Human ecology
Mathematics education
Psychology of learning mathematics
PLM - Notes - 28-29/1/00

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.

Scientific vs. Interpretative research (p.6) Paul E. used the word triangulation when talking about interpretative research. Note: triangulation is usually considered as a process in measuring an objective world (triangulation, parallax, aperture synthesis, binocular vision, etc)

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.

Bloom's taxonomy of the cognative domain (quoted Ernest, 2000, p.9) (1. Lowest to 6. Highest level of skill)
  1. Knowledge: Recal of knowledge items
  2. Comprehension: translation, interpretation, extrapolation
  3. Application: use of abstractions in concrete situations
  4. Analysis of elements, relationships, organisation principles
  5. Synthesis: production of unique communications, plans, sets of abstract relations
  6. Evaluation: judgements using internal evidence, external criteria
Paul W. - questioned the position of evaluation in this hierarchy. Proposed that Evalution should be a lower level skill than Analysis or Synthesis.

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

Learning outcomes... Bell, et. al. (1993) Ernest (2000, table 2.3, p.10) This is slightly different from the discussion (Ernest, 2000, pp.10-13) where the following classification is used: Task applying outcomes of school maths (Table 2.4, p.13) (Some altered notation to express mathematical formulae in HTML)
  1. Can reflect right angled triangle in y-axis Skill
  2. Knows cm stands for centimetre Fact
  3. Says maths is the study of pattern Attitude, concept, appreciation
  4. Knows 5/8 mile is 1km Fact
  5. Can calculate that +4--5=+9 Skill, (concept??)
  6. Knows how to find tan(x) for 0 deg < x < 180 deg with calculator Skill
  7. 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
  8. Can find number of ways of getting a total score of 10 on 3 dice General strategy, skill if fixed
  9. Knows that a.bcd * 100 = abc.d Concept, general strategy
  10. Knows how to test the formula x^2 + 2x - 17 to see if it will always give prime numbers General strategy
  11. Comes back at lunch time to work on GCSE maths project Appreciation
  12. Can identify prime numbers General strategy (using concepts or fact?)
  13. Knows that 2x^2 is 2*x*x Fact (notation)
  14. Says maths is the subject in which you always know if you're right or wrong Attitude
  15. Says maths is a subject of exciting challenge problems Appreciation
  16. Knows 75% = 3/4 Fact (also conceptual structure)
  17. Can draw an equilateral triangle with compasses Skill
  18. Decides to test multiples of six to see if they are perfect numbers Appreciation
  19. Knows 0.7 > 0.25 Concept, fact?
  20. Will not do matrix multiplication Attitude, (appreciation?)
  21. Prefers 100 simple additions to finding numbers expressible with four 4s combined Appreciation
  22. Can estimate angles to 20 deg Appreciaton and skill
  23. Can explain when 9.7 + 8.65 might be used Concept?
  24. Can calculate total area of classroom walls, floor and ceiling General strategy
  25. Knows tan(a)= O/A Fact?
Saturday 29/1/00
Chunking (pp.18-19)
Example: chemical formula of ethanol (ethyl alcohol). A chunk for chemists, lots of items for non-chemists. Other web friendly notation:


CH3- methyl group

CH3.CH2- ethyl group

-OH alcohol group

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

Skill chaining (p.22)
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.

Children's errors
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 - Mapping Children's Errors (pp.40-43)

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?)

Rote and Schema Learning (p.49)

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)?

Concept mapping

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


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...
Links at other sites...
Created 31/1/00
Last modified 17/2/00