The Reflective Educator

Education ∪ Math ∪ Technology

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Day: October 18, 2011

Should we teach the standard algorithms for arithmetic?

Just posted this comment on this article lamenting the loss of the standard algorithms in Mathematics classrooms.

Should we teach the standard algorithms for arithmetic? Absolutely, but they shouldn’t be the only algorithms kids learn.

Why exactly is the ability to add, subtract, divide and multiply large numbers so critical? It seems clear to me that these are useful skills for numbers we will encounter in our day to day lives, and that it is useful to know that algorithms exist to work with larger numbers, but your other connections seem tenuous to me at best.

You’ve argued that without practice using algorithms, students will not be able to remember them to use them later, and this I agree with. It is a basic tenet of education that spaced repetition helps students remember how to use knowledge.

The question is, what type of knowledge is critical for students to remember? Does knowing how to multiple 39835 by 2338383 or any other arbitrarily large number assist the typical person in their life? Does it even contribute to a greater understanding of advanced mathematics? Has the number of people completing advanced mathematics degrees dropped? Statistics Canada data from 2007 suggests that it has dropped very slightly (see http://www.statcan.gc.ca/pub/81-004-x/2009005/article/11050-eng.htm) but not by an alarming amount.

Regarding your achievements as a PHD in mathematics, don’t forget, the plural of anecdote is not data. You can’t generalize from your one experience to what is useful for all of society.

Understanding how to use the algorithm seems sensible to me, but I think it is even more important that people understand algorithms (emphasis on the plural) which is probably lacking in the current curriculum as it is constructed.

One problem is that all across our society, at many different age groups, we have a lack of people using any advanced mathematical thinking to solve problems. If you look at how people solve problems similar to what they learned in school, but in a different context (see Jean Lave’s work), you find that it is rare for people to use the standard algorithms they learned in life, despite the fact that the standard algorithms are much more efficient than the various algorithms people construct for themselves. This suggests that even though the standard algorithms are more efficient, they may still not be the best algorithms to teach.

It seems to me that if over the course of a lifetime, some knowledge is going to be forgotten, the skill of learning is more important than what specific knowledge is learned.

Update: I’ve had another conversation with the author of the blog post above, and it seems I’ve over-reacted a bit. We have more in common than we disagree about.

Mathematical Modelling for Real – Exploring Mathematics

This playlist has some very interesting videos from the Open University on their YouTube channel on the topic of mathematical modelling, and how it is used in different contexts.

Eric Mazur: Memorization or understanding: are we teaching the right thing?

I recommend this talk by Eric Mazur on why he switched his teaching from lecture based teaching to peer instruction based approach. It’s more than an hour long, but it really is worth it.

 

How does this change how we teach? How much of what students learn in our classes is actually learned? If a student can only apply the concepts they have learned to very familiar contexts, and are completely unable to apply them in different contexts, can we really say they have learned the concepts?

I tried the Khan Academy

As an experiment, I started out the beginning of this year and tried flipping my classroom, but with a slight twist: I have extra instructional time, so students were to watch the instructional videos (from the Khan Academy and IBVodcasting.com) during classroom time. We spent about 1/3 of classtime using the Khan Academy videos and exercises, about 1/3 doing problem solving activities (like what is available on www.mathpickle.com and projecteuler.net), and the rest of the time attempting to put the knowledge we were learning into a useful context for the students. While students were involved in these activities, I spent my time circulating the classroom and providing individual and small group support and instruction.

After a month I ended my experiment and am currently in a state of transition while I explore other possible ways of running my classroom. Here are some of the reasons I ended it.

  • Some students chose, despite repeated requests from me, to only watch videos and do exercises that were really easy for them, instead of advancing their knowledge. One student said "she liked the easy videos because it was easy to get points." Another student said she chose the easy exercises because "she was worried about getting problems wrong." These students were more focused on getting easy points and avoiding challenges than learning.
     
  • Some of my students ignored the point system of the Khan Academy and focused on learning, but found that the information from the Khan Academy wasn’t challenging enough. When given practice questions from the course content, they found that the Khan Academy style questions didn’t adequately prepare them. This was partially addressed for these students by switching to the IBVodcasting.com videos, since they are more difficult.
     
  • A few students were able to "master" the content in the Khan Academy exercises after watching a few of the videos, but were unable to transfer what they had learned to any other context, and when queried in more depth, lacked basic understand of what they were learning. For example, they could solve problems like log10 + log2 = log20, but had no idea how to find the value of log20 in terms of p and q when log10 = p and log2 = q.

I’m hoping to implement the RME model and looking for resources that will help support the course curriculum I’m required to cover in the International Baccalaureate program. If I can’t find resources to support this, I’m switching back to my style where I spend some time with students doing experiments in math, some time working on practice problems, and some time with me explaining mathematical concepts. I’m definitely not using the Khan Academy videos again (but I will probably use the IBVodcasting.com videos as additional support for students).

 

See this Slideshare presentation for a description of what the RME model looks like.

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