Computer algebra is a revolution. But 21st century skills ?

This present weblog text summarises this paper:

Computer algebra is a revolution. But 21st century skills ?
With protest against misrepresentation by Koeno Gravemeijer

Seminal revolution

We are living during a seminal revolution similar to the invention of the wheel, the alphabet and positional number system:

we can do mathematics on the computer – and it is called “computer algebra”.

We know this since 1963 and Project MAC.  This is doing mathematics, rather than mere programming or punching buttons. Namely compare:

  • The arrival of calculators is not much different from the invention of ruler and compass, or tables for trigonometry and logarithms (recovered exponent, rex rather than log). Those are techniques, with the didactic balance of drilling and understanding.
  • Doing mathematics on the computer however is a game changer.
Challenges for education including mathematics education

There are consequences for education, including mathematics education:

  • Since it is mathematics, computer algebra must also be studied in didactics of mathematics.
  • For education the challenge is to bring computer algebra into the textbooks and the schools in ways that work.
  • Applications of computer algebra to particular fields must be distinguished from those for learning mathematics proper.
  • Computer programming – like the new British curriculum – is another application of mathematics and should not be confused with doing mathematics proper.

In this, there is nothing special about the calendar, the year 2000 and the 21st century.

Ghosts from the past

Instead of a fruitful exchange of ideas and experiences on education and didactics on computer algebra, the decision making discussion is haunted by ghosts from the past.

Holland is a small country but it has had some influence on mathematics education, witness the Freudenthal Award at the International Mathematical Union (IMU / ICMI).

  • Hans Freudenthal (1905-1990) created “realistic mathematics education” (RME). This RME was not tested in experimental manner but still introduced generally in Dutch education.
  • RME appears to be a failure. It is rather not a theory but an ideology, given how it originated, was executed and monitored.
  • The Dutch government in 2015 has set up additional courses and exams for secondary education to correct for what now has gone lacking in primary education.
  • In 2014 it appeared that Freudenthal also committed intellectual fraud on RME by appropriating and misrepresenting ideas from Pierre van Hiele (1909-2010).

It is a horribly huge project to re-evaluate all publications by supporters of RME and separate science from ideology. Perhaps the best way is to annul everything. Anyone who refers to such a RME publication would have to treat it like astrology, and have to redo the claimed RME research from scratch in a proper way.

Who still supports RME is in a state of denial.

  • Koeno Gravemeijer (1946) has been promoting RME since around 1980 apparently without real interest in testing it, without discovering this (obvious) fraud, and has since 2008 not explicitly accepted its failure.
  • Since at least 2001 Gravemeijer argues for “21st century skills”, and uses the same arguments as for RME.
  • Gravemeijer has written on computer algebra and supervised the Paul Drijvers (2003) thesis on this subject. Yet, both of them are supporters of RME. Their wrong handling of didactics on RME makes their expertise on didactics of computer algebra questionable too.

Gravemeijer’s lecture for the November 2015 NVVW annual convention of teachers of mathematics in Holland:

  • neglected the failure of RME
  • was scare-mongering about the economic risks of the 21st century (greater dynamics on the jobs market)
  • and disinformative about the really interesting challenges with respect to computer algebra.
Screenshot of the Dutch website for 21st century skills in mathematics and arithmetic

Screenshot of the Dutch website for 21st century skills in mathematics and arithmetic

21st century skills

There is nothing particular about the year 2000 and the calendar, but perhaps one might forgive marketing people for using new labels. Technology develops faster than society indeed, and thus there are real challenges to select and further R&D those technologies that are useful. Marketing people also have clients, of course.

There is more than computer algebra. Computers allow adaptive testing and assessment. Internet technology allows more integration of functions than before. Chemistry and biology and nano have their impact. Cognitive, neuro and social psychology have their impact too. Economics and law are there too. Thus, obviously, there is more than computer algebra. One can understand that some educators create a platform like p21. The partnership for 21st century skills that includes the US Department of Education states:

“P21’s mission is to serve as catalyst for 21st century learning to build collaborative partnerships among education, business, community and government leaders so that all learners acquire the knowledge and skills they need to thrive in a world where change is constant and learning never stops.”  (

This however is tricky:

  • In general this is an issue for political economy, i.e. the theory of management of the state. Rather than getting lost in this particular p21, it is better to make the proper economic analysis.
  • One insight is to design institutions that deal with these issues structurally. One such ideas perhaps 200 year ago has been to create a Department of Education. Thus if DoE doesn’t handle the issue itself and needs such p21, then something is amiss with the institutional structure. In general: (1) at the micro level teaching is too specialised on teaching with insufficient time for research and development, (2) at the meso level teachers have insufficient  influence on design of the curriculum.
  • A key role remains for mathematics. When students get a proper training in mathematics then they are better equipped for the future. Other fields like physics and economics can prosper with mathematically competent students. The horror story of RME shows that something has been amiss in the institutional framework here too.
The power unbalance for teachers of mathematics

The current decision making framework puts teachers in a powerless position. Each nation should rather create a national institute for mathematics education, with a key role for teachers. For Holland I propose a Simon Stevin Institute (SSI), honoring the engineers rather than the abstract thinking mathematicians.

See this earlier weblog text on the power void. (Actually, unbalance is a better word than void, since the void is filled by educators and projects like 21st century skills.)

PM 1. Expertise and disclaimer

I write this as an econometrician (Groningen 1982) and teacher of mathematics (Leiden 2008). I also have some background in computer algebra: which may also be regarded as a vested interest.

  • My three books that use computer algebra are Voting Theory for Democracy (2001, 2014), A Logic of Exceptions (2007, 2011) and Conquest of the Plane (2011), all applications of Mathematica (developed by Wolfram Research Inc.).
  • Logic and analytic geometry and calculus are obvious mathematics, and voting theory comes with discrete mathematics and axiomatics. The objective of these books is to re-engineer mathematics in those fields, and they use only a small part of the functionality of Mathematica.
  • See also The Economics Pack. Applications of Mathematica.
PM 2. Conrad Wolfram and

At some time in the last decade there was a MathWire that stated that Conrad Wolfram took the initiative for what now has become (It is UK based but uses US “math” instead of UK “maths”.) I didn’t join since some aspects are too critically sensitive to local customs and regulations. Conrad’s recent weblog text is:

“I am not the slightest bit surprised at the recent OECD report that use of computers in education hasn’t improved PISA results − and indeed that many countries with the best technology provision have mediocre performance. Why? Because the world’s most transformative machines have been used for entirely the wrong purpose in most classrooms: automating pedagogy not changing the subject taught. Countries with the most attentive teaching are also likely countries where there is least pressure to computerise pedagogy for teaching today’s school subjects. They do best in PISA because they are best at helping students through those subjects.” (Conrad Wolfram, September 15 2015)

People not familiar with the issue will enjoy this talk at Wise Channel Doha, November 15 2012.

People familiar with the issue will enjoy the discussion at the end. At minute 23 moderator Jon Snow, doing an excellent job, puts two questions to Conrad. The first is how to deal with computers that are still fallible (as if the human brain is infallible). The second is that learning arithmetic also teaches about logic. Thus, replacing arithmetic with the magic of computer outcomes may destroy the learning of logic. Conrad starts with the latter and correctly answers that programming teaches you about logic too. However, it also causes him to question the use of teaching long division (at 24:30). Naturally it is an empirical question what works. However, as a teacher I would hold that it is useful to have been taught long division at an early age, in particular since it helps you to understand e.g. why 3 / 4 becomes decimal format 0.75 or 75%. It won’t do to let this remain a great mystery. (See the earlier weblog text that also mentions the dismal alternative to long division: partial quotients.) It is a good project in learning to program to actually write a program for long division: and then it is rather necessary to know what it is and why it works always. Why do I mention this ? The key observation is: Conrad has no formal training on didactics as a teacher. His views on education and didactics do not have such a base, however informative and stimulating as they are. The uplifting aspect is that he is developing the community at and we ought to expect evolution of ideas.

Conrad Wolfram on long division, Wise Channel Doha October 2013 (Screenshot)

Conrad Wolfram on long division, Wise Channel Doha October 2013 (Screenshot)


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