Tag: Teaching

Why Math Class Is Boring—and What to Do About It

There are two types of people in the world: those who enjoyed mathematics class in school, and the other 98% of the population.

No other subject is associated with such widespread fear, confusion, and even outright hatred. No other subject is so often declared by children and adults alike to be something they “can’t do” because they lack an innate aptitude for it.

Math is portrayed as something you get or you don’t. Most of us sit in class feeling like we don’t.

But what if this weren’t the fault of the subject itself, but of the manner in which we teach it? What if the standard curriculum were a gross misrepresentation of the subject? What if it were possible to teach mathematics in a manner naturally incorporating the kinds of activities that appeal to children and learners of all ages?

All of those things are true, argues Paul Lockhart, a mathematician who chose to switch from teaching at top universities to inspiring grade-schoolers. In 2002, he penned “A Mathematician’s Lament,” a 25-page essay that was later expanded into a book.

In the essay, Lockhart declares that students who say their mathematics classes are stupid and boring are correct—though the subject itself is not. The problem is that our culture does not recognize that the true nature of math is art. So we teach it in a manner that would just as easily ruin any other art.

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To illustrate the harms of the typical mathematical curriculum, Lockhart envisions what it would look like if we treated music or painting in the same dreary, arbitrary way.

What if music education was all about notation and theory, with listening or playing only open to those who somehow persevered until college?

“Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.

Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.”

And what if art students spent years studying paints and brushes, without ever getting to unleash their imaginations on a blank canvas?

“After class I spoke with the teacher. ‘So your students don’t actually do any painting?’ I asked.

‘Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations—dipping the brush into paint, wiping it off, stuff like that. Of course we track our students by ability. The really excellent painters—the ones who know their colors and brushes backwards and forwards—they get to the actual painting a little sooner, and some of them even take the Advanced Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they get out there in the real world and paint their kitchen they don’t make a total mess of it.'”

As laughable as we may find these vignettes, Lockhart considers them analogous to how we teach mathematics as something devoid of expression, exploration, or discovery.

Few who have spent countless hours on the equivalent of paint-by-numbers in the typical math class could understand that “there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.” Like other arts, its objective is the creation of patterns. The material mathematical patterns are made from is not paint or musical notes, however, but ideas.

Though we may use components of mathematics in practical fields such as engineering, the objective of the field itself isn’t anything practical. Above all, mathematicians strive to present ideas in the simplest form possible, which means dwelling in the realm of the imaginary.

In mathematics, Lockhart explains, there is no reality to get in your way. You can imagine a geometric shape with perfect edges, even though such a thing could never exist in the physical, three-dimensional world. Then you can ask questions of it and discover new things through experimentation with the imaginary. That process—“asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations”—is mathematics itself. What we learn in school is merely the end product.

We don’t teach the process of creating math. We teach only the steps to repeat someone else’s creation, without exploring how they got there—or why.

Lockhart compares what we teach in math class to “saying that Michelangelo created a beautiful sculpture, without letting me see it.” It’s hard to imagine describing one of Michelangelo’s sculptures solely in terms of the technical steps he took to produce it. And it seems impossible that one could teach sculpture without revealing that there is an art to it. Yet that is what we do with math all the time.

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If school curriculums fundamentally misrepresent math, where does that misrepresentation come from? Lockhart views it as a self-perpetuating cultural deficiency.

Unlike other arts, we generally don’t celebrate the great works of mathematics and put them on display. Nor have they become all that integrated into our collective consciousness. It’s hard to change the feedback loops at play in education because “students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely.”

In schools, mathematics is treated as something absolute that needs no context, a fixed body of knowledge that ascends a defined ladder of complexity. There can be no criticism, experimentation, or further developments because everything is already known. Its ideas are presented without any indication that they might even be connected to a particular person or particular time. Lockhart writes:

“What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources—beautiful works of art by some of the most creative minds in history—in favor of third-rate textbook bastardizations?”

Efforts to engage students with mathematics often take the form of trying to make it relevant to their everyday lives or presenting problems as saccharine narratives. Once again, Lockhart doesn’t believe this would be a problem if students got to engage in the actual creative process: “We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.” An escape from daily life is generally more appealing than an emphasis on it. Children would have as much fun playing with symbols as they have playing with paints.

Those whose mathematics teachers told them the subject was important because “you’re not going to have a calculator in your pocket at all times as an adult” have a good reason to feel like they wasted a lot of time learning arithmetic now that we all have smartphones. But we can imagine those who learn math because it’s entertaining would go out into the world seeing beautiful math patterns all over the place, and enjoying their lives more because of it.

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If the existing form of mathematics education is all backward, what can we do to improve it? How can we teach and learn it as an art?

Lockhart does acknowledge that the teaching methods he proposes are unrealistic within the current educational system, where teachers get little control over their work and students need to learn the same content at the same time to pass exams. However, his methods can give us ideas for exploring the topic ourselves.

An education in the art of mathematics is above all a personal process of discovery. It requires tackling the sort of problems that speak to us at that particular point in time, not according to a preordained curriculum. If a new direction seems of interest, so be it. It requires space to take our time with exploration and an openness to making judgments (why should mathematics be immune to criticism?) All of this is far from ticking boxes:

The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.

We should probably let go of the idea that doing math is about getting the right answer. Being creative is never about getting to a destination.

Above all, mathematics should be something we engage with because we find it to be a fun, challenging process capable of teaching us new ways to think or allowing us to express ourselves. The less practical utility or relevance to the rest of our lives it has, the more we’re truly engaging with it as an art.

Arguments Are For Learning, Not Winning

Despite his best efforts and long hours, Nobel-Prize winning physicist and professor Carl Wieman grew frustrated by his inability to teach and his students’ failure to learn.

When I first taught physics as a young assistant professor, I used the approach that is all too common when someone is called upon to teach something. First I thought very hard about the topic and got it clear in my own mind. Then I explained it to my students so that they would understand it with the same clarity I had.

At least that was the theory. But I am a devout believer in the experimental method, so I always measure results. And whenever I made any serious attempt to determine what my students were learning, it was clear that this approach just didn’t work. An occasional student here and there might have understood my beautifully clear and clever explanations, but the vast majority of students weren’t getting them at all.

In a traditional classroom, the teacher stands at the front of the class explaining what is clear in their mind to a group of passive students.

Yet this pedagogical strategy doesn’t positively impact retention of information from lecture, improve understanding of basic concepts, or affect beliefs (that is, does new information change your belief about how something works).

Alison Gopnik, says “I don’t think there’s any scientist who thinks the way we typically do university courses has anything to do with the best methods for getting people to learn. ”

Given that lectures were devised as a means of transferring knowledge from one to many, it seems obvious that we would ensure that people retain the information they are consuming.

Wieman mentions three studies, the last of which perfectly emphasizes the disturbing point that passive lectures do not seem to work.

In a final example, a number of times Kathy Perkins and I have presented some non-obvious fact in a lecture along with an illustration, and then quizzed the students 15 minutes later on the fact. About 10 percent usually remember it by then. To see whether we simply had mentally deficient students, I once repeated this experiment when I was giving a departmental colloquium at one of the leading physics departments in the United States. The audience was made up of physics faculty members and graduate students, but the result was about the same—around 10 percent.

Wieman argues these results are likely generic and make a lot of sense if you consider the extremely limited capacity of short-term memory.

The research tells us that the human brain can hold a maximum of about seven different items in its short-term working memory and can process no more than about four ideas at once. Exactly what an “item” means when translated from the cognitive science lab into the classroom is a bit fuzzy. But the number of new items that students are expected to remember and process in the typical hour-long science lecture is vastly greater.

The results were similarly disturbing when students were tested to determine understanding of basic concepts. More instruction wasn’t helping students advance from novice to expert. In fact, the data indicated the opposite: students had more novice-like beliefs after they completed a course than they had when they started.

We’re left with a puzzle about teaching. The teachers, unquestionably experts in their subjects, are not improving the learning outcomes: students are not learning the concepts. How can this be?

Research on learning provides some answers.

Cognitive scientists have spent a lot of time studying what constitutes expert competence in any discipline, and they have found a few basic components. The first is that experts have lots of factual knowledge about their subject, which is hardly a surprise. But in addition, experts have a mental organizational structure that facilitates the retrieval and effective application of their knowledge. Third, experts have an ability to monitor their own thinking (“metacognition”), at least in their discipline of expertise. They are able to ask themselves, “Do I understand this? How can I check my understanding?”

A traditional science instructor concentrates on teaching factual knowledge, with the implicit assumption that expert-like ways of thinking about the subject come along for free or are already present. But that is not what cognitive science tells us. It tells us instead that students need to develop these different ways of thinking by means of extended, focused mental effort. Also, new ways of thinking are always built on the prior thinking of the individual, so if the educational process is to be successful, it is essential to take that prior thinking into account.

This is basic biology. Everything that constitutes “understanding” science and “thinking scientifically” resides in the long-term memory, which is developed via the construction and assembly of component proteins. So a person who does not go through this extended mental construction process simply cannot achieve mastery of a subject.

This reminds me a lot of what Charlie Munger said on mental models:

What is elementary, worldly wisdom? Well, the first rule is that you can’t really know anything if you just remember isolated facts and try and bang ‘em back. If the facts don’t hang together on a latticework of theory, you don’t have them in a usable form.

You’ve got to have models in your head. And you’ve got to array your experience both vicarious and direct on this latticework of models. You may have noticed students who just try to remember and pound back what is remembered. Well, they fail in school and in life. You’ve got to hang experience on a latticework of models in your head.

What are the models? Well, the first rule is that you’ve got to have multiple models because if you just have one or two that you’re using, the nature of human psychology is such that you’ll torture reality so that it fits your models, or at least you’ll think it does…

It’s like the old saying, ”To the man with only a hammer, every problem looks like a nail.”

Students are not learning the basic concepts that experts rely on to organize and apply information. And they are not being aided in developing the mental framework – the latticework – they need to improve retrieval and application of knowledge. “So it makes perfect sense,” Wieman writes “that they are not learning to think like experts, even though they are passing science courses by memorizing facts and problem-solving recipes.”

Improved teaching and learning

A lot of educational and cognitive research can be reduced to this basic principle: People learn by creating their own understanding. But that does not mean they must or even can do it without assistance. Effective teaching facilitates that creation by getting students engaged in thinking deeply about the subject at an appropriate level and then monitoring that thinking and guiding it to be more expert-like.

So what are a few examples of these strategies, and how do they reflect our increasing understanding of cognition?

Reducing Cognitive Load

The first way in which one can use research on learning to create better classroom practices addresses the limited capacity of the short-term working memory. Anything one can do to reduce cognitive load improves learning. The effective teacher recognizes that giving the students material to master is the mental equivalent of giving them packages to carry. With only one package, they can make a lot of progress in a hurry. If they are loaded down with many, they stagger around, have a lot more trouble, and can’t get as far. And when they experience the mental equivalent of many packages dumped on them at once, they are squashed flat and can’t learn anything.

So anything the teacher can do to reduce that cognitive load while presenting the material will help. Some ways to do so are obvious, such as slowing down. Others include having a clear, logical, explicit organization to the class (including making connections between different ideas presented and connections to things the students already know), using figures where appropriate rather than relying only on verbal descriptions and minimizing the use of technical jargon. All these things reduce unnecessary cognitive demands and result in more learning.

Addressing Beliefs

A second way teachers can improve instruction is by recognizing the importance of student beliefs about science. This is an area my own group studies. We see that the novice/expert-like beliefs are important in a variety of ways—for example they correlate with content learning and choice of major. However, our particular interest is how teaching practices affect student beliefs. Although this is a new area of research, we find that with rather minimal interventions, a teacher can avoid the regression mentioned above.

The particular intervention we have tried addresses student beliefs by explicitly discussing, for each topic covered, why this topic is worth learning, how it operates in the real world, why it makes sense, and how it connects to things the student already knows. Doing little more than this eliminates the usual significant decline and sometimes results in small improvements, as measured by our surveys. This intervention also improves student interest, because the beliefs measured are closely linked to that interest.

Stimulating and Guiding Thinking

My third example of how teaching and learning can be improved is by implementing the principle that effective teaching consists of engaging students, monitoring their thinking, and providing feedback. Given the reality that student-faculty interaction at most colleges and universities is going to be dominated by time together in the classroom, this means the teacher must make this happen first and foremost in the classroom.

To do this effectively, teachers must first know where the students are starting from in their thinking, so they can build on that foundation. Then they must find activities that ensure that the students actively think about and process the important ideas of the discipline. Finally, instructors must have mechanisms by which they can probe and then guide that thinking on an ongoing basis. This takes much more than just mastery of the topic—it requires, in the memorable words of Lee Shulman, “pedagogical content knowledge.”

Arguments Are For Learning, Not Winning

Is arguing the path towards learning?

I assign students to groups the first day of class (typically three to four students in adjacent seats) and design each lecture around a series of seven to 10 clicker questions that cover the key learning goals for that day. The groups are told they must come to a consensus answer (entered with their clickers) and be prepared to offer reasons for their choice. It is in these peer discussions that most students do the primary processing of the new ideas and problem-solving approaches. The process of critiquing each other’s ideas in order to arrive at a consensus also enormously improves both their ability to carry on scientific discourse and to test their own understanding.

Ten Commandments For Living From Philosopher Bertrand Russell

Philosopher Bertrad Russell:

The Ten Commandments that, as a teacher, I should wish to promulgate, might be set forth as follows:

1. Do not feel absolutely certain of anything.

2. Do not think it worthwhile to proceed by concealing evidence, for the evidence is sure to come to light.

3. Never try to discourage thinking for you are sure to succeed.

4. When you meet with opposition, even if it should be from your husband or your children, endeavor to overcome it by argument and not by authority, for a victory dependent upon authority is unreal and illusory.

5. Have no respect for the authority of others, for there are always contrary authorities to be found.

6. Do not use power to suppress opinions you think pernicious, for if you do the opinions will suppress you.

7. Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.

8. Find more pleasure in intelligent dissent than in passive agreement, for, if you value intelligence as you should, the former implies a deeper agreement than the latter.

9. Be scrupulously truthful, even if the truth is inconvenient, for it is more inconvenient when you try to conceal it.

10. Do not feel envious of the happiness of those who live in a fool’s paradise, for only a fool will think that it is happiness.

Still curious? Russell is the author of many books including Why I Am Not a Christian, History of Western Philosophy, and The Problems of Philosophy.