Graphical displays about the math war

There is a math war between on the one side “realistic mathematics education” (RME) also known (for the present purposes) as “reform” and on the other side “traditional mathematics education” (TME). I take a third position: decent science (DS).

In Holland this math war is particularly dirty. The scientific approach is maltreated by adherents of both RME and TME. In Holland, RME is given by the Freudenthal Head in the Clouds Realistic Mathematics Institute, and such a TME adherent is Jan van de Craats (UvA) with his SGR foundation. The following gives a graphical display of the various positions, to clarify how people get lost when they replace science by ideology.

While we use the label ME this actually also includes ME research. My position of decent science may also be called “neoclassical mathematics education” (NME). The first step is to observe that both RME and TME actually don’t do mathematics but create so-called “mathematics”. When you repair this, then you get “classical mathematics education” (CME). This CME however is not very didactical. The next step is to re-engineer CME into a form that is didactically acceptable, which gives NME.

The earlier weblog article on the power void in mathematics education puts RME, TME, CME and NME into a table. The standard example is 2½ (RME, TME), 2 + ½ (CME) and 2 + 2^H for mathematical constant H = -1 (NME).

Dispute between RME and TME

In the 1950s teachers concentrated on teaching, and it were academics who trained the teachers who developed the RME ideas how things could be “improved”. It was only when RME was imposed on education, and education started failing, that TME came up as a defence. An example is the defence by the New York City HOLD group.

  • In the 1950s mathematics education research and training became more and more institutionalised at the universities. These academics were oriented on “theory”. That is, they created texts that claimed to be theory. But they looked less at practice and they hardly collected data. A scientific theory must be developed w.r.t. data, and thus RME cannot be seen as a scientific theory, and only counts as ideology. This conclusion may sound a bit harsh, but check out the English Summary on p10-16 of the 2009 KNAW report (pdf) on the empirical base in research on education in arithmetic in Holland. My problem with the report is that it likely neglects the relation of arithmetic to algebra (see here too), so that it downplays the importance of TME over RME. Obviously, the report also neglects my 2008 paper (see here in Dutch). PM. One can imagine RME as a philosophy of education, but philosophy also looks at different views.
  • This also explains that RME has a lot of texts and that this situation suggests a consistent body of claims, while TME consists more of simple rules and lines of defence. It seems as if TME is less developed while RME might appear to be attractive since it has such a body of texts and authors referring to each other. But we are comparing apples and oranges. Ideology does not quite compare with empirical science.

For Holland, there is also the Freudenthal fraud and disaster. The International Mathematical Union (IMU) has a committee on education (ICMI). They have created a “Freudenthal Medal” as a reward for mathematics education researchers. The 2015 medal has been awarded to Jill Adler. I wrote to IMU / ICMI in 2014, but they have not responded till now, and we may fear that they did not inform professor Adler about the discovery of the fraud.

To be sure:

  • Mathematics teacher Pierre van Hiele (1909-2010) observed in practice that students showed levels in insight. He presented in his 1957 thesis a general theory of levels of insight, with suggestions how education can help students to move from one level to the next one. He used geometry to demonstrate this, but said explicitly that the theory applied also to other areas of knowledge. His work can best be seen as NME. He investigated the use of 2 + 2^(-1). My suggestion to use H = -1 is an improvement on this. Using -1 directly suggests to students that they must do some calculation, which is distractive. Instead, 2^H indicates the inverse value of 2, which is precisely what is needed, with the rule that 2 2^H = 1.
  • Mathematician Hans Freudenthal (1905-1990) was the thesis supervisor of Van Hiele. Freudenthal lectured mathematics at university and  had hardly training and practical experience as math teacher for other levels of education. Apparently he put in hours in looking at such practices, but we can diagnose that he remained an abstract thinking mathematician and that he did not develop into an empirical scientist. His wife supported the Jena-plan schools, and may have influenced him in preferring understanding above drilling. Unfortunately, he stole ideas by Van Hiele, and distorted them, and abused his position as a professor to take the limelight. Freudenthal presented his RME and blocked the development of NME.
First graph: Cause and effect

We can make a first graph on the difference between RME and TME w.r.t. cause and effect in the relation between insight and skill. These notions meet with difficult questions on measurement, on which RME and TME may not agree. When insight is defined as “problem solving skills” then we must distinguish between different kinds of skills. The following remains a sketch that is targeted at understanding the difference in reasoning.

  • RME holds that students should understand issues much deeper than TME are willing to accept before they should be trained. Potentially this order might be reversed, like in behaviourism with Pavlov reactions, but doing so would not be educational (in the long run). Thus in the RME View, insight (grasp) drives the subsequent skill. RME will not drill on the tables of multiplication and on long division. By working with insight on sums, the required skills will sink in over time. There might be a trade-off, with little grasp causing little skill and a high grasp on theory with little practical skill (the distracted professor). It is not required to target for the highest level of skill.
  • The TME response is that this is too simple. It is obvious that exercises better be preceded by an introduction. But the introduction would be shorter than with RME. Once you understand what addition and multiplication are, you still have to exercise on tables of addition and tables of multiplication, and learn to know those by heart for the numbers 1-10. You must master long division for later algebra (e.g. divide x^2 -1 by x – 1). There are different levels of insight. At some point the development of skill can help to arrive at more insight. At some point training (like training to the test) can reduce understanding, but this doesn’t devalue the other notions. TME will have less assumptions about the form of the graph, and for simplicity I use a similar form.
Traditional versus reform views

Traditional versus reform views

Second graph: Combined

When we assume that the notions of insight and skill are measured in the same manner for both RME and TME positions, then we can combine above graphs. We indicate the tops of curves by labels T (TME) and R (RME) only to help distinguish them. Potentially, the curves can still be everywhere, either be distinct (left graph, T on the left of R) or overlap (right graph, T on the right of R). Overall, RME claims that it can be superior in insight than TME, with the graph extending to the right of T.

Views combined in the same space

Views combined in the same space

Third graph: Decision space

When the notions of insight and skill are measured in ways that we can agree upon scientifically, then it may also be that these processes like “too much training on skill destroys insight” or “too much time to develop insight destroys skill” take a special form. An economist is reminded of the Production Possibility Curve (PPC).

Production factors like time and energy create both insight and skill, with a concave trade-off given by the PPC through the points A and B. In the philosophy of RME there is A = T and B = R, so that the TME might have perhaps more skill but at the cost of insight. TME holds that the RME claim is false: RME might succeed in doing more context sums A = R (drilling that isn’t called drilling) but with little mathematical insight B = T.

PPCOf relevance is point C. This point is given by neoclassical mathematics education (NME). This point gives more skill and insight than both A and B. Whether RME or TME is at A or B does not really matter, because C is always better. This position of NME has not been proved empirically yet, but follows logically from its removal of crummy “mathematics”.


The argument of this weblog text is quite simple. NME beats RME & TME. Curiously RME & TME neglect NME, or attack it in ad hominem and not at rem manner. Dutch readers can check my letter to parliament.

  • There is no reason on content for RME & TME to neglect NME, since NME is better than both on both insight and skill.
  • That RME rejects a better analysis can be understood from RME’s genesis as an ideology. Ideologues like Koeno Gravemeijer are not interested in empirics but in writing about their ideology.
  • That TME neglects a better analysis can be understood from its focus on RME, but also from the property that mathematicians are trained in abstraction and not in empirical science (which also caused the RME disaster).
  • There is a curious attitude: “When you don’t agree with us then you must be against us and thus with the other side.”
  • The empirical operationalisation of insight and skill and the production factors that cause them will help to put flesh on these graphs. This will reduce the impact of ideology, and will shift the focus to competence in research (see the failure of psychometric research w.r.t. arithmetic and algebra).

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