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  1. Waren de Freudenthalers

    Waren de Freudenthalers kosmonauten ?


    Dit is een praktijkvoorbeeld van een eis van de Common Core Standards in de VS: Leerlingen moeten hun antwoorden ook verklaren. Zoals hier: 3 x 5 = 5 + 5 + 5. Iemand die hier als antwoord geeft 3 + 3 + 3 + 3 + 3 heeft er volgens de reformers niets van gesnapt.   


    Het FI  blijft beweren dat vroeger niemand begreep wat een vermenigvuldiging was.  Ze kunnen zich blijkbaar ook niet herinneren dat er vroeger in de wiskunde-lessen naast het bijbrengen van procedures, ook wel inzicht bijgebracht werd, het ging zelfs hand in hand. Ofwel de Freudenthalers leiden aan collectief geheugenverlies ofwel ze komen van een andere planeet.

    Reflecteren is dankzij het FI niet voor niets belangrijk bij de invoering van de Wiskundige Denkactiviteiten. Hoe geweldig het reflecteren in de VS gaat leren we van Barry Garelick, een wiskundedocent.  

    Barry Garelick [Explaining your Math]

    • 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 tadem 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.
    • But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way.  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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