Mathematics education

Psychology of learning mathematics

Notes

My notes from the first taught weekend of the course. These are largely a re-construction of bits that interested me. It is made from loose notes taken at the time and memories and further ideas from after the event. I apologise in advance for any mistakes or misquotations.

Teacher: Paul Ernest

Present: (anti-clockwise around table?)

Jane Gaffney,
Rebecca Sherwin,
Paul Wilson,
Amanda Bristow,
Nicola Klampfer,
Lindsey Ferrie,
George Said,
Paul Marshall,
Chris James,
Grant Sherman,
Judith McCullouch,
Marie Byrne,
Chris Haynes

These are also notes to the course notes (Ernest, 2000). All unreferenced page numbers will be to these course notes.

**Comment from Paul E. - 10/2/00**

The concept of triangulation originates in taking multiple
measuring view points/baselines in surveying, but the
metaphor of triangulation is the standard term in the interpretative
research paradigm for the equivalent of
'validity and reliability' in scientific paradigm research:
i.e., using multiple peoples' perspectives or research
gathering methods to check that your interpretation is
robust and not wholly a subjective construct. See the
literature.

**Knowledge:**Recal of knowledge items**Comprehension:**translation, interpretation, extrapolation**Application:**use of abstractions in concrete situations**Analysis**of elements, relationships, organisation principles**Synthesis:**production of unique communications, plans, sets of abstract relations**Evaluation:**judgements using internal evidence, external criteria

Paul E. - (GKS - oops. forgotten what he said. Compare this with Figures 7.5-7.7 p.77)

Amanda - Evaluation as part of a cyclic process (GKS - How does this relate to cyclic processes later in the course? Or to critical theory)

Grant - Other classifications put Evaluation in different places. If scientific methodology is assumed to be unchangeable, does Evaluation play a smaller part in science?

Further ideas and clarifications on this point 28-31/1/00

- I don't think hierarchies should be assumed between different aspects of cognition (a bias against hierarchies on my part). Levels of cognition may arise from the act of combining more than one aspect. (Did Paul E. mention that someone was researching this idea?)
- Having different classifications of cognition (such as Guilford's 'structure of intellect' model) doesn't have to mean that one classification is wrong and another is right. (Some notes on classification and the librarian's problem.)
- The science thing is a bit overstated; however there is the question - 'Is science value-free?' How is this similar / different to evaluation?

- Facts
- Skills
- Concepts
- Conceptual structures
- General strategies
- Attitudes

- Facts
- Skills
- Concepts and conceptual structures
- General strategies
- Attitudes
- Appreciation

**Can reflect right angled triangle in y-axis**Skill**Knows cm stands for centimetre**Fact**Says maths is the study of pattern**Attitude, concept, appreciation**Knows 5/8 mile is 1km**Fact**Can calculate that +4--5=+9**Skill, (concept??)**Knows how to find tan(x) for 0 deg < x < 180 deg with calculator**Skill**Knows how to find x given values for a, b, c, and a:x::b:c (a is to x as b is to c)**General strategy, skill**Can find number of ways of getting a total score of 10 on 3 dice**General strategy, skill if fixed**Knows that a.bcd * 100 = abc.d**Concept, general strategy**Knows how to test the formula x^2 + 2x - 17 to see if it will always give prime numbers**General strategy**Comes back at lunch time to work on GCSE maths project**Appreciation**Can identify prime numbers**General strategy (using concepts or fact?)**Knows that 2x^2 is 2*x*x**Fact (notation)**Says maths is the subject in which you always know if you're right or wrong**Attitude**Says maths is a subject of exciting challenge problems**Appreciation**Knows 75% = 3/4**Fact (also conceptual structure)**Can draw an equilateral triangle with compasses**Skill**Decides to test multiples of six to see if they are perfect numbers**Appreciation**Knows 0.7 > 0.25**Concept, fact?**Will not do matrix multiplication**Attitude, (appreciation?)**Prefers 100 simple additions to finding numbers expressible with four 4s combined**Appreciation**Can estimate angles to 20 deg**Appreciaton and skill**Can explain when 9.7 + 8.65 might be used**Concept?**Can calculate total area of classroom walls, floor and ceiling**General strategy**Knows tan(a)= O/A**Fact?

- 8. Jane raised the point that not many of these items are
concepts.
(GKS Some of the Skills and General Strategy need concepts but the problems don't seem to directly test the concepts. Are there any other aspects of school maths that might be described as concepts?)

- 22. Judith - RAF teaching requires angle estimation to 5 deg,
some people have the ability - some can't be taught.
- Judith - General structures are turned into skills through
teaching or learning about a particular problem.
(GKS - Example - sentence 10. above - I came across this problem in another book (can't find the reference). That formula had +11 at the end. I eventually found that the formula didn't generate a prime number when x = 11. So when I saw this example, I knew to try x = 17. Does this make this a skill?)

Example: chemical formula of ethanol (ethyl alcohol). A chunk for chemists, lots of items for non-chemists. Other web friendly notation:

CH3.CH2.OH

CH3- methyl group

CH3.CH2- ethyl group

-OH alcohol group

(Cross reference: formulae of amino-acids and l-systems)

Paul W. asked Lindsey if she thought skill chaining was similar to learning music.

Paul E. - multiple repetition might be needed for skill chaining (?)

Lindsey - only needs to see or hear the music twice to recall it. (Musical ability)

A slight diversion - some sounds of numbers. Number streams (from the hexa-decimal digits of pi and from iterations of the logistic map) converted into melodies.

Symbolic place value subtraction of 109-70 vs. verbal subtraction of seventy pounds from one hundred and nine pounds. How different are these processes? Do we tend to assume that 109 is the same mental concept as one hundred and nine?

Are Roman Numerals still taught in schools? I think we were taught them in 1st or 2nd year Comprehensive. What are the error patterns with Roman Numerals?

(See Butterworth (1999, pp.129-134) for why its easier to count in Chinese)

New 17/2/00 -

How does the pattern of untaught / learnt skills relate to Vygotsky's "Zone of Proximal Development"? (The zone of proximal development is the zone between what can be done alone and what can be done with the aid of others. Is this tested?)

Are Skemp's experiments more difficult if the explanations are not in your first language? Is this similar to the first and second language learning of mathematics in Orton (1992)?

Jane - how acceptable is the use of concept maps in mathematics research? (sections in Orton, 1992, pages numbers in the index. Has anyone got any other references?)

Grant - concepts maps are more acceptable in biology and geography.

Additional info

- An explanation of the use and construction of concept maps in ecology can be found in Miller (1996).
- Web pages are concept maps, and the topology of webpages can be described by graph theory.
- In geography, graph theory is used to assess transport routes and connectivity between towns. Whynne-Hammond (1979) "Elements of Human Geography", George Allen & Unwin, London

The following contains some notes on how I think constructivism is related to objectivity & subjectivity, absolutism & relativism. (Conjectures, working hypotheses, and cross references - nothing rigorous!)

Links at this site...

- Concept maps
- Classification
- Critical theory
- Guilford's 'structure of intellect' model
- L-systems
- Some sounds of numbers

Links at other sites...

Created 31/1/00

Last modified 17/2/00