What is what?
Discovery-based learning. Ook wel genoemd: ‘problem-based-‘, ‘inquiry-based‘, ‘experiential-based‘, of ‘constructivist-learning’ of ‘21st-century learning‘.
Long division: Staartdeling.
Math wars: strijd over wiskundeonderwijs tussen ‘traditionalisten’ en ‘reformisten’
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. Andere benamingen voor Reform Math: ‘Where’s the Math?’ , ‘Anti-Math‘, ‘Math for Dummies‘, ‘Junk Math‘, ‘No-Math Mathematics‘, ‘Fuzzy Math‘, ‘Everyday Math‘, ‘New New 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.
Aan het einde van de jaren 90 hebben de overheden van een aantal Canadese provincies besloten om het rekenen-wiskunde-onderwijs te baseren op ‘discovery-based learning’. Want rekendeskundigen hebben gewaarschuwd: door ‘drill-and-practise’, ook wel genoemd ‘drill-and-kill’, verliezen leerlingen hun motivatie, hun creatieve vermogens en hun probleem-oplossende vaardigheden. Ze wezen op de gevaren van het van buiten leren van de vermenigvuldingstafels. Wiskunde gaat over redeneren; dat staat haaks op het toepassen van de mechanistische standaardprocedures; deze staan begrip en inzicht in de weg en zijn de oorzaak van wiskunde-angst. Leerlingen moeten hun eigen oplossingstechnieken ontwikkelen, door hun uitdagende practische problemen voor te leggen; dat is beter dan expliciete instructie van de standaardtechnieken. Ook moet er rekening gehouden worden met de verschillende leerstijlen van de leerlingen en moeten ze meerdere strategieën kunnen bedenken, die ze dan aan klasgenoten uitleggen. Door leerlingen zelf de wiskunde te laten ontdekken onthouden ze het beter, begrijpen ze het beter, verkrijgen ze een dieper inzicht, is er meer transfer, ontwikkelen ze betere ‘problem-solving skills’, ‘kritisch denkvermogen’ en creatieve vermogens. Voortaan is niet de uitkomst belangrijk maar de weg daarheen. Het motto nu: “You do the math”.
Deze rekendeskundigen lieten zich o.a. leiden door een rapport uit 1998 van de onderwijskundigen Constance Kamii en Ann Dominick:‘The Harmful Effects of Algorithms in Grades 1-4’:
“Algorithms not only are not helpful in learning arithmetic, but also hinder children’s development of numerical reasoning. We have two reasons for saying that algorithms are harmful: They encourage children to give up their own thinking, and they unteach place value, thereby preventing children from developing number sense. The persisting difficulty with standard algorithms lay in the column-by-column, single-digit approach that prevents children from thinking about multidigit numbers. Children in the primary grades should be able to invent their own arithmetic without the instruction they are now receiving from textbooks and workbooks.“
De invloed van dit artikel is te lezen op *Wikipedia*: “Textbook curricula which were inspired by the NCTM standards fully carry out the recommendation of not only omitting instruction of traditional computation methods, but instructing teachers in the teacher guides to not permit students to use such methods as e.g. multiplying length by width by height to compute volume, even if learned outside the class from home.”
Voor kritiek op dit rapport zie *The Bogus Research in Kamii and Dominick’s Harmful Effects of Algoritms Papers*.
[You do the math. Because our kids cant] [New math doesn’t add up for Calgary parents] [Why is it your job to teach your kid math?] [This new math is stealing their confidence and their dreams] [The Great Canadian Math Debate]
Leraren durven zich niet uit te spreken. Velen hebben hun baan opgegeven. Kinderen komen huilend thuis. Ze worden neurotisch bij het horen van het woord ‘math’. De eenvoudigste berekeningen lukken niet. Zelfs kinderen die goed zijn in wiskunde haten nu de wiskundelessen. Ouders spenderen overmatig veel uren aan bijles aan hun kinderen, of betalen voor bijlessen, dit alles gestimuleerd door de scholen met de opmerking dat ze niet de standaardalgoritmen mogen gebruiken. Ook ouders in tranen. Ze begrijpen niets van de schoolboeken. Standaardalgoritmen zijn taboe, want blokkade voor begrip. De tafels leren mag niet, want betekenisloos. Wiskunde op de middelbare school lukt deze kinderen helemaal niet meer, ze raken hopeloos in de knoei met algebra. Breuken optellen, staartdelen, is ook aan universiteiten te hoog gegrepen.
Scholen hebben de keuze uit meerdere rekenboekjes: allemaal discovery-based en geschreven door niet-wiskunde-experts, meestal onderwijskundigen; de boeken bevatten nogal wat fouten en onduidelijkheden.
Bedrijven klagen over de slechte rekenvaardigheden van pas afgestudeerde werknemers. Ze weten niet meer wat een high-school diploma nog waard is.
Er is hoogconjuctuur bij onderwijsadviesbureau’s: allemaal preken ze het belang van leerstijlen en discovery-based-learning. Royal Oak School maakte zoveel gebruik van hun diensten dat ook aan de ouders een bijdrage gevraagd werd van $20.000. De ouders reageerden furieus; ze wilden juist af van dit reken-onderwijs.
In een poging het tij te keren hebben een aantal wiskunde hoogleraren *WISE Math* (Western Initiative for Strenghening Education in Math) opgericht.
We lezen o.a.:
“The most recent version of the WNCP math curriculum omits all standard algorithms for addition, subtraction, multiplication and division. Thus, students are left with insufficient and cumbersome methods for solving arithmetic problems which are often confusing for both parents and children. Furthermore, standard algorithms, which should be mastered in elementary school, have theoretical significance which is important for learning later mathematics. Standard algorithms must be taught to children early in elementary school, and they must be taught properly with the meaning behind the algorithms explained by knowledgeable teachers.”
In 2015 publiceerde het Howe Instituut een kritisch rapport over het Canadese wiskundeonderwijs: *What to do about Canada’s Declining Math Scores?*.
De provincie Quebec staat aan de wereldtop bij wiskunde; deze provincie heeft het zelf-ontdekkend leren niet ingevoerd!
Voorstanders van Discovery-based Math
Commentaren van voorstanders:
- The standard algorithms pose a serious threat to the retention of insight. (math education professor)
- In the current mathematics program of studies, students are expected to know and master number facts as they develop number sense. (team leider wiskunde ministerie van onderwijs)
- Students can learn through memorization, but it’s not enough. They need to learn through inquiry and discovery and learn a number of different ways to approach problems. (adviseur)
- By decreasing emphasis on rote calculation, drill and practice, and the size of numbers used in paper and pencil calculations, more time is available for concept development. (team leider wiskunde ministerie van onderwijs)
- It’s no longer sufficient to just pass information from teacher to student. (adviseur)
Marian Small (Stichter van ‘University of New Brunswick’s Mathematics Education Centre’)
- If math scores are down, it’s in part due to teachers who have not fully made the transition to the discovery method.
- I too believe in teaching the basics, it’s how children learn those basics that needs to be rethought. There used to be one way to do it and now we think there’s more than one way. I think we have to decide what are the basics in the year 2016 as opposed to what were the basics a bunch of years ago.
- Whether a kid can do that long multiplication, I think that’s a dead issue in the year 2016.
- I do not think we need a lot of people who can multiply three, two-digit numbers or two, three-digit numbers because if you’re going to do that, you get out a calculator.
Tegenstanders van Discovery-based Math
Commentaren van tegenstanders:
- This math curriculum looks like it was written by an elementary Language Arts professor. (Universitair docent)
- As a retired secondary math teacher, I am astounded at the imbecility of the new math curriculum.
- There are a fair number of consultants who do not think it’s important for kids to be fluent with calculations.(Hoogleraar)
- The notion that practice of basic skills interferes with understanding of math concepts is illogical and misguided and denigrating terms like “drill and kill” do not serve students or teachers well. Indeed, understanding and practice of basic skills go hand-in-hand. (hoogleraren wiskunde)
- “Mindless algorithms” are powerful tools that allow us to operate at a higher level. The genius of algebra and calculus is that they allow us to perform complex calculations in a mechanical way without having to do much thinking. One of the most important roles of a mathematics teacher is to help students develop the flexibility to move back and forth between the abstract and the mechanical.
- Students need to realize that, even though part of what they are doing is mechanical, much of mathematics is challenging and requires reasoning and thought. (Hoogleraar)
- How can the memorization of times tables harm understanding of numbers? (Ouder)
- Strategies and tricks are interesting for the brightest students, but the average student (including future professionals) will be better served by learning the time-tested algorithms for arithmetic computation that we learned as children and continue to use in our daily lives.(Hoogleraar)
- Multiple strategies and discovering their own strategies confuses most students. This is not to say that you don’t accept different ways of solving problems, but the ones who can do this are the same ones who master and memorize their basic facts, who master fundamental algorithms. You can show them why algorithms work, but they want the simplicity and ease of established algorithms – this has been proven to me time and again by students. (docent)
“It’s impossible to draw a circle with a circumference of exactly 33 cm. Pi never terminates or repeats. So the circumference will never be a whole number.”
Uit ‘Math Makes Sense’, een Canadees wiskundeschoolboek
Anna Stokke (Associate Professor at the Department of Mathematics and Statistics, University of Winnipeg)
Ze is Award Winner ‘National Teaching Fellows’ 2021:
“Dr. Anna Stokke is a mathematician and an ambassador for math education. Her passion for improving the way students at all levels learn math has inspired curriculum improvements provincially and nationally and led to higher university standards. Anna is determined to dispel the myth that some people are good at math while others are not.”
Medeoprichter van WISE Math (Western Initiative for Strengthening Education in Math).
Anna Stokke geeft uit noodzaak bijles aan haar 2 kinderen en een aantal van hun vriendjes. Toen ze de schoolboekjes bekeek schrok ze:
- Instead of teaching the standard methods of arithmetic, the emphasis had shifted to a wide range of alternative methods, such as using grids, blocks, or strips of paper to multiply. Even as a professor of math, I found the methods confusing. It was shocking. We’re talking about adding, subtracting, multiplying and dividing. It shouldn’t be so overly complicated that even parents can’t understand it. It’s absolutely ridiculous.
- You see a lot of convoluted methods for doing simple problems that end up being really confusing for kids. Overly complicated teaching techniques using blocks to solve arithmetic problems instead of doing it with pencil and paper are making it tough for kids to learn basic math. If you’re doing 36 plus 78, you shouldn’t be using blocks to figure that out, you should be able to do it quickly using pencil and paper or in your head.
- Relying on ad hoc strategies to calculate the number facts involved puts the student at an extreme disadvantage relative to one who can quickly recall a fact such as 7 × 8 = 56, since working memory becomes overburdened with the strategies involved to compute basic number facts.
- You know what’s the worst kind of instruction? The kind of instruction that makes kids feel stupid. And that’s what a lot of that discovery stuff does; their working memory gets overloaded, they’re confused.
- Giving people addition or subtraction or multiplication questions and then giving hem a calculator to do them is not testing them on the basics. It’s testing them on whether or not they can use a calculator.
- I have already seen a number of deficits in the math skills of my own university-aged students centred around basic learning, including the use of fractions. Since the introduction of discovery-based math, students are being overwhelmed with trying to figure out basic skills, like multiplication, which should simply be memorized. And when they go on to more complex problem-solving they struggle, they feel defeated and then quickly give up.
- Multiple strategies, open-ended problems and hands-on materials are overemphasized in classrooms and textbooks, causing children to become confused.
- Provincial governments use meaningless buzzwords like ‘problem solving’and ‘critical thinking’ when teaching math.
- It is a common misconception that basic skills and deliberate practice interfere with understanding. Students are left without the solid foundation needed to tackle more complex problems.
- You don’t acquire problem-solving skills by being thrown into a classroom with a bunch of blocks and given no help from a teacher.
- You acquire good problem-solving skills by being given a good foundation by a teacher and given the tools to actually solve those problems.
- Students need basic facts and techniques in long-term memory, developed through hours of practice, in order to become strong problem-solvers.
- There is no valid research that supports teaching multiple strategies to novice learners or that supports the elimination of standard algorithms, etc. At least, no one has been able to provide me with any reasonable studies and I’ve been asking for the past 2.5 years.
- Every time someone makes a “research shows” claim, I ask for articles. I haven’t received any yet.
- The research evidence strongly favours direct instruction over discoverybased instruction for nurturing understanding, deeper learning and better problem solvers.
- Where there any math experts (not math education profs) involved in the writing of the textbooks? As far as I know, there weren’t any involved and those texts contains some major mathematical errors.
Vladimir Troitsky (Hoogleraar Wiskunde aan de Universiteit van Alberta)
- Students show up in my university classroom unable to perform basis arithmetic without a calculator and are unable to add fractions.
- Knowledge is necessary for discovery.
- You cannot Google your way through mathematics.
Herbert Gaskill (Emeritus hoogleraar wiskunde aan de Memorial University van Newfoundland)
Schrijver van wiskunde-boeken en een boek over wiskunde-didaktiek
- When I was head at Memorial University I saw too many students coming in from high school who had very high grades in math and were not prepared for university,
- The pedagogical practice known as ‘discovery-based instruction’ is ineffective when applied to novice learners.
- The most effective curriculum in the world can be defeated by poor teaching practice. A teaching methodology that demands that our children develop for themselves the ideas and/or methods that it took our ancestors thousands of years to develop is such a practice.
- Mathematics is essentially procedural; that is, there is a short list of procedures that enable anyone to solve problems. The WNCP curriculum replaces the standard procedures of arithmetic with a multiplicity of strategies which are arguably ineffective in comparison. If our teachers are asked not to teach the time-tested algorithms to our children, then we certainly cannot expect them to learn these algorithms on their own.
- The most direct analysis of why unguided instruction does not work for novice learners is contained in a readable paper by Kirschner, Sweller and Clark, three eminent cognitive scientists. A central point of their paper is that discovery learning is in conflict with what is known from neuroscience about learning.
- As a math teacher for more than 40 years, I will simply say that the most effective instruction involves guiding students through well-chosen examples which students can then use as models in future problem solving. Guided instruction also is superior by every other measure as well, for example, self-esteem.
- The key result presented in the Stokke report was that Alberta students were unable to correctly identify an expression for (1/3-1/4) at a rate that was any better than guessing. It is tempting to dismiss this with ‘So what, we live in an age of calculators’. However, by making that dismissal we should understand that that act closes the door to all learning requiring any form of symbolic algebra and this list runs the gamut from science to business to many trades.
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.
Andrew Robinson (Docent Natuurkunde aan de Carleton University Canada. Award winning teacher)
- My first year engineering students sat their first ever university exam yesterday. While I was walking around answering questions and queries, the one question which was repeated over and over was the following: “The formula for the volume of a cylinder is not on the formula sheet. How do I calculate it?”.
- Even worse, they did not know how to deduce the volume from the picture of a cylinder as a height times the area of a circular base.
- Now, don’t for a minute think that I’m blaming the students for this. They are an industrious and smart group of students. They have been failed by their education system at secondary level.
- Desperate Alberta parents are now seeing the need for private tutoring. For example, there’s been a stampede to private Kumon math. Students there attempt to learn the basic skills that many of our schools have downplayed.
- This is an educational disaster, with our most vulnerable students hammered hardest, but not once have discovery math’s architects — an army of “21st-century learning” educational consultants, professors and gurus — been held to account.
- The “21st-century learning” gurus are now pushing for yet more focus on schooling with a massive, over-riding focus on the same concepts and buzzwords: discovery, inquiry, exploration, child and student-centred learning, 21st century competencies.
- In this new system, our children, the new masters of their own learning, are asked to somehow discover the ways of arithmetic by trying to figure out wordy math problems. Today’s math isn’t about numbers, it’s about words and theories, as if the curriculum was written by folks who hate the clear logic of pure mathematics.
- Grade 9 students here are required to use sticks, tiles, swatches of cloth and colouring to do complex math operations such as multiplying polynomials with monomials.
Tara Houle (Oprichter van WISEMathBC. Parent advocate)
Houle is initiator van een wiskunde-petitie gericht aan de Minister van Onderwijs van British Columbia om een einde te maken aan ‘Discovery/problem-based maths’.
- Ample evidence illustrates there has been a significant decline in our student’s math performance over the past 15 years, and we also know that the percentage of our top math students has fallen dramatically.
- Math has always been difficult, but it’s even harder today due to the convoluted ways it’s being taught. Even though it took thousands of years to develop successful, universal methods to teach arithmetic, the foundations of teaching this particular subject have now been relegated to the dustbin.
- Kids are encouraged to explain and develop multiple ways to find the answer, confusing and frustrating them along the way.
Fractional arithmetic is paramount to future success in mathematics, yet it’s not mandatory until Grade 8 in the new curriculum. This is creating massive panic for kids who are then being exposed to algebra in high school without fully mastering the fundamentals. And without a strong grasp of algebra, kids are denied the understanding of higher-order mathematics such as trigonometry and calculus.
- Tutoring rates have recently skyrocketed, as parents are now scrambling to ensure their kids learn the fundamentals properly — something that is lacking in today’s classrooms. This spike in enrolment correlates with an increased use of inquiry/problem based learning in our schools.
- Today, we have manipulatives such as fraction strips as opposed to learning fractional arithmetic, explaining one’s work rather than adding and subtracting in columns, creating multiple strategies to write addition and/or subtraction sentences, using calculators as early as Grade 2, and there is very little emphasis on mastering any arithmetic procedure, let alone ensure that kids memorize their times tables!
- Pure “discovery based math” methodology is ineffective and it has been statistically proven to cause cognitive overload in our kids. In fact, there is no single review or study of discovery-based math and methods that confirms the claims of its designers. Using pencil and paper, and practicing daily is still the most effective method for kids to learn math, yet these successful methods are being ignored.
- The familiar refrain of how the world is changing and the requirement to break the factory school model is as relevant now as it was in 1895. But what has changed is the content and the methods for teaching our kids. Previous lessons were rigorous, and they provided ample practice time and classroom guidance to ensure ALL kids had the opportunity to learn their math fundamentals. In contrast, today’s resources, such as “Math Makes Sense”, is chaotic in design, offers very few challenging problems for students to solve, and supports an increased dependence on calculators and manipulatives. “Math Makes Sense” is also riddled with errors and inaccuracies, yet the ministry continues to endorse this textbook for classroom instruction.
- ‘Math Makes Sense’ textbooks have been proven to be filled with inaccuracies, omissions and errors, yet they are still being used as a leading textbook in our province. SFU lecturer, Malgorzata Dubiel, spent a year researching this issue. Her conclusions outlined a number of errors in the math terminology, definitions and mathematical presentations of these textbooks. Dubiel stated that these errors and inaccuracies might also be contributing to the students’ misunderstanding of math concepts. She went on to reveal that these textbooks were written by teachers, and there was not a single consultation made by a mathematician either in reviewing or writing the text. In our own household, the issue wasn’t that our child could not do the math; it was that she didn’t have the proper tools to solve the problem.
Those teachers who try to use successful, straightforward methods are labelled dinosaurs, and some have been shuffled out of their districts if they do not conform.
There is now a rabid fervour promoting 21st-century learning, and insisting that inquiry-based learning take precedence over everything else. However, the basic fundamental principles of arithmetic are non-negotiable. Without mastering these crucial facts at the elementary level, any attempts at Math 10, pre-Calculus or entry-level university mathematics will end in failure.
Even more frustrating is the blatant disregard of thousands of teacher and parent concerns that our education partners in this province have demonstrated. Multiple requests to discuss the implications of years of poor math instruction in this province have been ignored. In fact, the only teaching methods endorsed in the flurry of multiple teacher workshops, conferences and parent information sessions, are the new inquiry/discovery/21st-century methods.
If you want your child to have a strong foundation of mathematics, you’ll have to pay for it at a tutoring centre.
- Yet education leaders don’t want to acknowledge the tutoring phenomenon. They are silent when asked to investigate this issue, to determine how many kids are using tutors, and if so, why?
- Send them your tutoring bills. Time for them to acknowledge the mess that they have created.
- Don’t let the educationalists be the only voice in this discussion. Fight for your kids, because it’s their future that’s at stake.
Mary Turpel-Lafond (‘Child and youth advocate’ in Canada)
- I’m very disappointed that there isn’t a single child in care that can take the Math 12 exam.
- The number one social inequity in our schools today is the continued disenfranchisement of at risk youth through application of learning fads and practices in modern math curricula.
- Why do our curriculum guidelines suggest that memorization is not intended at the Grade 4 level? Why do they suggest (without any shred of evidence) that memorization is harmful to students.
- Yet despite all the evidence, our education leaders insist on toting unproven educational resources and fads intent on making snake-oil salesman profit and dumb our kids down even more.
- There is nothing equitable about sacrificing level playing fields for children in the pursuit of edu- quackery and ignoring solid evidence that supports effective math instruction.
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.
- Scientists from developmental and educational psychology as well as cognitive neuroscience have been busy accumulating evidence regarding the ways in which children learn math and what factors influence their learning trajectories and achievement success. This evidence suggests that the dichotomy between discovery-based or conceptual learning, on the one hand, and procedural or rote learning, on the other, is false and inconsistent with the way in which children build an understanding of mathematics. Indeed, there is a long line of research showing that children learn best when procedural and conceptual approaches are combined.
- Moreover, children’s procedural and conceptual knowledge are highly correlated with one another, speaking against creating a dichotomy between them through instructional approaches. Researchers have demonstrated that an effective use of instructional time in math education involves the alternation of lessons focused on concepts with those concentrated on instructing students on procedures.
- Discovery math proponents argue strongly against the use of setting time limits for students to complete mathematical tasks, such as calculation. Here again, the empirical evidence speaks against the notion that speeded instruction necessarily has negative consequences.