Education ∪ Math ∪ Technology

Month: February 2011 (page 2 of 2)

Clayton Christensen on the shortcomings of math education

"The curricular activities are designed to make students feel like failures." Clayton Christensen

When Christensen talks about designing things as an interesting activity for students, I think it’s pretty clear that this is a real world application of mathematics. This problem he talks about, where students have to first learn a bunch of mathematics before doing anything really interesting or powerful with it, is not just confined to the United States. It’s a problem here in Canada too. I’m sure it’s a problem all over the world.

We can design curriculum so that students get to work on really interesting problems. In fact, it’s been done.

(I recommend checking out a series of textbooks published by the Consortium for Mathematics and Its Applications. I’ve looked through the books, and while they won’t replace a good teacher searching for resources, they are a terrific way to start with real problems and look at solving those problems using mathematics.)


The relationship between family income and FSA score

The Fraser Institute released their annual "report" on the Foundation Skills Assessment (FSA, a standardized exam given to 4th and 7th grade students in British Columbia) results. As usual there have been complaints about the validity of the results, and some interesting side stories. I decided to look at the Fraser institute’s results from a particular perspective. I took the FSA data and isolated the "parent income" and available FSA results (out of 10) from the Fraser Institute data. (I could have used the data from the BC government’s education site, but the Fraser Institute data was more convenient to work with.)

After writing a script to scrape the data (download the raw data here) from the PDF provided by the FSA, throwing away schools for which either FSA results or parental income was unknown, I graphed the data.

From the graph it seems clear that there is some sort of relationship between the two variables, but it is not clear how strong the relationship is between family income and FSA scores.

So I’ve calculated a couple of regressions for a linear fit and a logarithmic fit between the two variables. The value of the correlation coefficients are approximately 0.485 and 0.517 respectively (square roots of the r2 values given above), which given the large number of data points is statistically significance, showing that there is a moderate to strong relationship between the mean family income of a school, and the mean FSA scores for a particular school.

This type of analysis has been done before for other standardized tests, most notably the SAT exams in the United States. What this type of analysis shows is that standardized exams like the FSA are better measures of the wealth of a community than the strength of the schools in that community.

The Fraser Institute rankings are a flawed comparison of schools. It is not possible to fairly rank schools of different socio-economic status because of this strong relationship between scores and the family income. What would be more fair would be to re-rank the data and break it down into different sub-classes based on socio-economic status and not try to compare apples and oranges. 

What would be even more fair would be to throw out the exams all together. What we really want to know is how effective our schools are at ensuring that our students are successful. Given the enormous complexity of this issue, and the wide variety of variables involved, finding a solution to that issue will be extremely difficult. A standardized exam is like a fast-food approach to collecting data, it is cheap and fast, but not very filling. 

Blended Learning: The Importance of Face to Face contact

Here’s a great story (shared on the Huffington post) about a student who is attending his school remotely, through a robot. Watch the video below.

The robot has become a proxy for face to face communication, and this family considers face to face communication so important, they are ignoring other, probably easier, solutions for his education. Lyndon could be going to a virtual school and learning remotely through initiatives like the Khan Academy, but he’s not. Instead he and his parents have chosen to send a robot in his stead, which Lyndon controls via his computer. 

In any blended learning model, it’s important to remember stories like Lyndon’s and remember why we would use blended learning over pure e-learning. Although e-learning has the potential to allow for a much greater degree of personalization of learning, it is a poor substitute for face to face interaction. The ability to quickly communicate a lot more than just the course content is a critical aspect of face to face learning. Lyndon has joined this school virtually because he wants the emotional contact with other people his age. He’s not just content "knowing" stuff, he wants to know it through other people’s eyes as well, hence his comment on "the other student’s point of view."

As educators, we do this too. Although we have a thriving community on Twitter, many of us will jump at an opportunity to see each other face to face. We’ll spend collectively spend vast sums of money on going to conferences like Educon and ISTE, or spend hours planning local unconferences. The online interactions are great, but nothing beats a face to face conversation.

Mumbo Jumbo

Algebra is just mumbo jumbo to most people. Seriously.

If you asked 100 high school graduates to explain how algebra works, and why it works, I’d guess that 99% of them couldn’t, not in sufficient detail to show that they really deeply understand it. Remember that I am talking about high school graduates, so these people have almost certainly had many years of algebra and algebraic concepts taught to them. Most of these people will only be able to give you some of the rules of algebra at best, and some of them don’t even remember that much.

Algebra is an amazing tool for solving problems though! Formulate a problem as an equation, and unless the equation is too complex, there is an algebraic algorithm to solve that equation, and hence the problem you formulated.

Maybe it is such a useful tool that people don’t really need to understand how it works, maybe they can get by without a deep understanding, but still be able to follow the rules of algebra and use it to solve problems. I don’t really buy that argument though, simply because people who don’t understand something are prone to make mistakes, and not be able to check their work with a reasonable level of accuracy.

Computers are also mumbo jumbo to most people. If you asked people to explain how computers work, most of them cannot. There are actually very few people in the world who can explain from start to finish how a computer works, and there is no one that can explain every single piece of a computer. Computers are still amazing tools though, and give people the ability to solve problems that would otherwise be intractable.

I think computers are a useful tool despite our lack of understanding of how they work. Like algebra, computers are a block box in which we put our inputs and get outputs and don’t understand how the inputs are related to the outputs. Given this similarity, we should look at other reasons why using a computer might be superior to algebra.

There are some significant differences between using computers to do computation, and using algebra to do computation. The first is that using a computer, the error rate is much lower. Obviously you can still press the wrong buttons, enter the wrong information, read the information the computer gives back to you improperly, so there is error, but I’d argue that this error is much less than the standard error rate for algebra. The second benefit of using computers is that they are much faster than doing even moderately complicated algebra by hand, including entering the computation into the computer. In the case that doing it by hand is faster, then I’d say you should do the calculation by hand. 

The largest difference between using a computer to do the calculation and using algebra is that algebra is a single use tool. It can only be used to turn an equation into a solution. A computer can be used for so much more.

Granted we should consider computational mathematics to be a broader tool than just plain algebra, if we want a more fair comparison with a computer, but I’d argue that all of the same problems exist with other areas of computational mathematics. As we increase the scope of computations we can learn how to use, the power of the computer becomes even more evident. It takes much less effort to learn how to compute a broader scope of problems using a computer than learning all of the individual computational methods. Witness the power of Wolfram Alpha, for example. Enter in a search phrase and all sorts of useful information comes up.

So in the consideration of using computers for solving computations, over a by hand approach, we can see postulate that the computer will produce less errors, be generally faster, and is more multipurpose than the pencil and paper model is. Furthermore, the computers can do a lot more as a tool than what you can do with algebra.

Another issue I see is that our current mathematics curriculums leave very little time to learn more important skills than computation. As Dan Meyer (@ddmeyer) points out, the formulation of a problem is more important than the actual solution. Learn how to formulate problems and understand how to verify that what you are doing makes sense, then spotting errors in computation becomes that much easier. Furthermore, I’d like to see mathematics education be much more grounded in what is relevant, than be a collection of different types of math which are taught for historical purposes or because they are the ground-work for calculus.

The question for me is, why aren’t we using computers more to do mathematics in elementary and secondary education? It can’t just be because people are scared of change, can it?