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[ This is the same text as the former weblog (here), but now we follow Van Hiele’s argument for the abolition of fractions. The key property is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there. Thus we replace (y / x) with y xH with H = -1. ]

Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a bH “and” c dH kids are supposed to find (a d + b c) (b d)H instead of (a + c) (b + d)H.

Obviously this is a matter of definition. For “plus” we define: a bH + c dH = (a d + b c) (b d)H.

But we can also define “superplus”: a bHc dH = (a + c) (b + d)H.

The crux lies in “and” that might not always be “plus”.

When (a + c) (b + d)H makes sense

There are cases where (a + c) (b + d)H makes eminent sense. For example, when a bH is the batting average in the Fall-Winter season and c dH the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) (b + d)H. Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) (b + d)H makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

[ Some readers may not be at home in statistics. Let the weight of b be w = b (b + d)H. Then the weight of d is 1 – w. The weighted average is (a bH) w + (c dH) (1 – w) = (a + c) (b + d)H. ]

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses animal size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 female pets and 16 male pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 7H = 0.86, 2 10H = 0.20 and (6 + 2) (7 + 10)H = 0.47.
  • For the dogs: 8 10H = 0.80, 1 6H = 0.17 and (8 + 1) (10 + 6)H = 0.56.

A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.

It is straightforward to observe that the F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 ∼ 9, and (9 + 2) (10 + 10) H = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) (10 + 10) H = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.

Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).

Conclusions

Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-03-03-data-from-kleinbaum-2003-adapted

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Robert Siegler participates in the “Center for Improved Learning of Fractions” (CILF) and was chair of the IES 2010 research group “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” (report) (video).

IES 2010 key advice number 3 is:

“Help students understand why procedures for computations with fractions make sense.”

The first example of this helping to understand is:

“A common mistake students make when faced with fractions that have unlike denominators is to add both numerators and denominators. [ref 88] Certain representa­tions can provide visual cues to help students see the need for common denominators.” (Siegler et al. (2010:32), refering to Cramer, K., & Wyberg, T. (2009))

For a / b “and” c / d kids are supposed to find (ad + bc) / (bd) instead of (a + c) / (b + d).

Obviously this is a matter of definition. For “plus” we define: a / b + c / d = (ad + bc) / (bd).

But we can also define “superplus”: a / c / d =  (a + c) / (b + d).

The crux lies in “and” that might not always be “plus”.

When (a + c) / (b + d) makes sense

There are cases where (a + c) / (b + d) makes eminent sense. For example, when a / b is the batting average in the Fall-Winter season and c / d the batting average in the Spring-Summer season, then the annual (weighted) batting average is exactly (a + c) / (b + d). Kids would calculate correctly, and Siegler et al. (2010) are suggesting that the kids would make a wrong calculation ?

The “superplus” outcome is called the “mediant“. See a Wolfram Demonstrations project case with batting scores.

Adding up fractions of the same pizza thus differs from averaging over more pizzas.

We thus observe:

  • Kids live in a world in which (a + c) / (b + d) makes eminent sense.
  • Telling them that this is “a mistaken calculation” is actually quite confusing for them.
  • Thus it is better teaching practice to explain to them when it makes sense.

There is no alternative but to explain Simpson’s paradox also in elementary school. See the discussion about the paradox in the former weblog entry. The issue for today is how to translate this to elementary school.

Cats and Dogs

Many examples of Simpson’s paradox have larger numbers, but the Kleinbaum et al. (2003:277) “ActivEpi” example has small numbers (see also here). I add one more to make the case less symmetrical. Kady Schneiter rightly remarked that an example with cats and dogs will be more appealing to students. She uses size (small or large pets) as a factor, but let me stick to the idea of gender as a confounder. Thus the kids in class can be presented with the following case.

  • There are 17 cats and 16 dogs.
  • There are 17 pets kept in the house and 16 kept outside.
  • There are 17 female pets and 16 male pets (perhaps “helped”).

There is the phenomenon – though kids might be oblivious why this might be “paradoxical”:

  1. For the female pets, the proportion of cats in the house is larger than the proportion for dogs.
  2. For the male pets, the proportion of cats in the house is larger than the proportion for dogs.
  3. For all pets combined, the proportion of cats in the house is smaller than the proportion for dogs.
The paradoxical data

The paradoxical data are given as follows. Observe that kids must calculate:

  • For the cats: 6 / 7 = 0.86, 2 / 10 = 0.20 and (6 + 2) / (7 + 10) = 0.47.
  • For the dogs: 8 / 10 = 0.80, 1 / 6 = 0.17 and (8 + 1) / (10 + 6) = 0.56.

A discussion about what this means

Perhaps the major didactic challenge is to explain to kids that the outcome must be seen as “paradoxical”. When kids might not have developed “quantitative intuitions” then those might not be challenged. It might be wise to keep it that way. When data are seen as statistics only, then there might be less scope for false interpretations.

Obviously, though, one would discuss the various views that kids generate, so that they are actively engaged in trying to understand the situation.

The next step is to call attention to the sum totals that haven’t been shown above.

It is straightforward to observe that the F and M are distributed in unbalanced manner.

The correction

It can be an argument that there should be equal numbers of F and M. This causes the following calculations about what pets would be kept at the house. We keep the observed proportions intact and raise the numbers proportionally.

  • For the cats: 0.86 * 10 ∼ 9, and (9 + 2) / (10 + 10) = 0.55.
  • For the dogs: 0.17 * 10 ∼ 2, and (8 + 2) / (10 + 10) = 0.50.

And now we find: Also for all pets combined, the proportion of cats in the house is larger than the proportion for dogs. Adding up the subtables into the grand total doesn’t generate a different conclusion on the proportions.

Closure on causality

Perhaps kids at elementary school should not bothered with discussions on causality, certainly not on a flimsy case as this. But perhaps some kids require closure on this, or perhaps the teacher does. In that case the story might be that the kind of pet is the cause, and that the location where the pet is kept is the effect. When people have a cat then they tend to keep it at home. When people have a dog then are a bit more inclined to keep it outside. The location has no effect on gender. The gender of the pet doesn’t change by keeping it inside or outside of the house.

Vectors in elementary school

Pierre van Hiele (1909-2010) explained for most of his professional life that kids at elementary school can understand vectors. Thus, they should be able to enjoy this vector graphic by Alexander Bogomolny.

Van Hiele also proposed to abolish fractions as we know them, by replacing y / x by y x^(-1). The latter might be confusing because kids might think that they have to subtract something. But the mathematical constant H = -1 makes perfect sense, namely, check the unit circle and the complex number i. Thus we get y / x = y xH. The latter would be the better format. See A child wants nice and no mean numbers(2015).

Conclusions

Some conclusions are:

  • What Siegler & IES 2010 call a “common mistake” is the proper approach in serious statistics.
  • Teaching can improve by explaining to kids what method applies when. Adding fractions of the same pizza is different from calculating a statistical average. (PM. Don’t use round pizza’s. This makes for less insightful parts.)
  • Kids live in a world in which statistics are relevant too.
  • Simpson’s paradox can be adapted such that it may be tested whether it can be discussed in elementary school too.
  • The discussion corroborates Van Hiele’s arguments for vectors in elementary school and the abolition of fractions as we know them (y / x) and the use of y xH with H = -1. The key thing to learn is that there are numbers xH such that x xH = 1 when x ≠ 0, and the rest follows from there.

PM. The excel sheet for this case is: 2017-01-30-data-from-kleinbaum-2003-adapted.

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 (*):

  1. that N ⊂ Z ⊂ Q R,
  2. that the elements in R are called numbers,
  3. 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).

Conquest of the Plane, p78

Conquest of the Plane, p78

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 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 2a 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).

  1. If a rational number is equal to an integer, it is written as this integer, and otherwise:
  2. The rational number is written as an integer plus or minus a quotient of natural numbers.
  3. 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).
  4. The quotient part has a denominator that isn’t 0 or 1.
  5. The quotient part is not written when the numerator is 0 (and then the whole is an integer).
  6. The quotient part consists of a quotient (form) with an (absolute) value smaller than 1.
  7. The quotient part is simplified by elimination of common primes.
  8. When the integer part is 0 then plus is not written and minus is transformed into the negative sign written before the quotient part.
  9. 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 + 7a and no 74 + 7a.” (Skemp, p230)

H = -1

A new suggestion is to use = -1. Then we get 2 + ½ = 2 + 2H= 5 2H. Pierre van Hiele (1973) suggested to abolish fractions as we know them. He observed that 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 xH) and the dynamic quotient (y xD), in which the brackets may be required in the dynamic case to indicate the scope of the simplification process.

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 3H = (3 + 1) 3H = 1 + 3H translates as repeated subtraction (not for the classroom but for reasons of current exposition):

4 – (1 + 3H) – (1 + 3H) – (1 + 3H) = 4 – 3 (1 + 3H) = 4 – 3 – 3 3H = 4 – 3 – 1 = 0

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.

Jan van de Craats, Workshop 2010, page 28

Jan van de Craats, Workshop 2010, page 28

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

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)

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.

We arrive at a paradox.

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 but no contradiction

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.

2016-10-26-petervanwijk-smaller-englishDiscussion 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):

2016-10-26-petervanwijk-smaller-english-altA 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.)

 

 

 

 

 

 

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
CIDREE 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

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)

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

Event agenda for November 23

Listening to Roy Orbison – Pretty Women

Let me give a clear and unbiased assessment of the qualifications and work of “dr.” Michiel Doorman at “Freudenthal Institute” at Utrecht University.

Let me first present four neutral points and then follow Sherlock Holmes.

(1) Michiel Doorman defended his “thesis” in 2005: “Modelling motion: from trace graphs to instantaneous change” (online), written under the supervision of P.L. Lijnse and Koeno Gravemeijer.

(2) He might best be introduced by his cv on p243 of his “thesis”:

“Michiel Doorman was born on 1 october 1962 in Eindhoven, the Netherlands. He completed his secondary education in 1981 at the Minkema College in Woerden. In1988 the Utrecht University awarded him a masters in Mathematics for his thesis on the extension of a proposition in intuitionistic logic for automated theorem proving. He minored in Computer Science. From 1988 he has been working at the Freudenthal Institute. Until 1992 he was mainly devoted to software development. During the following years he has been involved in curriculum and teacher training projects, mainly concerning the role of information and communication technology in mathematics education. Since 1994, this work concentrated on upper secondary (pre-university) mathematics education in a research project on the integration of the graphing calculator, in a curriculum development project (Profi), and in a project that aimed at guiding the Biology, Chemistry, Physics and Mathematics departments in schools to cooperate. In 1998 he started his PhD research study.” [my emphasis]

(3) There is also his Utrecht University webpage, that states:

“Interests are context-based mathematics education, modeling as a lever for learning mathematics, inquiry based learning and coherency between mathematics and science learning.”

(4) Michiel Doorman was also invited as one of the keynote speakers at the 3rd International Conference on Research, Implementation and Education of Mathematics and Science (ICRIEMS) at Yogyakarta, Indonesia, in May 2016.

  • This is Doorman’s presentation pptx (on “inquiry based learning”) and
  • this is the proceeding pdf: “What Can Mathematics Education Contribute To Preparing Students For Our Future Society?“.
Michiel Doorman at ICRIEMS 2016 (fourth from left) (Sources: ICRIEMS website)

Michiel Doorman fourth from left (Source: ICRIEMS website 2016)

Why does Doorman in 2016 claim success for RME while it failed ?

Around 2005 there was much discussion in Holland – a real math war – about arithmetic in elementary schools. The academy of sciences (KNAW) set up a committee to look into this.

Recall the graphical display of the math war between RME and TME, and the solution approach of NME. These abbreviations are:

  • RME = realistic mathematics education
  • TME = traditional mathematics education
  • NME = neoclassical mathematics education

(i) This KNAW-committee concluded in 2009 (see the English summary on page 10 in the report) that pupil test scores for RME and TME did not really differ. Paraphrased: one cannot claim a special result for RME. Observe that many test questions contained contexts.

“Growing concern about Dutch children’s mathematical proficiency has led in recent years to a public debate about the way mathematics is taught in the Netherlands. There are two opposing camps: those who advocate teaching mathematics in the “traditional” manner, and those who support “realistic” mathematics education. The debate has had a polarizing effect and appears to have little basis in scholarly research.” [This neglects my third position with NME.]

“The public debate exaggerates the differences between the traditional and realistic approaches to mathematics teaching. It also focuses erroneously on a supposed difference in the effect of the two instructional approaches whereas in fact, no convincing difference has been shown to exist.” (KNAW 2009)

(ii) Doorman in Yogyakarta 2016 is unrepentingly for RME. He refers to key authors on RME, and takes a question of TIMMS 2003 with an international score of 38% and a Dutch score of 74% and claims, misleadingly:

“This cannot fully [be] attributed to the implementation of  RME, but it strengths [sic] the feeling that this approach contributes to the quality of mathematics education.” [my emphasis]

(iii) Subsequently, I criticised the KNAW report on these counts, and neither KNAW nor Michiel Doorman have responded to this criticism:

  • Before the report was published by alerting the committee chairman to Elegance with Substance (2009, 2015), that however is not included in the references.
  • In 2014 explicitly for the collective breach of research integrity, for either neglecting or maltreating my books Elegance with Substance (2009, 2015) and Conquest of the Plane (2011) and Dutch Een kind wil aardige en geen gemene getallen (2012) notably on issues of arithmetic (the present subject): the pronunciation of numbers and notation of mixed numbers.
  • In 2015 for neglecting the issue that TME prepares for algebra while RME doesn’t. The KNAW report uses the outcome of test questions and not the intermediate steps. Pupils who can only use RME will be very handicapped for algebra in secondary education.

See my 2016 letter and its supplement to the president of KNAW and director of CPB about the failure of the KNAW report and the neglect of criticism.

A repeat exercise that isn’t quite superfluous

I have explained, to boring repetition, that the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) should not be at a university. Please observe that first there was criticism on the failure of “realistic mathematics education” (RME) and only later it was discovered that Hans Freudenthal had actually abused the work by Pierre van Hiele. There also is a sound scientific explanation why it is a failure: namely a confusion of processes of learning with applied mathematics.

Thus it holds:

There is little advantage in repeating this analysis,
neither for each and every individual working at FHCRMI.

For example, stating that Michiel Doorman works at FHCRMI should be sufficient. That he is at FHCRMI does not imply that he can indeed be at university and that his “thesis” and “PhD title” are proper.

However, the following points cause that it isn’t quite superfluous to look into Doorman’s qualifications and work.

  • Michiel Doorman is member of the board of NVvW, the Dutch association of mathematics teachers. See my recent letter with a Red Card for this board. Thus it helps for the next annual meeting of NVvW in November 2016 to be specific.
  • Also, there is my letter of April 15 2016 to NRO, the Dutch organisation for the distribution of funds for research in mathematics education. I advise them to stop subsidising FHCRMI. It so happens that Michiel Doorman did a project ODB08008 for them in 2009-2012 on the “digital mathematics environment” and “efficient exercising mathematics” (DWO). It will be helpful for NRO to see that, for example, Doorman is an ideologue and no scientific researcher. This is related to the following.
  • There is the new impulse for “21st century skills” or in Holland “Onderwijs2032“. Part of the attention is for soft skills, part of the attention is for computer programming, part is elsewhere. ICT brings us to the work of Doorman too. There has already been a major disaster with the neglect of computer algebra since 1990. For example DWO at FHCRMI tends to present many Java applets that lack the flexibility of computer algebra. Don’t think that these issues are easy to resolve, but I do hold that the decisions have been driven by ideology and that the results are a disaster and a great waste of funds: penny wise pound foolish. See for example these two reports by the Inspectorate for Education: In 2002, mathematical topologist Hans Freudenthal is called a “pedagogue” while he had no education or training on this, and they assume that FHCRMI knows about ICT while the report doesn’t mention computer algebra but applets on “wisweb”. In 2006, the “waarderingskader” (inspection standards) doesn’t seem to realise that computer algebra can used in all subjects that use mathematics.

Above, I mentioned four neutral points. Following Sherlock “Google” Holmes I already debunked the event in Yogyakarta. Let us look at the other three points. Beware of confusion.

Ad 1. Doorman’s “thesis” of 2005

Michiel Doorman defended his “thesis” in 2005: “Modelling motion: from trace graphs to instantaneous change” (online), written under the supervision of P.L. Lijnse and Koeno Gravemeijer.

  1. In the “thesis”, Doorman basically refers to Paul Drijvers at FHCRMI for computer algebra, but Drijvers is no light on this either. For the subject of the thesis (“Furthermore, it has been examined what role computer tools could play in learning mathematics and physics.”) it would have made much sense to look deeper into computer algebra.
  2. Also check my analysis that Koeno Gravemeijer is no scientist but an ideologue for “realistic mathematics education” (RME) who misrepresents issues on “21st century skills” (in Holland “Onderwijs2032“), and who doesn’t see the revolution by computer algebra. (Dutch readers can look here too.)
  3. On p58-59 Doorman critically adopts RME, and remember that this was in 2005, while RME was under discussion, see (a) the discussion in 2006 between Robbert Dijkgraaf (who has no qualification for math ed at this level) en Kees Hoogland (a RME ideologue, see below), which report was written by RME supporter Martinus van Hoorn, and (b) while Jan van de Craats (who has no qualification for math ed at this level) was protesting about RME, and published “Waarom Daan en Sanne niet kunnen rekenen” in 2007. See my criticism w.r.t. Jan van de Craats (fighting his math war on the side of TME and neglecting NME since 2008).
  4. Doorman refers to Freudenthal for “guided reinvention”, but this is a wrong reference (he may only have coined the phrase but not the concept), and Doorman’s thesis does not refer to the true inventor of the concept (guide through levels of insight) Pierre van Hiele at all.
  5. I will not look at this “thesis” in detail because there is really no reason to so so now.
Ad 2. Curriculum vitae

Doorman’s cv shows that he got a mathematics degree and continued at Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI), thus without a teaching degree in mathematics and without proper training in research of mathematics education.

  • The KNAW report of 2009 showed that FHCRMI doesn’t do research on arithmetic education, and one should not suppose that this is different for other areas.
  • Thus Michiel Doorman is neither a teacher of mathematics nor a researcher in the education of mathematics.
  • We find no qualification for teaching and research, but immersion into ideology, and while he is involved in programming and the role of ICT for (mathematics) education, there are only perfunctory statements on computer algebra (for all subjects that use mathematics).
Ad 3. Webpage

Observe that “context-based mathematics education” is a rephrasing of “realistic mathematics education” (RME). Also “inquiry based learning” is basically RME, with contexts as the starting point for the “inquiry” (constrained by learning goals).

Someone really interested in didactics and RME would also have been interested in my analysis that shows that RME is a confusion and an ideology.

Observe also that the sciences are easy victims of RME. The sciences do not care much about mathematics education, and when RME people flock in to assist in the learning of the sciences, and when student learning time for mathematics is actually spent on the sciences, then professors of physics or biology might hardly object. For RME the sciences are useful window dressing, since those provide both contexts and an aura of respect and acceptance, and an argument that “students are learning something” (even if it isn’t mathematics). There is a curious historical link-up of mathematics with the beta sciences while there are also the humanities (alpha) and social sciences (gamma), see here. But is it really curious, and isn’t there a method, that the human and social scientists who know the techniques and who also do research by observation are kept out from this association between “mathematics education” and “science education” ?

Possible confusions that are triggered immediately

Stating the above might immediately trigger some confusions.

  • As member of the NVvW board Doorman might argue “not to look into the criticism on FHCRMI because of an interest there”. Instead, he should rather take the initiative and make sure that this criticism was answered in decent manner rather than burked. If his interest is so large that he cannot speak freely on science then he should rather not be in the board.
  • Doorman in a 2015 text in Euclides, the journal of NVvW, referred to (intellectually stealing) Freudenthal and not to (victim) Van Hiele. When asked to correct, he didn’t reply to this very question (see page 8) but talked around it, see my deconstruction of his “reply”. Potentially Doorman just didn’t have the relevant knowledge about didactics, for histhesis” refers to RME but not to Van Hiele. If Doorman had looked into this criticism of mine, he could have been a bridge of understanding for the other members of the NVvW board and readers of Euclides, but he wasn’t.
    PM 1. I don’t understand either why these people didn’t see that he dodged the question.
    PM 2. Doorman in Indonesia sheets 44-49 on RME repeats the reference on anti-didactic inversion to (intellectually abusing) Freudenthal at the cost of (victim) Van Hiele.  Thus I asked a correction, he dodged the question, and proceeds as if there would be no problem (and likely not informing the audience about the criticism).
  • Thus Doorman is neither teacher nor researcher: so what is he doing in the board of NVvW ? From qualifications, actually their lack, and work, actually the lack of answers, it is a small step towards wondering about motivation. A good hypothesis is: he is spreading the gospel of RME and blocking criticism. If Doorman wants to become professor at FHCRMI he must show that he is a true sectarian of RME. He is polishing up his cv and can now claim that he has been involved in the community of teachers, even when it was dodging questions. This is a hypothesis only, and one might also offer other explanations like blindness.
  • I wonder who paid for this trip to Yogyakarta. It is also known that Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI) still is busy in establishing footholds over the world even though RME has failed. See also FIUS.org, who apparently neglect the KNAW report or my criticism.
  • Yes, there is also NRO-project supervisor Frits Beukers, but he is professor of mathematics also without qualification for mathematics education research. Beukers presently is chairman of the Platform Wiskunde Nederland (PWN)-education committee, but would represent the universities since he has no qualification for primary or secondary education or the trade colleges. In that committee we also see Kees Hoogland, who abused the biography by John Allen Paulos for RME, and who has not corrected yet and who refuses to give an English translation of the key section.
Disclaimer: Can I be unbiased ?

I stated that I would give an unbiased assessment. Can I really do so ? Undoubtedly the reader will make up one’s own mind, but my perception is that I have been fair and unbiased.

Doorman’s “thesis” of 2005 is closely related to the education on the derivative. There was ample scope for a meeting of minds. When Doorman sticks with RME and Java applets, instead of NME and computer algebra, it is all of his own choosing. The differences in positions can be mentioned:

  • Check my proposal since 2007 for an algebraic approach to the derivative, see e.g. Conquest of the Plane (2011).
  • COTP was also programmed in Mathematica – a system for doing mathematics on the computer (a.k.a. “digital environment”) .
  • Also, I am a user of computer algebra since 1993, while Doorman tends to use other programs that don’t have the flexibility of computer algebra.
  • Doorman was editor of TD-Beta when I submitted a short note in 2012 on my invention of the algebraic approach to the derivative. This was maltreated. See here anonymised  and see here with full names (it is a scientific discourse and no private exchange).

At the NVvW annual convention of 2015 when Doorman was elected to the board, I had my doubts but opted still mildly optimistically for the benefit of the doubt. I had no experience with this TD-Beta journal and perhaps everything was an unfortunate misunderstanding. It doesn’t happen so very often that someone can propose a new approach to the education on the derivative. The performance of last eight months however gives evidence of the mindset of an ideologue.

Conclusion: Doors of perception

Any link to Doorman’s name is coincidental, and it is also a coincidence that the Dutch family name “Doorman” indeed is related to the English “doorman” (at least according to the Meertens institute).

The phrase “doors of perception” comes to mind when looking at Doorman’s presentation sheet “Aim of Primas” that states:

“A question not asked is a door not opened!”

  1. This implies that an opened door is a result of a question. (This need not be an open door.)
  2. This doesn’t imply that asking a question will open a door. (The question might e.g. be ignored.)

The message of this present weblog is, amongst others, that there are some crucial questions that Michiel Doorman refuses to look into and apparently doesn’t want to answer. He is employed at the Freudenthal Head in the Clouds Realistic Mathematics Institute, that should not be at a university. Apparently Doorman did not inform his hosts in Indonesia about the existing critique either.

The last discussion on teaching the quadratic function was fun, but it left me with some questions, and these clarify that there is a long road ahead.

The recipe for re-engineering didactics of mathematics is:

When you see a student struggling with a topic, ask two (introspective, sympathetic) questions:
– What causes that a (and/or this particular) student experiences this struggle ?
– What causes the mathematics community to create this didactic situation ?
Subsequently put your hunches to the test, and recycle till you have a satisfactory answer.
Then proceed with experimental design and hypothesis testing.

Conventional studies in didactics consider only the first question. They assume that the math is sound and that the struggle is caused by external factors like the use of the calculator or perhaps the order of topics in the curriculum. I agree that much can be improved on these external factors, but my diagnosis is that we better first solve the internal problems within “mathematics” itself. A student who struggles with a math question is often quite capable in dealing with complex issues (check what kids can learn) and a struggle indicates that the cards are rigged. The large number of examples of crummy “mathematics” that has already been identified in my book Elegance with Substance (pdf online) shows that “mathematics” education and its conventional research have lost their innocence and even the benefit of the doubt, and are actually the prime suspects.

Didactic divide from counting to addition

I am not qualified for elementary school and the following is only a hypothetical idea, to provide us with a useful example for below.

In elementary school some kids don’t quickly master addition, and get stuck in counting only. For sums like 7 + 2 = 9 they rely on their fingers, which eventually becomes awkward for a sum like 2 + 7 = 9. The didactic question is rather not why these kids get stuck but why others take the leap across this divide from counting to addition. Apparently, a majority of kids have a way of mastering both commutation that 2 + 7 = 7 + 2 and also the tables of addition, so that the outcomes are automated, and are available directly from permanent memory without the need to find the result by actually counting (and check and double check). Didactics helps to identify the points for a natural learning process: (1) awareness of the didactic divide, (2) diagnosis of what is the cause for slow progress (not math itself), (3) treatment, namely create the tables of addition, create awareness of these and the properties, and practice makes perfect. In this manner the teaching environment can help the slow kids to find the learning route they were unfortunate not to find themselves – and later they may be fast themselves again on another subject. (See Henk Boonstra on elementary school reform w.r.t. structural slow and fast.) Learning goals are, even at this age: (a) knowing how, (b) knowing why, (c) knowing about knowing. A child who has mastered addition has every right to be proud of it, and can be invited to explain what it has learned.

PM 1. For numbers above 10 see A child wants nice and not mean numbers.
PM 2. This also clarifies why the tables of multiplication must be known by heart, to cross the divide between addition and multiplication.

Quadratic functions revisited, conventionally

The last discussion on teaching the quadratic function caused a focused search on what others wrote about its didactics. We find a similar issue of exploration versus automation (counting versus addition). Perhaps we may speak about a didactic divide between quadratic equation solving (factoring or completing the square) and automation (quadratic formula). The situation is a bit more complex than with counting and addition (perhaps also because it happens in highschool and not in elementary school). The quadratic formula can be automated only for the standard polynomial form. Handling the quadratic function requires a certain level of algebra overall.

Let us first follow conventional didactics that assumes that the “mathematics” is sound. In conventional didactics, “completing the square” is a solution technique, and not the basic form. The quadratic formula is called an algorithm, a fancy word for recipe. However, the other solution methods follow recipes too, and the key notion is automation.

Colin Foster (2014), ‘Can’t you just tell us the rule?’Teaching procedures relational, distinguishes procedural knowledge (knowing how) from relational knowledge (knowing why), and finds two uses for the quadratic formula: (1)  to learn more about how and why, perhaps prove it, (2) to use it automatically with a focus on a different target. In itself this is sound, but this doesn’t generate a specific didactic strategy yet, except perhaps the order first (1) then (2). Foster laments the current “political” focus on teaching to the test, and thus (perhaps) emphasis on the quadratic formula. Foster usefully indicates that tradition has been strong and research has been weak:

“Vaiyavutjamai and Clements (2006) comment on the lack of research into students’ difficulties with quadratic equations, and since then a number of studies have explored this area (…).”

David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), Evolving a three-world framework for solving algebraic equations in the light of what a student has met before, suffer from Tall’s idiosyncratic verbosity on “embodied” versus “crystaline” notions (see related critique), but it is helpful to see some of the literature and experiments and student confusions. De Lima (2011), summarizes a study in Portuguese:

“In this paper, we present an analysis of 77 14-15 year-old students’ work with a non-familiar situation: the solution of a quadratic equation, written in a factorized form. Data is analysed in the light of a theoretical framework that considers three different worlds of Mathematics and the influence of ‘met-befores’ derived from learning experiences related to them. We show that having the quadratic formula as the only met-before to solve quadratic equations may not help the students to face all kinds of situations involving such equations. In addition, we claim that it is necessary to present to students learning situations that involve at least two worlds, the embodied and the symbolic, but in ways which also allow consideration of characteristics of the formal world, without which students may create their own inappropriate techniques.”

Alwyn Olivier (1989), Handling pupil’s misconceptionsis informative. Students have minds of their own and they may induce themselves to create a recipe like:

(x u)(x – w) = c solves as x u = c or x w = c, which works for c = 0 and hence for any c

One way to approach such frequent misconceptions is to explicitly discuss them, and for example let students prove a theorem that the above is true only for c = 0. Mathematicians create theorems for what they regard as relevant steps. At the level of students, such issues can be considered to be relevant steps that eliminate their confusions. PM. For this particular equation, it is a step further to analyse it more. We can observe that the vertex lies at = (+ w) / 2 and that solutions will be x1,2h ± d for some d. Substitution of generates = (c + (– w)2 / 4). Perhaps this must be mentioned too, perhaps this distracts, and the traditional approach has chosen the latter.

The traditional approach causes the following didactic issue.

  • When a student knows only the quadratic formula, and doesn’t know about factoring or the basic (vertex) form, then teachers will tend to see this as problematic. An equation (x u)(x – w) = 0 must first be expanded to fit the standard polynomial form, with b = -(u + w) and c = u w, to subsequently apply the quadratic formula, and then generate the solution that already could be seen at the start, namely x = u or x = w. Teachers may still opt for the latter level of competence, since the quadratic formula generates a solution for all cases.
  • Teachers may reason that mastery of the quadratic formula still involves some algebraic competence. The latter may also be a delusion, since not-seeing the direct solution to the factored form isn’t quite competent. Lack of algebraic competence can also result in other errors, like when expanding the factored form. A student may however still earn some points for “knowing the quadratic formula” and “making proper substitutions (even from the wrong derivation)”.
  • When a student is mathematically proficient and uses the quadratic formula, few teachers will object. However, such a student would tend to recognise the basic (vertex) and factored forms, and directly state their solution. It would only be an oversight (inattention) when such a student would still adopt the quadratic formula.
A work of art

Source: David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), p14

Quadratic functions revisited, re-engineered

In the re-engineered approach, teaching starts with the basic form. This is not mistaken as a solution method (“completing the square”). It is only a solution method when one starts from the standard polynomial form, but such is a wrong place to start from.

  • The three forms – basic, factored, standard polynomial – arise in structured order.
  • The relationships between parameters and solutions directly follow from this.
  • The relevant questions concern the turning point (vertex) and zeros. Above discussion concerns the quadratic formula and focuses on the zeros, but there is also the turning point.
  • Each form has its own solution method for turning point and zeros.
  • A 3 by 3 table can show how one can transform one form into the other. It can be emphasized that transformation need not be remembered since the questions in the upcoming test concern vertex and zeros. Special attention would be given to special forms like a x^2 = – b x. The latter can be solved directly for x = 0 or assuming x ≠ 0 as x = – b / a. It can be clarifying to also write it as a (x – 0)(x + b / a) = 0.
  • The quadratic formula is derived directly from the basic form, and is usefully remembered for automated application for the standard polynomial form only.

Essential for the handling of the quadratic function are the recognition of the graph and the three algebraic forms and the associated solution approaches. Essential is the attitude to check the solution and correct errors. Essential are the knowing how, knowing why, and knowing about knowing.

After the above has been clarified and assessed in examination, subsequent lessons can be spent on derived questions. The above has a specific learning goal, but the overall goal of competence in algebra and analytic geometry remains. For example, given two points and a particular parabola, state the formula in this form and / or that form. For example, check reactions to solving forms like x^2 h^2 + v = 0 or a x^4 + b x^2 + c = 0. It is fair that students have a stage during which the names of parameters help support memory, but the next step is to recognise forms and infer the role of parameters.

As said, this re-engineered form must be tested in experiments before it can replace current convention. Didactics remains an empirical science, and the students themselves must show what works for them.

There is one issue of design that needs close scrutiny. Consider students who didn’t get adequate arithmetic in elementary school, who didn’t quite learn the tables of addition and multiplication, and thus fail to directly recognise the factors of say x^2 – 10 x + 24 = 0. I always feel sorry for students who have diligently learned the quadratic formula and who lose time on tedious arithmetic, only because of this background, so that they are severely punished on their weak spot.

  • My null hypothesis is: It is only optional to factor a standard form by using the rules on adding and multiplying zeros. This is a different kind of competence. Focusing on this competence is derivative, inessential, and distracting from the real learning goals on the quadratic function. It is proper to show the method of direct factoring and train on this, since it helps algebraic insight and skill, but in another chapter. It should not be a prime element in the discussion of the quadratic function. Having test questions with functions in polynomial form that are easy to factor by students with that level of algebra, puts a bonus on a competence that is irrelevant here, and puts a malus on students that follow the longer route of the algebraic formula.
  • The alternative hypothesis: It might empirically still be possible that students are served by this repeat training on the tables of addition and multiplication and this factoring, perhaps since they better learn what factors are, and perhaps since this is adequate skill in arithmetic so that the quadratic formula can be put on the graphic calculator. (See also Filloy et al..)

There is a curious key remark by David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), p18-19, on the relevance of the didactic divide. It is correct that the form ‘quadratic expression = number’ doesn’t have a simple recipe, see Olivier, but this does not warrant a conclusion that students need not cross a divide. Also, Lima and Healy (2010) mention the basic (vertex) form of the quadratic function and are close to finding the re-engineered approach. Unfortunately, their frame of reference is that “mathematics” is sound and does not need to be questioned itself.

Tall-Lima-HealyPM 1. Relevant is also Jan Block (2016), Flexible algebraic action on quadratic equations, who advises “tasks which focus not on finding solutions of equations by on classifying different types of equations” (p397). Block uses dashed lines and particular shapes to indicate kinds of relations or operations. This might be useful for teachers. For students, it is indeed a reminder that recognition of shapes can by supported by icons. The basic (vertex) form can be identified with an icon of a vertex (U), the factored form by an eye since the zeros can be seen directly (ʘ), and the polynomial can have this (√) icon to remind of the quadratic formula.

PM 2. Here is a long discussion on how to prove the quadratic formula, which doesn’t quite use the easy one of the re-engineered approach (but of course the methods are the same implicitly).

Conclusion

The quadratic function is an example of the usefulness of the recipe for re-engineering. Do not take “mathematics” for granted but accept it as the prime suspect itself for blocking student learning.

Empirical testing of this issue requires the design of integrated lesson plans, with detailed learning goals and exam questions, and worked-out presentations and materials for practice.

Once the re-engineered method has been developed and field tested and shown successful, then publishers can step in and reap the profits from the hard work of the re-engineers.