Monthly Archives: November 2016

Mathematicians can be seen as lawyers of space and number.

Euclid wrote about 300 BC. Much earlier, Hammurabi wrote his legal code around about 1792-1749 BC. It is an interpretation of history: Hammurabi might have invented all of his laws out of thin air, but it is more likely that he collected the laws of his region and brought some order into this. Euclid applied that idea to what had been developed about geometry. The key notions were caught in definitions and axioms, and the rest was derived. This fits the notion that Pierre de Fermat (1607-1665) started as a lawyer too.

Left: codex Hammurabi. Right: a piece of Euclid 100 AD. Wikimedia commons

The two cultures: science and the humanities

In Dutch mathematics education there is a difference between A (alpha) and B (beta) mathematics. B would be “real” math and prepare for science. A would provide what future lawyers can manage.

In the English speaking world, there is C.P. Snow who argued about the “two cultures“, and the gap between science and the humanities. A key question is whether this gap can be bridged.

In this weblog, I already mentioned the G (gamma) sciences, like econometrics that combines economics (humanities) with scientific standards (mathematical models and statistics). Thus the gap can be bridged, but perhaps not by all people. It may require some studying. Many people will not study because they may arrogantly believe that A or B is enough (proof: they got their diploma).

Left and right hemisphere of the brain

Another piece of the story is that the left and right hemispheres of the brain might specialise. There appears to be a great neuroplasticity (Norman Doidge), yet overall some specialisation makes sense. The idea of language and number on the left hemisphere and vision on the right hemisphere might still make some sense.

“Broad generalizations are often made in popular psychology about certain functions (e.g. logic, creativity) being lateralized, that is, located in the right or left side of the brain. These claims are often inaccurate, as most brain functions are actually distributed across both hemispheres. (…) The best example of an established lateralization is that of Broca’s and Wernicke’s Areas (language) where both are often found exclusively on the left hemisphere. These areas frequently correspond to handedness however, meaning the localization of these areas is regularly found on the hemisphere opposite to the dominant hand. (…) Linear reasoning functions of language such as grammar and word production are often lateralized to the left hemisphere of the brain.” (Wikipedia, a portal and no source)

For elementary school we would not want kids to specialise in functions, and encourage the use of neuroplasticity to develop more functions.

Pierre Krijbolder (1920-2004) suggested that there is a cultural difference between the Middle East (Jews), with an emphasis on language – shepherds guarding for predators at night – and the Indo-Europeans (Greeks), with an emphasis on vision – hunters taking advantage of the light of day. Si non e vero, e ben trovato.

There must have been at least two waves by Indo-Europeans into the Middle-East. The first one brought the horse and chariot to Egypt. The second one was by Alexander (356-323 BC) who founded Alexandria, where Euclid might have gotten the assignment to write an overview of the geometric knowledge of the Egyptians, like Manetho got to write a historical overview.

Chariot spread 2000 BC. (Source: D. Bachmann, wikimedia commons)

It doesn’t actually matter where these specialisations can be found in the brain. It suffices to observe that people can differ in talents: lawyers would deal much with language, and for space you might turn to mathematicians.

Pierre van Hiele (1909-2010) presents a paradox

The Van Hiele levels of insight are a key advance in epistemology, for they indicate that human understanding itself is subjected to some structure. The basic level concerns experience and the direct language about this. The next level concerns the recognition of properties. Another level is the recognition of relations between these properties, and the informal deductions about these. The highest level is formalisation, with axiomatics and formal deduction. The actual number of levels depends upon your application, but the base remains in experience and the top remains in axiomatics.

Learning goes from concrete to abstract, and from vague to precise.

Thus, Euclid with his axiomatic approach would be at the highest level of understanding.

The axiomatic approach is basically a legal approach. We start with some rules, and via substitution and reasoning we arrive at other rules. This is what lawyers can do well. Thus: lawyers might be the best mathematicians. They might forget about the intermediate levels, they might discard the a-do about space, and jump directly to the highest Van Hiele level.

A  paradox is only a seeming contradiction. The latter paradox gives a true description in itself. It is quite imaginable that a lawyer – like a computer – runs the algorithms and finds “the proper answer”.

However, for proper mathematics one must be able to switch between modes. At the highest Van Hiele level, one must have an awareness of applications, and be able to translate the axioms, theorems and derivations into the intended interpretation. In many cases this may involve space.

Just to be sure: the Van Hiele levels present conceptual divides. At each level, the languages differ. The words might be the same but the meanings are different. This also causes the distinction between teacher-language and student-language. Often students are much helped by explanations by their fellow students. It is at the level-jump, when the coin drops, that meanings of words change, and that one can no longer imagine that one didn’t see it before.

Thus it would be a wrong statement to say that the highest Van Hiele level must have command of all the lowest levels. The disctinction between lawyers and mathematicians is not that the latter have command of all levels. Mathematicians cannot have command of all levels because they have arrived at the highest level, and this means that they must have forgotten about the earlier levels (when they were young and innocent). The distinction between lawyers (math A) and mathematicians (math B) is different. Lawyers would understand the axiomatic approach (from constitutional law to common law) but mathematicians would understand what is involved in specific axiomatic systems.

Example 1

I came to the above by thinking about the following problem. This problem was presented as an example of a so-called “mathematical think-activity” (MTA). The MTA are a new fad and horror in Dutch mathematics education. First try to solve the problem and then continue reading.

Discussion of example 1

The drawing invites you to make two assumptions: (1) the round shape is a circle, (2) the vertical x meets the horizontal x in the middle. However, why would this be so ? You might argue that r = 6 suggests the use of a circle, but perhaps this still might be an ellipse.

In traditional math ed (say around 1950), making such assumptions would cost you points. In fact, the question would be considered insoluble. No question would be presented to you in this manner.

In traditional math, the rule would be that the proper question and answer would consist of text, and that drawings only support the workflow. Also, the particular calculation with = 6 would not be interesting. Thus, a traditional presentation would have been (and also observe the dashes):

A quick observation is that there are three endpoints, and it is a theorem that there is always a circle through three points. So the actual question is to prove this theorem, and you are being helped with a special case.

Given that you solved the problem above, we need not look into the solution for this case.

The reason for giving this example is: In mathematics, text has a key role, like in legal documents for lawyers. Since mathematicians are lawyers of space and number, they can cheat by using supporting drawings, tables and formulas. But definitions, theorems and proofs are in text (formulas).

(Potentially lawyers also make diagrams of complex cases, as you can see in movies sometimes. But I don’t know whether there are particular methods here.)

Example 2

The second example is the discussion from yesterday.

In text it is easy to say that a line has no holes. However, when you start thinking about this, then you must define what such a hole might be. If a hole doesn’t belong to the line, what does it belong to then ? How would you know when you would pass a hole ? Might you not be stepping over holes all the time without noticing ?

These are questions that lawyers would enjoy. They are relevant for math B but can also be discussed in math A.

See the discussion of yesterday, and check that the main steps should be acceptable for lawyers, i.e. math A.

These students should be able to master the symbolism of predicate logic, since this is only another language and a reformulation of common text.

Conclusions

Thus, a suggestion is that students in math A should be able to do more, when better use is made of the legal format.

Perhaps more students, now doing A, might also do B, if their learning style is better supported.

(Perhaps the B students would start complaining about more text though. Would there still be the same question, when only the format of presentation differs ? Thus a conclusion can also cause more questions. See also this earlier discussion about schools potentially manipulating their success scores by causing student underperformance.)

The standard treatment of continuity in mathematics textbooks in schools tends to be a bit crooked.

• The continuum is first assumed, but it is not stated what is assumed.
• For the real line, the lack of holes is a key property of continuity, but it is called by a word that students might have no affinity with (“completeness” rather than “wholeness”).
• When continuity is actually discussed in analysis (if at all), then this concerns the continuity of functions, which is rather a different subject.
• A discussion of the continuum brings us to topology, but do we really need to start with topology before we can do analysis ? Do you want to start your junior highschool class by stating: “In the mathematical field of point-set topology, a continuum (plural: “continua”) is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space.” ?

Our research question for today is: What might be a more logical exposition ? (Didactics would be second phase.)

I will not be telling anything new here, but some students might benefit from the more explicit and straightforward discussion.

Continuity as a primitive notion that cannot be defined

The basic notion of continuity is the real line. One might also think about 3D space or time. L.E.J. Brouwer wouldn’t trust space (Euclidean or non-Euclidean ?) and take time as his intuition, and hence speak about “intuitionism”. My impression is that space is more easy to communicate about (measuring rods are easier to make than clocks), whence I adopt the real line.

Definition w.r.t. human experience: Continuity is a primitive notion, that you might grasp by considering a line (section) in space.

Definition for mathematics: The set of real numbers R can be defined in a particular way. Personally I prefer the method by Timothy Gowers to develop the real numbers as infinite decimals.

Once the real numbers have been defined, then we can say that they also satisfy the notion of continuity. Thus, continuity is either a human experience or defined as the real numbers.

Once we have done this, then we can find the “Cantor-Dedekind Axiom“:

“In mathematical logic, the phrase Cantor–Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.” (PM. This wikipedia page should link directly to Tarski’s axioms for geometry.)

I find the term “axiom” a bit problematic.

• Given the two properties mentioned above, this isomorphism rather expresses human experience from modeling practice. The real numbers would be a model for the line (section) in actual space.
• Likely though, the identification of R with space is best seen as a definition of what we regard as Euclidean space. It is a question whether actual space would be Euclidean. It is a question whether we can actually imagine space being non-Euclidean, since we imagine e.g. a sphere still in 3D Euclidean space.

But the isomorphism is explained rather easily – and for didactics we would likely begin with this. For the numbers we can look at a development of binary decimals between 0 and 1. The next decimal is 0 or 1, and again, and so on. For space we can make a cut and have left and right parts, make a cut again with new left and right parts, and so on. Thus this is the same structure. But these also are different realms: numbers and space. Thus it is not quite an identity but an isomorphism. Interestingly: when cutting in this manner, we will never meet a hole.

Continuity can be explained only for subsets

Subsequently, the key properties of continuity can be formulated w.r.t. subsets S of R, rather than w.r.t. R itself.

Definition: A subset S of R is called continuous, if between two elements (values) in S there is always another one (i.e. it is dense), and when there are no holes. Or in formulas:

(a) For each x, y in SR with x < y, there exists z in S such that x < z < y

(b) For each x, y in S R with x < y, there exists no z in R \ S such that x < z < y

The last property shows the difficulty for R. If one would want to specify that R has a hole, then one would have to specify what that hole belongs to. To some X ? What is X ? In the past, people had a problem imagining what a vacuum was: the horror vacui. For them, space could only exist if something occupied that space. Nowadays, mathematical space is understood merely as a set of co-ordinates, and the issue what physical space would be is left to physics.

Also observe that this definition essentially depends upon the fact that the real numbers have been given, i.e. the earlier section. Thus, continuity is a basic notion given for R and there is only a “proper” (explicit) definition for subsets: which definition relies on the use of R.

If you don’t assume R, you get into problems. For example, if you were to take the set of rational numbers Q rather than R, then (a-Q) could be satisfied for some S, say = [1/2, 3/2], and (b) would become:

(b-Q) For each x, y in S Q with x < y, there exists no z in Q \ S such that x < z < y

In that case, one might say that Q becomes Q-continuous, but this is not the continuity that we want, since there are elements in R still in that interval. (Contiguity comes to mind as a label, but already has some use.)

Further developments

Property (b) is called “(Dedekind) completeness“. It is true that “complete” is a proper translation of German “vollständig” (German wiki), but I would rather prefer “(Dedekind) wholeness”, since this better indicates the lack of holes. But let me admit that I am used to the phrase “completeness” as well, for the chapter of ordering, and thus my preference is weak. Perhaps it is best to speak about “completeness (wholeness)”.

Subsequently, when we forget about the reliance on R, and try for a more abstract formulation, then the notion of supremum (least upper bound) comes into play. We can look at some S independent from R, as the “linear continuum“. This is not intuitive and not feasible in junior highschool. Potentially this approach actually captures continuity in a definition, so that it isn’t just primitive, and can be defined, but (for me, yet) there is no clear connection between the notion of continuity and the property of having suprema. The switch to topology comes into play, see G.H. Moore “The emergence of open sets, closed sets, and limit points in analysis and topology” (and thanks to Dag Oskar Madsen for the reference, in a discussion about open sets that is closed now).

Continuous function

Obviously it helps to have clarity on continuity in the reals first before speaking about continuous functions.

The notion of a continuous function f uses both domain S and range f[S]. At stackexchange, many readers liked the view by Qiaochu Yuan (answer 17):

“One abstract way to think about continuity (…) is that it is about error. A function f:XY is continuous at x precisely when f(x) can be “effectively measured” in the sense that, by measuring x closely enough, we can measure f(x) to any desired precision. (…) This is an abstract formulation of one of the most basic assumptions of science: that (most of) the quantities we try to measure () depend continuously on the parameters of our experiments (…). If they didn’t, science would be effectively impossible.”

For Dutch readers, Vredenduin has a nice exposition in Euclides 1969 on the notion too, partly containing this intuition on the error too, but not so explicitly. He speaks about a small change in the domain and no dramatic change in the range, but it is more enlightening to explicitly speak about (measurement) error.  (And I would have a question on continuity of “f / g“, p14.)

Conclusion

The main argument is that this storyline is more straightforward for understanding continuity. All this suggests that school would benefit from a discussion of the reals. This would include issues like 0.9999…. = 1.0000…

I am supposing that junior highschool could manage the expressions of mathematical logic. The New Math tried and failed, but there should be more clarity why it failed.

PM. For completeness: there is always philosophy (and nonstandard analysis).

Horror vacui: no space without something (Wikimedia commons)

After my short stint as expert on national security in 2005, I now had another short stint, now as expert on mathematics education, STEM and the role of mathematics education in the whole curriculum. First Jos Tolboom of SLO had hinted that I could be invited for an expert meeting, but then this invitation didn’t actually materialise.

This is a bummer. I wrote several books and articles on the subject. One of the reasons why there is so little progress in the field is that there isn’t enough attention for my novel analysis.

The November event

This concerns the following event:

22 – 23 November 2016
STEM (science, technology, engineering, mathematics)  Expert Meeting

Utrecht, The Netherlands
Hosted by Freudenthal Institute and SLO
Topic: The position of mathematics education and informatics education in a coherent STEM curriculum
This meeting aims to create an international overview of innovations in mathematics education and informatics education, their relationship and the coherence from the STEM perspective, with a special interest in computer based mathematics and its relation with computer science (informatics). This goal has been determined with other CIDREE members as a follow up on the expert meeting in June 2015 in Trondheim, Norway.

The announcement in Dutch is here. Three questions for the meeting are (in my translation):

1. How can we create more coherence in STEM education overall ?
2. How can we create a mathematics curriculum with a strong component in computer based (mathematics) education ?
3. How can we create a curriculum for computer science (“informatica”) as part of the STEM curriculum ?

The schedule with the speakers is here. For example, Cambridge Mathematics will be present, as they also announce on their website:

Cambridge Mathematics will be represented at this event as we explore innovations in mathematics education and how we can plan for the future of the mathematics curriculum.

Key warning for STEM researchers

(1) Researchers on STEM should be aware that the researchers on STE may have little knowledge or interest in both Mathematics education (ME) and its research (MER). Every field tends to focus on itself, and coherence is secondary.

(2) A key difference is:

• STE fields have empirics as a judge of what works. This empirical mindset is also applied to the education in these fields.
• Mathematics is directed at non-empirics (abstraction). There is no external judge but only personal opinion. Thus mathematicians tend to regard power play and “math wars” as acceptable methods to get views accepted. (Examples of such thinking in Holland are mathematicians Jan van de Craats and Henk Broer.)
• See this discussion about the math war between “realistic mathematics education” RME and traditional ME (TME), and the scientific alternative of neoclassical NME. Look also for the explanation that the name “Freudenthal Institute” does not convey the true meaning of the institute, and that it is better to speak about “Freudenthal Head in the Clouds Realistic Mathematics Institute” (FHCRMI). Namely, RME is like astrology or homeopathy.

(3) In combination: STE are willing victims of “realistic mathematics education” (RME) ideology. STE provide “contexts” and they apparently appreciate the interest. However, it really requires a study of ME and MER to get rid of the RME ideology and their unscientific narratives.

My qualifications

Let me mention my qualifications as expert for this topic and these questions, and observe that my books are online:

1. I developed four books within Mathematica, a system for doing mathematics on the computer: Voting Theory for Democracy (2001), A Logic of Exceptions (2007), Conquest of the Plane (2011), and The Economics Pack. (since 1993). These books provide the coherence that the expert meeting is looking for, with text, formulas, graphs, tables, routines, programming (informatics) and interaction. (A missing element is assessment.) There is also the book Transport Science for Operations Management (2000) that has been supported by routines in Mathematica. It is likely the “not invented here” syndrome at Dutch universities that these books are not being used regularly in matricola. (There is also the breach of scientific integrity w.r.t. a “review” of COTP.) (The distinction between the popular vote for Clinton and the Electoral College for Trump might cause more attention for voting theory nowadays, but my expectation is that Dutch universities will continue to neglect VTFD.)
2. I also discussed “Beating the software jungle”, included in Elegance with Substance (2009, 2015). This explains about the current chaos in software for education and what an effective and proper approach would be to resolve this. (The STEM researchers at this meeting might not have enough background in economics to understand the argument on market structure.)
3. I clearly explained the failure by the Dutch organisers of the event, both SLO and the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI), in their dealing with these issues before. Thus the organisers would know that inviting me would give scope for a discussion that goes to the heart of matters (and not beating about the bush again). Let me discuss this in the subsequent sections. (If these institutes would be scientific, then not-inviting me amounts to blocking me, since I would like to attend. Blocking me is an abuse of power, made possible by the current power void in mathematics education and its research, see here. But these institutes might also argue that they are not scientific.)
Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI)

I find myself repeating again. This isn’t good.

The actual argument is quite elaborate. When I would fall into the trap of using one-liners, then readers might think that I am being simplistic and that I (over-) generalise. Let me refer to Elegance with Substance, Chapter XIII and the note on p114.

A new phase in the discussion is the breach of research integrity by psychometricians at Leiden University. They actually expose the unscientific nature of RME / FHCRMI ideology but they don’t do so adequately.

Let me also refer to the abuse of so-called “21st century skills”. This label is deliberately used as a Trojan horse for re-introducing “realistic mathematics education” (RME) ideology. The true revolution is computer algebra. See here.

News on Michiel Doorman

A new element – on which I am not repeating myself – is that I have now collected my documentation about the unscientific and ideological performance by Michiel Doorman, one of the employees of FHCRMI on STEM and one of the key organisers of above event. Much of this documentation is in Dutch, but I provided an overview in English.

Two elements are relevant for STEM:

• Doorman maltreated the new algebraic approach of the derivative. Obviously, for physics education it is a key discovery that it is a false mathematical argument that limits would be required. For the derivative, it suffices to use algebra.
• Doorman promoted Java applets instead of computer algebra.

A google on Doorman also generated this diagram within the EU project of Mascil. I regard this as simsalabim, a phrase used in magic tricks, with flash and smoke that hide what is really happening. For example, there are “inquiring minds” and there is reference to a “collaborative classroom culture”, but these RME / FHCRMI “experts” clearly close their minds and use their elbows.

Simsalabim, taken from Doorman et al. 2014 on the European Union Mathematics and Science for Life (mascil)

SLO – Dutch expertise center on curriculum

SLO would be the Dutch national expertise center on the curriculum. It started as a foundation and initiative by researchers, and it is gradually absorbed by government regulations, with work packages and subsidies. This particular QUANGO has no longer a transparant structure. Best would be a decent government body, with accountability, but at some distance of political decision making because of the scientific base. For mathematics education, each nation should have a national organisation with a key role for teachers and researchers, and this organisation would also supervise the curriculum. For Holland my suggestion is a Simon Stevin Institute, and it would give directions to SLO (instead of SLO telling teachers what to do).

The traditional approach in pedagogy looks at the triad of student, material and teacher. In this approach, the student features with both a personality and personal development. SLO however derives from the world of “education studies” that are at a distance of the traditional development of pedagogy. “Educational studies” tend to overlook the student. In the Van den Akker diagram about learning, students who would do the learning are not mentioned themselves. They are regarded as learning machines, and the personalities of students might only be considered from contacts w.r.t. “other students”, see the “spider diagram”, at the SLO “Europe: Mathematics and Science knowledge” and SECURE project page. Potentially SLO exports this spider diagram to other countries, and foreigners cannot check whether they listen to criticism. Dutch readers / viewers will benefit from this video interview with professor emeritus in traditional pedagogy Jan Dirk Imelman. (Interviewer Ad Verbrugge is an ideologue too, but Imelman makes it a useful interview.)

Spider web, by Van den Akker (SLO). (Replace the word “learning” with “human” or “humanity”.)

Letter to Dutch Parliament, September 27 2016, on Onderwijs2032

I wrote this letter in English to Dutch Parliament, about much of the same thing. As stated in the letter, I chose for the use of English because I wanted that OECD and CIDREE would be able to read the argumentation.

The minister of education is considering a transformation of Dutch education to “21st century skills” (“Onderwijs2032”). This would involve the abolition of traditional subjects (like physics or economics) and merging those into common labels (like nature or society). The idea is that education should increase understanding that surpasses the various subjects. This is also known as the “transfer” problem. However, the traditional approach is: one must first master a subject before one can surpass it. Thus Onderwijs2032 is created from rosy dreams.

What is crucial to know is that RME already belonged to that stream of rosy “21st century skills” way of thinking, and that it failed miserably. This is also why it is so curious that the Leiden psychometricians failed in reaching the proper analysis, even though they did show that RME claims are exaggerated. See this submission to the integrity board and this link on algebra as a troubling word.

Curiously, SLO has been supporting this Onderwijs2032 project. Though they oversee the curriculum, they did not notice that you need traditional algorithms in arithmetic in elementary school because you need those for algebra in secondary education.

News on Jos Tolboom

At SLO, Jos Tolboom (LinkedIn) is the other key organiser of the November event. He informs me that he read Elegance with Substance this last Summer and is impressed by its quality and relevance. I hope that he finds time to express this in a public statement that others can check (and that I would include on the EWS website). It is still possible that there are misunderstandings though, for we haven’t had a discussion on particulars. I also hope that he finds time to read COTP and then will protest against the abuse by Jeroen Spandaw in the journal Euclides in 2012.

Tolboom also alerted me to this CIDREE event and hinted at the possibility that he might invite me to attend. I am wondering now why he doesn’t. He gave me a reason but in my expert view not a convincing one, and lacking in respect for science. (Perhaps though he doesn’t regard SLO as a scientific organisation.)

I observed that Tolboom is giving video presentations on the new Dutch national exam on mathematics. A key element in the exam renewal are the “mathematical think-activities” (MTA). When I evaluated this MTA notion last month, I found it deficient. (1) MTA is severely confused w.r.t. didactics and testing, (2) MTA is a disguise of RME, (3) MTA is not proper mathematics (which would have a development towards deduction with definitions, theorem, proof). Another suggestion for Tolboom is to state a reply to this criticism. Hopefully, Holland finds a way to rewrite the national exam regulation.

The revolution is computer algebra

Let us return to the November event and the revolution of computer algebra. The problem with this revolution is that mathematicians are riding their hobby horses and creating chaos, e.g. by creating that software jungle. Perhaps there must be a jungle for a survival of the fittest, but in the past organised efforts like for Algol seem to have worked better.

Looking at the schedule, I don’t see a presentation that will explain that computer algebra forms the common core. Suggestions for a common core are “drawings” and “modeling”. You can check how the sessions of November 23 are geared to such a conclusion. Such a (prepared) inference would be deficient in understanding of didactics. Apparently Jos Tolboom didn’t read Elegance with Substance well enough. (Potentially, for mathematics, the “modeling” can be translated as MTA ?)

There will be a presentation on November 22nd by “Computer Based Maths” (CBM). This organisation was founded by Conrad Wolfram, and thus CBM is personally related to Wolfram Research Inc. (WRI), the makers of Mathematica, created by Stephen Wolfram. WRI apparently decided that it would not do to simply advertise the use of Mathematica. This might come across as a commercial enterprise. Conrad Wolfram set up CBM, and I suppose that they use more resources than only Mathematica (for example on assessment). See my earlier comments on computer algebra and Conrad, here.

Still, I find the situation needlessly complex and disinformative. The proper analysis is that computer algebra is the revolution, and that this provides the common core in education with the computer, not only for STEM but also for languages and the arts. There are other computer algebra implementations than Mathematica, but this remains the best system, consistently since at least 1993 when I started using it. We can allow for various implementations but as long as the same computer language is used for doing mathematics. The commercial venue by WRI is somewhat of a distraction. The creation of CBM is evidence of this too, and an admission by WRI too. It would be better that WRI is turned into a public service utility. The true question is what arguments would cause WRI to agree on this.

My impression is that it would help a great deal when the community of educators would agree that the common core can be found in mathematics itself, and, when the computer is used, in computer algebra. It is easier (also for CBM and WRI) to agree to help out when you are lauded than when your accomplishments are misunderstood and when you feel that you have to put in an effort to get recognition. If WRI would make Mathematica free for elementary and secondary education, then they can still earn their income on universities and research institutes as they are doing now.

When these CIDREE conference “experts” would arrive at the “STEM common core” of “modeling” then they do not understand didactics of mathematics and then their conference result is caused by an abuse of power by excluding a proper expert, duly signed, yours truly.

Event agenda for November 23