Ethan Demme

Building a Better Pennsylvania

Math Students: Sense Makers Not Mistake Makers

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lImage by wecometolearn CC BY 2.0

For many, if not all, students, achieving mastery in mathematics involves intense struggle. Often, that struggle is seen as a negative thing, by teachers and students alike, and becomes the source of frustration. But what if we were to shift our perspective on students as “mistake-makers” to “sense makers”? What if we celebrated the process, trials and errors and all, as much as the ability to solve a problem and get a right answer?

This perspective, students as sense-makers, is championed by math teacher David Wees. In an article in MindShift, Wee is quoted:

“I want to know the ways that they [the students] are thinking rather than the ways they are making mistakes . . . My interpretation that they’re making a mistake is a judgment and usually ends my thinking about what they are doing.”

This way of thinking is also celebrated by mathematician Paul Lockhart who writes:

Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.

Wees says that “Kids ask questions: 1) to find out if they did the problem right; 2) because the teacher is standing near them and they can, and; 3) occasionally they ask “I wonder what if” questions, which show they are thinking about the math.” In order to help his students develop their mathematical thinking abilities “Wees took to not answering the first two kinds of questions and encouraging the third. He found himself often asking the same question, whether a student had gotten the problem right or wrong. He’d ask them to explain their answer or how they could check to see if they were right or wrong.”

To explore more about this philosophy of mathematics, check out my series on parental engagement in math. And to read about how we can turn everyone into a math person, click here.

Let’s Turn Every Child Into A “Math Person”

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Are you a “math person”? Most likely, the way in which you answer this question determines how well you understand mathematical concepts, and just as important, how much you appreciate/enjoy mathematics. But what if that is the wrong question?

What if we are all math people, at least until we’re convinced that we’re not?

A writer for an article on Quartz explains that the research just does not support the notion that we are genetically predisposed to either understand mathematics or not. Instead, he proposes the “love it and learn it” hypothesis, which is more supported by the research and which has three elements:

  • For anyone, the more time spent thinking about and working on math, the higher the level of mathematical skill achieved.
  • Those who love math spend more time thinking about and working on math.
  • There is a genetic component to how much someone loves math.

So what are the implications if this hypothesis is true? The article goes on:

If the “love it and learn it” hypothesis is true, it gives a simple recommendation for someone who wants to get better at math: spend more time thinking about and working on math. Best of all: spend time doing math in the kinds of ways people who love math spend time doing math.

if a kid has a bad experience trying to learn math in school, or is bored with some bits of math, the answer isn’t to say “Well maybe you just aren’t a math person.” Instead, it is to find some other way to help that kid with math and to find other bits of math that would be exciting for their particular kid to help build her or his interest and confidence.

Why do so many people get convinced that they are just “not a math person”? If someone has an awful math teacher as a student, it’s common for the student to associate his/her frustrations with math, rather than the teacher. Another big problem is that classroom instruction is not set up to be student-paced. The article quotes blogger Cathy O’Neil who writes:

There’s always someone faster than you. And it feels bad, especially when you feel slow, and especially when that person cares about being fast, because all of a sudden, in your confusion about all sort of things, speed seems important. But it’s not a race. Mathematics is patient and doesn’t mind.

Being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine.

Click here to read the rest of the article. It’s fairly lengthy but well worth reading.

Toward A Theory of Instruction (Bruner) – Book Review

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Jerome Bruner is one of the foremost education psychologists. A graduate of Duke University (B.A.) and Harvard (Ph.D.), Bruner has contributed greatly to the study of development, cognitive ability, and pedagogy. I recently read a collection of his essays published in a book entitled “Toward A Theory of Instruction.”

Jean Piaget is perhaps one of the earliest well-known child psychologists. Piaget theorized that children move through distinct stages and are limited in what they can learn and understand by the stage that they’re in. Rather than distinct stages Bruner sees stages that a child moves continuously through. Bruner’s research suggests that even young learners are capable of learning any material if the instruction followed a sequence of action to icon to symbol and is adapted to the learner. Bruner writes,

“. . . There is an appropriate version of any skill or knowledge that may be imparted at whatever age one wishes to begin teaching – however preparatory the version may be . . .The deepening and enrichment of this earlier understanding is again a source of reward for intellectual labors.” (pg. 35)

Regarding sequence of learning, Bruner identified three systems of representation: enactive representation (action-based), iconic representation (image-based), symbolic representation (language-based) – (pg. 10-11)  He writes that

“. . . the nature of intellectual development . . . seems to run the course of these three systems of representation until the human being is able to command all three.” (pg. 12)

Bruner also used blocks to teach mathematics to children. He found that allowing them to build (action), and giving them imagery (the blocks illustrated the math problems and concepts), helped the kids learn the language (symbolism) of mathematics. He writes that,

“The children always began by constructing an embodiment of some concept, building a concrete model for purposes of operational definition. The fruit of the construction was an image and some operations that ‘stood for’ the concept. From there on, the task was to provide means of representation that were free of particular manipulations and specific images. Only symbolic operations provide the means of representing an idea in this way . . .”, the children “not only understood the abstractions they had learned but also had a store of concrete images that served to exemplify the abstractions. When they searched for a way to deal with new problems, the task was usually carried out not simply by abstract means but also by ‘matching up’ images.” (pg. 65)

One last really important contribution by Bruner in regards to education is his emphasis on sequential learning. He rightly points out that we learn at different paces and that there are personalized factors that are at play in determining the speed at which an individual can master a new concept.

“Instruction consists of leading the learner through a sequence of statements and restatements of a problem or body of knowledge that increase the learner’s ability to grasp, transform, and transfer what he is learning. In short, the sequence in which a learner encounters materials within a domain of knowledge affects the difficulty he will have in achieving mastery.” (pg. 49)

Our philosophy of education at Demme Learning, as seen in our Math-U-See and Spelling You See curriculum, is very similar in these regards to the research of Bruner. We too place importance on sequential learning that is student-paced and individualized, builds concept on concept, and moves from enactive (action, building the problem), to iconic (looking at the problem via the blocks), to symbolic (language, which is why have the children build then say or teach back.)

Towards the end of the book, Bruner writes with insight on the necessity of mastering the two major tools of thought, mathematics and the deployment of language.

“It was Dante, I believe, who commented that the poor workman hated his tools. It is more than a little troubling to me that so many of our students dislike two of the major tools of thought – mathematics and the conscious deployment of their native language in its written form, both of them devices for ordering thoughts about things and thoughts about thoughts. I should hope that in the new era that lies ahead we will give a proper consideration to making these tools more lovable. Perhaps the best way to make them so is to make them more powerful in the hands of their users.” (pg. 112)

Bruner’s book, “Toward a Theory of Instruction” is an excellent foray into his thoughts on education theory and is a must read for anyone interested in education. If you don’t have time for the book here is a five minute interview with Jerome Bruner to get you started.

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