It is no mystery that students are having increasing difficulty learning algebra in schools. One proposition referenced in
http://www.edutopia.org/blog/math-as-social-justice-lever-jose-vilson,
has been to eliminate some requirements, or make the content easier. Clearly they don't understand the value of learning algebra, or higher mathematics even when it isn't their area of study, but that is content of a future post.
By surrendering to this mind set, we are as educators, failing to give our students the credit they deserve. Where then, lies the problem?
In my experience it seems that implementation and delivery of the tools available to educators is a source of some of the difficulties our students have with mathematics. I believe our teachers would benefit from more training that would bring them up to date with the new and sophisticated tools we have now. Many of them are unprepared to deliver an effective mathematics education in this day and age where students rely on 'googling' and tools like mathematica to to hand them out answers to problems without even having to attempt them. I recently even had a new student ask me, "Why should I have to bother learning algebra, when Google gives me all the answers." Then there are also the 'accountability' requirements that forces educators to 'teach to the test', of which I talk more about more here:
Part I: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of.html
Part II: http://buildingblocks123.blogspot.com/2015/04/mathematics-problems-in-age-of_29.html
Before I mention what I believe is a possible solution, lets take a look at what type of questions our students today are having trouble with.
On the 2014 Texas end of course algebra I standardized exam question 20 was one of the top 2 most missed questions with only 32% of students answering correctly, was not multiple choice and read,
"20. There are 156 laptops and desktop computers in a lab. There are 8 more laptops than desktop computers. What is the total number of laptops in the lab?"
These is a surprisingly simple question, yet a large number of students missed it. What then can we do about this?
In Notices of the AMS Volume 61, Number 6, Dr. Igor Rivin discusses the principle that "A computer program IS a proof." . He goes on to write, "The justly acclaimed book 'Structure and Interpretation of Computer Programs' by H. Abelson and G. J. Sussman introduces the fundamentals of computer programming and, together with a companion book on 'Structure and Interpretation of Mathematics', this would constitute the core of a modern introduction to mathematics." I agree. Technology is all around us, our children use it everyday, and yet they don't really know how to use it outside of Facebook, YouTube, and 'googling'. Teaching mathematics is the perfect opportunity to conceptualize mathematical ideas, while at the same time teaching our students the true beauty of the technology that surrounds us.
I have successfully applied these methods while volunteering my time at local public housing projects helping troubled youth, living in poverty, with their schooling. One young man who had previously failed the algebra I eoc exam was my first student at the projects. I constructed a project from a problem I had seen, I taught him to write basic python code to help us conceptualize the problem. However, before he could write effective code, he had to understand the problem. Not long after he completed it with great success, I developed some groundwork and put together some notes for future use, and he went on to score 'Advanced' on the eoc exam. He now loves math and computer science and wants to study programming in college. I went on to have further success with this method getting much more mileage from this project, not only with troubled kids, but students of all ages, levels, and economic status. In case you are curious here are those notes(with my code at the end)--a bit crude but these are for my personal use. . .
Finally, I agree this is a, "difficult problem with nuanced roots and a hard uphill battle," but we have the tools to make a huge difference. Once again, I believe the problem is mostly in implementation and delivery, and not in the students' abilities. However, this takes hard work, dedication, and patience from educators who roll this out as they may likely not see immediate rewards. The biggest challenge however is, this takes a lot of work to make it work!
While a program can elaborate an algorithm, a program is not a proof.
ReplyDeleteWhile programming can complement a student's exploration of mathematics, the introduction of programming itself is not a panacea. Furthermore, the style of programming appears to matter as well. If the focus is functional relationships, then functional programming provides a great metaphor. If the relationship is proportional input-output relationships, then imperative appears to be better. More importantly, effective pedagogy requires that students be prepared and motivated to engage in analytic problem solving. That seems to generally require a skilled instructor.
First of all, thank you once again for your invaluable insight.
DeleteClearly, I have to be less liberal with my use of language--I should know better than to not be more precise in my language usage, especially in my field.
A computer program is of course not literally a proof, and I should have elaborated on my view point. However its design is analogous to a proof, and is this reason why it’s a great complement when exploring mathematics.
Coincidently you mentioned functional programming, which in essence, implement lambda calculus, and reminded me of the ‘Curry–Howard–Lambek correspondence’ which describes the isomorphism between t-lambda calculus, cartesian closed categories, and intuitionistic logic. Two nice reads on that here:
http://pages.cpsc.ucalgary.ca/~robin/class/617/projects-10/Subashis.pdf
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Berger.pdf
I don't pretend my ideas are an all-around solution, but rather a proposition to incorporate technologies that are advancing faster than we are adjusting to in the classroom. I will discuss this more in upcoming posts, but for now a few observations why I believe this is a good start.
Just yesterday a student asked me for help on her homework, but added she already found all the answers on wolfram alpha, and really wasn’t interested on explanations, instead just wanted me to fill in the steps because she wasn't subscribed to the pro-version which gives her the steps. A few minutes later another student asked me to check his work on an integral he had just finished solving. It was just a linear equation he had to integrate, but he converted from rectangular to polar coordinates, made the appropriate variable transformations before solving it, and then converting back 10 steps after he began (a one or two stepper). When I asked him why he did that, he told me it’s what seemed the most obvious and reasonable thing to do—turned out he is subscribed to the pro version.
I volunteer at local high schools. At one high school one of the teachers came to me because there was a brilliant young man he felt could no longer challenge since he solves all his challenges in minutes, and is super bored in his class. I accepted to help, and decided on some very elementary algebra and geometry problems after testing their searchability on the top three search engines. At the beginning I told him I would give him some easier stuff than his teacher to test his foundation. He was very vocal about how he felt he needed more challenging material, and this would only waste his time. An hour later he returned asking for hints, about fifteen minutes after that he emailed me saying the problems were impossible, and after about two wrote he had done some research(I assume this meant endless ‘googling’ yielded no results), and found out the problems likely required knowledge of material years beyond what he knew...I went on to have the privilege of working with this extremely bright young man personally, and eventually he went on to have several first place finishes at regional math competitions.
Solving math problems with smartphones in their hands and wolfram alpha or the 'Google' set and ready to go are only some of the things I see out in the field. We have to accept that technology is moving forward, and adjust accordingly, but I also feel we should teach our children to think for themselves. Though this is an exaggeration, I'd hate to think of a future where employees just sit in cubicles blindly ‘googling’ away, and copy pasting their search results as a product of their labor--reminds me of https://www.youtube.com/watch?v=VtvjbmoDx-I
Of course, this is only one aspect that improves as a consequence. I have applied these methods with great success in public classrooms, learning centers, and during private instruction. What gets me excited about these methods is I have in general seen students make a huge turn around, at an almost 100% rate--too high a rate of success to be a mere coincidence.