# Education, group theory and division

A number is what satisfies the axioms of its number system. For elementary and secondary education we use the real numbers *R.* It suffices to take their *standard form* as: sign, a finite sequence of digits (not starting with zero unless there is a single zero and no other digits), a decimal point, and a finite or infinite sequence of digits. We also use the isomorphism with the number line.

##### Thus a limited role for group theory

Group theory creates different number systems, from natural numbers *N, *to integers *Z, *to rationals *Q, *to reals *R, *and complex plane *C, *and on to higher dimensions. For elementary and secondary education it is obviously useful to have the different subsets of *R. *But we don’t do group theory, for the notion of number is given by *R.*

It should be possible to agree on this (*):

- that
*N*⊂ Z ⊂*Q*⊂*R,* - that the elements in
*R*are called numbers, - whence the elements in the subsets are called numbers too.

Timothy Gowers has an exposition, though with some group theory , and thus we would do as much group theory as Gowers needs. There is also my book *Foundations of mathematics. A neoclassical approach to infinity* (FMNAI) (2015) (pdf online) so that highschool students need not be overly bothered by complexities of infinity. FMNAI namely distinguishes:

- potential infinity with the notion of a limit to infinity
- actual infinity created by abstraction, with the notion of “bijection by abstraction”.

There arises a conceptual knot. When *A *is a subset of *B, *or *A* ⊂ *B, *then saying that *x *is in *A *implies that it is in *B, *but not necessarily conversely. Who focuses on *A, *and forgets about *B, *may protest against a person who discusses *B. *When we say that the rational numbers are “numbers” *because* they are in *R, *then group theorists might protest that the rationals are “only” numbers *because* (1) *Q* is an extension of *Z *by including division, and (2) then we *decide* that these can be called “number” too. Group theorists who reason like this are advised to consider the dictum that “after climbing one can throw the ladder away”. In the real world there are points of view. When Putin took the Crimea, then his argument was that it already belonged to Russia, while others called it an annexation. In mathematics, it may be that mathematicians are people and have their own personal views. Yet above (*) should be acceptable.

It should suffice to adopt this approach for primary and secondary education. Research mathematicians are free to do what they want at the academia, but let they not meddle in this education.

##### Division as a procept

The expression 1 / 2 represents both the *operation* of division and the resulting *number*. This is an example of the “procept“, the combination of process and concept.

The procept property of *y */ *x *is the cause of a lot of confusion. The issue has some complexity of itself and we need even more words to resolve the confusion. Wikipedia (a portal and no source) has separate entries for “*division*“, “*quotient*“, “*fraction*“, “*ratio*“, “*proportionality*“.

In my book *Conquest of the Plane *(COTP) (2011), p47-58, I gave a consistent nomenclature (pdf online):

“Ratio is the input of division. Number is the result of division, if it succeeds.” (COTP p51)

This is not a definition of number but a distinction between input and output of division. My suggestion is to use the word *(static) **quotient *also for the* form *with *numerator* *y *divided by *denominator* *x.*

(static) quotient[*y, x*] = *y / x*

This fits the use in calculus of “difference and differential quotients”. The form doesn’t have to use a bar. Also a computer statement Div[numerator *y,* denominator *x*] would be a quotient.

This suggestion differs a bit from another usage in which the quotient would be the *outcome* of the division process, potentially with a remainder. We saw this usage for the polynomials. This convention is not universal, see the use of “difference quotient”. However, if there would be confusion between *outcome* and *form,* then use “*static quotient*” for the *form.* This is in opposition to the *dynamic quotient *that is relevant for the derivative, as *Conquest of the Plane *shows.

##### Proportionality and number

Check also the notion of proportionality in COTP, page 77-78 with the notion of proportion space: {denominator *x*, numerator *y*}. Division as a process is a multidimensional notion. The wikipedia article (of today) on proportionality fits this exposition, remarkably with also a diagram of proportion space, with the denominator (cause) on the horizontal axis and the numerator (effect) on the vertical axis (instead of reversed), *as it should be* because of the difference quotient in calculus. In *Conquest of the Plane *there is also a vertical line at *x *= 1, where the numerators give our numbers (a.k.a. slope or tangent).

##### Avoiding the word “fraction”

My nomenclature uses the quotient and the distinction in subsets of numbers, and I tend to avoid the word fraction because of apparent confusions that people have. When someone gives a potential confusing definition of fractions, my criticism doesn’t consist of providing a proper definition for fractions, but I point out the confusion, and then refer to the above.

Below, I will also refer to the suggestion by Pierre van Hiele (1973) to abolish fractions (i.e. what people call these), and I will mention a neat trick that provides a much better alternative.

##### Number means also satisfying a standard form

Number means also *satisfying a* *standard form. *Thus “number” is not something mysterious but is a form, like the other forms, yet standardised.

For example, we have 2 / 4 = 1 / 2, yet 1 / 2 has the standard form of the rationals so that 2 / 4 needs to be simplified by eliminating common prime factors. The algebra of 2 / (2 2) = 1 / 2 can be seen as “rewriting the form”.

What the standard is, depends upon the context. We can do sums on natural numbers, integers, rationals, reals. In education students have to learn how to rewrite particular forms into a particular standard. Student need to know the standard forms, not the group theory about the subset of numbers they are working in.

The equality sign in *x *= *a* is ambiguous. Computer algebra tends to avoid ambiguity. For example in *Mathematica*: Set (=) vs Equal (==) vs (identically) SameQ (===). Doing computer algebra would help students to become more precise, compared to current textbooks. Learning is going from vague to precise.

The equality sign in highschool tends to mean “of equal value”, which is above “==”. But two expressions can only be of equal value when they represent the identically same value. Thus *x * == *a *would amount to *Num*[*x*] === *Num*[*a*]. The standard mathematical phrase is “*equivalence class*” for a number in whichever format, e.g. with the numerical value at the vertical position at line at *x *= 1 (also for the denominator 1).

The standard form takes one element of an “equivalence class” (depending upon the context of what numbers are on the table, e.g. 1 / 2 for the rationals and 0.5 for the reals). (See COTP p45-48 for issues of “approximation”.)

##### Multiplication is no procept

Multiplication is no procept. For multiplication there is a clear distinction between the operation 2 * 3 and the resulting number 6. When your teacher asks you to calculate 2 * 3 then the answer of 2 * 3 is correct but likely not accepted. The smart-aleck answer 2 * 3 = 3 * 2 is also correct, but then the context better be group theory.

It is a pity that group theory adopted the name “group theory”. My proposal for elementary school is to replace the complicated word “multiplication” by “group, grouping”. With 12 identical elements, you can make 4 groups of 3. (With identical elements this isn’t combinatorics.) See *A child wants nice and no mean numbers *(CWNN) (2015). If this use of “group, grouping” is confusing for group theory, then they better change to something like “generalised arithmetic”.

##### The hijack of number by group theory

The world originally had the notion of number, like counting fingers or measuring distance, but then group theory hijacked the word, and assigned it with a generalised meaning, whence communication has become complicated. Their use of language might cause the need for the term *numerical value. *I would like to say that 2 is identically the same number in *N, Z, Q *and *R, *but group theorists tend to pedantically assert that the notion of number is relative to the set of axioms. In the Middle Ages, people didn’t know negative numbers, and they couldn’t even think about -2. Only by defining -2 as a number too, it could be included as a number. This sounds like Baron von Muenchhausen lifting himself from the swamp. The answer to this is rather that -2 is still a number even though it wasn’t recognised as this. I would like to insist that we use the term “number” for the numerical value in *R, *so that we can use the word “number” in elementary school in this safe sense. Group theorists then must invent a word of their own, e.g. “generalised number” or “gnumber”, for their systems.

Changing the meaning of words is like that your car is stolen, given another colour, and parked in front of your house as if it isn’t your car. Group theorists tend to focus on group theory. They tend not to look at didactics and teaching. When group theorists hear teachers speaking about numbers, and how 2 is the same number in *N *and *R, *then group theorists might smile arrogantly, for they “know better” that *N *and *R *are different number systems. This would be misplaced behaviour, for it are the group theorists themselves who hijacked the notion of number and changed its meaning. When research mathematicians have the idea that teachers of mathematics have no training about group theory, then they better read Richard Skemp (1971, 1975), *The psychology of learning mathematics, *first. This was written with an eye on teaching mathematics (and training teachers) and contains an extensive discussion of group theory. (Though I don’t need to agree with all that Skemp writes.)

##### Quote on human folly

Peter van ‘t Riet edited Vredenduin (1991) “*De geschiedenis van positief en negatief*“, Wolters-Noordhoff, on the history of positive and negative numbers. Van ‘t Riet allows himself a concluding observation:

“Kijken wij er achteraf op terug, dan kan een gevoel van verwondering opkomen, dat begrippen die ons zo vanzelfsprekend en helder lijken, zo’n lange ontwikkelingsgeschiedenis hebben gehad waarin vooruitgang, terugval en nieuwe vooruitgang elkaar afwisselden. Opmerkelijk is dat begrippen zich soms pas echt ontwikkelen als zij bevrijd worden van een dominerende idee die eeuwenlang hun ontwikkeling in de weg stond. Dat is bij de negatieve getallen het geval geweest met de geometrisering van de algebra: de gedachte dat getallen representanten waren van meetkundige grootheden is eeuwen achtereen een obstakel geweest teneinde tot een helder begrip van negatieve getallen te komen. Achteraf vraag men zich af: hoe was het mogelijk dat eeuwenlang deze idee de algebra bleef domineren?” (p121)

Since we sometimes check Google Translate for the fun ways of its expressions, it is nice to let the machine speak again:

“If we look afterwards back, then bring up a sense of wonder that concepts which seem to us so obvious and clear, have had such a long history in which progress, relapse and further progress alternating. Remarkably concepts sometimes only really develop as they freed from a dominant idea that for centuries had their development path that is in the negative numbers was the case with the geometrization of algebra:. the idea that numbers representatives were of geometric quantities is centuries successively been an obstacle in order to achieve a clear understanding of negative numbers retrospect one question himself:. how was it possible that for centuries the idea continued to dominate the algebra?” (Google Translate)

Just to be sure: analytic geometry has the number line with negative numbers too. Van ‘t Riet means the line section, that always has a nonnegative length.

A step to answering his question is that mathematicians focus on abstraction, whence they are more guided by their own concepts rather than by empirical applications or the observations in didactics. I included this quote in the hope that group theorists reading this will again grow aware of human folly, and realise that they should support empirical didactics and not block it.

##### More sources for confusion on formats

More noise is generated by the different “number formats” that have been developed over the course of history. We have forms 2 + ½ = 2½ = 5 / 2 = 25 / 10 = 2.5 = 2 + 2^{-1} (neglecting the Egyptians and such). We should not forget that the decimals are actually also a form or result of division. Another example is 0.365 = 3 / 10 + 6 / 100 + 5 / 1000. Only the infinite decimals present a problem, since then we need an infinite series of divisions, yet this can be solved. The various formats have their uses, and thus education must teach students what these are.

An approach might be to only use numbers in decimal notation. However, the expression 1 / 3 is often easier than 0.33333…. Students must learn algebra. Compare 1 / 2 + 1 / 3 with 1 / *a* + 1 / *b.*

“But to understand algebra without ever really understood arithmetic is an impossibility, for much of the algebra we learn at school is a generalized arithmetic. Since many pupils learn to do the manipulations of arithmetic with a very imperfect understanding of the underlying principles, it is small wonder that mathematics remain a closed book to them.” (Skemp, p35)

The KNAW 2009 study on arithmetic education and its evidence and research is invalid. It forgot that pupils in elementary school have to learn particular algorithms in arithmetic in preparation for algebra in secondary education. It scored answers to sums as true / false and didn’t assign points to the intermediate steps, so that pupils who used trial and error also had the option to score well. In a 2011 thesis on the psychometrics of arithmetic, the word “algebra” isn’t mentioned, and various of its research results are invalid. There is a rather big Dutch drama on failure of education on arithmetic, failure of supervision, and breaches of integrity of science.

Irrational numbers started as a ratio. Consider a triangle with perpendicular sides 1 and then consider the ratio of the hypothenuse to one of those sides. The input √2 : 1 reduces to number √2.

##### Standard form for the rationals

There are students who do 2 + ½ = 2½ = 2 ½ = 1, because in handwriting there might appear to be a space that indicates multiplication, compare 2*a *or 2√2 or 2 km where such a space can be inserted without problem. See the earlier weblog text how Jan van de Craats tortures students. A proposal of mine since 2008 is to use 2 + ½ and stop using 2½.

Yesterday I discovered Poisard & Barton (2007) who compare the teaching of fractions in France and New Zealand, and who also advise 2 + ½. The German wikipedia has also a comment on the confusing notation of 2½. I haven’t looked at the thesis by Rollnik yet.

For a *standard form* for the rationals, the rules are targeted at facilitating the location on the number line, while we distinguish the operation *minus *from the *sign* of a negative number (as -2 = negative 2).

- If a rational number is equal to an integer, it is written as this integer, and otherwise:
- The rational number is written as an integer
*plus*or*minus*a quotient of natural numbers. - The integer part is not written when it is 0, unless the quotient part is 0 too (and then the whole is the integer 0).
- The quotient part has a denominator that isn’t 0 or 1.
- The quotient part is not written when the numerator is 0 (and then the whole is an integer).
- The quotient part consists of a quotient (form) with an (absolute) value smaller than 1.
- The quotient part is simplified by elimination of common primes.
- When the integer part is 0 then
*plus*is not written and*minus*is transformed into the negative sign written before the quotient part. - When the integer part is nonzero then there is
*plus*or*minus*for the quotient part in the same direction as the sign of the integer part (reasoning in the same direction).

Thus (- 2 – ½) = (-3 + ½) but only the first is the standard form.

PM 1. Mathematica has the standard form 5 / 2. *Conquest of the Plane *p54 provides the routine RationalHold[*expr*] that puts all Rational[*x, y*] in *expr* into HoldForm[IntegerPart[*expr*] + FractionalPart[*expr*]].

PM 2. Digits are combined into numbers, so that we don’t have 28 = 2 * 8 = 16 = 6. Nice is:

“For example, 7 (4 + *a*) is equal to 28 + 7*a* and no 74 + 7*a*.” (Skemp, p230)

*H *= -1

A new suggestion is to use *H *= -1. Then we get 2 + ½ = 2 + 2* ^{H}*= 5 2

*. Pierre van Hiele (1973) suggested to abolish fractions as we know them. He observed that*

^{H}*y*/

*x*is a tedious notation, and students have to learn powers anyhow. I agree that the notation

*y*/

*x*generates so-called “mathematics” which is no real mathematics but only is forced by the notation. Using the power of -1 can be confusing because students might think of subtraction, but the use of (abstract)

*H*for the inverse clinches it. See here and my sheets for a workshop of NVvW November 2016.

Above quotient form then becomes (*y x ^{H}*) and the dynamic quotient (

*y x*), in which the brackets may be required in the dynamic case to indicate the scope of the simplification process.

^{D}There are students who struggle with *a *– (-*b*) = *a *– (-1) *b,* perhaps because subtraction actually is a form of multiplication. Curiously, this is another issue of inversion that is made easier by using *H,* with *a *– (-*b*) = *a *– *H* *b = a + H H b = a *+* b. *See the last weblog entry that division is repeated subtraction. The only requirement is that each number has also an inverse, zero excluded, so that these inverses can be subtracted too. For example 4 3* ^{H}* = (3 + 1) 3

*= 1 + 3*

^{H}*translates as repeated subtraction (not for the classroom but for reasons of current exposition):*

^{H}4 – (1 + 3* ^{H}*) – (1 + 3

*) – (1 + 3*

^{H}*) = 4 – 3 (1 + 3*

^{H}*) = 4 – 3 – 3 3*

^{H}*= 4 – 3 – 1 = 0*

^{H}##### Group theory is for numbers. It is not for education on number formats

The last weblog entry on group theory showed that group theory concentrates on numbers, whence it (cowardly) avoids the perils of education on the various number formats.

Group theory mathematicians will tend to say that 1 / 2 = 2 / 4 = 50 / 100 = .. .are all member of the same “*equivalence class*” of the number 1 / 2, whence their formats are no longer interesting and can be neglected.

In itself it is a laudable achievement that mathematics has developed a framework that starts with the natural numbers, extends with negative integers, develops the rationals, and finally creates the reals (and then more dimensions). This construction comes along with algorithms, so that we know what works and what doesn’t work for what kind of number. For example, there are useful prime numbers, that help for simplifying rationals. For example 3 * (1 / 3) = 1 whence 3 * 0.3333… = 0.9999… = 1.000… = 1. (Thus the decimal representation is not quite unique, and this is another reason to keep on using rational formats (when possible).)

When these group theory research mathematicians design a training course for aspiring teachers of mathematics, they tend to put most emphasis on group theory, and forget about the various number formats. This has the consequences:

- Teachers from their training become deficient in knowledge about number formats (e.g. Timothy Gowers’s article), even though those are more relevant to teachers because these are relevant for their students.
- There is also conditioning for a future lack of knowledge. The aspiring teachers are trained on abstraction and they will tend to grow blind on the problems that students have when dealing with the various formats.
- All this supports the delusion:

“We should teach group theory so that the students will have less problems with the algebra w.r.t. the various number formats. (For, they can neglect much algebra, like we do, since most forms are all in the same equivalence classes.)” (No quote)

##### Bas Edixhoven chairs the delusion

Bas Edixhoven (Leiden) is chair of the executive board of Mastermath, a joint Dutch universities effort for the academic education of mathematicians. They also do remedial teaching for students who want to enroll into the regular training for teacher of mathematics but who have deficiencies in terms of mathematics. Think about a biologist who wants to become a teacher of mathematics. For those students the background in empirical science is important, because didactics is an empirical science too. Such students are an asset to education, and they should not be scared away by treating them as if they want to become research mathematicians. Obviously there are high standards of mathematical competence, but this standard is not the same as for doing research in mathematics.

- The “Foundations” syllabus for remedial teaching 2015 written by Edixhoven indeed
*looks at group theory with the neglect of number formats.*The term “fraction” (Dutch “breuk”) is used without definition, while there is also the expression “fraction form” (Dutch “breukvorm”). I get the impression that Edixhoven uses*fraction*and*fraction format*as identical. Perhaps he means the*procept*? The fractions are not the rationals since apparently π / 2 has a fractional form too. - At a KNAW conference in 2014 on the education of arithmetic Edixhoven presented standard group theory, presumably thinking that his audience had never heard about it and hadn’t already decided that its role for non-university education is limited. Edixhoven insulted his audience (including me) by not first studying what didacticians like Skemp had already said before about group theory in education.

I find it quite bizarre that mathematics courses at university for training aspiring teachers would neglect the number formats and treat these (remedial) student-teachers as if they want to become research mathematicians. Obviously I cannot really judge on this since I am no research mathematician so that I don’t know what it takes to become one. I only know that I have a serious dislike of it. Yet, the group theory taught is out of focus for what would be helpful for mathematics for teaching mathematics.

PM 1. The Edixhoven 2014 approach at KNAW fits Van Hiele (1973) who also suggests to have a bit of group theory in highschool. Yet, there is the drawback of confusion about the power -1 that students might read as subtraction. I would agree on this idea of having some group theory, but with the use of *H *= -1 and not without it. Let us first introduce the universal constant *H *= -1, thus also in elementary school where pupils should learn about division, and then proceed with some group theory in junior highschool.

PM 2. Edixhoven wrote this “Foundations” syllabus together with Theo van den Bogaard who wrote his thesis with Edixhoven. Van den Bogaard has only a few years of experience as teacher of mathematics. Van den Bogaard was secretary of a commission cTWO that redesigned mathematics education in Holland, with a curious idea about “mathematical think activities” (MTA). Van den Bogaard has an official position as trainer of teachers of mathematics but failed to see the error by the psychometrians in the KNAW 2009 study on education on arithmetic. I informed him about my comments on cTWO, MTA and KNAW 2009 but he didn’t respond. Now there is the additional issue of this curious “Foundations” syllabus. Four counts down on didactics and still training aspiring teachers.

##### Letter to Mastermath

These and other considerations caused me to write this letter to Mastermath.

The following indicates that research mathematicians can have their own subgroups or individuals who meddle with education. None is qualified for education, and one wonders whether they can keep each other in check.

##### Research mathematicians are at a distance from didactics

Research mathematicians may develop a passion for education and interfere in education, and then start to invent their own interpretations, and then teach those to elementary schools and their aspiring teachers. These mathematicians are not qualified for primary education and apparently think that elementary school allows loose standards (since they can observe errors indeed). Then we get the blind (research mathematicians) helping the deaf (elementary school teachers), but the blind can also be arrogant, and lead the two of them into the abyss.

A September 2015 protest concerned Jan van de Craats, now emeritus at UvA. For the topic of division, his name pops up again. In this lecture on fractions for a workshop of 2010 for primary education Van de Craats for example argues as follows (my translation). It is unfair to have criticism on this since *these are only sheets*. Yet, even sheets should have a consistent set of definitions behind them. These sheets contribute to confusion. Remember that I didn’t give a definition of “fraction”, and that I propose an abolition of what many people apparently call “fraction”.

- Sheet 3:
*“Three sorts of numbers: integers, decimals, fractions”.*

(a) The main problem is the word “sort”. If he merely means “form” (with the decimals as the standard form that gives “the” number) then this is okay, but if he means that there are really differences (as in group theory) then this is problematic. A professor of mathematics should try to be accurate, and I don’t see why Van de Craats regards “sorts of” as accurate.

(b) If he identifies fractions with the rationals (but see sheet 26) then we might agree that*Z*⊂*Q*⊂*R,*though there are group theorists who argue that these are different number systems, and it is not clear whether Van de Craats would ask the group theorists not to meddle in education as he himself is doing.

(c) My answer: for education it seems best to stick to “various forms, one number (for standard form)”. - Sheet 30:
*“A fraction is the outcome of a division.”*

(a) As fraction is a number (Sheet 3), presumable 8 : 4 → 4 / 2 might be acceptable: (i) It is an outcome, (ii) the answer is numerically correct (as it belongs to the equivalence class), (iii) there is no requirement on a standard form (here).

(b)This doesn’t imply the converse, that the outcome of a division is always a fraction. Then it is either an integer (but then also a fraction (Sheet 25)) or decimal (but then also fraction (Sheet 26)). Thus fraction iff outcome from division.

(c) PM. My definition was: “Ratio is the input of division. Number is the result of division, if it succeeds.” (COTP p51), which doesn’t define number but distinguishes input and output. - Sheet 8:
*“Cito doesn’t test (mixed) fractions anymore in the primary school final examination.”*As an observation this might be correct, but if Van de Craats had had proper background in didactics, then he should have been able to spot the error by the psychometricians in the KNAW 2009 report, which should have been sufficient to effect change, instead of setting up this “course in fractions” (that he isn’t qualified for). - Sheet 18:
*Pizza model.*Didactics shows that students find this difficult. Use a rectangle. - Sheet 25:
*“Integers are also fractions (with denominator 1).”*On form, students must know the difference between integers and fractions (whatever those might be, see Sheet 30). The answer of (3 – 1) / (2 – 1) = ? better be 2 and not 2 / 1 because the latter can be simplified. - Sheet 26:
*“Decimals are also fractions.”*Thus*fractions are not the rational numbers.*The example is that √2 is irrational,*also in decimal expansion*(a “fraction”). Van de Craats apparently holds fractions and the decimals as identical, only written in different form. Thus also an infinite sum of fractions still is a fraction. A fraction is not just the form of the quotient as defined in*Conquest of the Plane*and above (though perhaps it can be written like this ?). - Sheet 27:
*“However, not all fractions are also decimals.”*This is a mystery. There are only three “sorts of” numbers, and w.r.t. Sheet 30 we found that fraction iff division, and all numbers should be divisible by 1. Also, the real numbers contain all numbers we have seen till now (not the complex numbers). Thus there would be phenomena called “fractions” (but still numbers, not algebra) not in the reals ? It cannot be 0 / 0 since the latter would be a result that cannot be accepted. Division 0 : 0 might be a proper question with the answer that the result is undefined. Perhaps he means to say that “1 / 2” doesn’t have the form of “0.5”, and that the expressions differ ? But then we are speaking about form again, and Van de Craats spoke about “sorts of numbers” and not about “same numbers with different forms”. - Sheet 28:
*“This course doesn’t offer an one-to-one-model for discussion at school.”*It sounds modest but I don’t know what this means. Perhaps he means that the sheets aren’t a textbook. - Sheet 30:
*“A fraction is the outcome of a division.”*(I moved this up.) - Sheet 33:
*“4 : 7 = 4 / 7”.*Apparently the ” : ” stands for the operation of division and “4 / 7” for the result. Apparently Van de Craats wants to get rid of the procept. The equality sign cannot mean identically the same, because otherwise there would be no difference between input and output. Is only 4 / 7 the right answer or is 8 / 14 allowed too ? Perhaps one can teach students that 4 : 7 is a proper question and that 8 / 14 is unacceptable since this must be 4 / 7. However, 4 : 1 would be a proper question too, and then Van de Craats also argues that 4 / 1 would be a fraction (and result of division). - Sheet 65:
*“Actually 2 4/5 means 2 + 4/5.”*(Van de Craats read an article of mine.) It would have been better if he stated that the first is a horrible convention, and that he proceeded with the second. He calls the form a “mixed fraction” while the English has “mixed number“. Lawyers might have to decide whether “fractions are numbers” implies that a “mixed fraction” is also a “mixed number”.

If a professor of mathematics becomes confused on such an “elementary (school)” issue of fractions (I still don’t know that is meant by this), why would the student believe that anyone can master this apparently superhumanly difficult subject ?

##### Will the ivory tower stop the blind ?

Would research mathematicians who do group theory be able to correct Van de Craats ?

Let us consider Bas Edixhoven again, see again his sheets.

Or would Edixhoven argue that he himself looks at natural numbers, integers, rationals and reals, so that he has no view on “fractions”, as apparently defined by Van de Craats ? Though the “Foundations” syllabus refers to the word without definition and Edixhoven might presume that aspiring teachers of mathematics know what those fractions are.

Edixhoven in the 2014 lecture only suggests that there better be more proofs and axiomatics in the highschool programme, and he gives the example of a bit of group theory for arithmetic. He also explains modestly that he speaks “from his own ivory tower” (quote). Thus we can only infer that Edixhoven will remain in this ivory tower and will not stop the blind (but also arrogant) Van de Craats from leading (or at least trying to lead) the deaf (elementary school teachers) into the abyss.

However, professor Edixhoven also left the ivory tower and and joined the real world. At Mastermath he is involved in training aspiring teachers. Since February 2015 he is member of the Scientific Advisory Board of the mathematics department of the University of Amsterdam, where professor Van de Craats still has his homepage with this confusing “course on fractions”. I informed this board in Autumn 2015 about the problematic situation that Van de Craats propounds on primary and secondary education but is not qualified for this. I have seen no correction yet. Apparently Edixhoven doesn’t care or is too busy scaring aspiring teachers away. Apparently, when a teacher of mathematics criticises him, then this teacher obviously must be deficient in mathematics, and should follow a course for due indoctrination in the neglect of didactics of mathematics.