The Reflective Educator

Education ∪ Math ∪ Technology

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Day: April 25, 2012

First the work, and then the theory

I love this quote shared by Gary Stager via the Daily Papert.

“They first learn engineering, then from there they progress to learning the ideas behind it, and then they learn the mathematics. This would be inventing, it’s a little probe toward inventing a different kind of content. It’s not a different way of teaching; it’s not pedagogy. It’s different knowledge. It’s a good example of turning knowledge – turning learning – upside-down. Instead of starting with this abstract stuff we had from the nineteenth century, let’s start with stuff that’s really engaging for the children, out of which the deeper ideas can develop.” Seymour Papert, 2004

Instead of students learning a bunch of theory, and then being able to apply it to practice, they would engage in building and in creating, which would provide a need and a motivation for the knowledge behind the thing that they are building. One thing that is missing in great degrees from our school is motivation. Why do I need to learn this math when I can see no use for it?

It’s not to say that there aren’t fascinating ideas which are worth learning without something practical to support them, but this is not the general tendency of school. We do not usually teach (with obvious exceptions being good teachers) that which is interesting. We teach practical knowledge. We tend to say, "Oh, we’d like kids to be able to be doctors. Well, what do they need to know? Let’s teach them that." with the result that students do not see the connection between what we teach, and what they can do with it.

Learning mathematical ideas through literature

I looked through our school library today to see if we had any books which would tell mathematical narratives, and I found the following collection. Some of these stories are more "mathy" than others, but each of them has a narrative written around a mathematical concept. Some of these stories could be used to develop context for your students.

 

Anno’s Mysterious Multiplying Jar

This short book tells a very interesting story about a mysterious land with 2 countries, 3 mountains inside each country, and 4 kingdoms inside those countries, and so on, ending with 10 jars. The first pages are essentially describing what a factorial looks like, and if the story ended with question "how many jars are on the island?" I think it would have been an excellent lesson hook into factorials. Unfortunately, the book continues after describing this very interesting narrative with a fairly complete description of factorials. If you are a home-schooling parent, and you want to understand factorials better, and have a story to share with your students, this could be a fabulous resource. If you are a teacher, I recommend ending the story on the page where it first asks how many jars are in the boxes (read the story), and using this as an introductory activity with your students into factorials. Alternatively, this could be an interesting lead up to a less interesting arithmetic problem, wherein the students actually calculate how many jars there are.

 

The Number Devil

This is a novel about a boy who has a fantastic series of dreams full of interesting mathematical ideas, described in language he understands by a character called the Number Devil. The ideas in this story are very interesting to me as a mathematics teacher, and although I’m not sure every kid would enjoy this story, certainly those kids (and adults!) who are interested in mathematical ideas would find this story very interesting. Teachers may also find this book a useful resource for analogies and narratives to help students understand some complicated mathematical concepts. Disclaimer: I have not yet read this entire book, but have enjoyed the 1/3 of it or so that I have read.

 

The Fly on the Ceiling

This story is a historical account of how Rene Descartes may have come up with the idea of the Cartesian plane. According to some other sources I consulted, unfortunately the story is either not true, or incomplete, as it somewhat ignores the series of other inventions made by other mathematicians in advance of Descartes. That being said, it does have an excellent description of how one might make their own Cartesian plane, and teachers may find some inspiration for activities related to introducing the Cartesian plane from this book.

 

A remainder of one

This short story is an excellent description of remainders when doing division, and presents an interesting puzzle. How can 25 soldiers be divided into rows evenly with no remainders left over? One could easily find other such puzzles that are related, and make the concept of remainders much more tangible for students. I think that students may find this book interesting (even younger readers who have not yet learned about division) as well as parents and teachers looking for a lesson idea.

 

Sir Cumference and the Great Knight of Angleland

This story is an attempt to justify the use of angles to solve a problem of navigating through a maze. In terms of giving students some context for understanding angles, I think it does an okay job. One of the benefits of this book is that it does not show the same angles using exactly the same diagrams, which may help students understand that just because two angles have different size rays (or line segments) attached to them, they may still be the same size. I think some students will enjoy this, and teachers may find some useful activities for students to do related to the concept of angles.

 

One Hundred Angry Ants

One hundred angry ants is a short story about 100 ants trying to get to a picnic quickly, and trying different arrangements of rows of ants and the number of ants in each row. The story is appropriate for talking about factors of 100, and could be easily turned into a problem about factors of other numbers. Students will probably find this story interesting up until about 7 or 8, but teachers may find it a source of an idea about teaching that some numbers have multiple factors.

 

Among the Odds and Evens

This short story describes what happens when X and Y visit the land of the numbers. They find to their surprise that there is some strange relationship between the oddness or evenness of the parents, and their offspring. I think this book is interesting to help children remember the fact that odd numbers when added together always add up to an even number, and that even numbers always add up to be even, and that an odd and an even number add up to be odd. However, I suspect that this will be more interesting to student to discover this relationship between numbers (among the many other relationships out there).

 

One Grain of Rice

This is a retelling of the classic story where a peasant outwits the ruler of the land by asking for a doubling reward each day, and ends up with a much larger reward than the ruler expected. It would be good for introducing exponential growth. I would recommend stopping through the story occasionally and asking for predictions from the students about how good a deal the Raja gets. There is an interesting follow-up question for the students as well which is somewhat open-ended: how many years has the Raja been collecting rice? Is it possible for him to have collected 1 billion grains from the lands in his kingdom?

 

Two of Everything

In this story, an old woman and her husband discover a magical pot that allows them to double everything. It could lead to some interesting questions, like "how long will it take the couple to gain enough money from their pot to be comfortable for the rest of their lives?" I’d recommend this for parents who would like to develop more number sense in their children, and for teachers who would like a hook for a lesson around symmetry, doubling, or multiplication.


The Phantom Tollbooth

This book is a treasure trove of logical puzzles, mathematical ideas, and will get kids thinking about different ways of viewing the world. I remember reading it when I was a kid, and I thought it was excellent. Years later, I realized just how many mathematical ideas were in the book. I would recommend this as reading material for students, parents, and teachers.

 

If you know of more books like this, which have a mathematical concept (more interesting than counting books please, there are SO many of those) embedded within the storyline in some way, please share them.

Update: I saw this huge list of books with mathematical ideas shared via Twitter. No reviews, but each book has a very short word description of the math idea to which it links.

On incentives in the teaching profession

So I read some interesting research on student incentive progams which has a couple of very important paragraphs. Here’s the abstract:

This paper describes a series of school-based randomized trials in over 250 urban schools designed to test the impact of financial incentives on student achievement. In stark contrast to simple economic models, our results suggest that student incentives increase achievement when the rewards are given for inputs to the educational production function, but incentives tied to output are not effective. Relative to popular education reforms of the past few decades, student incentives based on inputs produce similar gains in achievement at lower costs. Qualitative data suggest that incentives for inputs may be more effective because students do not know the educational production function, and thus have little clue how to turn their excitement about rewards into achievement. Several other models, including lack of self-control, complementary inputs in production, or the unpredictability of outputs, are also consistent with the experimental data.

Notice the sentence in bold. Incentives for outputs (like let’s say student test scores) does not improve performance. Incentive for inputs (like increased collaboration, more training, working in high needs schools) does. Of course, the benefits gained by the incentives are not terribly strong (with the exception in the study of students being paid to read more) and so any benefit from paying teachers for changing the inputs to education may be minimal.

The merit pay for teachers movement has it all wrong, and the very people in promoting merit pay for teachers have access to this research. In fact, the list of acknowledgements at the beginning of the paper is a veritable who’s who of the leading education reformers in the US.

Converting degrees to radians

One of my students came up (with some help) this procedure for converting between degrees and radians.

  1. Memorize the fact that 60° is π/3 and that 30° is π/6.
  2. Note that 10° is therefore π/18 and that similarly 1° is π/180.
  3. You can then take any degree measure and convert it by converting the number of degrees into sums of degrees where you know the conversions. For example, 70° is equal to 60° + 10° = π/3 + π/18 = 6π/18 + π/18 = 7π/18.

Obviously this procedure is not by any means the most efficient way to convert between radians and degrees. Although I showed a much more efficient algorithm for converting between degrees and radians, it didn’t make sense for this student, and so he and I came up with this procedure (which I drew out of him by asking him questions about the angles), which he does understand.

In general, I’d prefer students use inefficient techniques that they understand completely than highly efficient techniques that they do not understand. Hopefully this student will continue to work on his procedure to make it more efficient as he has to use it over and over again, but if not, at least he will be thinking with something that makes more sense in his head.