second effort

Posted on July 15, 2012

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There’s a “Khan-test” now to critique a Khan Academy video in an entertaining and enlightening way.  (That’s “contest,” with cash prizes.)
When I saw my first two Khan videos, I was inspired  to sign up for the next courses in Parkland College’s Graphic Design curriculum.   When  the course ended in May, I realized I didn’t need another course — I needed practice applying those skills. So, I fished out my class notes, watched “An Artist” for inspiration, and put together a minute or so about exponential powers.
Here’s a draft of my #mtt2k entry, which I’ll be revising between now and the August 15 entry deadline.  This is the flash version; YouTube per contest, with “critique” pending.
http://www.resourceroom.net/2012math/july14exponents.swf

Update:   the actual entry with critique of Khan’s treatment of videos is at http://www.youtube.com/watch?v=8DGoUkhLBIY

I have been excited at recent developments in curriculum and pedagogy in teaching math to older students. People are actually asking students about what they understand,  instead of simply giving them math problems to do and wondering why our pass rate in math courses is so abysmal.

It seems people are figuring out that we should put less emphasis on procedural knowledge and a whole lot more on conceptual knowledge in math.

The Khan Academy goes off like a bullet in the opposite direction.
The Khan videos talk *about* the idea of concepts… but the actual instruction? Procedure, procedure, procedure. The belief is that if students practice procedures enough, they will intuit the concepts (Khan says as much in some of the videos and interviews).

Welp, that works for some of us… but leaves entirely too many students behind, believing that if they don’t learn that way, they “aren’t a math person.”

See http://www.resourceroom.net/Math/perspectives_dyslexia_math.pdf  –“Why Can’t Students Get the Concept of Math?”  — and whilst the article is in a periodical dedicated to dyslexia, the research it’s talking about is not.   (That issue of Perspectives just talked about students struggling with math, not students with any kind of diagnosis of disability. Yes, that says something about how normal it is to struggle with math.)

I believe that this is what Salman Khan finds frustrating.   He says people are accusing him of wanting the opposite of what he wants.  He *wants* people to understand the concepts.  I believe that — but I also believe that he doesn’t know how to teach concepts.    He doesn’t even try.  There’s a huge gap between what he says is important and those videos.

I firmly believe that a whole lot more people could be “math people” if we actively taught them the concepts, and used computer technology to *show* the connections between the symbols of math and their concepts and meanings, and then consciously have the students build a bridge between words, symbols, and both the concrete and abstract ideas that mathematical symbols and equations stand for.
I taught at The New Community School, where we focused on starting where the student was and building bridges between the concrete and the abstract, and consciously connecting language to those concepts. I watched students get smarter before my eyes. Many of them were bright and gifted students who were struggling in school. I wondered then, as I wonder now, why we don’t use our technology tools to facilitate that connection between visual learning and symbols.

Many of them, though, *weren’t* particularly bright or gifted.   Guess what?   They got smarter when the teaching was better. I don’t just mean they learned better.  I mean they got better at learning.

Instead of deciding that if somebody doesn’t get “simple” concepts, that it’s not worth teaching them, perhaps we should look for better ways to convey the ideas.  I don’t mean to dumb them down and walk the person through everything — I mean to scaffold things and then build the cognitive muscles and then ease the scaffolding away.

Yes, that’s a big task — but I think technology tools can make it possible and even feasible.  Just imagine what we could do if the folks we relegated to “can’t do math” actually learned to do math, *and* to figure out how to do new stuff. We’d be unstoppable.
The other apparent belief at Khan Academy is that time is better invested in the trappings of the videos — badges and the like — than in the educational content. My students deserve better and I’m not going to pretend that it doesn’t make me  angry that millions of dollars are funneled into that. Instead of saying, “well, nothing’s perfect,” why not spend the time to make something good?

Can I back up what I say?
Here’s some of the evidence that “practice some more, you’ll figure it out!” is what we need to leave behind, even if it’s dressed up in nice technology, and why I think we need to explore more concrete & visual connections to math:

A. Research & evidence of the need
http://www.learner.org/resources/series26.html “Minds of their own” — these documentaries demonstrate that misconceptions are, actually, hard to uproot even in very bright individuals. (Hint: saying that to solve an exponent we “don’t just multiply the numbers!” many, many times does not uproot the misconception.)

What Community College Developmental Mathematics Students Understand About Mathematics ( http://www.carnegiefoundation.org/carnegie-perspectives/what-were-learning/what-community-college-developmental-mathematics-students-understand-about-math ) This includes an analysis of test scores showing  common errors were, demonstrating a rather profound lack of basic understanding of concepts. Extensive interviews showed that students relied on memorized procedures (even when they got answers correct), rather than comprehension, and that K-12 instruction emphasizes procedures. Key quote:
“…these students who have failed to learn mathematics in a deep and lasting way up to this point might be able to do so if we can convince them, first, that mathematics makes sense, and then provide them with the tools and opportunities to think and reason.”

Building on Foundations for Success: Guidelines for Improving Adult Mathematics Instruction http://lincs.ed.gov/publications/pdf/AdultNumeracyReportFinal2011.pdf :
“Mathematics content should emphasize a consistent link between math concepts learned and their use in context and form a coherent progression of learning.”

The Components of Numeracy: http://www.ncsall.net/fileadmin/resources/research/op_numeracy.pdf
Discusses issues with adults learning about math, three components of numeracy:
“In total, we found 29 appropriate or informative frameworks applicable to adult numeracy. From these documents and from our understanding of the existing body of related research, we propose three major components that form and construct adult numeracy:
1. Context — the use and purpose for which an adult takes on a task with mathematical demands
2. Content — the mathematical knowledge that is necessary for the tasks confronted
3. Cognitive and Affective — the processes that enable an individual to solve problems, and thereby, link the content and context ”
and 4 ways adults use math:
Numeracy for Practical Purposes … addresses aspects of the physical world to do with designing, making, and measuring.
• Numeracy for Interpreting Society … relates to interpreting and reflecting on numerical and graphical information of relevance to self, work or community.
• Numeracy for Personal Organization …focus is on the numeracy requirements for the personal organizational matters involving money, time and travel.
• Numeracy for Knowledge …deals with mathematical skills needed for further study in mathematics, or other subjects with mathematical underpinnings and/or assumptions (Butcher et. al., 2002, p. 215). ”

(Bloggers note: Number four is where we spend a great majority of our time and effort, and could have been rewritten as “Doing Required Math Classes for School.” )

http://pact.cs.cmu.edu/koedinger/pubs/BoothKoedingerCogSci2008.pdf Key Misconceptions in Algebraic Problem Solving Julie L. Booth (juliebooth@cmu.edu) Kenneth R. Koedinger (koedinger@cmu.edu) Human Computer Interaction Institute, Carnegie Mellon University Pittsburgh, PA 15213 —
Abstract: The current study examines how holding misconceptions about key problem features affects students’ ability to solve algebraic equations correctly and to learn correct proceduresfor problem solution. Algebra I students learning to solve simple equations using the Cognitive Tutor curriculum (Koedinger, Anderson, Hadley, & Mark, 1997) completed a pretest and posttest designed to evaluate their conceptual understanding of problem features (including the equals sign and negative signs) as well as their equation solving skill.Results indicate that students who begin the lesson with misconceptions about the meaning of the equals sign or negative signs solve fewer equations correctly at pretest, and also have difficulty learning how to solve them. However,improving their knowledge of those features over the course of the lesson increases their learning of correct procedures.

Blogger’s Note: This is especially interesting because, essentially, their “Cognitive Tutor” curriculum was shown to be reasonably effective at teaching the math procedures, but if students didn’t understand concepts such as what equals signs and negative signs mean at a conceptual rather than procedural level, that this adversely affected their learning from the software. “However, improving their knowledge of those features [the equals & negative signs] increases their learning of correct procedures.”

(One of the authors also wrote an article about this research that appears in teh International Dyslexia Association’s _Perspectives_ which can be found at http://www.resourceroom.net/Math/perspectives_dyslexia_math.pdf .)

http://www.caalusa.org/NumeracyChallenge.pdf
“FACING THE CHALLENGE OF NUMERACY IN ADULT EDUCATION is
CAAL’s final report on its two-year Adult Numeracy project. This topic is one of the
most complex, neglected, and extremely important areas of adult education. We hope the
report will generate further discussion of it and action….
Facing the Challenge lays out the case for reform. It explains the differences between
traditional math and “numeracy” and why we should care. It presents and analyzes the
best ideas currently available for shifting from traditional math instruction to a more
comprehensive numeracy curriculum. It looks, often for the first time, at the key
challenges and issues we must face if we are to reform math education. For example,
major attention is given to several disturbing articulation problems that exist between
ABE preparation for the GED and between the GED and college placement tests
based on COMPASS. The paucity of math instruction for adult ESL students with low
levels of prior education, who make up the bulk of the adult ESL population, is another
key area of concern. The report calls for changes in curriculum design, measures of
assessment, teacher training and recruitment, and other elements of reform.” (from page 5 in the Foreword)
http://www.ncsall.net/?id=481 – Focus on Basics article by Dorothea Steinke about teaching “Parts and Wholes.”

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