Year: 2012 (page 8 of 14)

May 26, 2012 / 12 Comments

Mathematical notation is broken

Having spend the last ten years teaching students mathematical notation (while simultaneously teaching the mathematical concepts described by these symbols), I have often reflected on how efficient and amazing it is, and how unfortunately broken it often is.

Some notation shows off some of the power of mathematical thinking (for example, algebra), but some notation has clearly not been designed for clarity. In fact, my suspicion is that much of mathematical notation has been invented to save space.

Of course, a reason why one might one want to save space with mathematical symbols is because paper used to be expensive but I suspect this is not the main reason mathematical symbols are so tightly packed with information. It is also time-consuming to use more clear mathematical notation, and mathematicians love to be concise. In fact, I have often noticed that mathematicians often equate the length of a mathematical proof with its elegance, which over time may have supplied pressure to reduce the notation used to describe these proofs. A few mathematicians have contributed heavily to mathematical notation, most notably Leonard Euler, and these few mathematician’s desire for brevity has defined the notation we use today to communication mathematics.

Look at sigma notation for example. What does the letter sigma from the Greek alphabet have to do with finding sums of things? ~~Absolutely nothing as far as I can tell.~~ According to Dave Radcliffe, Sigma (∑) is short for summa (probably because they start with the same sound), which is the Latin for sum. Euler invented the symbol to use for summation, and we’ve been using it ever since. Essentially, we are using ∑ to mean sum for historical reasons.

$Summation from 3 to 6 of i^2$

The portion of this equation to the left of the leftmost equals sign is summation notation, which I have taught for years. I usually have to spend a class, sometimes two, explaining this specific set of notation. The brevity of the summation notation contributes little to the comprehensibility of this statement. It is essentially equivalent to the following:

Summation (i, 3, 6, i²) = 3² + 4² + 5² + 6² = 86

Unfortunately this notation requires us to memorize the order of the parameters in the summation function, but this is functionally the same as the previous notation, except one more piece of information is given to us; we know we will be doing a sum of some kind without having to memorize the meaning of sigma. With some work, we may be able to improve upon this notation more, and provide even more clarity.

Summation (index: i, start: 3, end: 6, function: i2) = 32 + 42 + 52 + 62 = 86

This notation is somewhat more clear the second option I suggested, since the parameters are defined within the notation. It is significantly longer to write than the original notation (takes up twice as much space) but it has a huge benefit of being significantly clearer. Further, one could imagine that if I were entering this notation into a computer, that the autocomplete function (which is common to code editors) could suggest parameters for me, as well as show me the definition of the parameter as I enter it. Finally, this notation is similar to how we define functions in computer programming (in some languages), and so when we teach mathematical notation, we will also be giving our students some ability to read computer programming code.

This issue about notation is not a trivial concern. The notation used to explain mathematical ideas is often a barrier to some students learning how to communicate mathematical ideas. Quite often students (and sometimes teachers) confuse learning notation for learning mathematics.

Furthermore, notation which is excellent on paper may be somewhat less useful on a computer. I have spent many hours looking for solutions to make adding mathematical symbols to websites more convenient and have discovered that there is no easy way to do this. Every method has drawbacks, and no method is as convenient as adding the same symbols to paper. My conclusion in terms of using mathematical notation with computers is that one of two things (or both) will happen. Computers will develop more touch senstitive interfaces, and software developers will create software that recognize the current mathematical symbols, or we will start to change mathematical notation to be more easily inputted into a computer.

The one huge advantage of our current notation is that it is somewhat universal. Essentially the same notation is used around the world, and by choosing a more amateur friendly notation, we will be creating localized versions of the notation for each language which is obviously problematic. In a computer, this is easily resolved by making the names of mathematical objects translatable so that whomever is viewing a mathematical document can select their language of choice. In print, this is more of an issue, and so we should reluctantly continue to use our existing notation until we have more fully transitioned from our traditional print medium, but the more we use computers to communicate mathematics, the more likely it is that we should fix mathematical notation.

Update:

Here are a couple of critiques of this post:

May 25, 2012 / 0 Comments

Video explanations using animation

Derek Muller sent me this link to a very popular video animation that attempts to explain fundamental forces in nature. You can watch it for yourself below.

The video uses analogy and some cute animations to attempt to explain how forces in nature come from difference between measurements of those forces in different parts of the university. For example look at the screen-shot taken from the video shown below.

Forces of nature causes by ruler

If you look at this picture, does it accurately represent the statement given by the narrator? It seems to me that if you are going to use a visual to explain a concept, it should be clear from the visual what you mean. Visuals should support your explanation, and if your analogy strays too far from the concept you are trying to explain, your visuals do more harm than good. What was the first thing you thought of when you looked at this visual? I bet it wasn’t "Measurement by itself is meaningless, but as surprising as it sounds, that meaningless is exactly what causes the fundamental forces in nature" which is what the narrator says at this moment.

Here’s another screen-shot.

Ignore quantum effects

This visual says two things. The first is not stated by the narrator, but is suggested by the equation shown, specifically that what we are going to look at next is very complicated. The second is suggested by the crossing out of the word Quantum. In this case, the visual definitely describes what the narrator is going to do in the rest of the video – ignore quantum effects on the four fundamental forces. The bad news here is that ignoring quantum effects means that whatever follows is going to be out of date by 100 years of science, and not necessarily a very good representation of the apparent strangeness of the universe. In other words, what follows is a bad model that one will probably not understand.

Now let’s see what happens next.

Economic model of currency exchange explained

My question here is, what of the previous 1 minute and 30 seconds do you remember? I’m going to suggest that you probably do not remember much. This new model is so vastly different than the old model the narrator starts with (and that previous model was not well explained, as you may recall) that the transference of the introductory model to this new model is not likely to happen. If you happen to be an expert in the area of the fundamental forces of nature, you may not notice this effect, since the earlier model is (maybe) describing something you already understand, and have already internalized. If you are not an expert, I very much doubt that a 1 minute explanation is going to make you one.

Further, if you look at this section, you may notice the model for currency transaction (which looks a lot like a function machine, an analogy mathematics teachers often use to explain functions to students) in the middle of the currency exchange. The currency portion of this implicit analogy probably makes sense, but the symbol in the middle may be lost on a lot of people, particularly since the narrator doesn’t take the time to explain what this symbol means.

Explanation complete

Now this is where the narrator makes a huge assumption. He assumes that people have been able to make a somewhat over-arching generalization from his single example. He says, "Hopefully now you can see why measuring things differently in different places inevitably gives rise to a long range interaction, mediated by a particle." I doubt that anyone would be able to make that generalization without a fair amount of expertise in long-range interactions themselves.

It is a form of cognitive bias to assume that because an analogy makes sense to you, that it will make to other people. Analogies are useful as a sense making activity when the analogy describes a shared experience between two people, and very few people have an experience of currency exchange (surprisingly, only a small percentage of any population travels to other countries). In other words, using an analogy that people lack experience with is unlikely to lead to further understanding of a more complicated phenomena.

This particular video, when I watched it, had over 156, 000 views, and over 5000 likes, which suggests to me that one cannot take the popularity of a video and use it to gauge the effectiveness of the learning from the video. I recommend reading the comments on the video. You will see more than a few people who are confused by the video, or who add messages which are essentially unrelated from the video itself. The most popular discussion point I saw, in the 100 or so comments I read, was that this "minutephysics" video was in fact longer than a minute.

My complaints while directed at this one video are generalizable. Analogies used in videos should be related to common experiences, where possible. Visuals matter – using visuals which are confusing, or even wrong, not only distracts from the intended objectives of the videos, it can introduce other possible misconceptions. Avoiding people’s misconceptions in the videos, and attempting to present clear explanations means that people will, in general, incorporate the new information into their existing schema, leaving their current misconceptions intact.

This comment on the video essentially summarizes my main point (notice how many people agree with it).

When I'm watching the video, I feel like a genius. When the video ends, I don't remember anything.

May 23, 2012 / 0 Comments

Disease simulation

Yesterday, our learning specialist for science, Ana, read an article about how games are used to help simulate the spread of disease. She suggested that we could turn this into a collaboration between biology and math, and create a game so that students learn some of the principles of the spread of disease (which is a biology topic) from a mathematical perspective.

I created a simulation so we could test what parameters we may want to use in the classroom so that students are most likely to see that the spread of a disease can be modelled effectivelyh, and see the probability of the infection being spread from person doesn’t change the type of mathematical infection curve much. Try the simulation here.

Some assumptions I’ve made with this simulation:

Individuals once infected, stay infected.
Each individual has an equal probability of being infected by anyone else in the population.
The probability of anyone being infected remains constant over time.
Individuals can be re-infected.

I don’t know if we will end up using this simulation with students, but if we do, I’d like it to be fairly clear so they can get started using the simulation without much intervention from me.

May 22, 2012 / 3 Comments

Research on word processors in student writing

I was looking for research on whether word processors are effective when students are learning to write. So far the research is supportive, but I can’t find any research done recently. I suspect there must be research that is current and supports students using word processors. Please let me know if you have any research more recent than what I have below.

Bangert-Drowns, R., (1993). The Word Processor as an Instructional Tool: A Meta-Analysis of Word Processing in Writing Instruction, Review of Educational Research, p69-93, doi:10.3102/00346543063001069

Abstract: Word processing in writing instruction may provide lasting educational benefits to users because it encourages a fluid conceptualization of text and frees the writer from mechanical concerns. This meta-analysis reviews 32 studies that compared two groups of students receiving identical writing instruction but allowed only one group to use word processing for writing assignments. Word processing groups, especially weaker writers, improved the quality of their writing. Word processing students wrote longer documents but did not have more positive attitudes toward writing. More effective uses of word processing as an instructional tool might include adapting instruction to software strengths and adding metacognitive prompts to the writing program.

Lewis, R., Ashton, T., Haapa, B., Kieley, C., Fielden, C., (1999). Improving the Writing Skills of Students with Learning Disabilities: Are Word Processors with Spelling and Grammar Checkers Useful?, Learning Disabilities: A Multidisciplinary Journal, retrieved from http://www.eric.ed.gov/ERICWebPortal/detail?accno=EJ594984 on May 22nd.

Abstract: A study involving 106 elementary and secondary students with learning disabilities and 97 typical peers found that students who used spelling and grammar checkers were more successful than transition group students in reducing mechanical errors, particularly non-real-word spelling errors, and in making positive changes from first to final drafts.

Owston, R., Murphy, S., Wideman, H., (1992). The Effects of Word Processing on Students’ Writing Quality and Revision Strategies, Research in the Teaching of English, Vol. 26, No. 3 (Oct., 1992), pp. 249-276

Abstract: This study examines the influence of word processing on the writing quality and revision strategies of eighth-grade students who were experienced computer users. Students were asked to compose two expository papers on similar topics, one paper using the computer and the other by and, in a counterbalanced repeated measures research design. When students were writing on the computer, "electronic videos” were taken of a subsample of students using an unobtrusive screen-recording software utility that provided running accounts of all actions taken on the com- puter. Papers written on computer were rated significantly higher by trained raters on all four dimensions of a holistic/analytic writing assessment scale. Analysis of the screen recording data revealed that students were more apt to make microstructural rather than macrostructural changes to their work and that they continuously revised at all stages of their writing (although most revision took place at the initial drafting stage). While the reason for the higher ratings of the computer-written papers was not entirely clear, student experience in writing with computers and the facilitative environment provided by the computer graphical interface were considered to be mediating factors.

May 22, 2012 / 1 Comment

Student brings typewriter to class

In this video, shared with me by Philip Moscovitch, a student has brought a type-writer into class. Is this perhaps, as Philip suggested, a protest against the use of an old pedagogy by bringing in an old technology? Does the use of a typewriter to record notes seem a bit ridiculous? Is it even more ridiculous that the student, as he states at the end of the video, can download the notes for the course?

A well motivated, literate student can learn as much or more from a good set of notes (or a decent textbook) for a course. Why come to class at all if all that is going to happen is a repetition of the notes?

May 16, 2012 / 6 Comments

Can you teach thinking?

Derek Muller: "Can you teach a general thinking skill?"

John Sweller: "I don’t believe you can. It can be learned, it is learned, and it is biologically primary…If you are talking about a teachable thinking skill, one you have to specify it, you have to provide evidence that it has been taught and learned and that you get a different response from people who have learnt that skill and been taught that skill and people who haven’t been."

So here’s my challenge. Can anyone find evidence of a "general thinking skill" that has been taught and then learnt by students?

May 16, 2012 / 1 Comment

Scientific method

Science lab
(Image credit: Jack Amick)

When many people think of science, they think of the tools of science, much like the photo of a traditional science lab above shows. They think of beakers, and hypotheses, and labs, and think that this is science. Playing with the tools of scientists does not make one a scientist, or become a scientist. Thinking like a scientist does.

Science is a way of thinking, a way of reasoning about the world. People who reject science, reject reason. Science is not a linear process, it is a dynamic way of thinking and collaborating about the world.

There are flaws with this way of thinking, as there are with all ways of knowing. Science cannot answer ethical questions. Scientific results get fabricated, exaggerated, and misunderstood all the time, since they are produced and understood by human beings. However, the process of reproducing results with additional experiments ensures that, over time, bad ideas get weeded out of what we know to be true about the world. Ideas which are correct get re-inforced by additional experiments.

Teaching science as a series of facts someone else has discovered about the world does not give them the opportunity to learn about the process through which those "facts" were discovered. The process, in this case, is far more important than the result. Our schools need to spend far more time dealing the messiness of the process of science, and less time focusing on the results of the scientific process. Students learn process through practicing it.

We also need to recognize that the standard science lab write-up emphasizes a linear process of science, which does not exist anywhere in the scientific community. Following someone else’s lab to learn how to use the tools of science is fine, but one must actually design experiments for oneself in order to learn the process. We need to de-sitcom science education.

May 15, 2012 / 1 Comment

Toxins in schools

No peanuts allowed
(Image credit: Schockwellenreiter)

It occured to me today that schools spend an enormous amount of effort to ensure that they are free of toxins for students. We ban common allergens from the school that are life-threatening for some students (like peanuts) and we build our schools so they do not contain asbestos insulation or lead pipes. Some schools are very concerned about the effects of wifi on students, and so have banned wifi from their schools. When we have a belief as a community that something is toxic for our students, schools rally to protect students from that toxin.

So why are so many schools toxic places for LGBTQ youth?

Obviously many schools have made an effort to develop cultures which are supportive of all of their students, but there are places where physical toxins are banned, and emotional ones are encouraged and even nurtured.

May 14, 2012 / 2 Comments

Paulo Freire reflects on his life

Interview from 1996 World Conference on Literacy, organized by the International Literacy Institute, Philadelphia, USA.

I watched this interview of Paulo Freire, and I thought what he had to share is so important that I took the time to transcribe the interview, which you can read below.

A conversation with Paulo Freire

"If you ask me Paulo, what is in being in the world, that calls your own attention to you? I would say to you that I am a curious being, I have been a curious being, but in a certain moment of the process of being curious, in order to understand the others, I discover that I have to create in myself a certain virtue, without which it is difficult for me to understand the others; the virtue of tolerence.

It is through the exercise of tolerance that I discover the rich possibility of doing things and learning different things with different people. Being tolerant is not a question of being naive. On the contrary, it is a duty to be tolerant, an ethical duty, an historical duty, a political duty but it does not demand that I lose my personality.

On a critical way of thinking

Even so it is for me, it should be a great honor to be understood as a specialist in literacy. I have to say, no because my main preoccupation since I started working 45 years ago had to do with the critical understanding of education. Of course, thinking of education in general, I also had to think about literacy which is a fundamental chapter of education as a whole.

Nevertheless, I also had strong experiences in this chapter of adult literacy, for example, in Brasil and outside of Brasil. The more I think about what I did and what I proposed the more I understand myself as a thinker and a kind of epistimologist proposing a critical way of thinking and a critical way of knowing to the teachers in order for them to work differently with the students.

On language and power

Who says that this accent or this way of thinking is the cultivated one? If there is one which is cultivated is because there is another which is not. Do you see, it’s impossible to think of language without thinking of ideology and power? I defended the duty of the teachers to teach the cultivated pattern and I defended the rights of the kids or of the adults to learn the dominant pattern. But, it is necessary in being a democratic and tolerant teacher, it is necessary to explain, to make clear to the kids or the adults that their way of speaking is as beautiful as our way of speaking. Second, that they have the right to speak like this. Third, nevertheless, they need to learn the so-called dominant syntax for different reasons. That is, the more the oppressed, the poor people, grasp the dominant syntax, the more they can articulate their voices and their speech in the struggle against injustice.

In the last moments of my life

I am now almost 75 years old, sometimes when I am speaking like right now, I am listening to Paulo Freire 40 years ago. Maybe you could ask me, but Paulo, look then you think you did not change? No, I change a lot, I change everyday but in changing, I did not change, nevertheless some of the central nucleus of my thought. The understanding of my own presence in the reality. How for example, could I change the knowledge or the experience which makes me know that I am curious? No, I was a curious boy, and I am a curious old man. That is, my curiousity never stops. Maybe in the last moments of my life, I will be curious to know what it means to die.

My philosophical conviction is that we did not come to keep the world as it is. We came into the world in order to remake the world. We have to change it." Paulo Freire (1921 – 1997)

Of course, Paulo’s arguments on language and power can be adapted to not just apply to the indigenous people to whom he was referring, but to any group without power. Teach your students that words have power, and that you respect their words, whatever their source, but to learn the "dominant" culture’s words is to empower yourself, and to give yourself a voice.

May 13, 2012 / 5 Comments

Teaching probability

My colleague found an activity to do with his 5th grade class, similar to this one. Basically, he gave the students 10 coins each, and asked them to put the 10 coins on a number line (with numbers from 1 to 12) with a partner. Each round they roll 2 six-sided dice, find the total, and remove a coin from their number line if it matches the roll. They keep going until one of the two students has no coins on the number line.

At first, most student’s starting positions looked like this:

Student 1 - flat distribution

or this:

Student 2 - another flat distribution

At the end of the lesson, we played a 5 coin version of the game, and one student’s paper (after 1 round) looked like this:

Student 3 - All 4 coins on number 7

Unfortunately, although most of the students did notice that some numbers came up more frequently than others, as evident by their distributions looking a bit less flat, and bit more centred on 7 on the number line, many of the students still had obvious misconceptions of probabilty. Students made comments like:

"If I spin around twice before rolling, I get a more lucky roll."

"I got a few 11s last game, so I’m going to put a few more coins on 11."

"8 is my lucky number! I’m going to put 3 coins on 8."

"I need to spread out my numbers so I have more chance of getting a coin taken on each roll."

Our plan for next class is to have students switch up groups, discuss insights they’ve had on the game, play a couple of rounds, switch them up again, while we walk around and see what strategies they use. Hopefully we’ll see less students spinning around in order to improve their dice luck…

I also asked for resources on Twitter for using a probability game as part of a lesson on probability, and had the following 5 games recommended (all of these look good because they are relatively easily produced and used in an elementary school classroom).

Pig via @delta_dc
Can’t stop via @Fad23
Decimal race via @mathhombre
Farkle via @bennettscience and @mathhombre
Press your luck via @mathhombre

Update:

I wrote a simulation to test to see what distribution of coins is the best. I am somewhat surprised by the results. Check it out here.

Why am I surprised by the results? My intuition about how this game works failed me. I thought that the likelihood of each number coming up was the most important factor in deciding a good strategy for the game. It turns out that one has to balance out the knowledge of the likelihood of each number being rolled with having a selection of numbers available. It may be that the probability of a 7 being rolled is 6/36 but the probability of a 6 or a 7 being rolled is 11/36.

In fact, it turns out that having a selection of different rolls available is fairly important. I updated the simulation so I could choose the number of coins to place, and with 3 coins placed, choosing 4, 5, 6 is better than having all three coins on the number 7. With 2 coins, it is better to choose 6 and 8 than to put both of the coins on 7 (although 6, 7 is better than 6, 8 – but only slightly).

There are three messages I get from running this simulation.

One should, in general, not make too many intuitive assumptions about how probability works, particularly with somewhat complicated examples.
One should be careful how one uses even simple games to teach probability. All of the students saw that 7 was the most common number, but their intuition of "choosing a wide spread" is a valuable one for this game, but it doesn’t help us get at the idea of some numbers being more likely than others. I think we’d need a different game for that.
It is probably a good idea to build the simulation before you play the game with students, if at all possible.