Mathematics: skills, understanding or

both? Christian Bokhove

researchEd maths and scienceOxford

11 June 2016

Twitter: @cbokhovePresentation 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.

Both

Claims1. ‘Understanding’ is important2. ‘Skills’ are important3. 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.

• 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.

Wolfram“Get the Basics first”“Computers dumb math down”“Hand calculating procedures teaches understanding” Whole story: http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

I wrote a response blog:

https://bokhove.net/2012/08/05/teaching-kids-real-math-with-computers-a-comment-on-wolfram/

Quantitative Literacy vs. Calculus Preparation

Theory vs. Applications

Rote vs. Constructivism

Tracking vs. Mainstreaming

Etc.

The Math Wars

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

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

In own research

‘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).

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.

They defined conceptual understanding as ‘‘thecomprehension of mathematical concepts, operations, andrelations’’ and procedural fluency as the ‘‘skill in carryingout procedures flexibly, accurately, efficiently, and appropriately’’ (p. 116).

Furthermore, ‘‘the five strands areinterwoven and interdependent in the development ofproficiency in mathematics’’ (ibid.). Kilpatrick et al. (2001) • Very influential with NCTM

• No very tight definitions

Adding it up (Kilpatrick et al.)

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.

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

Another aspect: salienceAttractiveness 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

Star’s reconceptualisationStar (2005) described in terms of ‘knowledge quality’ (De Jong and Ferguson-Hessler 1996) and knowledge type.

Star, J. R. (2005). Reconceptualizing procedural knowledge. Journalfor Research in Mathematics Education, 36(5), 404–411.

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

Rethinking role of algorithms

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).

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’.

International perspectives

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

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/

Some rankings here

Source: http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

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

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.

TIMSSThe 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.

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)

Reading demand

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!

Example questions• PISA toevoegen

http://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf

Roads into skills AND understanding

1.2.

3.4.

Bokh

ove

(in p

ress

)

Conflict

Example how Asia might link memorisation and understanding: variation

Might facilitate schema building. Not an automatic process

Slides from Clare Hill, Twynham School, Dorset

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)

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.

Questions/discussionMathematics: skills, understanding or both?

Both

• Questions/criticism/comments/discussion• Thanks• Twitter: @cbokhove• Presentation on www.bokhove.net

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

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 PressOhlsson, 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.