Math in Real Life

Take what you find interesting and turn it into something challenging, something provocative for someone else.
–Dan Meyer (2011): [anyqs] Hurricane Irene Edition

I’m looking for a good reference for project-based math. Where students face the real-life problems, and learn math as they try to solve them, yet covers the entire curriculum in a complete way.

What I’m considering right now is to swap in some of the real-life questions for some of the sections in the text that consist of rather pedantic word problems, things like: the sum of two numbers is three times less than the square root of the second plus the reciprocal of the first.

Instead, I’d rather do problems like determining the height of a tsunami, which can be treated in different ways depending on which math class you’re teaching, and tie into the science classes (like Physics) as well.

Dan Meyer is a proponent of the project based approach, and he has a lot of interesting problems on his blog.

Learning from Multiple Perspectives Works Better

In fact scientists have found that variety boosts both attention and retention.

–Patti Neighmond on NPR’s Morning Edition (2011): Think You’re An Auditory Or Visual Learner? Scientists Say It’s Unlikely

Morning Edition has an excellent piece that points out that there is little or no actual experimental data supporting the idea that teaching should be individually tailored for different learning styles.

So presenting primarily visual information for visual learners has no proven benefit.

This is something we’ve seen before, however, this article points out that providing each student with the same information in different ways makes it much more interesting for them, increasing their motivation to learn and their retention of what was taught.

Which is fortunate because it means that if you were trying to teach in multiple ways, hoping that the more vocal stuff benefits the auditory learners and the pretty diagrams resonate more with the visual learners, even if this principle is all wrong, all of your students would still have gotten the benefits of variety.

Another key point is that:

Recent studies find our brains retain information better when we spread learning over a period of time versus cramming it into a few days or weeks.
–Patti Neighmond on NPR’s Morning Edition (2011): Think You’re An Auditory Or Visual Learner? Scientists Say It’s Unlikely

So the educational psychologist, Doug Rohrer, recommends giving less math problems at a time but spreading the work out over a longer time. Our block schedule, with three weeks on and three weeks off, ought to work well for this, since students will be studying math intensely on the on-blocks and doing revision assignments on the off-blocks.

The article is below:

Letting Students Personalize their Grading Scheme

How do you know if a student has mastered a subject? How do you get students to better understand how they learn and take more control of their education? I’ve been thinking that giving them more control of their grading might be the answer.

Test grades give some information, but experiments can be just as, if not more, informative. Much depends on the learning style of the student and how they express themselves. Verbally oriented might be good at processing written information and putting what they learned on paper. Kinesthetic-oriented students are likely to do better with practical demonstrations and labs that require movement and coordination.

Since there’s some merit to both exams and laboratory experiments – tests are good for checking the understanding of basic facts, while good labs require application of concepts – they have to be somehow added together to determine if and how well as student has mastered the topic.

Usually, the different types of assessment are combined with different weights. 60% of the total grade for a class might come from exam scores, and 40% from labs. But, given the different talents of different students, might it not make more sense to adjust the weights based on the specific student.

In fact, it would probably be even better to have the students decide for themselves on their own personal grading scheme. It could be part of a classroom contract.

Students would have a strong incentive to come up with their own most beneficial grading system, and, if you gave them a little time to understand the exam and lab requirements (say half a semester) before coming up with the weights, they’d have a lot of incentive to really try to understand how they learn best, and how to demonstrate that knowledge.

Once they’d made a decision on grading weights, they could then focus more energy on the parts of the class they find interesting, which, if we’re lucky, make them more motivated to learn the subject. Then they could set out to acquire the same information and concepts from what is to them a more interesting perspective, without having to worry so much about the stress of struggling through those activities they find difficult and tedious.

A student who is good at experiments might learn the facts in the textbook better if they were looking up information for an experiment – a big picture to little picture perspective – while a student who’s read and understands the text might find the experiments a lot easier to deal with (and so perform better) if they’re less worried about getting the perfect grade.

There would probably have to set some limits as to how much they could play with the weights, say plus or minus 15%, but individualized, self-assigned weights could be a very powerful way of tailoring education, especially in a context where grades are necessary.

Adolescents Versus Their Brains

The part of the brain responsible for logic and reasoning is slow to develop compared to the rest during our adolescence. As a result, adolescents are driven way too much by their emotions and instincts. This means that a lot of the time someone else, teachers and parents usually, have to provide that rationality for them, and help them develop those thinking skills for themselves.

That, at least, was my take-home message after reading David Walsh’s excellent book Why Do They Act That Way. He does an excellent job explaining how the brain develops during adolescence, how it affects the way teenagers behave, and some of the best approaches to dealing with it.

There are a lot of excellent details about how brain development interacts with hormones to create many of the behaviors we find typical of teenagers. Since puberty proceeds differently for girls and boys, Walsh also highlights the differences in the timing of development, and the contrasting results of the different hormones released.

Yet, he also recognizes that adolescent behavior is not solely the result of biology. The effects of neurological and hormonal changes are amplified in industrial societies where kids spend less time with parents, and more time with peers, than in non-WIERD cultures (see The Myth of Adolescent Angst) which leads into his approach to dealing with teens.

To address this unfortunate combination of nature and culture, Walsh advocates a structured approach to parenting, where rules are clear, reasonable, and enforced. This, however, needs to be balanced with the need to keep lines of communication open, which is not an easy trick. Teenagers will want to push you away, but it’s necessary to keep connected to them anyway.

He also emphasizes the need for mentoring good behavior and rational thinking, because, as we’ve seen before, while the developing pre-frontal cortex provides the capacity for formal thinking, it needs practice and training to work well. And, after all, two of the key things we ultimately look for in adults are self-control and the ability to think rationally.

This book is an extremely useful read for parents and teachers (though the first chapter is a bit slow for the impatient). It does a great job of explaining how biology affects behavior, and how to deal with them. I particularly like fact that Walsh has found that teens find it useful to know all this biology stuff too, and it affects how they behave.

Molly Backes on How to Be a Writer

Molly Backes, an author of young adult fiction, considers the question from a mother about her teenager, “She wants to be a writer. What should we be doing?”

Her first answer was, “You really do have to write a lot. I mean, that’s mostly it. You write a lot.”

But then she thought about it, and that’s where it gets really interesting:

First of all, let her be bored. …

Let her be lonely. Let her believe that no one in the world truly understands her. …

Let her have secrets. …

…

Let her fail. Let her write pages and pages of painful poetry and terrible prose. …

Let her make mistakes.

…

Let her find her own voice, even if she has to try on the voices of a hundred others first to do so. …

Keep her safe but not too safe, comfortable but not too comfortable, happy but not too happy.

Above all else, love and support her. …

— Bakes (2011): How to Be a Writer

At the end she posts a picture of her collection of forty-two writer’s notebooks.

It’s a wonderfully written and well considered post that I’d recommend to anyone trying to teach writing and language, particularly if you take the apprentice writer approach. And, I’ve always been a great believer in the power of boredom.

Backes’ advice more-or-less summarizes my interpretation of the Montessori approach: create a safe environment and give students the opportunity to explore and learn, even if it means a certain amount of struggle and failure.

Jungle play area at the Skudeneshavn Primary School in Karmøy on the west coast of Norway is another great example of creating an environment that offers students the opportunity to explore.

It’s also interesting to note how differently writers and other experts think, yet how much their practices overlap. Mathematician Kevin Houston also recommends writing a lot when he explains how to think like a mathematician, but his objective is to use full, rigorous sentences to clarify hard logic, and less to explore the beauty of the language or discover something profound about shared humanity.

Scaffolding and Peer-learning: Thinking about Vygotsky’s “Zone of Proximal Development”

When a student is struggling with a problem, and they just need that little boost to get them to the next level, they’re in Vygotsky’s Zone of Proximal Development, and it’s appropriate for the teacher to give them that crucial bit of help. The idea implies that students really have been trying to solve the problem so the help they get will be useful.

It also implies that the teacher can recognize precisely the help they need and deliver it, which is often easier than it sounds. As an adult, from a different generation and culture, and with more experience with these problems, I see problems very differently from my students. Indeed, experts solve problems by developing rules of thumb (heuristics) that speed problem solving by amalgamating large volumes of information. Unfortunately, for these heuristics to be meaningful, students often have to arrive at them themselves. Thus the student looking at the details is unable to communicate effectively with the expert who sees the big picture.

Peer-Teaching

One remedy Vygotsky advocated was peer-teaching. By letting students of similar but differing abilities work in groups, they can help each other: often a lot more effectively than a teacher would be able to. The teacher’s main interventions can be with the more advanced students who do not have anyone more knowledgeable to help, but who are best able to communicate with the teacher because of a smaller knowledge gap.

Practically, this suggests multi-aged classrooms, and a high level of vertical integration of the subject matter. Consider, for example, which topics from algebra, geometry and calculus might be appropriate for students from middle to high school to be working on together at the same time in the same room.

Scaffolding

Another, more typical, approach to this problem would be to provide all the extensive scaffolding – all the information including explicit demonstrations of ways of thought – that students need to get started, and then gradually take the scaffolding away so that they have to apply it all on their own.

In a high school laboratory science class, a teacher might provide scaffolding by first giving students detailed guides to carrying out experiments, then giving them brief outlines that they might use to structure experiments, and finally asking them to set up experiments entirely on their own.

Slavin (2005) (online resources): Classroom Applications of Vygotsky’s Theory.

In Combination

Elements of both these approaches are necessary – and they’re not mutually exclusive. The scaffolding perspective is most important when introducing something completely new, because they’re all novices at that point. But as you build it into the classroom culture in a multi-aged classroom where there is institutional memory and peer-teaching, then the job of the teacher evolves more into maintaining the standards and expectations, and reduces (but does not eliminate) the need for repeatedly providing the full scaffolding.

You Value Learning More if You Discover Things for Yourself

A key tenant of Montessori is that students have an innate desire to learn, so, as a teacher, you should provide them with the things they need (prepare the environment) and then get out of the way as they discover things themselves.

In the book, The Upside of Irrationality, Dan Ariely explains from the perspective of an economist how people tend to value things more if they make it for themselves. He uses the example of oragami (and Ikea furniture that you have to assemble yourself), where he finds that people would pay more for something they made themselves, as opposed to the same thing made by someone else.

Just so, students value things more, and remember them better, if they discover them themselves.

(video via The Dish)

How to Think Like a Mathematician

The epistemological approach to education suggests that the best way to learn a subject is to learn how to think like the experts in the field: how to think like a scientist; how to think like an historian; how to think like an engineer; etc.

How to think like a mathematician is Kevin Houston‘s attempt to explain how one mathematician at least approaches problems. To whet your appetite, he has a free pamphlet, 10 Ways To Think Like a Mathematician, which starts off with:

Question everything, and
Write in sentences

Logic is, apparently, quite important.

If you want to understand mathematics and to think clearly, then the discipline of writing in sentences forces you to think very carefully about your arguments.

— Kevin Houston: 10 Ways To Think Like a Mathematician

It’s an interesting introduction to how mathematicians see the world, and its a useful reminder that many of the ways of thought that apply to any field can be useful in other places, or even in life in general.