:
- The effect of rates being quoted in terms of being effective or nominal is taken into account.
- An additional Assessment Criteria for Specific Outcome 3:
- The effect of the use of computational technology on the manner in which budgets are dealt with is assessed critically

Math 3003

- The range statement does not agree with the manner in which the December SAQA Unit Standards develop an approach to stepping up the range of functions. The latter approach looks at the power of working with transformations in an integrated way and provides some mathematical meaning at a deeper level to transformations that is lost in this rendition.
- More attention and specifics will be necessary in Range Statements and Assessment Criteria to distinguish this level from M2003.
- The modeling Unit Standard at level 2 (M2004) does not have a counterpart in Level 3. Specific Outcome 3 of M3003 could be looked to emphasise to a greater extent this crucial aspect of Mathematical Literacy.

Math 3004

- The title must be repaired—see earlier comments
- The Assessment Criteria of Specific Outcome 1 could include:
- The correct use of congruence and similarity in practical situations.
- The Assessment Criteria for Specific Outcome 2 could clarify that the contexts are practical rather than mathematical per se by replacing "geometric contexts" with "practical contexts" in 2.3 and 2.4.
- Could enlargements not be included in the range of transformations so as to fit in with similarity?
- Specific Outcome 3 could be stepped up to include volume. The fitting of data to a wider range of functions than in Level 2 should be indicated.

Math 3005

- The last bullet of the general Range Statement could drop the "determine how well" as this implies features of regression analysis which could be too demanding at this level.
- More attention could be given to emphasising that the concepts in statistics and probability mentioned need to be related to practical contexts and not solely for their mathematical usefulness. (For example, the use of permutations and combinations in texts often indulge complexity and nuance to the detriment of appreciating their efficiency in genuine practical situations.)

Math 4001

- The general Range Statement is not appropriate and does not link to the title
- Assessment Criteria 2.3: "Are given" could be changed to "are dealt with"
- The range of the types of sequences and of applications of sequences and series needs some specification.
- The embedded knowledge needs attention.
- Specific Outcome 3 does not seem necessary for a course in Mathematical Literacy

Math 4002

- Some of the aspects of this Unit Standard are possibly too mathematically demanding and too similar to the Core and Elective Unit Standards. The dilemma is that they are relevant to daily life and the workplace.

Math 4003

- The inclusion of vertical and horizontal shifts seems to be unnecessary for Mathematical Literacy.
- To "convert flexibly" should not be allowed to be interpreted as meaning lots of algebraic manipulation for the sake of manipulation.
- There are useful applications of trigonometric functions with angles of the form (bθ)—b a variable which would warrant them being included in the range statement.
- The "fitting" of functions to data deserves a mention here as well as in M4005 with the reminder that we should avoid hints at regression analysis

M4004

- The title needs attention similar to that suggested for M2005 and M3004
- Volume appears as a concept for the first time at this level—care needs to be taken to include it from NQF level 2.
- Does Specific Outcome 1 Assessment Criteria 1.5 imply formal geometry i.e. theorems etc? We would question the role of formal geometry in a Mathematical Literacy course.

M4005

- The emphasis on "critically interrogate" is appropriate even though being critical is a feature that carries right through mathematical literacy. The age of the learners at this stage will affect the relevance and insight that increasing maturity brings to being incisively critical as being a central aspect of education for democracy.
- Again here the mathematical demands implied in range statements and elsewhere need to be thought through in terms of the intentions of Mathematical Literacy in the SA context.

ADDENDUM B

DETAILED COMMENTS ON UNIT NO: MATHS 2001

By Moses Mogambery

Detailed comments are provided for Math 2001 to indicate the difficulty the educator/trainer will experience in interpreting the unit requirements. Time does not permit the writing of detailed comments for all the other units.

1.1 **Title***: Demonstrate understanding of rational and irrational numbers, and number systems.*

*There is nothing in the specification to suggest what sort of an understanding of "number systems" is required. Maybe some of the number systems required should be listed, possibly in the range statement*.

1.2 **Specific Outcome 1: ***Demonstrate understanding of rational and irrational numbers*

This is no more specific than the title.

Range

- use technology such as calculators

How does this relate to rational numbers () and irrational numbers ()? For what activities in and should be technology be used? Should one add, subtract, multiple, divide, exponentiate in and ? Should one solve practical problems (numerical and geometric) with numeric data in and ? When using the technology to perform calculations in should answers be returned in the form *a/b* where *a* and *b* are integers, or will decimal answers be good enough always? Should there be activities in and in which the use of technology should be disallowed?

Without this sort of information, the educator/trainer, learner and assessor will be kept guessing as to what the minimum standards are.

- Demonstrate understanding of mathematical relationships and principles involved in computation.

It is difficult to determine what computations are required with rational numbers and what computations are required with irrational numbers. This statement does not appear to point to the outcome at all.

*Find rational approximations to irrational numbers*.

This statement is clear enough and it the most elementary activity for this outcome, I think. Unless of course, the use of calculators is not allowed for examples like , where rationalizing the denominator may be required.

Assessment Criteria

1.1 *Rational numbers are used to develop set of odd, even and prime numbers*.

This is somewhat vague. The question we need to ask is how do we set assessment tasks to determine whether this criterion is satisfied. Are learners expected to use the set of rational numbers as a basis to generate the sets of odd, even and prime numbers?

1.4 *Algorithms are executed appropriately in calculation*.

What algorithms are learners to use, and for what purposes? There is no mention of any algorithm in the range statements.

1.5 *The viability of selected algorithms is verified and justified*.

An account of the lack of clarity in the range it difficult to determine whether this criterion refers to an "operation" or "algorithm".

**Specific Outcome 2**:*Know and understand relationships among numbers and number systems, and represent numbers in different ways*.

The first question that comes to mind is: does this outcome have anything to do with the title? The title refers to rational and irrational numbers.

Range

*Work with rational and irrational numbers*.

This says very little about what should be done with rational and irrational numbers.

*Explore repeating decimals and convert them to common fraction form*.

This is clear and requires the use of an algorithm or a calculator.

*Use scientific notation for small and large numbers*.

This fits in more with the third range statement of specific outcome 1, where approximation to irrational numbers can be expressed in scientific notation. As an extension, learners could apply the same rules to very small and very large numbers.

**Assessment Criteria**

- Methods of calculations and approximations are appropriate to the problem types.

It is not clear what the "problem types" are.

*Numbers and quantities are represented using rational and irrational numbers as appropriate to the context*.

What numbers need to be represented as rational **and** irrational numbers? This needs to be made more specific. Is this possible?

1.4 **Specific Outcome 3**:** ***Apply base 10 number systems on technology to demonstrate understanding of scientific notation and rounding off numbers*.

This outcome is covered in the third range statement of specific outcome 2. Base 10 is not an issue when other bases are not considered.

Range

This is a repetition of the range for Specific Outcome 1. The range is supposed to give the scope (depth and breadth) of an outcome. This implies that one of the outcomes is redundant.

1.5 **Concluding Remarks on Maths 2001**

The specifications in this unit gave the impression that the intended title is: "Use rational and irrational numbers in computations and practical problems".

Reference is made to "number systems" in the title. It is not clear what is meant by a number system – no clarity is given in either the range or the assessment criteria. It is also not clear as to how many number systems should be studied and compared in order for learners to "demonstrate understanding of … numbers systems".

In my opinion, there is no need for the 3 specific outcomes (none of which is particularly specific). A clearer title, followed by __minimum__ standards that need to be assessed will suffice. What needs to be stated clearly is that educators could go beyond the minimum and explore ideas that are appropriate to their local circumstances, provided the minimum standards are met.