Wednesday, April 17, 2013

Vedic Maths: Lipstick on a Pig

I was alerted to this video by a pre-service teacher that helps in my room every week. Before this post makes any sense, you should watch the video below. Try to watch the whole thing--I found that task very difficult.


As I watched, I found myself becoming increasingly annoyed with the topic. The presenter claims that the problem with current  mathematics is the algorithms that we teach. Ironically, he couples this solution with a boring lecture complete with lack luster audience polling, inadequate wait times, and dry humour. I imagine that a lot of what is wrong with math education can be pulled from his very presentation.

While I was watching, I was reminded of two phrases that have become part of my every day worldview.

The first is a quote from President Obama during his first run for election. (Yes, even us Canadians follow some American politics.) Although education only makes a cameo in the speech, the message rings true for change in any capacity. Obama accuses Republican promises as identical policies with different names and goes on to inform the audience that "you can put lipstick on a pig." (retrieved from YouTube http://www.youtube.com/watch?v=58FVeYjHpM8 )

To truly create a measurable gain in holistic numeracy across the world, educators need to undercut the surface gimmicks and stop putting makeup on pigs.

The second quote comes from a small but potent book on mathematics education written by Paul Lockhart called A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. This book is a staple on many educators' shelves. 

While musing on the tendency of school to get caught up in notation, he remarks that "it's like rearranging the deck chairs on the Titanic!" (p. 37). Education so often becomes a battle between frivolous things; in this case, it has become a war over which mindless--and preset--algorithms produce the best math scores.

With all of that said, there are three reasons why I feel that Vidic Maths is not revolutionary for math education:

It's spearheaded in a corporate manner

People with business degrees trying to push products in education scare me. It seems--from this presentation--that those behind the "system" of Vidic Maths are out to distribute their algorithms for profit. What scares me further is the fact that he bothered to mention the legal struggles he went through to gain the right to mathematics. Mathematics belongs to no one; You cannot take a patent out on deep understanding. He has attempted to take base-10 manipulation and claim that he invented its patterns and caveats. I chalk them up to nothing more than the hidden and beautiful patterns that emerge throughout mathematics. 

The presentation spends very little time on meaning
I put myself in a student's shoes while he flashes the slide with the number theory logic behind the trick. The slide was shown for under two seconds. He does the same thing with the other "trick". The focus of Vidic Maths is not understanding. In fact, the focus is on the exact opposite. The collection of tricks is designed to be memorized and correctly implemented. The whole time, comments like, "easy, isn't it?"(2:21) push instruction forward. The description of the video calls the method the "High Speed Vidic Maths" method. Same empty understanding, just faster. (Even that is debatable).

The method feigns numerical flexibility
This one stings me the most. People will look at this guy and think that he has a great understanding of how numbers work and interact, when really he is parroting. All that Vidic Maths accomplishes is the replacement of a long, yet robust, algorithm with a series of shorter, less versatile ones. Actually, it eliminates what little mathematical sense we did have in the old multiplication algorithm. Now we have many more mindless manipulations to complete. Also notice that the presenter still uses the language of the "inefficient" system. He speaks of "carrying" (2:13) in his example. A topic that is born from the understanding of the traditional algorithm. 

At the six minute mark, he mocks a student's method of multiplying eight by seven. Although the student miscounted their circles, they show an understanding of what multiplication is by drawing out sets of circles and then counting. Such learning is much more valuable, applicable, and transferable to a society that is in a mathematical crisis. Vidic Maths doesn't teach the beauty of our base-10 system, it exploits it and renders it unthinkably rigid. 

I believe that basic math skills need to be built up through the deep understanding of numerical flexibility. In other words, I would love students to understand why the processes they use to arrive at answers work. Vidic Maths creates more problems than solutions. Instead of focusing on the mechanisms behind the methods, they promote the rote memorization of more facts. Unfortunately, these rules work in very limited capacities. (Multiplying two numbers close to a power of ten, or multiplying two digit numbers by eleven, etc.)

Unless taught with a focus on understanding the mechanisms, Vidic Maths is reduced to yet another way we can teach students to follow directions in stead of creating directions. 

NatBanting

Note: My full impression of the system comes from this video. I can envision a system of education which uses understanding of these methods as a launchpad into deep understanding. Such a system is not portrayed in any shape, way, or form in the video. 

Tuesday, April 9, 2013

Rubricized: Thoughts Provoked by Skemp

This week I had the privilege of chatting with other math educators about an article written by Richard R. Skemp in 1976. We have formed a sort of ad hoc reading group built around reading classic and contemporary pieces of mathematics education research and discussing their application to our daily crafts. The inaugural meeting (so to speak) consisted of Raymond Johnson (@MathEdnet), Chris Robinson (@absvalteaching), Nik Doran (@nik_d_maths), Joshua Fisher (suspiciously un-twitterable), and myself(@NatBanting).

The full conversation--facilitated through Google Hangouts--can be viewed on Raymond Johnson's blog here.

It must first be said that I loved the article; it would be a great starting point for any math educator to begin to wrestle with the distinctions obnoxiously present in the field of math education. It also holds timeless value for those more experienced in the digestion of such literature. Full reference can be found at the end of this post.

Moving forward...

In a nutshell, Skemp proposes that there are two types of understanding when it comes to the field of mathematics. The first--relational understanding--describes the process of knowing what to do and why you are doing it. The second--instrumental understanding--describes the process of applying rules to arrive at answers. It must be said right that these two understandings are not mutually exclusive. In fact, most of our conversation revolved around their classification, necessity, and interrelation.

The group--following Skemp's lead--addresses the issue of assessment in light of relational understanding. How can we begin to assess deep understanding of mathematics, especially understanding that is not our own. The methods of portfolios and interviews were suggested, but dismissed as regular assessment pieces due to the extensive time burden. 

Here is my thought on the difficulty of assessing relational understanding:

Relational understanding is difficult to assess because teachers use an instrumental approach in all assessment.

Allow me to briefly explain...

A lot of  teachers match instrumental--or algorithmic--assessments to their instrumental teaching. That seems like the correct thing to do. Recent work in mathematics education suggest that including lessons that build relational understanding has certain benefits for student learning. The only problem is that teachers need to be able to spot and reward that learning. 

Let's say you are working through a task where a student has generalized a problem, worked effectively with others, deduced possible pathways toward a solution, checked the reasonableness of their solution, and even posed further problems or abstracted universal truths. In order to judge their mathematical progress, you naturally start looking for key indicators of success. 

Things like:
  • Did they use a diagram?
  • Did they consider all possible cases?
  • Did they link the problem to a previous one?
  • Did they check their answer?
  • Did they persevere throughout the process?
Teachers are using an algorithm to assess work that is designed to empower students to see the larger picture. I believe that, over the years, teachers develop their own set of rules that govern assessment and then apply them instrumentally to assess students. (Now this is a very young belief, and I am open for counter-beliefs). 

I also think that within this relational-instrumental struggle exists the birth of the infallible (**cough**) assessment tool known as the almighty rubric. All rubrics do is attempt to instrumentally describe relational skills, yet they are upheld as the ideal way to achieve relational assessment. As teachers, our entire assessment framework has been rubricized; everything must fit into a nice box in a nice grid.

This is the portion where I should suggest a solution, but I am far from it. I have only just begin to realize that even my observations during otherwise "relational" tasks are of an instrumental nature. Algorithms are a part of mathematics, and assessing them is--by its very nature--algorithmic. I am now wondering what a non-algorithmic assessment looks like.

NatBanting

Reference:
Skemp, R. R. (1976/2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Originally published in Mathematics Teaching. Retrieved from http://www.jstor.org/stable/41182357