As we all know, the world ended recently (December 21, 2012). Being mathematicians and scientists, we will use the more specific statement: After December 21, 2012 (on the typical Western calendar), human life on Earth changed in a basic way. I think we’d all agree with this statement as being so obvious a statement that it might seem trivial.
However, what is not so obvious is the information I have recently acquired concerning the nature of this change. Through the sophisticated work of quarks, photons, and Marvin the Martian, everybody will now ‘get math’. We will no longer have students ask ‘when will I use this’, because they will understand the math and appreciate the innate value of this understanding. No longer will we have students say “I’ve always been terrible at math”, though a few might have a nagging feeling that they weren’t always really good at math.
In this new world order, math will not filter students from any field of study or life work. This will not mean that all students will have STEM majors; this is okay … we need some people who are not geeks or nerds about math & science, who choose to learn the truly hard stuff that normal people do not get (like arts, language, and psychology).
We will, of course, have a difficult period of adjustment. In math classes, we are accustomed to spending a great deal of energy on motivation and confidence; it will take time for us to change, and we can just hope that our students will be patient with us as we struggle with their competence with math.
The only serious point in this post is this:
Sometimes, we expect most students to have trouble understanding math; perhaps we would be better served by a possibly baseless optimism that most students can “get math”.
“Go math in 2013!!”
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Some years ago, we had an extended discussion about college credit for developmental courses (math in particular). The proposal being discussed was eventually superseded by other policies; however, strong opinions were voiced. During one commentary, a colleague was decrying students getting credit for such courses (though he had nothing against faculty who teach them. Our Divisional Dean leaned over to me and said “some of my best friends are developmental math teachers”, which I thought was quite funny (though the situation prevented me from laughing at the time).
When I hear some colleagues talk about calculators, I am reminded of that comment rephrased … “some of my best friends are calculators”. Calculators have their place, such colleagues say; calculators are not bad … it’s how students use them, so we need to prevent students from using calculators in a math class (as they say). In fact, I once took the position that graphing calculators not be allowed in a first algebra course (back in 1993). Since 1995, I have taught in an environment where graphing calculators are required starting with our first algebra course; although there are days when I find this frustrating, I have become a supporter of using calculators.
Unfortunately, the problem is much more complex than a ‘no calculator’ policy could solve; nor does a ‘required calculator’ policy solve these problems. Here are some of the problems that we can avoid discussing by focusing on a calculator policy issue:
- Students want a calculator for basic operations for a reason — they feel ‘dumb’ at math; that’s a major issue.
- Students view correct answers as being a valuable commodity, instead of seeing correct answers as suggesting good understanding
- Numeracy leads to feeling smarter; having a sense of how quantities ‘behave’ is possible for almost all humans (just like language literacy).
- Reasoning about quantities is a natural human endeavor, though we communicate this with language systems that are artificial (a necessary condition)
- A single math class tends to be very ineffective at changing long-held beliefs and habits; data suggesting an impact normally are measuring temporary conditions.
- The big picture ideas are more important than how a student calculates a particular value; the big picture includes their self-image about mathematics.
I like requiring a calculator in math classes, to provide a better venue to discuss these issues with students. Sometimes, a student ‘gets it’ (what we are talking about) and they change their math trajectory; for most students, it’s not that much of an issue either way — it took them 12 or more years to get to this point, mathematically, and a short-term experience is not likely to hurt them any more. Using the calculator, it seems, at least opens the doors to possible positive changes over a longer period.
This conversation with myself started when somebody reminded me of an article I wrote for the 1993 AMATYC journal; reading that article was an awkward experience, as I could see errors in my own thinking. Perhaps this post will encourage readers to examine their own position on calculators in math classes from a different perspective, one reflecting my course correction on the use of technology in mathematics.
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I am finally going to be able to attend the Achieving the Dream conference (Anaheim, February 2013) … see http://www.achievingthedream.org/DREAM2013 . My college is sharing a workshop time with Muskegon Community College, and part of this workshop will be on our new Mathematical Literacy course (MLCS in the New Life model).
For others who are attending, I am thinking of having a “Birds of a Feather” on New Life and basic reform efforts in mathematics (Thursday, February 7 at 11am). If you are attending the conference and interested, please let me know!
I’m looking forward to seeing a diverse crowd at “DREAM 2013”.
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Like most faculty, I encounter times in the semester when I have to wonder “how did we get to this point?” — such as when a student in a course like intermediate algebra does not recognize a product versus a sum, or can not recognize a right-triangle distance problem in context. I could follow the path of blaming previous bad teachers (all of them except me [:)] ), or on students who do not study; there might even be some truth in these explanations. However, the professional response is to explore how my course enabled these problems to survive until the end of the semester.
I am concluding that we (and I) stop working on ‘basics’ too soon; I (and we) presume that a passing score on an assessment like a chapter test shows that a student has the basics. However, I suspect that I depend too much on closed-task items on assessments, which enables some students to simulate appropriate knowledge without its presence. In addition, I am concluding that I need to design classroom interactions to constantly build literacy and analysis of mathematical objects.
People often say ‘mathematics is a language’, and promptly teach mathematics as if it was a set of mainline cultural artifacts. We can learn much from our colleagues in foreign language instruction, who tend to constantly use basic literacy into all work in a language and to deliberately address the cultural components of the language. I see most of my student’s basic failures within mathematics to be cultural issues (context, norms) along with language literacy within mathematics.
The implication I see for my own teaching is that classroom time needs to deal with ‘sum or product’ as an issue every day; nothing is more basic than this issue. In algebraic classes, there is an added layer of work on symbols and syntax which needs a similar focus (sum or product). I’m also seeing a need to deliberately address reading skills applied to a math textbook, and hope to coordinate these types of efforts.
I am constantly reminded of this notion: Novices do not automatically see the critical features and structures that experts see without effort. Our students are capable of more, and can reason mathematically. We need to deliberately show the features and structures we see, and provide scaffolding for students to become more expert. We do students no good if they leave a math class in the same novice mode as they started, with some limited problems they can solve.
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humor | Jack Rotman | 
December 28, 2012 12:31 pm | 
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Jack Rotman
NOTE: This blog will become 'inactive' on January 1, 2020.