# Pierre van Hiele and Gerald Goldin (1)

*Listening to Izaline Calister “Mi Pais”
“Atardi Korsou ta Bunita”, or Willem Hendrikse,
or Rudy Plaate “Dushi Korsou” or IC & CR “Mi ta stimabu“,
and Frank might also have liked Las Unicas “Ban Gradici Senjor” from Aruba
*

We need two weblogs to discuss Gerald Goldin (2003), *Developing complex understandings: On the relation of mathematics education research to mathematics*.

We namely start with Goldin (1992), *Towards an assessment framework for school mathematics* (pages 63-88 in Lesh & Lamon, *Assessment of authentic performance in school mathematics*).

The *Framework* gives a baseline and also allows us to see the long standing involvement of Goldin in mathematics education and its research. We all know about the *New Math *disaster in the 1960s and about Morris Kline (1908-1992) who in 1973 wrote *Why Johnny Can’t Add: the Failure of the New Math*. I wasn’t aware however of the heavy involvement in the USA in the 1970s with behaviourism. This US phenomenon wasn’t copied to such extent in Europe. Perhaps the European attention for the *New Math *partly came from the European (Bourbaki) origins. With these two assaults on mathematics education in the USA, both *New Math *and behaviourism, we may better understand the flower power reaction movement, partly captured by Hans Freudenthal’s scam of “realistic mathematics education” (RME), in which students are left free to discover mathematics for themselves, with perhaps at most “guided reinvention”.

##### Recovering from New Math and behaviourism

Thus in 1992, after two decades of struggling to recover from *New Math *and behaviourism, Goldin writes the following Introduction to this *Framework.*

The role of the general public in understanding processes and content is that Goldin wants parents to understand what Johnny is doing in his proposed approach.

##### Test questions, purposes and methods

Goldin (1992) discusses various possible test questions, and then he questions what they are supposed to test. This is a nice example of a problem that Johnny might be asked to handle mathematically.

While you are solving this problem, you must also wonder whether this fits the RME philosophy, and what would count as a successful answer. Are you allowed to guess yes/no/maybe or would you have to give a proof ?

Goldin shows how the question can be used for various didactic goals with also various possible interventions, ranging from exploration (my suggestion now too: perhaps just give the egg timers and not pose any questions) to standardized solution technique (that flower power calls *boring* and behaviourism *exciting*).

##### A cognitive model to prevent disasters from new fashions

Goldin cannot avoid Freudenthal’s RME (applied mathematics), and gives an answer that fits Van Hiele’s view that mathematics is directed at higher levels of insight (abstraction).

Goldin gives the obvious warning that when we don’t know what we are doing, then we may make all kinds of investments in training and computer programs, that later appear to be useless.

##### A traditional question

Subsequently, Goldin discusses a traditional question.

RME would allow students to find their own solution strategies, and students perhaps run afoul when they don’t find any (or arrive at the right answer in mysterious ways). Traditional methods would provide a single algorithm that always generates an outcome, but perhaps students run afoul when the question is slightly changed. I refer to Goldin’s paper for these and more angles.

##### Conclusion

This discussion is only a rough introduction to Goldin (1992) “Framework”. The idea of this text (1) is to set the stage for weblog text (2). It helps to be aware of what Goldin was already proposing in 1992 when we are going to look at his view in 2003.

My problem with Goldin (1992) is that he does not distinguish the Van Hiele levels of insight. For example, it seems somewhat obvious that traditional Problem 2 requires a different approach for novices than for experts. But it helps to be also aware that the same words may have a different meanings depending upon the level of insight.

For novices Problem 2 may require much creativity and thus can also give a lot of fun. It is only when students become experts that problem 2 becomes boring. The discussion about assessment should not be *burdened* by the situation that mathematics education can create the paradox that students find Problem 2 boring and still cannot answer it properly. Like in Item-Response Theory, a test does not only say something about students, but students also say something about the test (environment).