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?)
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
Bloom's taxonomy of the cognative domain (quoted Ernest,
2000, p.9) (1. Lowest to 6. Highest level of
Paul W. - questioned the position of evaluation in this hierarchy. Proposed
that Evalution should be a lower level skill than Analysis or Synthesis.
- Knowledge: Recal of knowledge items
- Comprehension: translation, interpretation, extrapolation
- Application: use of abstractions in concrete situations
- Analysis of elements, relationships, organisation
- Synthesis: production of unique communications, plans,
sets of abstract relations
- Evaluation: judgements using internal evidence, external
Paul E. - (GKS - oops. forgotten what he said. Compare this with Figures
Amanda - Evaluation as part of a cyclic process (GKS - How does this relate
to cyclic processes later in the course? Or to
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
- 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
- The science thing is a bit overstated; however there is the
question - 'Is science value-free?' How is this similar / different
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:
- Conceptual structures
- General strategies
Task applying outcomes of school maths (Table 2.4, p.13)
(Some altered notation to express mathematical formulae in HTML)
- Concepts and conceptual structures
- General strategies
- Can reflect right angled triangle in y-axis Skill
- Knows cm stands for centimetre Fact
- Says maths is the study of pattern Attitude, concept,
- 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
- 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
- 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
- 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
- Knows 0.7 > 0.25 Concept, fact?
- Will not do matrix multiplication Attitude,
- 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
(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
- 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
Example: chemical formula of ethanol (ethyl alcohol). A
chunk for chemists, lots of items for non-chemists. Other web friendly
CH3- methyl group
CH3.CH2- ethyl group
-OH alcohol group
(Cross reference: formulae of amino-acids and
Skill chaining (p.22)
Paul W. asked Lindsey if she thought skill chaining was similar to learning
Paul E. - multiple repetition might be needed for skill chaining (?)
Lindsey - only needs to see or hear the music twice to recall it. (Musical
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
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
(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)?
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.
- 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
- 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...
Links at other sites...
Last modified 17/2/00