Konstruktivisme #5 – a case study


Hierso is ‘n artikel oor konstruktivisme in die laerskool-klaskamer:

In the article titled “A Constructivist Perspective on Teaching and Learning Mathematics”, the author, Deborah Schifter, contrasts two mathematics lessons which she feels offers an understanding of the difference between a lesson based on constructivism and one based on the traditional didactic approach to learning.

In the traditional approach, the teacher has noticed that the students are very excited to find out that blue whales can grow as long as 100 feet so she decides to have the students measure this length in the hallway. Here’s how she went about it:

I told the children exactly how we would go about measuring the whale’s length. We would take the yardstick, which we hadn’t explored, and we would put it down and keep track of where it ended and then place it there and keep counting till we reached where it ended and then place it there and keep counting till we reached 100 feet. (Schweitzer, 1996)
Although the children were quite impressed by the length of the whale, the teacher recounts that the lesson seemed unsatisfying, and wondered what the students had actually learned about measurement.

In the constructivist approach, the teacher had a measurement activity concerning Thanksgiving. She laid out a model of the Mayflower on the floor in the center of the room using masking tape. Then she prepared a scroll or edict for the students to read, telling them that the ship could not sail until they told the king how large the boat was. After the edict was read, she waited for the students to figure out how they could measure the ship and be on their way. Here’s how she described what happened after the reading of the scroll:

“Well, what should we do? Who has an idea?” I asked. Thus our discussion of measurement began… or I thought it would begin. But there was a period of silence-a long period of silence.
What do young children know about measurement? Is there anything already present in their life experiences to which they could relate this problem? I watched as they looked from one to another, and I could see that they had no idea where to begin. Surely, I thought, there must be something they could use as a point of reference to expand on. Someone always has an idea. But the silence was long as the children looked again from one to another, to Zeb, and to me. (Hendry, 1996)

After some confusion about the word Edict on the scroll (some students thought the boat was three feet long because the E in edict looked like a three) the following interaction occurred:

I felt we were back to square one again with more silence, until Tom raised his hand and said, “Mrs. Hendry, I know it can’t be three feet because the nurse just measured me last week and said that I was four feet, and this boat is much bigger than me!”
From Tom’s initial observation, our discussion on measurement was basically off the ground. Hands immediately went up. The children now realized that they knew a little about measurement, especially in relationship to their own size and how tall they were.

“Let’s see how many times Tom can fit in the boat,” someone suggested. Tom got down and up several times along the length of the boat: the children decided that the boat was four “Toms” long.

“How can we tell that to the King, since he does not know Tom?” I asked. “Send Tom to the King,” was their easy solution, while others protested that they wanted Tom to stay on the boat for the trip. I was really hoping that they would relate to the information Tom had already given us about his size. I thought someone might add four feet, four times, presenting us with a quick solution to the problem. But this was not the route they decided to take.

Mark raised his hand and suggested that we could measure the boat with our hands like they do with horses. His neighbor had a horse that was 15 hands. “Then we could tell the King how many ‘hands’ long the boat was.” The children agreed that this might be a better idea.

“All right,” I said. “Since it was Mark’s idea, he can measure the length of the boat with his hands.” Mark was also the biggest child in the class.

At first, Mark randomly placed his hands on the tape from one end to the other, but when he double-checked, he came out with a different answer. The children were puzzled for a while as to why this happened. It took several more tries and much discussion before they came to an important conclusion. The children decided that it was necessary for Mark to make sure that he began exactly at the beginning of the boat and did not leave any gaps in between his palms and his fingers as he placed them on the tape. Measuring this way, he discovered the boat was 36 hands long.

Great! We decided to tell the King this, but just to be sure, I suggested we have Sue, the smallest child in the class, measure the other side. She did and related to the class that her side was 44 hands long. Now there was confusion.

“Why are they different?” I asked. “Can we use hands to measure?” “No,” the children decided, this would not work either, since everyone’s hands were not the same size.

Al suggested using feet. We tried this, but once again, when someone else double-checked with their feet, we found two different measurements. The children at this time began to digress a little to compare each other’s hands and feet to discover whose were the biggest and smallest.

Finally, our original discussion continued, while the children explored various concepts and ideas. Joan sat holding a ruler, but, for some reason, did not suggest using it. Perhaps, I thought, it might be that her experience with a ruler was limited, and she may not have been quite sure how to use it.

Our dilemma continued into the next day when the children assembled again to discuss the problem with some new insights. One child suggested that since Zeb knew the King, and everyone knew Zeb, that we should use his foot. ‘Measure it out on a piece of paper and measure everything in ‘Zeb’s foot.”‘ Using this form of measurement, the children related to the King that the boat was 24 “Zeb’s foot” long and 9 “Zeb’s foot” wide.

Curiosity began to get the best of them and the children continued to explore this form of measurement by deciding to measure each other, our classroom, their desks, and the rug using “Zeb’s foot.” I let them investigate this idea for the remainder of the math period.

On the third day of our exploration, I asked the children why they thought it was important to develop a standard form of measurement (or in words understandable to a first grader, a measurement that would always be the same size) such as using only “Zeb’s foot” to measure everything. Through the discussions over the past several days, the children were able to internalize and verbalize the need or importance for everyone to measure using the same instrument. They saw the confusion of using different hands, bodies, or feet because of the inconsistency of size. (Hendry, 1996)

Schifter uses these two examples to contrast the methodology that would be applied in a traditional versus a constructivist classroom. She points out: “we do not see Hendry engaged in the commonest of traditional teaching behaviors-giving directions and offering explanations. Instead, we observe her questioning her students, the questions sometimes coming minutes apart. And when they do come, more often than not they appear to elicit, rather than allay or forestall, confusion.” (Fosnot, 1996, p.76) Schifter goes on to point out the following differences:

Similarities between the two lessons are easily identified. Both Hendry and Schweitzer were responsive to what had captured their students’ imaginations-Hendry’s class had been fascinated by a cutaway of the Mayflower they had made; Schweitzer’s, by the length of the blue whale. Both teachers decided to engage the class in measurement activities connected to those topics. And both teachers set up their lessons to involve the children in the actual measuring-their lessons were hands-on.
From the point of view of this discussion, however, the salient difference is that while Schweitzer told her class exactly how to perform the task she had devised, Hendry posed a problem with the expectation that her children would find their own way to a solution. Schweitzer crisply demonstrated the use of a yardstick; Hendry watched her students messily struggle to figure out what the inconsistencies in their results would tell them about the concept of measurement. In addition, while Schweitzer could have demonstrated the procedure to ten, five, or even one student, indifferently, Hendry’s lesson depended on her students interacting among themselves. What can we infer from these two units on measurement about the epistemological assumptions they enact? Hands-on though it may have been, Schweitzer’s lesson is nonetheless consistent with beliefs about learning that still order most of our classrooms-that people acquire concepts by receiving information from other people who know more; that if students listen to what their teachers say, they will learn what their teachers know; and that the presence of other students is incidental to learning. However, although Schweitzer’s students might now have been able to picture just how long a blue whale can get, most, as she would come to realize, had probably learned very little about the concept of measurement. For they had not had an opportunity to think through together what a yardstick is, or why they were supposed to lay it down exactly as Schweitzer prescribed. (Fosnot, 1966, p. 77)

* * *

Konstruktivisme word al dekades lank aangeprys as die aangewese leerteorie vir wiskunde-onderwys; is die messy maths manier van klasgee egter toepasbaar op ander leerareas ook?

7 responses »

  1. Goeie vraag 🙂

    Hoe meer abstrak die wiskunde, hoe makliker neig die onderwyser na konseptuele kennis-oordrag. En dan oefen die leerders slaafs die nuwe patrone in (ook belangrik).

    Ongelukkig verstáán hulle dan die werk selde en vergeet hulle dit direk ná matriek.

    * * *

    Ons almal weet “‘n min maal ‘n min is ‘n plus”, maar wie van ons verstáán regtig wat (-4) x (-5) beteken?

    Netso vermoed ek min leerders “verstaan” langdeling.

  2. Wel ek dink die grootste gros van leerder sal eenvoudig nooit verstaan nie. Nogtans moet hulle dit darem kan doen. Al vergeet hulle dit na matriek sal hulle dit makliker weer optel as hulle weer daarmee gekonfronteer word.

    Terloops om leerder ‘n konsep van (-4)x(-5) te gee kan mens mos vir hulle fisiese voorbeelde gee.

  3. Wiskunde is een van my ander leerareas. En ons doen baie patroon herkenning, waar daar egter nie gefokus word op die antwoorde nie, maar eerder die manier hoe jy die antwoord gekry het. Ons werk op ‘n onderrig metode met die naam: Wiskundige kennis vir onderrig. ‘n Opvoeder moet altyd bereid en ingelig genoeg wees om te kan se hoekom iets is soos dit is.

    ‘n Vraag wat egter bly maal in my kop is of dit regtig noodsaaklik is om te kan langdeel? Sodra mens ouer word, ruk mens ‘n sakrekenaar of deesdae jou selfoon uit en druk vinnig ‘n paar knoppies en siedaar daar is vir jou die regte antwoord.

    Ons fokus in Wiskunde meer op strategiese kennis, logiese kennis ens van somme om ons eie kennis en begrip tot wiskunde uit te brei.

    Ek hoop dit maak sin!

  4. Wiskunde is soos ‘n toring waar mens stelsel matig een laag stene op die vorige een bou. As een van daardie lae swak gebou is stort die toring inmekaar. As mens dus argumenteer dat leerders eerder sakrekenaars moet gebruik in stede van om te verstaan wat hulle doen dan ontneem mens hulle een van daardie lae stene. Miskien is langdeling as sulks nie so belangrik nie, maar die getal begrip wat daarmee saam kom help die leerders om te kan verder gaan.

  5. @mnr.muller:

    Jou vraag oor “‘n min maal ‘n min is ‘n plus” het my vir ‘n oomblik laat dink. Ek het probeer terugdink aan die eerste keer wat ek met hierdie konsep te doen gekry het, maar dis helaas al te lank gelede. 🙂 So, toe dag ek ek dink maar van voor af daaroor. Dit laat dink my toe ook weer aan die “instrumental” (ek weet -1x-1=+1) vs. “relational understanding” (wat is -1?) konsepte.

    In elk geval, hier volg my verstaan daarvan (vir die wat belangstel).

    (-4)x(-5) = (-1×4)x(-1×5) – (1)
    = (-1)x(4)x(-1)x(5) – (2)
    = (-1)x(-1)x(4×5) – (3)
    = (-1)x(-1)x(4 + 4 + 4 + 4) – (4)
    = (-1)x(-1)x(+20) – (5)
    = (-1)x(-20) – (6)
    = 20 – (7)

    In woorde:
    Konsep 1: Die getal se teken
    Vir my is dit die eenvoudigste om aan ‘n getal se teken as ‘n rigting te dink: Stel jouself voor jy staan op ‘n lyn, dan kan + die stuk lyn voor jou en – die stuk lyn agter jou beteken. (M.a.w. ons praat van ‘n vektor, maar ek wil die woord vermy, aangesien dit sover ek onthou nie op skool gebruik word/is nie.) Plus en minus is dus tekens om teenoorgesteldes aan te dui: As ek aan bv. 4 as ‘n afstand (bv. 4m) dink, dan is +4m die afstand 4 meter voor my en -4m *dieselfde* afstand agter my (die teenoorgestelde). As ek aan geld dink, dan kan +R4 beteken dis die geld wat ek in my beursie het om lekkergoed te koop teenoor -R4 wat sou beteken dis die geld wat ek kort om lekkergoed te koop. -4 is dus net ‘n wiskundig kompakte manier om te sê 4 is my kwantiteit en – is my rigting. – op sy beurt is eksplisiet eintlik -1 en dus skryf ons -1×4 = -4.

    Konsep 2: Vermenigvuldiging
    Meeste kinders leer baie gou hoe om op hulle vingers te tel, so optelling is ‘n relatief intuïtiewe begrip. As ek dus 2 + 2 + 2 = 6 skryf, dan kan ek daaraan dink as “ek het drie twees”. Dit vereis natuurlik ‘n abstrakte begrip van 2 (of 3 of 6), wat ons weereens letterlik kan kwantiseer (2 appels of in die konteks van die oorspronklike artikel). Die ding om te begryp is dat die vermenigvuldigingsteken ‘n kompakte manier van optelling is. Nou, wat beteken -4 dan? In konsep 1 het ek aangeneem vermenigvuldiging word reeds verstaan. So, ek kan -4 sien as -1×4 wat ek aan kan dink as vier minus-ene — (-1) + (-1) + (-1) + (-1). M.a.w. ek weet -1 is die (abstrakte) kwantiteit 1 agter my (minus as ek op my lyn staan.) Nou is dit maklik (hoop ek) om te verstaan wat -4 beteken: vier ene agter my. Anders gestel: vier eenhede in die negatiewe rigting.

    As ‘n mens nou na die eersgenoemde berekening kyk, dan ontkoppel stappe 1 tot 3 die teken van die kwantiteite. Ek konsentreer dus eers daarop om 4×5 te bereken (stappe 4 en 5) en dan bring ek die tekens terug in die prentjie (stappe 5 tot 7). Let op dat daar ‘n teken voor die 20 in stap 5 is wat sê 20 eenhede (ene) voor my. Die effek van een minus is dus om dieselfde kwantiteit te neem en dus net die rigting te verander (-20). Dieselfde reël pas ek dan net weer toe en so maak ‘n minus maal ‘n minus ‘n plus.

    Hierdie baie eenvoudige berekening was hoe nou gekoppel verskillende wiskundige begrippe met mekaar en met taal is! Ander dinge wat belangrik is, is om die abstrakte te kan visualiseer en om ook te besef dat wiskunde net nog (alhoewel meer formele) taal is.

    Laastens: Ek sou dink die kern tot langdeling is ‘n goeie verstaan (?) van breuke, maar nou is my tyd eers op.

  6. Terug pieng: Sinlose skooltake grens aan kindermishandeling « mnr.muller

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