Een rijke betekenisvolle uitdagende les over ggd en kgv

Een multi-dimensionale les boordevol 'problem-solving-activities'. Deze '21-st century skills'  zijn veel belangrijker dan het leren van formules en procedures, het Freudenthal Instituut heeft ons er telkens weer op gewezen. ‘Memorizers‘, leerlingen die formules en procedures van buiten leren, presteren het slechtst op de PISA-toetsen; dat heeft Jo Boaler zelf ontdekt na analyse van de PISA-gegevevens van 13 miljoen leerlingen. Niet voor niets worden binnenkort Wiskundige Denkactiviteiten geëxamineerd. 
 

Boaler:  “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 sense-making instead of just calculating without any thought.”

"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."

 

 
Boaler heeft een aantal voorbeeldlessen op haar website uitgewerkt, inclusief werkbladen en een handleiding voor de leraar. Zo ook een les over de ggd (grootste gemeenschappelijke  deler van 2 of meer getallen) en het kgv (kleinste gemeenschappelijke veelvoud 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'; 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 inspirerende les: [Counting Cogs from 'youcubed.org'].
 

Bryan Penfound:

"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 odd-toothed 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 6-toothed and 9-toothed 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 alway "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 pattern-recognition (skip-counting 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."

 

Meer over Jo Boaler vindt U hier (inmiddels flink geupdate) www.beteronderwijsnederland.nl/content/uitspraken-rekenen-deel-2

1 Reactie

  1. It makes me wonder if Jo

    It makes me wonder if Jo Boaler has the best interests of the students/teachers in mind?

    Ze zou natuurlijk ook uitsluitend geïnteresseerd kunnen zijn in haar eigen hersenspinsels. 

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