My Encounter with Stewart
In his book, The Unschooled Mind, psychologist Howard Gardner suggests that children enter school with strong yet imperfect mathematical theories. However, rather than embracing and expanding upon these existing theories, schools according to Gursky (1991) often disregard them and introduce new ones. This approach fails to acknowledge that these theories can coexist within a child’s mind, even if they appear contradictory. While students may be able to regurgitate information taught in the classroom upon request, the flawed ideas they hold onto are not abandoned. Consequently, Gardner argues that when faced with new situations outside the school environment, students tend to revert back to the simplistic theories they held at the age of five.
It’s important to acknowledge that students possess a wealth of understanding (though sometimes flawed), far beyond what may be initially assumed. In fact, their breadth of knowledge is quite extensive. This paradox creates a significant challenge for the process of learning. Before delving deeper into this intricate matter, allow me to illustrate the situation through an enlightening example.
As part of the work that I did with CIESE at Stevens Institute he would visit with teachers and observed their math classes. One day I was visiting a sixth grade teacher who was working on division with decimals with her students. Stewart was one of her students. In the course of the conversation that the teacher was having, Stewart responded with an interesting comment about division of whole numbers: you can’t divide a bigger number into a smaller number. A pause ensued. The teacher seemed stumped as to how to answer.
So I raised my hand and asked the teacher if he could follow up with a question or two for Stewart. She agreed. Here’s how the conversation went. (I am the I in this conversation and S is Stewart.)
I: What does 4 ÷ 8 mean?
S: It means 4 out of 8. 4 parts out of 8. If you cut a pizza into 8 slices, 4 of them would make 4 ÷ 8.
I: What else can 4 ÷ 8 equal?
S: (after some thought) 1/2!
I: Good. Now, can you divide a smaller number by a larger number?
S: No.
I: You mean you can’t do this: 4 divided by 8?
S: Oh, yeh, you can do that. That’s equal to 2.
I: Wait a minute. Is that the answer to 4 divided by 8 or 8 divided by 4?
S: Doesn’t matter. Either one. They are both equal to 2.
I: Really? But what if you have $4 shared among 8 people, how much would each person have?
S: 2 dollars.
I: Well, let’s think about that for the second. If 8 people each had two dollars, would they collectively have $4?
S: No. (Stewart then writes on paper $4 / 8 then crosses it out and writes 8 / $4.) Oh I got it! You do 8 | $4.00. (He proceeds to do the division.) The answer is 50 cents.
I: But isn’t that a bigger number going into a smaller one?
S: No, it isn’t. 400 is bigger than 8!
For a moment, I stopped and thought about what I should do or say next, but nothing of consequence came to mind. And since I was concerned that the teachable moment was slipping away, I just responded with the obvious question.
I: But isn’t 4.00 the same as 4.0? And isn’t 4.0 the same as 4?
S: Yeh…I guess so.
I: Then you are dividing a smaller number by a larger number!
S: Ok. If you say so.
Up until that moment I was having this interesting mathematical conversation with Stewart, and then suddenly, it came to a screeching halt. “Ok. If you say so,” jarred his brain. Fortunately, I had an out. This was not my class. So I graciously thanked the host teacher and walked back to my chair with my tail between my legs. End of teachable moment. Stewart didn’t get it.
What I was asking him to do was to look at some familiar ideas in a new, related way and hope that he would try to make sense of how the various pieces fit together. Stewart resisted because he was not comfortable with this kind of thinking, especially in math class.
Stewart’s beliefs about division were interesting. He made it very clear that he believes that you can only divide a smaller number into a bigger number. However, sometimes the division problems are written “incorrectly” (like when the smaller number is in the dividend position). When this happens, he can do one of two things. Either ignore the reversal and divide the smaller number into the bigger one or just simply make the dividend “bigger” than the divisor by appending two or more zeros (with a decimal point thrown in).
This did not play havoc with his initial notion about division because when you divide 8 into 4.00 what you are really doing is dividing 8 into 400. The placement of the decimal point is the window dressing that somehow justifies this “hocus pocus”. Now you would think that this would cause Stewart some cognitive discomfort and would motivate him to try to understand how these ideas were connected. But, alas, Stewart is perfectly comfortable living with this ambiguity, as long as he knows what his teacher expects of him on the next test. Here the teacher and student make an agreement (a compromise) that if Stewart can perform to a certain level on the test, then he “knows” the material. Stewart breathes a sigh of relief, comforted by the realization that he will not bear the sole responsibility of comprehending the intricacies of the situation at hand.
For Stewart doing math was a process of coming up with the right formula at the right time. It was like reaching into a bag of math facts and coming up with the right one. What made math hard is that you had to know which math fact to pull up at the right time.
Stewart suffers from “the little into big syndrome” which is well documented. (Graeber, 1992). What can be done with Stewart in this situation? One thing not to do is to reinforce what is usually provided in textbooks or stated by the teacher: a summary of what they should know. Apologies for stating something that appears as a contradiction, but apparently the hold of the 5 year old mind is too strong to be easily overcome by a simple correction. Stewart needs to have an “aha” moment when he realizes that dividing 8 into 4 is the same as dividing 8 into 4.0 because 4 and 4.0 are equal. Indeed, gaining a deep understanding of this concept which appears obvious often requires effort and perseverance.
Alternative conceptions (misconceptions) can really impede learning for several reasons. First, students generally are unaware that the knowledge they have is wrong. Moreover, misconceptions can be very entrenched in student thinking. In addition, students interpret new experiences through these erroneous understandings, thereby interfering with being able to correctly grasp new information. Also, alternative conceptions (or misconceptions) tend to be very resistant to instruction because learning entails replacing or radically reorganizing student knowledge. Hence, conceptual change has to occur for learning to happen. This puts teachers in the very challenging position of needing to bring about significant conceptual change in student knowledge. Generally, ordinary forms of instruction, such as lectures, labs, discovery learning, or simply reading texts, are not very successful at overcoming student misconceptions. For all these reasons, misconceptions can be hard nuts for teachers to crack. However, several instructional strategies have proven to be effective in achieving conceptual change and helping students leave their alternative conceptions behind and learn correct concepts or theories. (https://www.apa.org/education-career/k12/misconceptions)
The hope is that that bit of learning will help students to overcome various characteristics, such as having misconceptions and misunderstandings about numerical concepts. These students often rely rigidly on rules without grasping the underlying principles. Additionally, they tend to blindly obey authority figures, as demonstrated by Stewart’s comment, “If you say so.” Moreover, these students may hold the expectation that their efforts will be rewarded with partial credit from the teacher. All this leads to what Howard Gardner calls fragile knowledge which needs major attention to fix. This is particularly challenging because Stewart doesn’t know he is afflicted!
Misconceptions or naïve conceptions are commonly held ideas or beliefs that are contrary to what is formally acknowledged to be correct and acts like a malaise. Howard Gardner offers a candidate that would be an antidote for this ailment. He calls it a “Christopherian Encounter” named after the famous explorer. The prevailing notion that the earth was flat was beginning to fade at the time of Columbus and Columbus’s trips proved for the first time that the earth was indeed round. Gardner noted in the Unschooled Mind that his 6-year old son knew that the earth was round, yet when asked where on earth he was now, he would say “the flat part.”
Gardner states that the goal of an encounter is to:
(1) put conflicting ideas into sharp focus and so students can reflect on them and hopefully change their orientation to them.
(2) have students explore relevant mathematical semantic domains where important concepts are imbedded in the encounter and and are interesting to explore. They can be real life or fantasy.
To summarize: An encounter is a misconception “vaccine” where the virus is injected in order to have the student reflect on it. Reflective Assessment is necessary for the student to internalize the conflict between ideas such as “Big does not go into little” but 8 does divide into 4 when it comes to money. Now since numbers are “masters of disguise” the 4 magically becomes 4.0 which is one of 4’s forms. This is an appropriate form for solving the problem.
When Stewart stares at 4 ÷ 8 he knows that 8 does not go into 4. However, if he would only know that 4.0 is another form of 4 then he would know that division is possible. This is the thinking he should have. However, it will take an encounter — an Rx activity — to get him to “see” this.
At the moment that I left Stewart he still believes that it is “hocus-pocus” that converts 4 to 4.00 but after the appropriate encounter he should see the light which may take the form of an “aha moment” and then will be able to move from having this fragile, provisional knowledge to a higher level of understanding.
In future posts I will be sharing an Rx activity for Stewart as well as other examples.
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Gursky, Daniel (1991) The Unschooled Mind https://www.edweek.org/education/the-unschooled-mind/1991/11
Graeber, Anna O. and Baker, Kay M. (April, 1992, Arithmetic Teacher, NCTM) Little into Big is the Way it Always is.