*Overige ‘Uitspraken rekenen’blogs*
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Jo Boaler is hoogleraar ‘Maths Education’ aan de Stanford Universiteit. Ze is inmiddels de meest invloedrijke en aangehaalde wiskundeonderwijsvernieuwer, ze is razend populair met kritiekloze volgers.
Boaler is afkomstig uit GrootBrittannië en studeerde er psychologie en ‘math education’. Ze is vanaf 1998 hoogleraar ‘Maths Education’ aan de Stanford University met een onderbreking van 20072010; toen was ze ‘Marie Curie professor of mathematics education’ aan de Universiteit van Sussex (GB).
Boaler streeft naar radicale hervorming van het wiskundeonderwijs: projectgebaseerd, samenwerking tussen leerlingen, heterogene klassen, creatief denken i.p.v. procedures leren/toepassen. Volgens Boaler kan iedereen in wiskunde presteren op het hoogste niveau, een kwestie van geloven in jezelf; iedereen kan met plezier bezig zijn met wiskunde; wiskundeangst is een gevolg van de traditionele manier van antwoordgerichte lessen en de testcultuur.
Boaler heeft veel enthousiaste aanhangers; voor het Freudenthal Instituut behoort ze tot de grote helden, haar werken worden regelmatig aangehaald. Ze komt op voor vrouwen en minderheden die, volgens haar onderzoek, onderdrukt worden bij wiskunde.
Ze heeft talrijke wetenschappelijke publicaties op haar naam; ze is auteur van een aantal bestsellers, waaronder ‘What’s Math Got To Do With It? ‘, ‘The elephant in the classroom’ en ‘Mathematical Mindsets’. Ze is adviseur van het PISAteam van de OECD. Ze adviseerde het Witte Huis en Downing Street. Ze verschijnt regelmatig op de Amerikaande radio en TVzenders. Regelmatig verschijnen artikelen van haar in kranten en tijdschriften, waaronder New York Times, Time Magazine, The Telegraph, The Atlantic, Wall Street Journal . Ze heeft talrijke onderscheidingen ontvangen.
Kritiek
Er is veel (vaak vernietigende) kritiek op Jo Boaler. Veel van haar onderzoek verdient het stempel frauduleus. Haar statististische onderzoeksmethoden zijn niet goed opgezet, haar conclusies kloppen vaak niet, ze maakt zich schuldig aan misleidende datamanipulatie, ze maakt veel gebruik van (emotionele of persoonlijke) anecdotes, veel van wat ze beweert is niet of onvoldoende voorzien van referenties of het blijken zelfreferenties te zijn. Bij sommige referenties wordt zelfs het omgekeerde beweert van dat wat Boaler vertelt. Veel van haar beweringen zijn strijdig met wat bekend is. Ze reageert niet inhoudelijk op kritiek, maar doet critici af als pesters of als mensen die de statusquo in stand willen houden.
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Jo Boaler, ‘Maths Education Professor’ aan de Stanford Universiteit
“Viva la Revolution”
Jo Boaler, te lezen op haar website [youcubed.org]
“By the way math is mostly taught, you don’t get critical thinking or problemsolving abilities. You have to memorize meaningless rules. In the best classrooms, like Jo Boaler’s, it’s the opposite: there you are learning critical thinking and do solve problems.”
Steven Strogatz (Hoogleraar wiskunde)
“Jo Boaler is justifiably tired of all the uninformed BS about math education. Too many children’s lives are being harmed. In this book [Mathematical Mindsets], she takes the gloves off and comes out fighting. Her weapon? Scientific data. Lots of it. A readable, engaging, compelling case for revolutionizing math education. Ignore her message and you (or your children, or your students) will be locked out of much of the 21st Century.”
Prof. Keith Devlin, collega van Boaler; hij is auteur van “The Math Gene”
“One student said to me: ‘Math appeared to me as the enemy of creativity and social interaction, and the refuge of ruleloving, closedminded people’ .”
Jo Boaler, hoogleraar ‘Maths Education’
Jo Boaler:
 Students have told me that thought is not required, or even allowed, in maths class.
 Children begin school as natural problem solvers and many studies have shown that students are better at solving problems before they attend maths classes. After a few hundred hours of passive maths learning students have their problem solving abilities knocked out of them. They abandon their common sense in order to follow the rules.
 We really need a revolution in math education. The mathematics taught now is onedimensional math that means the teacher explains a method and kids copy and practice. There is only one way for being successful. It’s a very narrow version of what mathematics is. It is anticreative. It is boring and inaccessible for most kids.
 Top employers at Silicon Valley are exasperated about what happens at school. This is not what they need. They will tell you: the one thing we do not need is kids calculating. For that we have machines. We need flexible problem solvers, who reason about different mathematical directions, who can set up mathematical models.
 Some people don’t want everyone doing well in math. When we show that girls and minorities do just as well, they don’t like that because they want math just for some people. They are going to great lengths, even to squash the evidence we have. They want math to stay the elite preserve for some people. That if you can do math you are better than other people. It is all feed into this elitist production of math. People use math as tests whether you can pass the doors in this part of society where you do well. All that is wrong. Math appears harder than other subjects because of the terrible way it is taught. We teach it as if math is just for some students.
 The attacks that are being directed at scholars in mathematics education, all of them women and people of color, are important to consider. They are coming from right wing organisations.
 If you are not getting pushback you are probably not being disruptive enough. The system is failing students and we need to disrupt the status quo. Pushback means you are making a difference!
 Students are overtested to a ridiculous degree, which is damaging to schools, teachers and most importantly, to the health, hearts, and minds of students.
 Teachers always know how well kids are doing, so you really don’t need to test them. The kids themselves can also selfassess and tell if things are strong or not. They do that with extreme reliability.
 Most tests used do not assess what’s important anymore. They might assess whether you are computationally fast — but that’s the one thing computers do and we don’t need humans for.
 An important new study shows that direct instruction caused young children to learn less. It inhibited their curiosity and creativity.
 We can all learn from Canada’s innovations in maths teaching that have rocketed them to world success. (2018)
[Kort tevoren werd bekend dat het onderwijs in Canada heel goed is, behalve bij wiskunde, waar de resultaten slecht zijn]
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Het ‘Amber Hill’onderzoek
Voor dit onderzoek, verricht in GrootBrittannië in 19951998, onderzocht Jo Boaler twee scholen gedurende 3 jaar: Phoenix Park en Amber Hill (de echte namen werden niet openbaar gemaakt). De leeftijd van de leerlingen liep van 13 jaar naar 16 jaar. Een belangrijk element uit dit onderzoek bestond uit gesprekken met de leerlingen.
Phoenix Park: Hier was het wiskundeonderwijs gebaseerd op ‘openended projects’. Nieuwe begrippen en procedures werden uitsluitend d.m.v. authentieke activiteiten geïntroduceerd. Leerlingen werkten samen in ‘mixed ability’ groups (begaafden en onbegaafden bij elkaar).
Amber Hill: Een traditionele school, waar de wiskundeinhoud en examens centraal stonden.
De onderzoeksresultaten:
Jo Boaler [Open and closed mathematics]:
 At Amber Hill, the students developed an inert, procedural knowledge that was of limited use to them in anything other than textbook situations. Many students at Amber Hill found mathematics lessons extremely boring and tedious. There were many indications that the traditional mathematics approach of Amber Hill was ineffective in preparing students for the demands of the real world.
 At Phoenix Park the understandings and perceptions seemed to lead to increased competence in transfer situations. The Phoenix Park students were able to achieve more in test and applied situations than the Amber Hill students.
 Some students from Phoenix Park did not like neither the openness of the approach nor the freedom they were given. They said that they preferred working from textbooks [herkenbaar]. Most of these students were boys, and often they were disruptive, not only in mathematics classes but throughout the school [niet herkenbaar].
 The girls at Amber Hill consistently demonstrated that they believed in the importance of an open, reflective style of learning, and that they did not value a competitive approach or one in which there was one teacherdetermined answer.
Kritiek
Kritiek op het ‘Amber Hill’onderzoek
Zie o.a. [The case of Amber Hill and Phoenix Park], [Boaler’s Bias]:
De populatie is te klein om conclusies uit te trekken. Een belangrijk deel van dit onderzoek bestond uit vragen gesteld aan de leerlingen. Boaler is duidelijk vooringenomen t.a.v. het onderwijs op Phoenix Park en ook t.a.v. meisjes t.o.v. jongens; jongens krijgen regelmatig een veeg uit de pan. Amber Hill presteerde vòòr dit onderzoek zeer slecht en werd waarschijnlijk daarom gekozen als referentieschool. Beide scholen presteerden slecht op de officiële examens. De NFERtoets aan het einde van het experiment (na die 3 jaar) werd door Phoenix Park slechter gemaakt dan aan het begin, het vernieuwde onderwijs heeft de leerlingen dus blijkbaar geen goed gedaan. Een aantal conclusies van Boaler volgen niet uit de data. Wat wèl duidelijk uit de data volgt, maar wat ze niet benoemt, is dat ‘lowability’ leerlingen slechter af zijn met het ongestructureerde onderwijs op Phoenix Park dan op Amber Hill. De prestaties aan het einde op de GCSEexamens waren zeer slecht, zowel bij Phoenix Park als bij Amber Hill.
Boaler heeft na 8 jaar een vervolgonderzoek ingesteld: hoe succesvol zijn de leerlingen geworden? Je verwacht het niet: de oudstudenten van Phoenix Park (met zo ongeveer de slechtste prestaties van heel GrootBrittannië) “had significantly moved up the socialclass scale, compared to their parents, whereas those of Amber Hill had stayed at the same socialclass levels.” (Boaler)
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Het ‘Railside’onderzoek
Het ‘Amber Hill’onderzoek werd overgedaan in de VS: het ‘Railside’onderzoek. In 2008 publiceerde Jo Boaler samen met Megan Staples hun onderzoeksresultaten in het ‘Railside’rapport. Railside, een problematische school, een groot percentage minderheden, slechte wiskunde resultaten, maakte de overstap naar ’Reform’ Math’.
‘Reform Math’ is een leermethode, waarbij leerlingen zich nieuwe wiskundige begrippen eigen maken d.m.v. realistische onderzoeksprojecten. Ze gaan zelf op zoek naar oplossingsmethoden. De nadruk ligt op communicatie, sociale vaardigheden, samenwerken, het uitleggen aan andere leerlingen en het presenteren van de gevonden resultaten. Men werkt in heterogene groepen onder het motto: iedereen is wel ergens goed in.
De resultaten van haar onderzoek waren onthullend. Na een paar jaren zorgde Railside voor een spectaculaire ‘outperformance’ t.o.v. de 2 controlescholen waar traditioneel wiskundeonderwijs gegeven werd; ook kozen de leerlingen massaal voor (ook de zware) wiskundevakken; ze hadden meer plezier en ze bereikten hogere niveau’s, dan bij de controlescholen. Vooral vrouwen en minderheden gingen er flink op vooruit. Maar ook de goede leerlingen presteerden beter dan leerlingen in topklassen op traditionele scholen. Aldus Boaler. Als belangrijke reden voor dit succes zag Boaler de samenwerking binnen de wiskundeafdeling, de gedeelde verantwoordelijkheid tussen studenten, heterogene klassen, en de realistische multidimensionale open opdrachten. Boaler beweerde dat ze om privacyredenen de echte namen van deze 3 scholen niet kon noemen ‘but trust me’. Het rapport sloeg in als een bom, ook bij politici.
Jo Boaler [The case of Railside school] [A Multidimensional Mathematics Approach with Equitable Outcomes]:
 The high achievers at Railside also learned significantly more than the high achievers who went into top sets in the traditional schools. Many people worry about the high achieving students who may be held back in mixed groups, but we found that the students were advantaged because they spent time explaining work, which helped their own understanding, and they were able to think more deeply about maths, rather than rushing through more and more work, as typically happens in top set classrooms.
 In analysing the success of the Railside approach, we concluded that many more students were successful, because there were many more ways to be successful. A student who may not have been the fastest or best at following and executing methods, could be very successful if they asked good questions, or saw problems in different ways, explaining them well to others. One of the messages that the teachers frequently gave was that “noone is good at all of these ways of working, but everyone is good at some of them”.
Kritiek
Twee wiskundehoogleraren, James Milgram (Stanford University, dus collega van Boaler) en Wayne Bishop, samen met de statisticus Paul Clopton, onderzochten de claims van Boaler: [A Close Examination of Jo Boaler’s Railside Report]. Ze wisten de namen van de scholen te achterhalen. Ze ontdekten zeer ernstige tekortkomingen in haar onderzoek o.a.:
 Statistische fouten. De meest relevante data leiden juist tot omgekeerde conclusies
 Ze vergeleek de slechte leerlingen van de controlescholen met de goede leerlingen van Railside
 Ze gebruikte gegevens van studenten, die niet bij dit experiment betrokken waren
 Boaler gebruikte voor de leerlingen van Railside aangepaste testen; deze waren niet curriculumdekkend, het niveau lag ver beneden het niveau dat verwacht had mogen worden, veel vragen waren geen echte wiskundevragen en sommige vragen bevatten fouten (b.v. driehoeken die niet kunnen bestaan)
 Op de staatsexamens presteerden de Railsideleerlingen heel slecht
 Niemand op Railside deed de ‘Advanced Calculus Test’. Dit staat haaks op Boaler’s beweringen dat er ook massaal gekozen werd voor de zware wiskundevakken
 Relatief veel meer leerlingen kregen later grote problemen bij vervolgstudies
 Railside is later teruggegaan naar het traditionele wiskundeonderwijs vanwege de slechte resultaten met ’Reform’math’
Milgram’s en Bishop’s conclusies werden daarna bevestigd door andere ‘nacheckers’.
[Educational malpractice for the sake of reformmath]: “A high official in the district where Railside is located called and updated me on the situation there in May, 2010. One of that person’s remarks is especially relevant. It was stated that as bad as Milgram et al’s original paper indicated the situation was at Railside, the school district’s internal data actually showed it was even worse. Consequently, they had to step in and change the math curriculum at Railside to a more traditional approach. Changing the curriculum seems to have had some effect. This year (2012) there was a very large (27 point) increase in Railside’s API score and an even larger (28 point) increase for socioeconomically disadvantaged students, where the target had been 7 points in each case.”
Boaler verweerde zich door te stellen dat het artikel van Milgram en Bishop geen waarde had omdat er geen ‘peer review’ geweest was (die was er wel geweest), ze deed een beroep op de privacy (geen enkele wet verbood haar om de namen van de scholen bekend te maken) en klaagde over de ‘brutaliteit’ en ‘agressiviteit’ van Milgram en Bishop, maar tot op de dag van vandaag is ze niet in gegaan op de ernstige beschuldigingen over haar onderzoek. Wel klaagde ze Bishop aan bij Stanford Police.
In 2012 publiceerde Boaler op haar website een aanklacht tegen Milgram en Bishop: ‘When academic freedom becomes Harassment and Persecution’. Dit leidde tot wereldwijde reacties en een petitie voor steun aan Jo Boaler. Namens Nederland tekenden o.a. Kees Hoogland, Prof. Eijkelhof en medewerkers van het Freudenthal Instituut. Zie hierover [De JoBoaler petitie].
Voor de reacties van Milgram en Bishop op Boaler’s aanklachten, zie [Milgram on Boaler], [Bishop on Boaler].
Kritiek op haar Railsiderapport heeft haar invloed allerminst aangetast, integendeel. Ze speelt de slachtofferrol, haar volgelingen steunen haar blindelings en veroordelen unaniem Bishop en Milgram.
Jo Boaler:
 In 2006 Milgram claimed that I had engaged in scientific misconduct. This is an allegation that could have destroyed my career had it been substantiated.
 Milgram and Bishop would not have harmed me if I was a man. It has also to do with mathematics being seen as a male domain.
“Steun Jo Boaler in haar strijd voor goed rekenwiskundeonderwijs en tegen ‘Academic harassment and personal attacks.’ ”
Kees Hoogland, twitter
“In the Netherlands we also have a group mathematicians who repeatedly unfoundedly malign the research done at the Freudenthal Institute.”
Marja van den HeuvelPanhuizen, Hoogleraar rekenwiskundedidactiek, in de steunbetuiging aan Jo Boaler
“I have long had great respect for Jo Boaler and her work, and I have been very disturbed that it has been attacked as faulty or disingenuous. The critiques by Bishop and Milgram of her work are totally without merit and unprofessional. I’m pleased that she has come forward at last to give her side of the story, and I hope that others will see and understand how badly she has been treated.”
Jeremy Kilpatrick, hoogleraar ‘maths education’
“Ik ken Jo Boaler en Jeremy Kilpatrick persoonlijk. Beiden zijn in de internationale wiskundeonderwijswereld hoog gewaardeerd. Jo Boaler wordt in haar werk tegengewerkt door een paar ‘mathwar’ personen binnen haar eigen universiteit. Ik vindt de activiteiten van de ‘math war’ people in Amerika in veel opzichten vergelijkbaar met de manier waarop Joost Hulshof, Jan van de Craats, Ben Wilbrink en wellicht nog vier andere personen menen een land/ de wereld te kunnen inrichten volgens het model van een zeer aggressief werkende, selecte, absolute minderheid van achterhoedevechters.”
Henk van der Kooij (Freudenthal Instituut)
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Boaler over ‘memorisation’
In October 2016 verscheen het OECDrapport: [‘Ten Questions for Mathematics Teachers and How PISA Can Help Answer Them’].
Dit rapport bevatte de claim dat memorisers (leerlingen die de nadruk leggen op het van buiten leren van procedures) tot de slechtste presteerders behoren. Dit werd al jaren eerder beweerd door Boaler. Voor dit onderzoek, verricht door Jo Boaler en Pablo Zoido, werd gebruik gemaakt van vragenformulieren die leerlingen moesten invullen bij de PISA2012toetsen. Deze data werden nu pas beschikbaar gesteld.
Jo Boaler schreef samen met Pablo Zoido een artikel voor de november 2016editie van Scientific American over dit onderzoek:
[Why Math education in the U.S. doesn’t add up]
 I am excited to share with you the release of a Scientific American article I wrote with Pablo Zoido on the dangers of memorization approaches in maths. Together we analyzed data from 13 million students worldwide.
 In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment.
 In no country were memorizers in the highestachieving group, and in some highachieving economies, the differences between memorizers and other students were substantial.
Jo Boaler:
 The PISAtests (OECD) show that the lowest achieving kids in the world are those that are memorizers. The highest achieving kids are those looking for big ideas, who think about connections; they are approaching math more conceptually.
 What we do in classroom: we give kids 50 questions that they have to do in 3 minutes. Being a better memorizer does not mean anything, but in classroom if you are a better memorizer you get a push forward.
 Many kids go down a faulty pathway, a pathway that is very damaging to them early in their career. That pathway is one where they think math is a subject where you have lots of rules to remember. They don’t see a role for thinking because they’ve been taught very procedurally. They see math as a long ladder, if you will, of rules upon rules upon rules – They’re not connected, they don’t mean anything. You just have to remember them.
 The more we emphasize memorization to students, the less willing they become to think about numbers and their relations and to use and develop number sense.
 The UK and Ireland top the world in maths memorisation. Teachers and students deserve better.
 In a recent brain study scientists examined students’ brains as they were taught to memorize math facts. The researchers found that the students who memorized more easily were not higher achieving, they did not have what the researchers described as more “math ability”, nor did they have higher IQ scores.
 Brain researchers found that the two approaches, strategies and memorization, involve two distinct pathways in the brain. The study also found that those who learned through strategies achieved “superior performance” over those who memorized.
 The brain can only compress concepts; it cannot compress rules and methods. Therefore, students who do not engage in conceptual thinking, and instead approach mathematics as a list of rules to remember, are not engaging in the critical process of compression, so their brain is unable to organize and file away ideas; instead, it struggles to hold onto long lists of methods and rules. This is why it is so important to help students approach mathematics conceptually at all times.
 Students develop a connected view of mathematics when they work on mathematics conceptually and blind memorization is replaced by sense making.
 You become successful by seeing that there are just a few big ideas in math that you need to link together and think about in depth. And once you’ve understood the core ideas in math, everything kind of comes together. There’s really very little to remember.
 Math facts are a very small part of mathematics and problably the least interesting part. Conrad Wolfram speaks publically about the need to stop seeing mathematics as calculating.
Kritiek
Greg Ashman is Wiskundedocent uit Australië en onderwijsonderzoeker.
Ashman is bekend om zijn veelgeprezen blogs waarin hij onderwijsmythes ontmaskerd, ondersteund met veel literatuurverwijzingen.
Ashman onderzocht deze data. Hij had er geen 4 jaar voor nodig, binnen een paar dagen, publiceerde hij een aantal blogs, waaronder: [PISA data on maths memorisation].
Een aantal kritiekpunten op dit OESOonderzoek:
De manier van toetsen kan wetenschappelijk niet door de beugel. De vragenlijsten die leerlingen moesten invullen bevatten vragen waarvan niet duidelijk is waarom ze van invloed zijn op de ‘index of memorisation’. Zo werd gevraagd hoe men proefwerken/examens leert. Het betreft hier dus thuisstrategieën, en het zegt dus niets over het onderwijs dat men gehad heeft. Of misschien toch wel: iemand die op de ‘studentcentered’ manier onderwijs heeft gehad, kan wel eens bij het leren van een proefwerk meer de nadruk gaan leggen op het leren van de procedures.
De correlatiecoëffient bij de plot is zo laag dat hier niets uit te concluderen valt. Ook Prof. Daniel Ansari heeft een en ander nagerekend: er is eerder bewijs dat er geen enkele correlatie bestaat tussen ‘memorisation’ en wiskundeprestaties.
Ashman bekeek de data bij ‘studentoriented instruction’ versus ‘teacherdirectedinstruction’ en de behaalde wiskunderesultaten. En nu is er wel een grote correlatie: ‘TEACHERDIRECTED INSTRUCTION LEIDT TOT BETERE WISKUNDEPRESTATIES’!!!. Hoe hadden Boaler en OESO dit kunnen missen? Ashman: “It is extraordinary that the authors highlight memorisation and teacherdirectedness when this elephant is occupying the parlour, stamping its feet and trumpeting the tune of La Marseillaise.”
De volgende landen voldoen het beste aan de hier geadviseerde OESOcriteria voor goed wiskundeonderwijs: Thailand, Jordanië, Quatar, Maleisië en Tunesië. GrootBrittannië en Ierland staan helemaal onderaan. Toch waren de PISAresultaten van de eerste groep landen slecht. OESO probeert in het rapport ons wijs te maken dat de landen in het verre oosten niet meer zoveel doen aan memoriseren. Canada en Finland voldoen al geruime tijd aan deze OESOadviezen; de wiskunderesultaten zijn daar sindsdien gekelderd.
Reacties op het Scientific Americanartikel van Boaler en Zoido door:
Greg Ashman [Why the Scientific American article on maths education doesn’t add up]
Paul Bennett (Hoogleraar Onderwijskunde Nova Scotia, Canada. Directeur en Lead Researcher ‘Schoolhouse Institute’)
[PISA Mathematics Lessons: Why ZeroIn on “Memorization” and Minimize TeacherDirected Instruction?]
Paul Bennett:
 Memorization has become a dirty word in teaching and learning laden with so much baggage to the point where it conjures up mental pictures of “drill and kill” in the classroom.
 Simply ignoring research that contradicts your ‘metabeliefs’ is common on the Math Education battlefield. Recent academic research on “memorization” that contradicts Boaler and her entourage, is simply ignored, even that emanating from her own university. Two years ago, Shaozheng Qin and Vinod Menon of Stanford University Medical School led a team that provided scientificallyvalidated evidence that “rote memorization” plays a critical role in building capacity to solve complex calculations.
 Ontario is the only Canadian Province not to have registered gains in maths over the past decade with Jo Boaler advising Ontario’s Ministry of Education.
Andreas Schleicher (hoofd PISA) en Jo Boaler hebben tot op de dag van vandaag niet gereageerd op deze kritiek en kritiek van anderen op dit onderzoek.
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Boaler over Breinresearch
Boaler heeft zich verdiept in het breinonderzoek.
Jo Boaler:
 Science of the brain shows that anyone can do well in maths and there is no such thing as a ‘maths person‘.
 Any child can excel in maths at all levels in school. Anybody can be brilliant at maths.
 There’s no such thing as a math person, no such thing as a math brain. But the idea that only some kids have a strong math brain is strong in our society.
 People have to understand that any math trauma or anxiety that’s set up, has come from the math experiences they’ve had. It’s not you, it’s something that has been done to you or happened with you. And then I show them the evidence on the brain and on how we learn, that shows them that anybody can do good at math.
 Researchers now know that when people with math anxiety encounter numbers, a fear center in the brain lights up — the same fear center that lights up when people see snakes or spiders.
 New scientific evidence showing the incredible capacity of the brain to change, rewire, and grow in a really short time suggests that all students can learn mathematics to high levels with good teaching experiences. Traditional educators believe that some students do not have the brains to be able to work on complex mathematics, but it is working on complex mathematics that enables brain connections to develop.
 Another study that is very important for educators is a threeweek training program, where people working for 10 minutes a day changed the permanent structure of their brains.
 What we are finding from brain research is that when students make a mistake in maths they grow a new synapse. That is brain growth that come from the sparks in the brain, that does not happen when people get work correct. Making mistakes in maths is the most useful thing students can do. Researchers also find that the brain growth was greater in growthmindset individuals that in fixedmindset individuals.
 Synapses do not fire when your answer is correct. So we actually want kids to make mistakes. When you don’t fail, you are not learning.
 Good teachers have said this for a long time — we can learn from mistakes. But this is a much more powerful message: that we can learn only from making mistakes. We need kids making mistakes. If kids are not making mistakes we’re limiting their brain growth.
 A research study found that when people make mistakes their brains grew more than when they got work right. Synapses are fired: the first comes when you make a mistake and the second comes if and when you’re aware that you’ve made a mistake. So the first occurs before the participants knew that they had made a mistake!!
 Our brains are like prediction machines. They predict ahead of us what will happen. When we make a mistake dopamine is released.
 Many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible. Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development.
 In fact, the quality of the 6yearold’s finger representation was a better predictor of future performance on math tests than their scores on tests of cognitive processing.
 Researchers have found that the students who do best on math tests are those who use the connections between the brain hemispheres. The perfect brain crossing is when kids interact with numbers while also thinking visually.
 Researchers also found that when students were working on arithmetic problems, the highest achievers were those who exhibited the strongest connections between the two sides of the brain.
 Sian Beilock and her colleagues studied people’s brains through MRI imaging and found that math facts are held in the working memory section of the brain. But when students are stressed, such as when they are taking math questions under time pressure, the working memory becomes blocked. They start to develop anxiety and their mathematical confidence wears away.
Kritiek
Er is veel kritiek op Boaler’s beweringen, ook van neurowetenschappers.
“Error helps learning, but only when it is a near miss.”
Uit ‘ScienceDaily’ juni 2018
Dr. Daniel Ansari (Hoogleraar Psychologie aan de Western University Canada)
Ansari doet onderzoek naar de wiskundige ontwikkeling van kinderen en individuele verschillen in cijfer en wiskundige vaardigheden, op cognitie niveau en neuraal niveau.
 The representation of neuroscientific evidence in this article [‘Not a math person‘, Jo Boaler] is shocking, to say the least.
 Sorry to disappoint, but the great research on brain plasticity for London Taxi Drivers tells us nothing that speaks to Math Education.
 The conclusion: “When a student made a mistake, a synapse is fired, even if the student wasn’t aware of the mistake” is plain wrong. There is no evidence linking mistakes to actual ‘growth’ of the brain.
 At a presentation Jo Boaler told : “Brain research shows the danger of relying on memorisation and speed.” Interesting, but where is the brain research in this?
Dr. Yana Weinstein (Psycholoog. Cofounder van ‘Learning Scientists‘).
Weinstein doet onderzoek naar geheugenprestaties en cognitieve functies.
 Why are not more people calling bullshit on neuromyth spreading á la Jo Boaler?
 We can’t just stand by and let her brainwash.
 Are we just that desperate to help kids learn math that we are happy to sacrifice reality and make up science?
Greg Ashman:
 Jo Boaler claims: “A research study found that when people make mistakes their brains grew more than when they got work right”. Boaler had mentioned the paper on Twitter when asked about it by Daniel Ansari. It is a 2011 article by Moser et. al. and is freely available online so you can read it for yourself. In this experiment, 25 participants have to pick out whether the central letter in a string of letters is congruent with the surrounding or ‘flanking’ ones. For example, in the first trial, “M” and “N” are used. The participants are under severe time pressure. The participants had electrodes attached to their heads to measure electrical activity known to occur in the processing of mistakes. Note that they do not measure other brain activity. This means that the fact that they found more activity when participants made mistakes is both unsurprising and a finding that does not rule out the possibility that there was even more activity of other kinds in other areas of the brain when participants got the answers correct. Also, recording a voltage in this way is not the same as concluding that the brain has ‘grown’. Yet brain growth is the repeated claim. The fact that one of these electrical bursts occurs before participants are aware of their mistake now seems less mysterious. These particular experiments seem very far removed from somebody solving a typical school maths problem, making an error, not realising they have made an error and having their brain grow in response.
David Didau (Exdocent; hij heeft een ‘education’blog ‘The Learning Spy‘ ; schrijver van boeken over onderwijs)
 It doesn’t help when highprofile and influential academics insist in passing on misinformation about the brain: Your brain does not “grow when you are challenged” nor is it “like a muscle”. A firing synapse does not constitute brain growth, and ‘brain growth’ does not equate to learning. It’s not as if she [Boaler] doesn’t know that her interpretation is, to put it politely, disputed so one wonders why she continues to put it about as uncontested fact.
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Boaler over ‘Mathematical Mindsets’
Boaler werkt samen met Stanford collegahoogleraar Carol Dweck i.v.m. Dweck’s populaire leer over de ‘Growth‘ vs. ‘Fixed Mindset’: wie gelooft in aangeboren vaardigheden waar verder weinig aan te veranderen valt, de Fixed Mindset, presteert minder, dan wie gelooft in de Growth Mindset.
Boaler’s boek ‘Mathematical Mindsets’, met een voorwoord van Carol Dweck, werd een bestseller met vrijwel uitsluitend 5sterren reviews.
[Developing Mathematical Mindsets]
Jo Boaler:
 Growth Mindset for me is always believing you can do anything!
 When you believe in yourself your brain operates differently.
 Students with a “growth” mindset perform at higher levels in math and in life.
 To teach students a growth mindset and general positive messages about mathematics learning, teachers should abandon testing and grading as much as possible. If they do continue to test and grade, they should give the same grade, or higher for mistakes, with a message attached that the mistake is a perfect opportunity for learning and brain growth.
 If you’re giving kids math lessons which are a series of short, closed questions with right and wrong answers, they won’t develop a growth mindset because they won’t see any room for growth and learning in those questions.
 Students with no experience of examinations and tests can score at the highest levels because the most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problemsolving mathematical tools to equip them for any mathematical situation.
 Research has shown that students only have to think they’re being graded for their achievement to go down.
 In a recent summer camp the youcubed team taught mindset and brain messages to local 6th and 7th grade students. After 18 days of math teaching the students improved their scores on standardized tests questions by an average of 50%.
 If brains can change in 3 weeks, imagine what can happen in a year of math class.
 PISAdata coming from 13 million 15yearold kids all over the world show a huge difference between kids with a growthmindset and those with a fixedmindset. Years of math separated them.
 Research on the brain tells us that the difference between successful and unsuccessful students is less about the content they learn and more about their mindsets.
 In an important study researchers found that when mothers told their daughters they were not good at math in school, their daughter’s achievement declined almost immediately.
Kritiek
Dweck’s leer is omstreden, lees o.a. [Mindset revolution based on shaky science].
Veel onderzoeksconclusies van Dweck blijken statistisch niet significant te zijn; de rapporten van Dweck en medewerkers bevatten vooral zelfreferenties. Andere onderzoeksgroepen hebben deze resultaten niet kunnen reproduceren.
“People with a growth mindset don’t cope any better with failure. If we give them the mindset intervention, it doesn’t make them behave better. Kids with the growth mindset aren’t getting better grades, either before or after our intervention study.”
Timothy Bates, psycholoog aan de Universiteit van Edinburgh
”Wat wel duidelijk is, is dat de noodzaak in het onderwijs rekening te houden met de mindset van de leerling, en energie te steken in het omvormen ervan tot een growth mindset, op dit moment niet door onderzoeksdata ondersteund wordt.”
Casper Hulshof (Onderwijskundige)
“I’ve read Boaler’s book, Mathematical Mindsets, and I found it very creative in the sense that the author seemed to have a rare ability to make arithmetic far more complicated than it needs to be.”
Greg Foley (Chemical engineer, Lecturer at Dublin City University)
Tussen alle euforische reviews op Amazon op Boaler’s boek: ‘Mathematical Midsets’, vinden we ook een interessante 1ster review:
“What about the kids that love math? Group think and group discussions, group problem work does not appeal to these children, who are most likely to become the next engineers and scientists. Write an essay explanation to describe your thinking on how you solved a problem? Give me a break. Math is a precise language of its’ own. Indeed, it is the language of engineering and science. There is a right and wrong answer and the lack of subjectivity and exactitude is why mathematics and science appeals to some. I have a real tough time with College of Education professors expounding on how to teach math when they themselves never take an advanced math class above basic calculus. How many teachers at the middle school and high school level even take Calculus? Check out the math class requirements for teaching majors within the college of education at most campuses — it is shockingly basic. Trying to make math appeal to all will inevitably discourage those kids drawn to the subject; the same kids we need to be pursuing engineering and science.” Suxie B
Een analyse van Boaler’s boek ‘Mathematical Midsets’ lezen we hier: ‘The Many Myths in Mathematical Mindsets‘.
Bryan Penfound (Docent wiskunde aan de lerarenopleiding van de Universiteit van Winnipeg Canada)
[Missing Messages from Jo Boaler’s Maths Video]
 Boaler’s Message: “Believe in yourself”. Let’s consider the argument of intrinsic motivation. Recently Greg Ashman, citing a longitudinal study, reminded us, that it is actually achievement in mathematics that predicts intrinsic motivation, and not the other way around. Believe in yourself, yes. But belief in yourself can only take you so far. Eventually you will need to develop content knowledge in mathematics, and developing this knowledge takes time, hard work and effort. The more content knowledge you have in mathematics, the more likely it is that your achievement in mathematics will increase. Higher mathematics achievement may then lead to higher intrinsic motivation, selfefficacy and confidence – leading you to have more belief in your abilities.
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Nog meer beweringen/’onderzoeksresultaten’ van Jo Boaler
Jo Boaler:
[Fluency without fear] [Beautiful Maths] [Youcubed] [Mathematical Modelling and New Theories of Learning] [Memorizers are the lowest achievers] [Elitist Math] [Jo Boaler, Revolutionalizing math education] [Why Kids Should Use Their Fingers in Math Class] [Solving our math problem]
 Giving children tests on their times tables is creating huge damage.
 I have never memorized my times tables. I still have not memorized my time tables. It has never held me back, even though I work with maths every day.
 I was never forced to even memorize times tables and still do not know them all by heart to this day. I have number sense, which we know is much more important. I have a feel for numbers and can work them out quickly.

The education minister for England insisted that all students in England memorize all their times tables up to 12 x 12 by the age of 9. This requirement has now been placed into the UK’s mathematics curriculum and will result, I predict, in rising levels of math anxiety and students turning away from mathematics in record numbers.
 Mathematician Francis Su describes the memorization of the 12×12 multiplication table as one of the most meaningless activities possible.
 Rote learning times tables is one of the worst things we do to kids. Nothing tells us rotelearning is good. One thing it does is turn kids off.
 Automaticity does not equal memorisation, it equals number sense.
 When students focus on memorizing times tables they often memorize facts without number sense, which means they are very limited in what they can do and are prone to making errors.
 We now know that what separates low achievers from high achievers in math is not how much they know, it’s how willing they are to engage with numbers flexibly.
 Time tables tests cause the early onset of maths anxiety.
 In fact, we know that timed tests cause a lot of the early onset math anxiety felt by students, especially girls.
 Why o why do Governments think they know better then mathematics educators? The British Government will be responsible for future math anxiety.
 We have a nation of mathtraumatized people. Even students who are successful are learning an antiquated set of methods they probably will never use.
 What we do with kids when we prioritize times tables is that we give kids the believe that this is what maths is: it’s about memorising facts quickly. That isn’t what maths is and it’s a terrible message to give to kids and that is part of the problems we have.
 Number sense is inhibited by overemphasis on the memorization of math facts.
 When students fail algebra it is often because they don’t have number sense.
 The key to success in math is having something called “number sense,” and number sense is developed through “rich” mathematical problems. Too much emphasis on rote memorization inhibits students’ abilities to think about numbers creatively. In my own study I found that low achieving students tended to memorize methods and were unable to interact with numbers flexibly.
 Lack of number sense has led to more catastrophic errors, such as the Hubble Telescope missing the stars; the telescope was looking for stars in a certain cluster but failed due to someone making an arithmetic error in the programming of the telescope.
 As Keith Devlin reflects: ‘Mathematical notation no more is mathematics than musical notation is music’.
 If you look at most math classrooms today, they are not that different from Victorian days.
 Most of the mathematics taught right now at school is over 400 years old and it is not the mathematics that students need.
 Students are taught content that often appears as a long list of answers to questions that nobody has ever asked.
 Multidimensional math is the math that mathematicians use, and it is the math that is needed in the world. It’s about asking questions, about problemsolving, about reasoning, about creativity, about communication.
 Algebra is taught in a way that is against all the knowledge we have gained from research about effective learning. Many students will say that algebra is just an act of manoeuvring meaningless symbols.
 In many places an algebra class starts with solving for X. So that’s finding a particular value of X in an equation. That’s actually a very narrow piece of algebra and the key idea of algebra is that this X thing represents a variable which isn’t captured in this solving for X. So we lead kids down the wrong path early on and they solve for X and then they’re told, X can actually be anything. It can be a variable. And that causes a huge conceptual block for them.
 Good algebra tasks are the ones that are more open, where they have different ways of seeing the math, different pathways through the problems, different representations. They involve sensemaking instead of just calculating without any thought.
 More knowledge was created between 19902003 than the history of the world before that. Preparing kids means preparing flexible maths thinking.
 Academic research has consistently found homework to either negatively affect or not affect achievement.
 PISA conducted a survey of 13 million students to study the relationships between homework, achievement and equity, and found that homework is inherently inequitable, and that it didn’t seem to raise achievement for students.
 Can we please start a movement to stop homework? It creates stress, inequity and it kills childhood.
 Replace homework with reflection questions:
– what did I learn today?
– what good ideas did I have today?
– where could I use the knowledge I learned today?
– what new ideas do I have that this lesson made me think about?  Group work is a strategy I regard as critical to good mathematics work.
 Many parents have asked me: what is the point of my child explaining their work if they can get the answer right? My answer is always the same. Explaining your work is what, in mathematics, we call reasoning. And reasoning is central to the discipline of mathematics.
 If you engage kids with particular needs, or kids who really have difficulty with remembering facts, in a problem solving task and get them to use their thinking, they can be better than kids who are traditionally high achievers.
 Timed tests evoke such strong emotions that students often come to believe that being fast with math facts is the essence of mathematics.
 Let’s work together to stop the performance culture in schools. It stole maths from us and ruined our kids’ relationships with maths.
 Students believe that the best mathematical thinkers are those who calculate the fastest—that you have to be fast at math to be good at math. Yet mathematicians are often slow with math. I work with many mathematicians and they are simply not fast math thinkers.
 We dissuade many children who are slow deep thinkers in maths.
 We want kids really doing in classroom what mathematicians do. School mathematics is a strange set of rituals which is nothing like the maths in the world and the maths that the mathematician choose. At school there is a large amount of long calculations by hand. We still see highschoolers sitting in classrooms going through long written solutions about quadratic equations. They are learning to calculate. But for one thing: we don’t need kids calculating. To repeat questions 40 times with different numbers.
 There is a large body of research that says evaluative feedback does not enable learning and in fact often causes the student to stop learning. Feedback that is based on next steps for improvement has proved far more effective.
 When flower seeds grow in spirals they grow in the ratio 1.618:1. Remarkably, the measurements of various parts of the human body have the exact same relationship. Examples include a person’s height divided by the distance from tummy button to the floor; or the distance from shoulders to fingertips, divided by the distance from elbows to fingertips. [In wiskundelessen hoort men juist te waarschuwen voor deze populaire mythes.]
jongens/meisjes
 The lovely thing is when you change math education and make it more about deep conceptual understanding, the gender differences disappear. Boys and girls both do well.
 Gender differences appear when we teach math by drill and practice, in an impoverished way when there is no room for asking questions. Girls are happier when we teach mathematics in a conceptual way in a more connected way, as a broad multidimensional subject.
 Research shows that girls need to explore subjects in depth, while boys are more prepared to accept rote learning.
 I’ve found in my own research that woman are more likely to reject subjects that do not give access to deep understanding.
 For many girls the identities they see on offer in mathematics and science classrooms are incompatible with the identities they want for themselves. They see themselves as thinkers and communicators and people who can make a difference in the world; in procedural classrooms they come to the conclusion that they “just do not fit in”. This relates in part to the lack of good role models but it also relates to the forms of knowledge that are privileged in many mathematics and science classrooms that leave no room for inquiry, connections or depth of understanding.
 I can’t tell you how many women, undergraduates at Stanford and others I’ve spoken to, said to me, ‘I was going to go into science, engineering, whatever, but I didn’t because of math’.
 A lot of woman choose out of maths because at university maths is often a very unwelcoming environment for young woman.
 Milgram and Bishop would not have harmed me if I was a man. It has also to do with mathematics being seen as a male domain.
 Claude Steele, a former colleague of mine at Stanford, found that when girls take math tests, just marking off their gender makes them more likely to underachieve. The stereotypes about women and math are so strong that they’re in the air all the time.
minderheden
 Some people don’t want everyone doing well in math. When we show that girls and minorities do just as well, they don’t like that because they want math just for some people. They are going to great lengths, even to squash the evidence we have. They want math to stay the elite preserve for some people. That if you can do math you are better than other people. It is all feed into this elitist production of math. People use math as tests whether you can pass the doors in this part of society where you do well. All that is wrong. Math appears harder than other subjects because of the terrible way it is taught. We teach it as if math is just for some students.
 The attacks that are being directed at scholars in mathematics education, all of them women and people of color, are import to consider. They are coming from right wing organisations.
 Math teachers may not be intentionally discriminating by race or ethnicity, but if they use other criteria, such as homework completion, that impact students of color more than other students, they are breaking the law.
 So many people have been damaged by the elitism in maths – women and people of color being made to feel they don’t belong is just the beginning.
 When mathematics is taught as a connected, inquirybased subject, inequities disappear and achievement is increased overall.
Kritiek
“It has been frequently reported in newspapers that there is a strong association between speeded math and math anxiety. Contrary to these claims, there does not, at present exist evidence to support this popular claim. And indeed, it has been shown that speeded practice can help 1st grade students at risk of developing mathematical difficulties to compensate for weak reasoning abilities.” ‘Math Anxiety: An Important Component of Mathematical Success‘, Daniel Ansari, Erin Maloney, Jonathan Fugelsang
Robert Craigen (Hoogleraar Wiskunde aan de Universiteit van Manitoba)
 I wish Boaler and her ilk would stop referring to retrieval practice as “testing”. Apparently that’s what she’s referring to, as the standard way to memorize times tables is to set up a frequent feedback cycle to give the student the rewarding experience of watching their own progress in a lowpressure environment. I fail to see how that can cause math anxiety. Seeing one’s own progress, in fact, is a pretty good way to improve selfesteem when faced with the daunting process of mastering a complex and abstract discipline covering a huge body of knowledges.
 We don’t look for original thought as much as we look for welldisciplined thought. The latter often reflects greater understanding, especially when the subject is highly technical.
Greg Ashman:
[Jo Boaler is wrong about multiplication tables] [Can Jo Boaler grow your brain?] [Math anxiety]
 The fact that Boaler never uses times tables as a maths education professor tells us something but I’m not sure it tells us much about the value of tables in solving maths problems. I am prepared to accept that you can get by without times tables. My question would be; why would you want to? Why make life hard?
 A new paper by Cambridge University researchers suggests that the negative relationship between maths anxiety and maths achievement is cyclical. This basically posits that maths anxiety is caused by a lack of ability: the deficit theory. The teaching implication of this is that we should teach maths in the most effective way possible in order to improve competence and reduce anxiety. We might also consider giving students experience of success with some relatively straightforward work. Jo Boaler is a prominent populariser of the debilitating anxiety theory amongst maths teachers and she advises that we avoid timed tests because they have been shown to induce anxiety. However, she also advocates openended problemsolving which is likely to overload working memory and which would not provide the routine competence that the deficit theory implies students need.
Kris Boulton (Wiskunde docent)
[The idea that it is wrong to memorize times tabels is a Zombie myth]
 It’s happened again: another wouldbe educational revolutionary has decried the memorisation of times table facts. The message Jo Boaler puts out there as a “Stanford professor” is old, tired and terrifying.
 The fact is, we already have moved away from times table memorisation in schools, and the results have been catastrophic; those of us in secondary education bear witness to this every day.
 The key to this debate lies in understanding how working memory functions. Boaler argues that she doesn’t need to know her times tables because she has processes for calculating the results quickly, as needed. For a child with smaller than average working memory capacity, trying to process any kind of midlevel arithmetic or basic algebra becomes a hellish nightmare, as they are still left painfully trying to process basics that should have been embedded from a very young age.
 The call to demonise times table memorisation further misses an important and fundamental component of mathematical development; the idea of relational understanding. If you had to recalculate the value of 7 x 8 every time you needed it, you would never appreciate that there is a relationship between the numbers 7, 8 and 56. The damage this does is evident every time we have to watch a child struggle in pain to do something as straight forward as simplifying the fraction 49/56, or factorising 28x + 56. For someone who knows their times table facts, this process is simple because the numbers 28, 49 and 56 are all intimately connected to the number 7; it’s painless, even obvious, that 49/56 should become “something over 8”, since 56, 7 and 8 are connected as a related triad of numbers. Being able to “notice” this renders such operations facile; being unable to notice this makes such operations painful and frustrating.
 Ironically, it is being abandoned to this mental abyss that imprisons a person in the very state of anxiety that Boaler would have us avoid.
Charlie Stripp (Directeur National Centre for Excellence in the Teaching of Mathematics, GrootBrittannië)
[It is wrong to tell children that they do not need to memorise their times tables]
 It is not the learning of times tables that is causing anxiety but rather it is lack of times table knowledge that is causing the anxiety. It should be an educational entitlement that all children are helped to learn their times tables.
 I’ve been in a primary school classroom recently where some of the previously lowerattaining children are making great conceptual progression. They understand the problems to be solved, they have strategies for tackling them. However, there is still one thing getting in the way: they don’t know their times tables and this is seriously inhibiting their progress, preventing them from using maths to solve problems.
 I have also experienced primary classrooms where all children do know their times tables. The confidence this gives them in tackling other areas of mathematics and in solving mathematical problems empowers them and enables them to really enjoy maths.
 My experience of working with adults who did poorly in maths at school suggests that they find working with numbers and other aspects of maths needlessly difficult because they lack the automatic recall of basic number facts. Their opportunities are restricted because they fear working with numbers. If only they had learned their times tables at primary school!
Uit: The Many Myths in Mathematical Mindsets
 Boaler: “When mathematics is taught as a connected, inquirybased subject, inequities disappear and achievement is increased overall.” Let’s see how true that is. Boaler explains that she takes her undergraduate class each year “on a field trip to Life Academy, a public school in Oakland that is committed to disrupting patterns of inequity on a daily basis.” She states that “The accomplishments of Life Academy are many; the school has the highest college acceptance rate of any high school in Oakland, and the proportion of students who leave ‘college ready’ with California’s required classes is an impressive 87%, higher than at the suburban schools in wealthy areas close to Stanford.” Okay… but… those look like cherrypicked stats… those are the types of numbers that can be manipulated, simply by giving students easy grades, or graduating them when they haven’t earned it. How much do they actually learn? Let’s see what GreatSchools.com has to say, based on actual performance:
Yet… here’s what the standardized tests show: How do 85% of students have a C or higher on AG classes, when only 14% of them passed standardized testing? This should be a red flag. And how does equity rate at this school? Not good, either… the only “equitable” thing at this school is that they perform equally poorly. What does this highlight? It actually highlights a serious problem: schools can “fudge” success very simply through grade inflation. While this might help students get into college, it sets them up for failure once they arrive, and in the years beyond. You want equity? This is the opposite of equity.  The reason the (influential) results of the (questionable) Railside study are so concerning, is that several districts and schools have bought into the “inquirybased” math curricula used, such as CPM (College Preparatory Math)… and the results have not been pretty. For one, Boaler claims that enjoyment and motivation are increased… but when I searched for reviews and opinions about CPM online, the response is quite different than what Boaler suggests. Many parents, teachers, and students have complained about the program — about its avoidance of direct instruction, about its insistence on group work for all aspects, and about its (apparent) lack of efficacy. One group of parents was so concerned that they created a “Fairfield Math Advocates” group to combat the adoption of CPM, and has a whole lot of data to highlight problems with it, as well as a list of myriad other advocacy groups that have arisen to rail against it. They also highlight how it caused fewer and fewer students — a disproportionate number of boys, especially — to complete the math program as years went on. So much for equity. In fact, I tried to find positive reviews of CPM, and the only positive messages I could find were by Boaler and her colleagues, or people who had direct professional connections to CPM.
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Boaler’s ‘Openended projects’
Jo Boaler:
 If we teach maths in the way mathematicians work on it, many more kids engage.
 The story of Andrew Wiles is fascinating. One clear difference between the work of mathematicians and schoolchildren is that mathematicians work on long and complicated problems that involve combining many different areas of mathematics. This stands in stark contrast to the short questions that fill the hours of maths classes and that involve the repetition of isolated procedures. Long and complicated problems are important to work on for many reasons, one of them being that they encourage persistence.
 The maths that millions of school children experience is an impoverished version of the subject that bears little resemblance to the mathematics of life or work, or even the mathematics in which mathematicians engage.
 Third graders can be fascinated by the notion of infinity, or the fourth dimension, but they do not need a race through procedural presentations of mathematics.
Typische ‘openended questions’ door Jo Boaler in een van haar boeken beschreven:
 An object has a volume of 216. What could it be? What would be its dimensions ? What would it look like?
 Find shapes with an area of 36.
 How many rectangles can you make with an area of 24?
 ∆PQR is a right triangle with hypotenuse PQ measuring 10 units. Find one pair of possible values for the lengths of PR and RQ.
Kritiek
Bovengenoemde ‘openended questions’ tonen een pijnlijk gebrek aan wiskundig inzicht.
Opdracht 1: An object has a volume of 216.
Verwachte antwoord: 216 = 6³, dus het is een kubus, dimensie 3.
Een ernstig misleidende opdracht. De getalwaarde 216 zegt helemaal niets over de dimensies. De getalwaarde is trouwens afhankelijk van de gekozen eenheid, door deze te veranderen kan men iedere gewenste getalwaarde verkrijgen. Maar ook bij een gegeven eenheid kan men uit het getal 216 niets over de solid afleiden: ook een bol kan een volume 216 hebben en dat geldt voor iedere solid met eindig volume.
Opdracht 3: How many rectangles can you make with an area of 24?
Verwachte antwoord: 8 want 24 = 1 x 24 = 2 x 12 = 3 x 8 = ……
Boaler ziet niet in dat een rechthoek bij gegeven oppervlakte iedere lengte kan hebben. Leerlingen krijgen dus weer verkeerde ideeën aangeleerd. Men had in plaats hiervan een interessante som kunnen maken over het splitsen van 24 in twee factoren; met behulp van priemfactorontbinding had men een systematische oplossing kunnen geven.
Jo Boaler heeft een aantal voorbeeldlessen op haar website uitgewerkt, inclusief werkbladen en een handleiding voor de leraar, zie [Low Floor High Ceiling].
Eén van deze lessen gaat over de ggd (grootste gemeenschappelijke deler van 2 of meer getallen): [Counting Cogs].
Leerlingen gaan aan de slag met tandwielen. Werkbladen worden bijgeleverd, leerlingen kunnen de wielen uitknippen. Er zijn 9 tandwielen met resp. 4, 5, 6, 7, 8, 9, 10, 11 en 12 tanden. Ook een handleiding voor de leraar is er te vinden. De les is geschikt voor ‘grade 5’ (1011 jarigen); er wordt geen voorkennis verwacht van de begrippen factor, priemgetal, ggd en kgv. Deze les werd in de praktijk gebracht op een school in Canada. Bryan Penfound (docent wiskunde aan de lerarenopleiding van de Universiteit van Winnipeg) hielp mee. Hier zijn verslag van deze les: [Counting Cogs from ‘youcubed.org’].
 The working memories of the grade five students quickly got overloaded. It was not that the students were not listening at the beginning of the lesson, it was because they would read the lesson, be overwhelmed by the number of tasks they had to do, and lose their indication of where to begin.
 So now the students were learning how to cut the cogs along the lines properly, rather than focusing on the mathematics that the lesson was supposed to be engaging the students in. I felt more like a babysitter for the first half of this class than a mathematics teacher.
 Think about it: students were expected to colour in one tooth, move the cogs around, follow where the coloured tooth would land, keep track of all the places where the coloured tooth landed, know what a cog was, know what a “tooth” (in reference to a cog) was, and then start to find patterns all while working with a partner. Talk about working memory overload!
 Before we even get too far into this section, what is meant by the word “work”? It is such a vague word in this context. How will a grade five mind interpret this question, especially if he/she has no schema for cogs? Does “work” mean the coloured tooth enters the same gap each time? Does it mean that the coloured tooth enters only a few gaps? Maybe all the cogs “work” in the sense that they can rotate around each other! So there is a lot of potential confusion around this vague term.
 I know the big ideas being promoted here are the ideas of the GCF [ggd] and factors. Two cogs will “work” when the GCF of the number of teeth is 1. For example, I know that GCF(5,7) = 1 [de grootste gemeenschappelijke deler van 5 en 7 is 1] so the cogs with 5 teeth and 7 teeth will “work” in the sense that all gaps will be filled by the full rotation of one coloured tooth. I also know that GCF(4,8) = 4, so the cogs with 4 teeth and 8 teeth will not “work.” This knowledge comes from a deep understanding of prime numbers [priemgetallen] and their relationships with other numbers – knowledge that a novice student does not have.
 Next, go back and look at the choices we have for number of teeth. Do you notice a major flaw? The only odd number [oneven getal] that is not prime on our list is 9 and we do not have a cog with 3 teeth, so GCF(5,7,9,11) = 1. Thus, all of the oddtoothed cogs are going to “work.” So a common misconception of the students was to state: “If two cogs have an odd number of teeth, then they work.” This is certainly true of the activity (because it has a major design flaw).
 Similarly, since the only pair of cogs sharing a factor of 3 are the 6toothed and 9toothed cog, most students didn’t even get to this pair. So they, again, often included incorrectly that a pair of cogs, one with an even number of teeth and one with an odd number of teeth, would always “work”.
 When introducing such an important idea as factors and the GCF of two numbers, does it not make more sense to use explicit instruction followed by discussion of some worked examples? This would not be a difficult thing to do using patternrecognition (skipcounting and making lists for example).
 Did you notice that my discussion of the GCF [ggd] and LCM [kgv] are not on the teacher handout? There is no connection to mathematics at all on this handout! Do you not find it odd that the most interesting aspects of this activity are not even mentioned to the teachers who would be using the activity? It makes me wonder if Jo Boaler has the best interests of the students/teachers in mind. How can students/teachers develop “rich and meaningful connections” to mathematics if the proper mathematical vocabulary isn’t even introduced? Very bizarre indeed.
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Reactie op de aanval van Boaler of het Brits onderwijs.
Jane Imrie, Charlie Stripp (Medewerker resp. directeur NCETM (National Centre for Excellence in the Teaching of Mathematics), GrootBrittannië)
 Jo Boaler is wide of the mark in much of her analysis of what’s currently going on in English classrooms.
 First….’the routine advice…of primary school teachers to young girls that “maths might not be for them.” ‘ paints a depressing and crass picture. But it’s a false one. Perhaps she has heard of one such instance, but to extrapolate and suggest this message is delivered in approaching 20,000 primary schools, without quoting any evidence, is rash in the extreme, and does not in any way match our current, first hand observation and experience in English primary schools.
 Even worse is the assertion that ‘In our maths classrooms today, students do not make conjectures, or learn creatively,’ twinned with a sideswipe at ‘ traditional, narrow, procedural mathematics that fills our classrooms, (and that) is particularly unattractive to women and girls.’ Again, this black and white view of mathematics teaching is both unhelpful, and unrepresentative of reality. The truth is that procedural fluency (being able to do calculations quickly and efficiently, mentally and on paper) and conceptual understanding must go hand in hand, and do go hand in hand in increasing numbers of classrooms. And, by the way, we see no difference here between boys and girls.
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10. Jo Boaler op het BONforum
Zie ‘When academic freedom becomes Harassment and Persecution‘ met uitgebreide commentaren.
Ik schrik van de regel: “projectgebaseerd, samenwerking tussen leerlingen, heterogene klassen, creatief denken i.p.v. procedures leren/toepassen”. Ik denk graag eerst over iets na voordat ik met personen in discussie ga.Een persoon die tegen me praat is een stoorzender bij mijn denken. Nadat ik het probleem doordacht heb, meen een oplossing te hebben gevonden of een weg naar een oplossing of zie dat ik er niet uitkom wil ik graag met anderen erover praten. Ik blink nu eenmaal niet uit in multitasking en er zullen wel meer mensen zijn zoals ik. Te heterogene klassen belemmeren de goede leerling en procedures toepassen doe je vaak in de aanloop van een probleem.