Wiskunde. De VS

*Overige ‘uitspraken-rekenen’-blogs*

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De VS

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What is what ?

Common Core State Standard in Mathematics (CCMS): Hierin worden richtlijnen aangegeven over het wiskunde-onderwijs. Deze richtlijnen zijn sterk beïnvloed door ‘Reform Math’. De nadruk ligt op begrip en kritisch denken; het van buiten leren van procedures is minder belangrijk, zelfs het verkrijgen van het juiste antwoord is van ondergeschikt belang. Leerlingen moeten meerdere strategiën kunnen toepassen. Leerlingen moeten aantonen dat ze het begrepen hebben (‘Explain the math’) in gewoon Engels. De CCMS zijn van toepassing op 45 staten van de USA sinds 2009.

De-tracking: Bij ‘de-tracking’ worden leerlingen niet meer in klassen geplaatst op basis van hun prestaties en capaciteiten. Men kiest dus bewust voor klassen met leerlingen die erg verschillen in hun mogelijkheden.

K-12: Periode vanaf basisonderwijs t/m voortgezet onderwijs.

Kumon: Een huiswerkbegeleidingsdienst met vestigingen in de hele VS en Canada. Heel veel leerlingen maken er gebruik van, vooral nadat ze door vernieuwde lesmethoden in problemen kwamen. Op Kumon wordt op de traditionele manier de stof behandeld.

Long division: Staartdeling.

Maths: In de VS, Canada en het VK wordt geen aparte naam gebruikt voor het schoolvak rekenen. Met maths wordt rekenen en wiskunde bedoeld.

Math wars: strijd over wiskundeonderwijs tussen ‘traditionalisten’ en ‘reformisten’

NCTM: National Council of Teachers of Mathematics (USA),  de vakbond voor wiskundeleraren. De NCTM is groot voorstander van hervormingen.

Reform Math: Leermethode, waarbij leerlingen worden uitgedaagd om zich nieuwe wiskundige begrippen eigen te maken d.m.v. onderzoeksprojecten, meestal in een realistische context. Ze gaan zelf op zoek naar oplossingsmethoden. De nadruk ligt op mondelinge en schriftelijke communicatie, sociale vaardigheden: samenwerken met andere leerlingen, het uitleggen aan andere leerlingen en het presenteren van de gevonden resultaten. Niet het resultaat is belangrijk maar het proces. Er is minder nadruk op pen-en-papier methoden, oefeningen (drill-and-kill genoemd) en het leren van algoritmen en procedures. Reform Math werd vanaf de negentiger jaren ingevoerd op veel Amerikaanse scholen, gestimuleerd door de NCTM.

Where’s the Math?‘ , ‘Anti-Math‘, ‘Math for Dummies‘, ‘Junk Math‘, ‘No-Math Mathematics‘, ‘Fuzzy Math‘, ‘Everyday Math‘, ‘New New Math‘: Andere benamingen voor ‘Reform Math’.

STEM: Opsomming van alle studies in ‘Science’, ‘Technology’, ‘Engineering’ en ‘Mathematics’

Word problem: Wiskundige opgave gepresenteerd als text, vaak een verhaal, zonder wiskundige notaties.

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Voorstanders van Reform Math

Jo Boaler, hoogleraar ‘Maths Education’ aan de Stanford University en wereldwijd de meest invloedrijke vernieuwer, is *hier* besproken.

 

Keith Devlin wordt ook in Nederland vaak aangehaald door onderwijsvernieuwers.

Prof. Keith Devlin

Devlin is mede-oprichter en directeur van het aan de Stanford-universiteit verbonden onderzoeksinstituut van menswetenschappen en technologie. Collega en supporter van Jo Boaler. Hij is schrijver van een aantal populair wetenschappelijke boeken, waaronder ‘The Math Gene’.

[ProfKeithDevlin] [All the mathematical methods I learned became obsolete] [Using video games to brake the Symbol Barrier]

  • The “standard algorithms” for numerical computations sacrificed ease of understanding in favor of computational efficiency, and that made sense at the time. But in today’s world, we have cheap and readily accessible machines to do arithmetical calculations.
  • The standard arithmetical procedures were never at the heart of mathematics. Anyone who says this should not purport to be sufficiently expert to pontificate on mathematics education. For mental calculation, left-right algorithm are far more efficient. But this is a red herring, since the focus in education should be learning and understanding, and there are algorithms that are far more efficient in achieving those goals.
  • Except for a few students, the classical teaching methods simply did not work. If they worked, you would not find so many adults who say they cannot do math!
  • It’s possible to do algebra without symbols. Formulas and equations are no more algebra than a page of musical notation is music.
  • School algebra does not have to be symbolic. It was done in a rhetorical fashion for thousands of years until the 16th century.
  • Everybody can do everyday math. The problem is symbolic representation. It’s not a math problem, it’s a language problem.
  • Algebra, and not arithmetic, should now be the main goal of school mathematics instruction. With a spreadsheet, you don’t need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can.
  • I am stressing the distinction between math-as-procedures and math-as-thinking because it is now extremely relevant to the way we educate our next generation of citizens. The complexity of 21st Century life is such that ordinary citizens now need to upgrade their mathematical knowledge and abilities the same way the professional mathematicians did in the mid nineteenth century.
  • When I graduated with a bachelors degree in mathematics, I had acquired a set of skills. Just over thirty years later, those skills were essentially worthless, having been very effectively outsourced to machines that did it faster and more reliably.
  • The most basic of today’s new mathematical skills is number sense. Number sense is a crucial 21st Century life-skill for everyone.
  • It has been well demonstrated that children who do not acquire number sense early in their mathematics education struggle throughout their entire subsequent school and college years, and generally find themselves cut off from any career that requires some mathematical ability.
  • The human brain compares miserably with the digital computer when it comes to performing rule-based procedures. But that human mind can bring something that computers cannot begin to do, and maybe never will: understanding.

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Tegenstanders van Reform Math

My kids used to love math. Now it makes them cry. Thanks standardized testing and common core!
The Washington Post

Parents hate Common Core Math. Parents have been speaking out about how they can no longer help their children with their homework because they don’t understand the methods. Last week, the New York Times‘ wrote about a mother who decided to homeschool her kids because she disagreed with the new math methods. The Common Core Standards require a different method of solving math problems—and proving you have the right answer.”
Business Insider

The way mathematicians learn is to learn how to do it first and to figure out how it works later.”
Stephen Wilson, Hoogleraar wiskunde aan de John Hopkins University

The sense is that all reasoning students attempt is valuable and should be celebrated. The trouble with this approach is that it is exactly status quo; we seem to have a mindset that, ‘Gee, Johnny reasoned’; it doesn’t matter that his actual reasoning is flawed.
James Milgram, Hoogleraar wiskunde aan de Stanford University, over de NCTM-guidelines

Only in word problems can you buy 60 cantaloupes and no one asks what the heck is wrong with you.”
Sally Brown (cartoon)

For all their sophistry, demagoguery and platitudes, fuzzy-math defenders cannot repair schools’ problems with math education by using a louder megaphone and a more polished sales pitch.”
Robert Craigen, hoogleraar wiskunde

 

*Green Hope students walk out, say Wake way of teaching math doesn’t add up*:

200 leerlingen van Green Hope High School lopen de klas uit n.a.v. het vernieuwde wiskunde-curriculum. Dit vernieuwd onderwijs is gebaseerd op het ‘Mathematics Vision Project’ (MVP). De nadruk ligt niet meer op formules en memoriseren; leerlingen krijgen taken en moeten het samen oplossen, de leraar is coach.  “Met MVP leer je een breder begrip van het waarom achter een vergelijking”. MVP is ontworpen op basis van de Common Core Maths Standards en heeft uitsluitend goede kritieken van de vernieuwers, de bestuurders en de politiek. Reactie van MVP op het protest: “We kunnen helaas niet in de klaslokalen kijken.”
Brendan Gipp, leerling en organisator van het protest: “Het is zo frusterend. Je stopt er eindeloos veel tijd en energie in, toch krijg je je werk terug en je hebt het gevoel dat je niets gedaan hebt”. Op Facebook worden de beweringen van MVP door leerlingen en ouders scherp bekritiseerd. Meerdere ouders: “Onze zoon/dochter was uitstekend in wiskunde, alleen maar A-levels. Maar na de invoering van MVP alleen nog maar onvoldoendes.”  Ouders investeren kapitalen in bijlessen, maar het blijkt niet eens te helpen. Ouders sturen hun kinderen naar lagere schooltypen, want daar is geen MVP.”

 

Barry Garelick (Docent wiskunde aan een High School. Schrijver van het boek ‘Confessions of a 21st Century Math Teacher’)

[Mathematics Education: Being Outwitted by Stupidity]

  • Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class. The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes math memorization and the other traditional methods for teaching the subject that have been decried by reformers as having failed millions of students.
  • Without sufficient skills, critical thinking doesn’t amount to much more than a sound bite.
  • A glance at the textbooks that were in use over the 40’s and 50’s shows that mathematical algorithms and procedures were not taught in isolation in a rote manner as is frequently alleged. In fact, concepts and understanding were an important part of the texts.
  • Absolutely no one has argued or is arguing for memorization without understanding, and that caricature of traditional methods is one of the bigger stumbling blocks in the debate.
  • I get a bit tired of the trope that students today are subjected to boring math with boring procedures and boring problems. Although I must say, I find the real-world problems that are supposedly interesting to be quite tedious and boring.
  • Over the past several decades students have been so “protected” with the goal of eliminating the so-called “achievement gap.” The result is that the achievement gap is being eliminated by eliminating achievement.
  • I continue to maintain that many of the difficulties we see students having in math may be attributed to insufficient and ineffective instruction. To put it as simply as I can, they may not be learning math because they aren’t being taught math.
  • Polya says: guessing and patterns are important. But he also says: you ultimately have to prove your guess. That’s the part that doesn’t get played up too much by the “math is all about patterns” crowd.
  • Also, in articles of this type, there is no attention paid to how students who go on to STEM majors learn their math. Many students receive help from home, from tutors, or from learning centers–something that didn’t occur that much in the days when “traditional math failed thousands of students” as the reformers like to say. It was possible for many students to make it all the way through calculus without aid of tutors or help at home–something that even today’s brightest kids are finding hard to do.

Over de Common Core Math Standards

  • A set of guidelines adopted by 45 states this year may turn children into “little mathematicians” who don’t know how to do actual math.

 

Carmen M. Latterell (Assistant Professor of Mathematics at the University of Minnesota)

Latterell is schrijver van het boek: ‘Math Wars: A Guide for Parents and Teachers’

  • Although procedures are not all of mathematics, procedures are a very important part of mathematics. Reform movements in mathematics should call for better teaching of procedures, not for procedures to be removed from curricula.
  • The basic skills that are being omitted in NCTM-oriented curricula form the foundation of mathematics. The NCTM-oriented curricula build their concepts without a base.
  • Calculators are replacing thought processes that students require. Long divison, for example, explains the base number system. Concepts that calculators replace are fundamental to mathematics and to a future understanding of higher mathematics.
  • Some concept in mathematics do have to be memorized. The world would not run smoothly if people were not willing to memorize anything. There is nothing inherently bad about memorizing.
  • Another mistake that NCTM-oriented curricula make is in giving fuzzy definitions. Mathematics is a precise science. Definitions must be detailed. NCTM-oriented curricula try too hard to decrease the distance between school mathematics and every day mathematics. In so doing, the science of mathematics is lost. Mathematics is not supposed to be so imprecise as to be an everyday occurence. It is better to teach mathematics and then show a few applications.

 

Jamie Gass (Director ‘Center for School Reform’ at Pioneer Institute), Ze’ev Wurman (Executive with a semiconductor startup in Silicon Valley; former senior adviser at the U.S. Department of Education)

[The gravitational pull of math education bureaucrats]

Over de invoering van de Common Core Standards in de staat Massachusetts:

  • Since Massachusetts adopted Common Core math, its NAEP scores have fallen. Nationally, 2015 math scores were the worst in years.
  • Polls show that Common Core is widely unpopular with the public, while infuriating parents with its nonsensical abandonment of algebraic fluency.
  • Decades of dismal international results indicate that American K-12 public education has been busier validating students’ math and science phobias than teaching academic content.
  • American K-12 math and science reform efforts fail decade after decade, because D.C. educrats consistently ignore the timeless wisdom of universal geniuses and the best examples of what works in schools.
  • Research shows that math and science are “ruthlessly cumulative,” requiring automatic recall of facts learned in the early grades. But memorization remains a dirty word in America’s schools.

 

Robert Craigen (Hoogleraar wiskunde aan de Universiteit van Manitoba)

  • Although we are now two decades into the push to maximize “understanding” while de-emphasizing “procedure”, I have seen a significant and continuing decline in student capability in this area, and seeing NO increase of note in their analytical prowess in this domain.

 

Barbara Oakley (Hoogleraar ‘Engineering’ aan de Oakland University in Michigan. Schrijfster van het boek ‘A Mind for Numbers: How to Excel at Math and Science’)

[The building blocks of understanding are memorization and repetition]

  • Students have been reared in elementary school and high school to believe that understanding math through active discussion is the talisman of learning. If you can explain what you’ve learned to others, perhaps drawing them a picture, the thinking goes, you must understand it.
  • In the current educational climate, memorization and repetition in the STEM disciplines are often seen as demeaning and a waste of time for students and teachers alike. Many teachers have long been taught that conceptual understanding in STEM trumps everything else.
  • The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t.
  • As one (failing) engineering student recently told me: “I just don’t see how I could have done so poorly. I understood it when you taught it in class.” My student may have thought he’d understood it at the time, and perhaps he did, but he’d never practiced using the concept to truly internalize it. He had not developed any kind of procedural fluency or ability to apply what he thought he understood.
  • In the same way, once you understand why you do something in math and science, you don’t have to keep re-explaining the how to yourself every time you do it. You memorize the idea that you simply add exponents when multiplying numbers that have the same base. If you use the procedure a lot, by doing many different types of problems, you will find that you understand both the why and the how behind the procedure very well indeed. The greater understanding results from the fact that your mind constructed the patterns of meaning. Continually focusing on understanding itself actually gets in the way.
  • Understanding doesn’t build fluency; instead, fluency builds understanding. In fact, I believe that true understanding of a complex subject comes only from fluency.

 

Ouders, help Uw kinderen niet

James Milgram (Hoogleraar wiskunde)

[Some Comments on the Teaching of Common Core “Mathematics” in the Lower Grades]

  • I don’t often get furious about the idiocies that daily emerge from the Common Core universe, but a recent statement by a lead writer of the Common Core math standards, Jason Zimba, got me there.
  • The developers and promoters of Common Core have perpetrated a gigantic fraud on this country. Now Zimba wants parents to sit back and stop trying to minimize the damage. In an article, Zimba addressed nationwide parental frustration at nonsensical math assignments by warning parents to basically shut up and let the teacher follow the standards. “The math instruction on the part of parents should be low,” Zimba said. “The teacher is there to explain the curriculum.”
  • So when a child sits for hours at the kitchen table struggling over math strategies that are counter-intuitive, inefficient, and blindingly stupid, his parents should not ease his pain and improve his education by showing him the simple and efficient way to work the problem. Rather, they should remind him that the teacher is the “expert” and then let him flounder through the rest of his K-12 career without ever learning how to actually do math.
  • Once I understood the idiotic and inappropriate things my son was being asked to do, I told him to do what his teachers wanted, to the best of his ability, but as soon as possible to come to me and I would explain what was really going on. My son finished his education with a Ph.D in molecular biology and worked for years on projects such as the human genome. If I had acted as Zimba advises, my son almost certainly would have had trouble even getting to the level of doing real college work.
  • History repeats itself in my own family. Just a few weeks ago, my son had been working with a child for whom he’s the guardian, a fourth-grader, to make sure she understood some basic math concepts about place value and how they work in the standard algorithm for long (stairstep) multiplication. Her teacher would not let her use it. Instead, she was required to draw pictures of lines, points, and squares, and then laboriously count them up to achieve the product of two whole numbers, each less than 100. She complained to my son, understandably, that she was totally confused and didn’t understand why she wasn’t allowed to use the standard method that she both understood and realized was tremendously more efficient.
  • If children waste a year ensnared in mathematical idiocy, they won’t be able to pick up what they need when they need it later. So asking parents to sit back and watch the sabotaging of their children’s mathematical future rather than intervene to straighten it out is asking them to abdicate their responsibilities as parents.

 

Verklaar je antwoord

Barry Garelick, Katherine Beals (Garelick is wiskundedocent aan een High School. Beals is lecturer aan de University of Pennsylvania Graduate School of Education)

[Explaining your Math]

Eén van de eisen van de Common Core Standards is dat leerlingen hun oplossingsmethoden kunnen verklaren:

  • What “understanding” in mathematics means has long been a topic of debate. One distinction popular with today’s math-reform advocates is between “knowing” and “doing.” A student, reformers argue, might be able to “do” a problem (i.e., solve it mathematically) without understanding the concepts behind the problem-solving procedure. Perhaps he or she has simply memorized the method without understanding it and is performing the steps by “rote.” “Students who lack understanding of a topic may rely on procedures too heavily”, states the Common Core website.
  • The underlying assumption here is that if a student understands something, he or she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: What constitutes a satisfactory explanation?
  • In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation.
  • In a middle school observed by one of us, the school’s goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them.They were instructed on how to write explanations for their math solutions using a model called “Need, Know, Do.” Students were instructed to use “flow maps” and diagrams to describe the thinking and steps used to solve the problem, after which they were to write a narrative summary of what was described in the flow maps and elsewhere. They were told that the “Do” (as well as the flow maps) explains what they did to solve the problem and that the narrative summary provides the why. Many students, though, had difficulty differentiating the “Do” section from the final narrative. But in order for their explanation to qualify as “high level,” they couldn’t simply state “100% – 20% = 80%”; they had to explain what that means. For example, they might say, “The discount rate subtracted from 100 percent gives the amount that I pay.” For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. As the above example shows, the explanations may not offer the “why” of a particular procedure.
  • It was not evident that the process of explanation enhanced problem solving ability. In fact, in talking with students at the school, many found the process tedious and said they would rather just “do the math” without having to write about it. In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way.
  • Math learning is a progression from concrete to abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, can impede the “doing” of actual mathematics.
  • Math reformers often fail to understand that conceptual understanding works in tandem with procedural fluency. Doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math because the very learning of procedures is, itself, informative of meaning, and the reptitious use of them conveys understanding to the user. Explaining the solution to a problem comes when students can draw on a strong foundation of content relevant to the topic currently being learned. As students find their feet and establish a larger repertoire of mastered knowledge and methods, the more articulate they can become in explanations. Children in elementary and middle school who are asked to engage in critical thinking about abstract ideas will, more often than not, respond emotionally and intuitively, not logically and with “understanding.” It is as if the purveyors of these practices are saying: “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills.
  • At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.

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White supremacy

Er is er een brede beweging in de VS, die stelt dat het wiskundeonderwijs discriminerend is voor minderheden en vrouwen. Een aantal hoogleraren ‘maths education’ hebben zich gespecialiseerd in ‘onderzoek’ hiernaar. Ook de NCTM (vakbond van wiskundeleraren) ondersteunt deze beweging. Zie hiervoor *White supremacy*.

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