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.

Chronologically:

  • In October 2014 I explained that Edward Frenkel abused “love” with his book “Math & Love“. As a mathematician he has no training in the empirical science of mathematics education, and what he states about mathematics education is often delusional. The situation is rather typical of the arrogance that is so common among mathematicians. The attitude is “the best way to teach and learn mathematics is to do mathematics”, but this confuses the context of mathematics research with the context of education. See here for a recent comment on the USA Common Core for mathematics.
  • In November 2015, Dutch NRC Handelsblad science journalist Margriet van der Heijden discussed the book, labeled him a “glamour-nerd” and expressed doubts about his book as a “cocktail of superficial superlatives, incomprehensible math and touching memories of his youth”. Unfortunately, Van der Heijden did not explicitly mention the gap between mathematics research and education in mathematics.
  • In April 2016, Dutch NRC Handelsblad USA correspondent Diederik van Hoogstraten recycled the story, flew from Los Angeles to San Francisco and drank wine with Frenkel, resulting in this interview. The catchy title of the interview is “Math anxiety ? Don’t be afraid anymore !Apparently professor Frenkel has found the medicine that we have all been waiting for. The label “rockstar-mathematician” is used (in quotes), and while there is proper reference to the criticism by Margriet van der Heijden, Frenkel gets the last word to state the good intentions of his book.

“It has to go wrong only once. You must solve a problem in arithmetic in front of the class. You can’t do it. You are scolded, because we are focused on the result and the result was wrong. You feel dumb. A nightmare. (…) Then you are afraid for the rest of your life. Afraid to make an error and be “dumb”. This is a shame, for it is a wonderful subject. In this way we deny millions of people this knowledge, wisdom and beauty. (…) I want to find a way to the heart of all those people with youth trauma’s. It is okay, I say, you don’t have to be afraid.” (Edward Frenkel, NRC 2016-04-23, my translation)

It may well be that there are still many math teachers who abuse their students, but we must wonder whether Frenkel bases his view upon statistics or folklore. Normal teachers tend to ask up front only those students who are likely to get the result (e.g. have shown this already on paper). The normal discussion is on method and not merely on result. Thus Frenkel’s view on “math anxiety” is rather folklore than a result of research in current mathematics education. Subsequently, people with math anxiety will not read and be cured by a book that has so much mathematics, however much “love” you will put in the title. Both diagnosis and treatment are delusional.

Dutch is a language sink. Why was it considered useful to translate this delusional book from English into Dutch ? Would there really be readers who would grasp Frenkel’s math but not be able to read English ? Translators from English into Dutch have an easy job, and publishers can piggy-back upon international bestsellers. It is more difficult to translate from Dutch into English, and market this in the English reading world, but it would be more relevant to open up Holland to the world. Why did Van Hoogstraten want to interview Frenkel ? Perhaps the Dutch publisher of the Dutch translation sent him a copy of the book with Frenkel’s telephone number ? Alongside money laundering there is also delusional book laundering (that sells for cash too).

This is not without cost.

  • NRC Handelsblad has misinformed its readership twice now. Readers who have the idea that mathematics is inaccessible actually have seen this idea confirmed.
  • There is nothing in their reports on Frenkel that explicitly shows that mathematics is accessible to more people than commonly thought, albeit that this can be found elsewhere and not with Frenkel.
  • NRC Handelsblad has never reported on my books since 2009 on mathematics education. The newspaper likely finds mathematics sexy and mathematics education both boring and no science.

Holland is not an open minded country. Holland is a country where books are sold that have been translated into Dutch.

Edward Frenkel in 2010 (Source: wikimedia commons)

Edward Frenkel in 2010 (wikimedia commons)

My book Elegance with Substance (EWS) (2009, 2015) (pdf online) has the theme that mathematics education requires fundamental re-engineering. Mathematicians are trained to think abstractly and are not trained for the empirical science of didactics. When they meet real life students in class, mathematicians suffer from cognitive dissonance, and resolve this by sticking to traditional ways of teaching, which tradition has not been designed for optimal didactics. In this manner, mathematics education is in shambles for some 5000 years.

One of the reasons why complex numbers are considered difficult – or difficult to teach – is that the traditional presentation of the square root is crummy. For this, see EWS p26 & p30.

  • An author may presume that readers of some chapter in a book (EWS) have read the preceding chapters. In a weblog, such an assumption is rather risky.
  • Please read EWS, for this already contains subtle discussion. When you have read EWS you better understand my intentions and what steps have been taken already. This will allow me to build upon what has been said before.
  • In this weblog I am forced to repeat issues nevertheless, for I cannot assume that you have read EWS or earlier weblog entries.
  • Let us make the best of this. For now, at least check out the earlier discussion of the quadratic function.
Metaphor

Math teacher John goes to school by vehicle, either car or bike. Since he uses his car more often he calls it “the” vehicle, and his bike “the other” vehicle. His class knows that he does so. One day, John goes to school by bike and hits a rock that bends the front wheel. Arriving late in class he explains: “Today I came by bike and had an accident. The vehicle was severely damaged and I had to walk.” Most students accept this explanation but some are inclined to mathematics or litigation, and they ask: “How can it be that you came by bike and that your car was severely damaged ?”

The idea that the word “the” could be reserved only for one vehicle conflicts with reality. The term “the” is used much more flexible in human discourse than one can really fix differently.

Correct in wikipedia

Wikipedia is not quite an encyclopedia but rather a portal to proper sources. In this case however it provides a proper definition of square root. I only edit the text to get the common {x, y} notation.

“(…) a square root of a number y is a number x such that x2 = y, in other words, a number x whose square (…) is y. (…) For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16.” (wikipedia, edited)

Note two aspects:

  • This is solving an equation. In Wolfram Alpha Solve[y = x^2, x].
  • This can be done algebraically. If you want a numerical result, use N[Solve[y = x^2, x]]
Confusing in wikipedia

Subsequently, wikipedia correctly restates what mathematicians are doing. Both generate confusion.

  • “Every non-negative real number y has a unique non-negative square root, called the principal square root, which is denoted by y, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 × 3 = 9 and 3 is non-negative.”
  • “Every positive number y has two square roots: y, which is positive, and −y, which is negative. Together, these two roots are denoted ± y (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation “the square root” is often used to refer to the principal square root.” (wikipedia, edited)

When teachers wonder why students have problems with square roots then teachers should look at themselves. It are they themselves who have introduced ambiguity by distinguishing the square root versus any square root. This distinction is okay for students who like litigation or mathematics, but it is too subtle for students who are still struggling to understand what the discussion is about.

Note two aspects:

  • y is a function from the set of nonnegative real numbers to itself.
  • This can be in algebra while a numerical result is N[√y]. For example √2 ≈ 1.414…
Function or correspondence

Students in highschool learn only about functions. For each input in a function there is only one outcome. There are also correspondences such that an input can have more outcomes. Solving an equation x2 = y can have more outcomes and thus it is a correspondence. Since textbooks do not want to discuss correspondences, the distinction is hushed up.

It is better to be explicit about it. Thus, let the correspondence be defined as follows:

Do√y = ± √y   since √y)2 = y .   This merely rewrites Solve[y = x2, x].

The inverse of function f is function g such that subsequent application returns the original input, thus g[f[x]] = x. The inverse can be found by mirroring alongside the line y = x. The distinction between function and correspondence also explains why the inverse of y = x2 gives the correspondence y = Do√x = ± x (check the switch in the labels of y and x). 

Function y = x^2 and its inverse correspondence y = DoSqrt[x]

y = x^2 and inverse correspondence y = DoSqrt[x]

Verb and noun

EWS links above notions to the linguistic distinction between verb and noun.

  • The correspondence Do√y is like a verb and the function √y is like a noun.
  • The algebraic solution √2 is a noun and using the square root sign as an instruction for the calculator is like a verb (outcome 1.414…).
  • See Gray & Tall on the notion of the “procept” (process & concept) (verb and noun). The idea of the procept is that mathematics deliberately uses few symbols but with different meanings depending upon context. Teachers tend to treat those contexts implicitly, leaving students to guess what is happening. It is better to introduce more symbols that are explicit about the meaning, like the distinction between function √ and correspondence Do√.
Solution of the abuse of “the”

We diagnose that mathematicians introduce confusion themselves. In highschool they explain what a function is and what solving an equation is, but they do not dwell on the difference between function versus correspondence (and employ such for solving an equation). They encode the hidden intention by introducing an artificial distinction between “the” and “any” root. The latter doesn’t work because language doesn’t not co-operate. For some 3000 years students have been suffering because math teachers have been behaving like Humpty Dumpty in Alice in WonderlandThrough the Looking-Glass, holding that language must adapt to their needs rather than the reverse. Source: wikipedia:

 “When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”
    “The question is,” said Alice, “whether you can make words mean so many different things.”
    “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

The obvious solution is to ditch this artificial distinction between “the” and “any” root.

Instead of saying that √4 = 2 is “the root of 4“, it is better to say that it is “the root function value of 4“. Since square roots may also be taught in elementary school, another expression is “rootsign 4“.

Subsequent drama of complex numbers

Complex numbers are not difficult, hardly different from the system of co-ordinates itself, and they are actually quite enlightening since they allow working in the plane directly. Teaching the square root for real numbers however is bodged up in traditional didactics, and hence there is the subsequent drama for complex numbers. The square root of -1 becomes incomprehensible, and the train runs into a mountain.

Merely giving the following definitions is begging for problems.

  • Let H = -1, see here. The negative numbers can be created by turning the positive numbers a half turn.
  • Let have two roots: i^2 = H and  (-i)^2 = H.
  • Let √ now be a function from all real numbers y to the complex numbers = x + y’ i.
  • Let i = √H.

Our earlier discussion of the complex numbers mentioned the danger. Consider:

-1 = H = i²  = (H) (H) = (H H) = 1 = 1 = –H            (*)

Obviously, -1 = 1 doesn’t work for real numbers. Common discussions of this deduction are fraught with problems. They explain that it doesn’t work but not why, or give a curious reason why.

Developing better didactics

When students better understand square roots, then above definitions are easier to extend upon.

The reference to William Rowan Hamilton (1805-1865) and the unit circle is standard, but I have not yet seen the following explanation for above conundrum (*). Rowan Hamilton provided a proper development for complex numbers, namely with the model that z = {x, y} and i = {0, 1}. The crux lies in a particular rule for multiplication which neatly fits the theory of angles (trigonometry). A multiplication by i can be interpreted as a quarter turn.

  • Starting from {1, 0} and turning counterclockwise along the unit circle in steps of i, we find the outcomes of i, H, -i, and return to 1.
  • Clockwise in steps of –i, we find the outcomes of –i, (-i)² = H, (i)³ = i and back to 1.

unitcircleThe conundrum (*) can now be explained:

  • In terms of reals, H = –H only works when H = 0.
  • When H means a half turn counterclockwise then –H means a half turn clockwise, and both generate the same result. Thus H = -H makes sense in terms of turning. (Creating the negative numbers from positive numbers x by using –x = H x can be done, but we may also allow for -x = –H x. To make this consistent, we require a more developed theory on rotations.)

Now that we understand why the H = –H outcome arises, we still find the situation paradoxical (without such a theory of rotations), and we decide to obliterate this outcome, so that we do not have to worry about it. This can be done with these definitions:

  • When y < 0 then √y = i √ |y|  where |y| gives the absolute value.
  • When y < 0 and w < 0 then y w > 0 and √(y w) ≠ √y √w.   PM. The latter is i√|y|  i√|w| = – √ |y w|
  • When y < 0 then Do√y = ± i √ |y|

Now deduction (*) is blocked: the step in the middle violates the rule for y < 0 and w < 0. The blockage is no mystery: we decided to choose it in this manner.

The rule for y < 0 and w < 0 causes the main conceptual issue for complex numbers. For example, what to do with √(-2) √(-3) ? Does it solve into √6 ? No, we have chosen to block this solution. The general rule now is to first substitute for i before doing something else. Then √(-2) √(-3) solves as i√2 i√3 = -√6. (There are various expositions online that show the proper calculation steps, e.g. this video. But showing how it is done is not explaining why it is done so.)

With this explanation of (1) what = –means and (2) that we block it deliberately,  I wonder whether Rowan Hamilton only gave a different model, so that there is no true difference between x + i y and {x, y}. The models may be fully equivalent, and choosing i = {0, 1} only turns equivalence into identity.

Some PM points

PM 1. This discussion extends on the discussion of complex numbers in EWS p43. (That page uses {1, 2} + {3, 4} = {4, 6}, and the subsequent discussion has an awkward typing error.)

PM 2. Observe that wikipedia has a long lemma on complex numbers and that the rule on y w only appears in notes to another lemma on square roots. Wikipedia is a portal and no textbook, but the rule still is an elementary part of the definition of i = √H. Perhaps the rule seems less relevant when you start from i = {0, 1} but you are bound to need it.

PM 3. A discussion here confuses √y with Do√y, and it argues that we have only:

  • Do√Do√H = (± i) (± i) = ± (-1) = ±1
  • Do√(H H) = Do√1 = ±1.

There is this equality for Do√ indeed, but this doesn’t explain why it fails for √. The subsequent discussion at that website is a bit clearer: “Of course, you can choose to define the square root of a negative number to be a positive multiple of i; but if your book hasn’t done that, it doesn’t have a right to ask this question.” It is okay to establish the freedom for such a definition. The crux however is why one would make it. That website leaves this open. (Our own explanation above is that it avoids confusion on H = –H, which is true for rotations but not for real values.)

PM 4. A reader alerted me to it that the common discussion of the complex solution to the quadratic function uses a projection from the Argand plane onto the Cartesian plane. I was aware of this, but let me pass on the links, provided in this alert, to these two papers by Ansie Harding and Johann Engelbrecht: one and two.

PM 5. Dutch readers can benefit from a discussion by J.H. Wansink, “De complexe getallen. Een algebraisch onderwerp op dood spoor“,  Euclides 51e jaargang no 4, 1975/1976, p127-149. Wansink is very positive about the way of presenting complex numbers as analytic geometry as had been done by H.J.E. Beth (father of E.W. Beth). I employed this same method in “Conquest of the Plane(2011). A more traditional and in my view less transparant presentation as mostly algebra is by J. van de Craats, here, who also doesn’t fully explain (*), see question 1.7 on p4 and the answer on p71.

PM 6. Rotation in 2D can be modeled by a 2 x 2 matrix, see a bit of theory: step1 and step2. A complex number is equivalent to a matrix, and the multiplication of two complex numbers can be represented as matrix times vector. Hence there is a relation between the number -1 and the matrix H for a half turn.

isomorphismLet Q be a quarter turn counterclockwise, then Q 1 = i. A half turn H = Q2 = – I, where I is the unit matrix. Then H 1 = -1. A clockwise turn is QT, the transpose of Q , and we find Q3 = QT. We can find H also as the square of this transpose.

RotationThe discussion above about H = –H thus can be resolved by the distinction between matrix H and number -1, and we find that actually H = HT. There remains a point of doubt though. There is an isomorphism between complex numbers and the matrix approach, and thus the explanation of rotation in the complex plane by the use of matrices is rather like begging the question.

My book Elegance with Substance (EWS) (2009, 2015) (pdf online) has the theme that mathematics education requires fundamental re-engineering. Mathematicians are trained to think abstractly and are not trained for the empirical science of didactics. When they meet real life students in class, mathematicians suffer from cognitive dissonance, and resolve this by sticking to traditional ways of teaching, which tradition has not been designed for optimal didactics. In this manner, mathematics education is in shambles for some 5000 years.

EWS documented this with a long list of examples and hence I should not be surprised anymore to find another example. However, I was very surprised to discover the following for the quadratic function. This function is so familiar for at least some 3000 years, and you would expect that mathematicians had optimized its teaching. However, be amazed as well.

A starting point for all approaches

The quadratic function is introduced in all cases by presenting the square of x. The formula is supported by text (explaining what the function does), table (first row for effect y, second row for cause x, namely allow for Δy / Δx), and graph. Write functions with square brackets.

f[x] = x2

Properties are:

  • The vertex or turning point can be found at the origin {0, 0}.
  • There is symmetry for left and right along the line x = 0 because f[x] = f[-x].
  • There are no outcomes f[x] < 0, unless by use of the complex plane with = i2 = -1.
  • Also f[x] and –f[x] mirror, but one would say that this requires a coefficient a = -1.
Continue re-engineered: a focus on understanding

Changing and moving (transforming and translating) the function gives the basic form.

f[x] = a (x h)2 + v

  • The vertex or turning point can be found at {h, v}.
  • Coefficient a stretches or squeezes, while a negative value flips or mirrors upside down. Making coefficient a negative turns a convex form into a concave form. (Seen from the origin, concave is hollow or h-shaped, and convex is bulging or b-shaped.)
  • The value h shifts the parabola horizontally. For example, x must have a higher value if some positive h is substracted from it. If h is negative, then the graph moves leftwards.
  • The line x = h is also the mirror-axis for left and right.
  • The value v shifts the parabola vertically. If v is negative, then it shifts downwards.

A key question for a parabola is where it intersects with the horizontal axis. Even when we want to find the points of intersection with a line y = d, then this only means an adjustment to v’ = v d. Finding the solution to this quadratic equation is straightforward.

f[x] = a (x h)2 + v = 0

(x h)2 = – v / a

x1,2 = h ± √(- v / a)

It is easy to check that all solutions must have this symmetric form, since substitution back first eliminates h and then squares the remainder, and reduces to zero. This solution thus also gives the rule:

  • When v = 0 then there is a single solution, or it is touching the horizontal axis.
  • When – v / a > 0 then there are two solutions.
  • When – v / a < 0 then there is no intersection with the horizontal axis. A solution can be found in the complex plane however. There is no need to be squirmish about complex numbers because these would have been discussed before, when discussing the system of co-ordinates (see here).

These graphs show the solutions for f[x] = 1 (x – 0)2 + v, for values v = -2, 0, 2. The relevant intersection value is √2 ≈ 1.414…. The complex solution can be found by flipping the function, giving g[x] = -1 (x – 0)2 + 2, solve for the intersections, and circle these a quarter turn. Here x1,2 = ± i √2.

parabola

Given the (two) solution(s) x1,2 = h ± √(- v / a), let us write for ease or w, and observe that the function can be written in the factored form: f[x] = a (x – u)(x – w). The vertex or turning point lies in the middle, so that = (u + w) / 2. Substitution of x = h gives v =  – a (u – w)2 / 4.

The basic form can be expanded into the standard (polynomial) form.

f[x] = a (x h)2 + v

= a (x2 – 2 h x + h2) + v

= a x2 – 2 a h x + a h2 + v

f[x] = a x2 + b x + c in which b = – 2 a h and c = a h2 + v

Thus, conversely, whenever one meets the form f[x] = a x2 + b x + c then h and v can be solved as:

h = – b / (2 a)

v = c – a h2 = – (b2 – 4 a c) / (4 a)

Acknowledgements.

  • After re-engineering the above, I knew what to look for, and found this video by MIT graduate Nancy (mathbff) who calls the basic form the “vertex form” but who still has it in 2nd place instead of 1st place. There is also this lesson plan at MARS that mentions the three forms, but their order is standard, factored, basic, and the latter is called “completed square form”. I would rather put basic first and express this in its name. The lesson is what you can directly recognise in each form, which is a fair question, but not the most relevant first question on the graph of the quadratic function.
  • The complex graph comes from Norton & Lotto 1984, reproduced by Francis Su et al. here. Su et al. don’t use the basic form whence it is difficult to see how to flip the function and why complex solutions have the real part h and the imaginary part √(- v / a). The complex solution is also mentioned by wikipedia here, but this doesn’t flip the graph and doesn’t explicitly make the quarter turn, so the solution remains mysterious.
The shamble tradition: starting from theory

Mathematicians have developed a theory of polynomials. The traditional approach in teaching the quadratic function is to treat it as a special case of such polynomials.

Thus students are presented with the polynomial format, called the standard form.

f[x] = a x2 + b x + c

The standard form is intransparant. Teaching thus proceeds by presenting tricks, only to recover what is already obvious for the basic form.

(1) Students are presented with the quadratic formula and must learn it by heart. This is the fail-safe approach for students without much interest in or understanding of mathematics.

x1,2 = (-b ± √D) / (2 a) with discriminant D = b2 – 4 a c.

The possible solutions are cataloged with D < 0, D = 0 and D > 0.

  • It is not clear where D comes from and what it means. (It is only a freak result of the polynomial form.)
  • The traditional approach takes so much time in general that there often hasn’t been time to explain about the complex plane, and thus it is often said that D < 0 has no solution rather than that it has a complex solution.

(2) More interested or advanced students can be shown where the quadratic formula comes from. This generates the ritual called derivation of the quadratic formula. There are a number of approaches (wikipedia), but, “surprisingly” the following approach may not be mentioned (not in wikipedia today), and the reason must be that the tradition has lost track of the basic form.

a x2 + b x + c = a (x h)2 + v

For x = 0: c = a h2 + v hence v = c – a h2

For x ≠ 0: a x2 + b x + c = a x2 – 2 a h x + a h2 + v

b x = – 2 a h x and hence h = – b / (2 a) and substitute this in v again

Subsequently resort to above disucssion, and find x1,2 = h ± √(- v / a)

Didactic conclusions

Didactic conclusions are:

  • The basic form is transparant, provides a clear path to the standard (polynomial) form, and also provides clarity for the return path from the standard form back to the basic form.
  • Starting with the standard form is convoluted. It starts from some distant theory about polynomials in general and creates clutter and mystery about solutions. This tradition has lost track of the basic form and thus also requires more intricate solution methods or reliance on memory for the “quadratic formula”.
  • Making math opaque for such simple issues puts a heavy burden on more intricate issues.
  • Making math opaque for such simple issues also causes a flight into alternative approaches, such as reliance on applications (“realistic mathematics education” (RME)). Interesting applications are here, but my impression is that students will study those applications with more interest once they have mastered the basic form first. The general format for teaching is: (a) psychologically prime notions with a basic example (here the square of x), (b) develop the theory, (c) apply the theory.

PM. Wikipedia is not a didactic environment but a portal. When you want to know what the “quadratic formula” is, then wikipedia presents it, and wikipedia doesn’t give a didactic presentation of the underlying issue. Wikipedia has other entries like on the quadratic equation, but following these links doesn’t generate a didactic exposition. Thus beware: mathematics in wikipedia is created by math students who copy their textbooks.

Examples: two English websites and a Dutch textbook for grade 9 (age 14-15)

A traditional presentation of the quadratic function that “completes the square” without giving the general form is here. Another website is here, and it gives the basic (“vertex”) form somewhere at the end.

A Dutch textbook for grade 9 at subtop-level (“Moderne wiskunde”, HAVO 3ab, together 400 pages, edition 8, 2005) includes the following approach to the quadratic function.

  • Ch 3 (some 25 pages): Transformation and translation of functions (linear, quadratic, hyperbolic, powers and roots). It makes sense to review the different functions. However, for the quadratic function only a and c are manipulated. Given above discussion, it would make more sense to discuss each function separately with its transformations and translations.
  • Ch 5 (some 25 pages): Factoring polynomials. This is a basic algebraic skill. It is enlightened in the textbook with graphs of rectangles and indeed a parabola. When the treatment of the parabola is as convoluted as it is now, then this order can be understood, for the solution of a quadratic equation can be found by factoring. A more didactic approach however would be to (a) discuss factors using rectangles, (b) discuss the parabola in the re-engineered fashion, and (c) only later practice also on the skill of factoring. The advantage of the re-engineered approach is that we can have text, formula, table and graph in a single review at a much earlier stage, which enhances understanding.
  • Ch 8 (some 30 pages): Quadratic formulas. The traditional f[x] = a x2 + b x + c with vertex and symmetry, quadratic formula, the use of D, graphical form, and solution of quadratic equations. In the re-engineered approach, this would be integrated in the earlier chapters 3 and 5.
  • Ch 9 (some 25 pages): Manipulating more functions: Addition and multiplication of graphs. Periodic formulas. This deepens earlier notions and provides more practice. This might be less needed in the re-engineered approach when issues are transparant.
  • Ch 11ab (some 50 pages): Graphs and equations. Intersection of graphs and finding solutions. However, finding the intersection of parabola a x2 + b x + c and line b’ x + c’ reduces to a x2 + b x + c = b’ x + c’ or solving a x2 + (b – b’) x + (c – c’) = 0, which is just solving for another parabola, which has already been discussed. This split chapter only practices earlier notions, which might be less needed when the presentation was transparent enough.

The conclusion is that this textbook provides key ingredients, but in less didactic format and order, such that the textbook resorts to a lot of practice to allow students to replace insight by routine. (Routine should not be called skill when insight is lacking.)

Conclusion

There is scope for re-engineering the didactics of quadratic functions into a much more transparant manner. The focus must be on what is essential to understand (at this level) and on starting from some abstract general theory (relevant for understanding issues at another level).

This re-engineered approach can likely already be used in elementary school. See the other book A child wants nice and no mean numbers (CWNN) (2015) (pdf online).

It remains to be tested whether the re-engineered approach indeed is as transparant as suggested here, and whether pupils would have a fast road to insight and skill and improved attitude. It always are the students themselves who show what works for them.

President Obama can do little other than teach, in the last year of his presidency and with a majority opposition. Obama just advised the British to vote for the EU on the Brexit referendum. He is at risk of infringing upon national sovereignty, the very thing that the referendum is about.

The Brexit referendum stay / leave question is, and let me include the FT poll score,

Should the United Kingdom remain a member of the European Union or leave the European Union?
Remain a member of the European Union [  ] (44%)
Leave the European Union [  ] (42%)

Leaving the EU still allows various alternatives. If the vote would split over those options, perhaps one better stays. There is no way of knowing. Referenda tend to be silly and dangerous.

  • Referenda work only well when there are two options only, with a clear-cut Yes / No answer. This kind of question occurs only by exception.
  • Normal issues have more options and grades of grey. With at least three options, there arises the Condorcet paradox. For such issues, there better be representative government, with a Parliament selected by proportional representation (PR), and which Parliaments uses more complex methods for bargaining and voting – see Voting Theory for Democracy.
  • The pitfall is that a question might seem to have a clear-cut Yes / No answer while it actually has other options and such grades. Check how the Brexit question masks the other options. It often is an issue of political manipulation to reduce a complex issue to seeming simplicity, and to create a situation such that the political leader who drafts the question might argue to have the backing of the people.
  • Referenda belong to populism and not to democracy.

In this case, UK prime minister David Cameron has to overcome a rebellion in his own party and the threat of defection to UKIP. Check this report on Cameron’s bargaining with the EU. Given this bargaining result Cameron now argues for the EU and he hopes to secure peace in his party. It is somewhat curious that the whole of the UK is called to the ballot box to resolve such internal strife, but the same happened in 1975 with Harold Wilson and the Labour Party.

An advantage of the Brexit referendum is that the BBC now had two broadcasts “Europe: Them or Us“, that review the relation of the UK to the EU. It has been awfully nice to see the ghosts of the pasts perform their part in this drama. See also here and youtube. Some key points that struck me were:

  • Churchill argued for a united Europe.
  • The UK 1975 referendum caused people to complain a decade later: “We voted for a Common Market and later we got something else.”
  • Margaret Thatcher started out as a European, supported Europe in 1975, actually initiated and signed the 1986 Single European Act, with the change from a Common Market (with veto power by country) to the (Europe 1992) Single Market (replacing veto power by qualified majority), and whisked it through the UK Parliament without proper discussion about this abolition of national sovereignty. Only later came the 1988 Bruges speech.

The key point for Cameron has been to restore a shadow of that veto power. Britain cannot block others from having an ever closer union, but it has an opt-out:

“Assessment: Mr Cameron has secured a commitment to exempt Britain from “ever closer union” to be written into the treaties. He has also negotiated the inclusion of a “red-card” mechanism, a new power. If 55% of national parliaments agree, they could effectively block or veto a commission proposal. The question is how likely is this “red card” system to be used. A much weaker “yellow card” was only used twice. The red-card mechanism depends crucially on building alliances. The sceptics say it does not come close to winning the UK back control of its own affairs – and Mr Cameron is set to announce further measures which he claims will put the sovereignty of the Westminster Parliament “beyond doubt”.” (BBC Feb 20 2016)

Some points that I missed in these two “Europe: Them or Us” broadcasts:

  • There is no recognition for Bernard Connolly whose The Rotten Heart of Europe helped the British to stay out of the euro and to keep the pound. There is still room for a better approach to the notion of an optimal currency area.
  • There is little clarity about what the economic discussion really has been about. “Economic union” and “political union” are vague words, and it seems relatively easy to make a political speech or TV broadcast with these. Details matter however. Details help to keep out the ideologues. It is said that Britain has the best economists (Marshall, Keynes, Hicks) but Germany the best economy. Margaret Thatcher would have been much more effective when she had proposed good economics rather than banging the handbag. The relation of the UK to the EU would have been far better had the UK shown better economic analysis and an economy to prove it. Thatcher came to power during a time of stagflation when economists were in disarray and neoliberalism seemed the only way out. This neoliberalism however contributed to the global financial crisis and the economic crisis of 2007+. See my analysis since 1990. Of the core issue, a recent turn is the myth about German decentralised labour market bargaining. Britain has an impact on the European economy via the City and its banks (a fair reason to stay in), but why doesn’t Britain have more impact ?
  • Democracy in the UK suffers from district representation (DR), and it would be better to have proportional representation (PR). There is too little awareness in the UK that much of their political mayhem is caused by their rather unresponsive electoral system. See the comparison of Holland and the UK, and see how Nick Clegg shot his own foot (and destroyed the LibDems).

PM. After writing this, I discovered this review of “Europe: Them or Us” by Sean O’Grady and he says much of the same thing.

The recent Dutch referendum on the Association treaty with the Ukraine is another example of how referenda can be silly and dangerous. I voted against that treaty because of the military section hat would involve the EU in helping secure the Ukrainian borders, which would effectively move NATO’s borders eastward. Government propaganda did not pay much attention to the military section and emphasized the section on free trade. Even there the propaganda didn’t draw the parallel with the economic collapse in East Germany (DDR) when it was merged with West Germany (BRD). In this case, representative democracy failed, for it created this Association treaty, and the Dutch referendum was a freak event that might actually do some good. It still confirms that referenda tend to be silly and dangerous, since the proper answer would have been a better informed discussion in Parliament, notably by having (a) an Economic Supreme Court, (b) annual elections.

Reproduced with permission by Jos Collignon

Thanks to Jos Collignon for permission to reproduce this

Last February, fellow economist John Quiggin tried to make sense of political developments by relating those to the economic crisis of 2007+ a.k.a. the Global Financial Crisis (GFC). Quiggin warned about his “amateur political analysis” and proceeded to identify a “three party system“: with leftism, neoliberalism and tribalism (see his definitions). The neoliberals are soft (“reforming” the welfare state) or tough (abolishing it). Dutch readers may check the column by Wouter de Been which alerted me to this, and who argued: “It’s still the economy, stupid.

A first comment is that Quiggin actually has four groups, namely leftism (Bernie Sanders), soft neoliberals (Hillary Clinton), hard neoliberals (Donald Trump) and tribalism (Ted Cruz). See the diagram below.

A second comment is that Quiggin uses the economic term “neoliberalism” as a political label. However, economics is a tool and not an objective. Instead we should categorize politics by values, such as (in-) equality and freedom. It must be admitted that economics has much influence on political discussion but it remains necessary to distinguish ends and means.

  • As discussed before, Emmanuel Todd distinguishes the stable categories of equality / inequality and authoritarian / liberal. A common distinction is between the political “left” and “right”, but this distinction is vague, and it is more informative to distinguish Todd’s two dimensions.
  • For Trump, the label rather isn’t “liberalism” but “neoliberalism” (“freedom”). Trump pursues more freedom but does so in an authoritarian manner. Over time his manner could be more important than his stated goals and then he would shift down.

Quiggin holds that there is instability because of the Condorcet paradox and other reasons. A third comment is that these four can actually be arranged on a left-to-right line and then fit the Duncan Black single-peakedness and the median voter theorem.

  • In the diagram, the conceptual gap between Sanders and Trump is rather large, and they are only comparable in radicalism. Clinton and Cruz are closer on caution and conservatism.
  • In Europe, we might perceive of a coalition that excludes the inequality & authoritarian group and includes the others on liberty, equality and fraternity (in the colour map: exclude green and include the others). This coalition might work if there is gradual change and no French Revolution.

A fourth comment however is that there better be electoral reform, such that parties can collaborate on issues rather than fight about ideology.

2D to 2D

Other charts on political views

Sanders and Trump seem to score higher on their radicalism than on clarity about a position on the scale from authoritarian to liberalism / freedom / neoliberalism. We could include a third dimension of the attitude towards change. There is a range from conservatism to radical adaptation. The term “conservative” itself is ambiguous: it may mean that one wants to maintain the status quo (e.g. with current liberal properties), but it may also mean that one wants to return to some ideal past (reactionary, e.g. with king and aristocracy (billionaires)).

A fifth comment thus is to maintain caution on these dimensions.

The Nolan chart has the axes of economic freedom (low to high) and personal freedom (low to high). This compares to the chart above, when we replace equality / inequality with economic freedom, and replace authoritarian / liberalism with personal freedom. The Nolan chart however again confuses political values with economics as a tool. There is also the problem of balancing opposite effects.

  • A minimum wage reduces the economic freedom of the entrepreneur and the outsider unemployed, but enhances the economic freedom (bargaining position, income) of the insider employees. See DRGTPE for a better approach to the economics of the minimum wage. This is economics and not an issue of political values, though politics are relevant when deciding what policy to adopt.
  • One might presume a natural state and then hold that any measure that is to the disadvantage of someone counts as the reduction of economic freedom, even when it would advantage others. However, there is no clear specification of such a natural state. (Todd’s scheme identifies at least four types of family structure.)

A classification of Dutch political parties by Andre Krouwel (“kieskompas“) still uses the political left-right distinction, but this distinction has been based upon programme details and thus might still be relevant.

The lifestyle consultancy firm Motivaction identifies eight groups in Dutch society, from traditionals to post-modern hedonists. Potentially these link up to political parties.

Mentality_EnglishReturn to economics

I tend to agree with Quiggin:

“But the more fundamental problem is that none of the competing forces has an obviously compelling solution to the problems we face. Neoliberalism has manifestly failed to deliver the prosperity promised by triumphalists like Thomas Friedman in the 1990s. Tribalism is already a lost cause, given the massive migrations that have already taken place, and can at most be slowed in the future. The left needs to rebuild institutions and policies that have been in retreat for decades.”

The true problem is one for economic analysis. The derived problem for the political leaders and parties and their voters is to select which economic advisors to listen to. The sixth comment is that there is censorship of economic science since 1990 by the directorate of the Dutch Central Planning Bureau (CPB). Governments didn’t get proper advice, and the economic crisis of 2007+ a.k.a. GFC is evidence of this. Boycott Holland till this censorship is solved. See the About page. What the censored analysis fully is can only be seen after the censorship has been lifted of course.

I am no expert on terrorism and wonder whether the supposed experts aren’t either.

Dutch historian and “expert on terrorism” Beatrice de Graaf gave a lecture on Dutch TV on March 11 (or see Utrecht University) about David Rapoport‘s four waves of terrorism (his original article).

Her main message was that people might find some comfort in the idea that waves die out. March 22 saw the bombs in Brussels.

This theory of four waves of terrorism appears to be rather silly. Below gives my common sense rejection.

De Graaf is not the only academic who regards the theory of the four waves as serious. The West is vulnerable to terrorism when its “experts on terrorism” are academics lost in theory. It is okay to sooth people not to worry too much, but intellectuals should present effective approaches rather than fairy tales.

The so-called “four waves”

Jeffrey Kaplan summarizes (and then proceeds in adding his own fifth wave) (while Dutch readers can check Edwin Ruis’s review of March 13):

“Rapoport’s theory, first published on the web before finally finding a home in a printed anthology, posited four distinct waves of modern terrorism (anarchist, nationalist, 1960s leftist, and the current religious wave). Each wave had a precipitating event, lasted about 40 years before receding, and, with some overlap, faded as another wave rose to take center stage. Most terrorist groups would gradually disappear, a few (the Irish Republican Army for example) proved more durable. Rapoport’s theory was elegant, simple, inclusive, and had a high degree of explanatory power. In short, it provides a good academic model.” (Kaplan 2008).

Jeffrey D. Simon holds (and wonders about a fifth wave too):

“David Rapoport’s “The Four Waves of Modern Terrorism” is one of the most important pieces ever written in the vast literature on terrorism (Rapoport 2004).  What Rapoport did in his classic study was take the complex phenomenon of terrorism and put it in a historical context that not only explained different periods of international terrorism, but also set forth theories and concepts that can be used to attempt to anticipate the future of terrorism.  That is no easy task.  There haven’t been many assessments and articles written about Rapoprt’s “Four Waves” theory, although this volume of papers initiates a discourse about his important thesis (See Thompson and Rasler, this volume).  Despite the numbers of scholars, policymakers, and others who have joined the field of terrorism studies after the 9/11 attacks, there does not appear to be a great deal of interest in the history of terrorism.  In today’s instant access and information-overload society, we are inundated with analyses of current affairs but pay scant attention to what we may learn from what has transpired in the past.” (J.D. Simon on the Lone Wolf, likely 2010)

I googled to find some criticism, but didn’t see much, though perhaps I didn’t google well. I noticed a critical text by Ericka Durgahee. I didn’t have time to look into this, and the following are my own common sense short remarks.

The anarchists 1880-1920

The dynasties of Hohenzollern, Romanov and Habsburg collapsed. Perhaps the anarchists didn’t really win because we don’t have anarchy now, but those anarchists were replaced by communists and fascists, and we ended up with two world wars, which isn’t quite “die out”.

Anti-colonialism 1920-1960

The anti-colonialists won. Winning isn’t quite “die out”.

Leftists 1960-1989

Leftism became impopular because of the Great Stagflation (unfavourable unemployment and inflation) and the collapse of the Berlin Wall. Young radicals were more motivated by Ronald Reagan and Margaret Thatcher.

In Germany, the police managed to isolate the Rote Armee Fraktion (RAF). In another article, Beatrice de Graaf explains how the Dutch radicals (Rode Jeugd, Krakersbeweging) lost their motivation by incompetence of the Dutch police. The Dutch police intended to adopt the tough German approach, but mismanaged this, and both radicals and the general population got the impression of an atmosphere of tolerance and dialogue. In that atmosphere, potential supporters saw no need for radicalisation, and radicals had the example of the dead-end street in Germany.

These events rather concern the transformation of European society after World War 2. There are pockets of terrorism, but there doesn’t seem much difference between RAF and other groups like IRA and ETA: except that each group requires specific attention for its idiosyncracies.

Religious terrorism 1979-now

Religious violence is of all times. There is no reason to predict that it will pass. This is no wave.

Alternative approach

Terrorists tend to be higher educated who are frustrated w.r.t. opportunities in society. They may feel sympathy with the unprivileged. They may adopt any ideology to recruit others in the resistance against the establishment. To counter this, one must look at society as a whole, create fair opportunity, and encourage people to participate. My own work contains aspects that are key to reduce terrorism.

  • Create a social welfare state that works. See DRGTPE.
  • Make democracy work. See VTFD.
  • Provide for good education, e.g. re-engineer mathematics. See EWS.
  • Let people learn how to deal with the human capacity for abstraction. See SMOJ.
Beatrice de Graaf, soothing Dutch viewers that a wave dies out

Beatrice de Graaf, soothing Dutch viewers that a wave of terrorism dies out

 

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