Presentation for researchED maths and science on June 11th 2016. References at the end (might be some extra references from slides that were removed later on, this interesting :-) Interested in discussing, contact me at C.Bokhove@soton.ac.uk or on Twitter @cbokhove I of course tried to reference all I could. If you have objections to the inclusion of materials, please let me know.

1. Mathematics: skills,
understanding or both?
Christian Bokhove
researchEd maths and science
Oxford
11 June 2016
Twitter: @cbokhove
Presentation on www.bokhove.net
I of course tried to reference all I could. If you have
objections to the inclusion of materials, please let me know.

2. Both

3. Claims
1. ‘Understanding’ is important
2. ‘Skills’ are important
3. If you are already annoyed I mention
‘understanding’ first, you may switch around
claims 1 and 2
4. You might even not like my use of the word
'understanding'. Now stop bickering. Both are
important. :-) Let's see how we can reinforce
both.

4. • Christian Bokhove
• Maths and computer science teacher in the
Netherlands from 1998-2012
• PhD maths education on use of technology for
algebraic skills
• Now lecturer in Southampton
• Secondary education, maths education, large-scale
assessment, etc.

5. Wolfram
“Get the Basics first”
“Computers dumb math down”
“Hand calculating procedures teaches
understanding”
Whole story:
http://www.ted.com/talks/conrad_wolfram_t
eaching_kids_real_math_with_computers.ht
ml
I wrote a response blog:
https://bokhove.net/2012/08/05/teaching-
kids-real-math-with-computers-a-comment-
on-wolfram/

6. Quantitative Literacy vs.
Calculus Preparation
Theory vs. Applications
Rote vs. Constructivism
Tracking vs. Mainstreaming
Etc.
The Math Wars

7. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286.

8. Action
http://www.stanford.edu/~joboaler/ Petition
Reaction
http://mathforum.org/kb/servlet/JiveServlet/download/206-2420462-7936509-
788386/Jo-Boaler-reveals-attacks-AccusationsResponse.pdf

10. In own research

13. ‘Understanding’ difficult term
• Learning that adds new knowledge to the learner’s
memory with little or no change in prior knowledge
is monotonic.
• If the learner must override or reject his or her
prior knowledge, his or her knowledge base shrinks
as well as grows, so learning is nonmonotonic
(Ohlsson, 2011).
• Cognitive change of this latter sort is variously
referred to as attitude change, belief revision,
conceptual change, restructuring, theory change,
and deep learning (Ohlsson, 2011).

14. Hiebert and Lefevre
• Procedural knowledge (knowing how)
• Formal mathematical language, algorithms and rules for solving
mathematical problems.
• Generates problem solving behaviour which is not always
meaningful and generalizable.
• Conceptual knowledge (knowing why)
• Product of a linking process, which creates relationships between
existing knowledge and information that is newly learned.
• Can be implicit or explicit, however it is flexible, not tied to specific
problem contexts and is therefore generalizable.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory
analysis. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: the Case of Mathematics (pp. 1-27).
Hillsdale, N. Jersey: Lawrence Erlbaum Associates.

15. They defined conceptual understanding as ‘‘the
comprehension of mathematical concepts, operations, and
relations’’ and procedural fluency as the ‘‘skill in carrying
out procedures flexibly, accurately, efficiently, and
appropriately’’ (p. 116).
Furthermore, ‘‘the five strands are
interwoven and interdependent in the development of
proficiency in mathematics’’ (ibid.). Kilpatrick et al. (2001)
• Very influential with NCTM
• No very tight definitions
Adding it up (Kilpatrick et al.)

16. The ‘iterative’ model
(Rittle-Johnson, Siegler, & Alibali, 2001)
• Procedural and Conceptual knowledge develop in a gradual, hand-in-
hand process.
• Causal, bidirectional relations.
• Need to consider the processes that underlie performance not just the
outcomes.
Rittle-Johnson, B., Siegler, R.S., and Alibali, M.W. (2001). Developing
conceptual understanding and procedural skill in mathematics: an
iterative process. Journal of Educational Psychology, 93, 2, 346-362.

17. • Arcavi (2005)
• Bokhove & Drijvers (2010)
• Structure sense, e.g. Hoch & Dreyfus (2004)

18. Another aspect: salience
Attractiveness of certain actions e.g. ‘work out
brackets’. This also depends on prior knowledge
• (x+2)(x+4)=-1
• (x+2)(x+4)=7
• (x+2)2=16
• Root signs
• ax2+bx+c=0

19. Star’s reconceptualisation
Star (2005) described in terms of ‘knowledge quality’
(De Jong and Ferguson-Hessler 1996) and knowledge
type.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal
for Research in Mathematics Education, 36(5), 404–411.

20. Rittle-Johnson,Schneider and Star (2015)
• Review
• Two directions
• In 1998, Rittle-
Johnson and Siegler
concluded that there
is no fixed order in
the acquisition of
mathematical skills
versus concepts (p.
80).
• In 2015, order
unclear
• More research

21. Rethinking role of algorithms

22. Another example: algoritmes
• Fan & Bokhove (2014)
• (Standard-)algorithms often unfairly linked to ‘rote’,
‘lower’, no understanding. Givvin et al. (2009):
‘‘very algorithmic’’, ‘‘rule-orientated’’
• In Canada at one point even not mentioned at all
• Kamii and Dominick (1997) ‘‘Algorithms are harmful
to children’s development of numerical reasoning
for two reasons: (a) they ‘unteach’ place value and
discourage children from developing number sense,
and (b) they force children to give up on their own
thinking’’(p. 58).

23. But…
• Dahlin and Watkins (2000), link memorization and
understanding through ‘‘repetition’’.
• Meaningful repetition can ‘‘create a deep
impression’’ which leads to memorization, and it
can also lead to ‘‘discovering new meaning’’ which
in turn leads to understanding (Li, 1999).
• And also: "many different kinds of procedures, and
the quality of the connections within a procedure
varies" (Anderson, 1982 cited in Star, 2005).
• So we wrote we’d love to see standard algorithms
‘reinstated’.

24. International
perspectives

25. Source: http://www.bbc.co.uk/news/education-25187997

26. Does PISA really say this?
• Not really
• Depends on what you look at
• Strong relationship ‘memorisation and
understanding’
Source: http://hechingerreport.org/memorizers-are-the-lowest-achievers-and-other-
common-core-math-surprises/

27. Some rankings here
Source: http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

28. OECD report
“Our analyses suggest
that to perform at
the top, students
cannot rely on
memory alone; they
need to approach
mathematics
strategically and
creatively to succeed
in the most complex
problems.”
http://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=EDU/WKP
%282016%294&docLanguage=En

30. TIMSS - cognitive domains
• The first domain, knowing, covers the facts,
concepts, and procedures students need to know.
• Applying, focuses on the ability of students to apply
knowledge and conceptual understanding to solve
problems or answer questions.
• The third domain, reasoning, goes beyond the
solution of routine problems to encompass
unfamiliar situations, complex contexts, and multi
step problems.

32. TIMSS
The good scores of East Asian students seem to
contradict the assumption that Asian classrooms are
traditional and aimed mainly at low-level cognitive
goals (e.g. emphasizing memorization), which has
been called a ‘‘Paradox of the Asian learner’’ (Biggs
1994, 1998).
Memorisation AND understanding.

33. Differences between PISA and
TIMSS
• PISA more problem solving
• Both very much influenced by IQ, but PISA more so
than TIMSS (Rindermann, 2007).
• TIMSS more curriculum oriented. (Rindermann &
Baumeister, 2015)

34. Reading demand

35. Language
• TIMSS reading
demand
• Number of Words
• Vocabulary
• Symbolic Language
• Visual Displays
• TIMSS-PIRLS
relationship report:
poor readers did
worse than better
readers but were
doubly disadvantaged
by more reading.
• Be aware of this!

36. Example questions
• PISA toevoegen
http://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf

37. Roads into skills AND
understanding

38. 1.
2.
3.
4.
Bokhove(inpress)
Conflict

39. Example how Asia might link memorisation
and understanding: variation
Might facilitate schema building. Not an automatic
process
Slides from Clare Hill, Twynham School, Dorset

40. Where to go next?
• Focus on skills AND understanding
• Research linking fMRI with algebra, for example
Cognitive Tutor
• Research on reading demand and assessment items
(e.g. student looking at Canada Ontario v England)
• Relation between ‘memorisation and
understanding’ (Asia hype, schemas?)
• Strong relations between education,
(neuro-)psychological research and computer
science (e.g. Rotation skills)

41. Advantages/disadvantages?
Tenison, C., Fincham, J. M., & Anderson, J. R. (2014). Detecting math problem solving
strategies: An investigation into the use of retrospective self-reports, latency and fMRI
data. Neuropsychologia, 54, 41-52.

42. Questions/discussion
Mathematics: skills, understanding or both?
Both
• Questions/criticism/comments/discussion
• Thanks
• Twitter: @cbokhove
• Presentation on www.bokhove.net

43. References (1)Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological Review, 89(4), 369–406. doi:10.1037/0033-
295X.89.4.369.
Arcavi, A. (2005). Developing and Using Symbol Sense in Mathematics, For the Learning of Mathematics. 25(2), 50-55.
Biggs, J. B. (1994). What are effective schools? Lessons from East and West. The Australian Educational Researcher, 21(1),
19–39.
Biggs, J. (1998). Learning from the Confucian heritage: so size doesn’t matter? International Journal of Educational
Research, 29(8), 723–738.
Bokhove, C. (in press) Supporting variation in task design through the use of technology. In, The role and potential of using
digital technologies in mathematics education tasks. Berlin, DE, Springer.
Bokhove, C., & Drijvers, P. (2010). Symbol sense behavior in digital activities. For the Learning of Mathematics, 30(3), 43-
49.
Chen, Z. (1999). Schema induction in children's analogical problem solving. Journal of Educational Psychology, 91,703-715.
Dahlin, B., & Watkins, D. (2000). The role of repetition in the processes of memorizing and understanding: A comparison of
the views of German and Chinese secondary school students in Hong Kong. British Journal of Educational Psychology, 70,
65–84.
De Jong, T., & Ferguson-Hessler, M. (1996). Types and qualities of knowledge. Educational Psychologist, 31(2), 105–113.
Echazarra, A., Salinas, D., Méndez, I., Denis, V., & Rech, G. (2016). How teachers teach and students learn : successful
strategies for School. OECD Education Working paper No. 130.
Fan, L. & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: a conceptual model with focus on
cognitive development. ZDM-International Journal on Mathematics Education, 46(3), 481-492. (doi:10.1007/s11858-014-
0590-2).
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J.
Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Erlbaum.
Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effect of brackets. In M. J. Høines & A. B.
Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics
Education (Vol. 3, pp. 49-56) Bergen, Norway: PME.
Jerrim, J. (2013). The reliability of trends over time in international education test scores: is the performance of England’s
secondary school pupils really in relative decline? The Journal of Social Policy

44. References (2)
Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. Journal of Mathematical Behavior, 16(1), 51–61.
Li, S. (1999). Does practice make perfect? For the Learning of Mathematics, 19(3), 33–35.
Mullis, I.V.S., Martin, M.O., & Foy, P. (2013). The Impact of Reading Ability on TIMSS Mathematics and Science
Achievement at the Fourth Grade: An Analysis by Item Reading Demands. In M. O. Martin, & I.V.S. Mullis (Eds.), TIMSS and
PIRLS 2011: Relationships among reading, mathematics, and science achievement at the fourth grade — Implications for
early learning (pp.67–108). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.
NRC (2001): Adding It Up . (J. Kilpatrick, J. Swafford, and B. Findell, (Eds.)). Washington, (DC): National Academy Press
Ohlsson, S. (2011). Deep learning: How the mind overrides experience. Cambridge, UK: Cambridge University Press.
Rindermann, H. (2007). The g-Factor of international cognitive ability comparisons: The homogeneity of results in PISA,
TIMSS, PIRLS and IQ-tests across nations. European Journal of Personality, 21(5), 667-706.
Rindermannm H, & Baumeister, A.E.E. (2015). Validating the interpretations of PISA and TIMSS tasks: A rating study.
International Journal of Testing, 15(1), 1-22.
Rittle-Johnson, B., Siegler, R.S., and Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in
mathematics: an iterative process. Journal of Educational Psychology, 93, 2, 346-362.
Rittle-Johnson, B., Schneider, M., & Star, J.R. (2015). Not a one-way street: bidirectional relations between procedural and
conceptual knowledge of mathematics, Educational Psychology Review, 27(4), 587-597.
Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–
411.
Sweller, J., Chandler, P., Tierney, P., & Cooper, M. (1990). Cognitive load and selective attention as factors in the structuring
of technical material. Journal of Experimental Psychology: General, 119, 176-192.
Tenison, C., Fincham, J. M., & Anderson, J. R. (2014). Detecting math problem solving strategies: An investigation into the
use of retrospective self-reports, latency and fMRI data. Neuropsychologia, 54, 41-52.