On August 25, Sol Garfunkel and David Mumford ignited a firestorm with their provocative piece in the New York Times on "How to Fix our Math Education.” I’d like to use this column to respond to one of their conclusions, the comment near the end of their article that, “In math, what we need is `quantitative literacy,’ …”
I’ve written many of my columns about Quantitative Literacy (QL), including as recently as this past January (Mathematics & Democracy + 10). I was one of the leaders in developing a QL program at Macalester, have served on the board of the National Numeracy Network, and continue to promote QL whenever and wherever I can. Yet I’m very bothered by the suggestion that QL is what we need in math.
This past winter, I visited Lehman College in New York to consult on the creation of a program in QL that is being developed by faculty from several disciplines. Their mathematics department was viewing this emerging course with some uneasiness. Would it replace their developmental courses in algebra? Were their students even ready for QL? In response to their concerns, I wrote:
A QL requirement should be independent of a mathematics requirement. If your students need algebra, QL should not replace that. In the other direction, algebra is not a substitute for QL. The mathematical and statistical skills needed for QL are basic. Algebra need not be a pre-requisite. What makes this college-level material is that these skills are applied and interpreted in messy, real-world situations, using quantitative approaches to aid analysis of complex social issues. In many respects, the natural home for QL is in the social sciences, but I believe that math and stat departments have an important role to play in keeping the mathematics of QL honest and encouraging quantitative thinking as one of the important tools for studying social issues.
This has been the guiding principle behind Macalester’s QL program. I am very proud of the strong inter-disciplinary nature of the QL program that we have created here, and I am suspicious of any program that claims to be QL but is taught exclusively by mathematicians. I also should add that Macalester has no mathematics requirement for graduation, but it does have a QL requirement. I heartily endorse this choice. I do not see a need for all students to study college-level mathematics, but I do see a need for improving their ability to apply quantitative reasoning.
This past summer, I had the chance to review a new textbook in QL that uses ratio and proportion as the unifying theme. The book presents a well-written course that can help students gain understanding of the power and uses of these basic mathematical tools. Many college students, some graduates of Macalester included, never achieve this level of understanding of ratio and proportion and would benefit from such a course. Yet I would hate to see this classified as a course in mathematics. It is not just that the mathematics is what should have been mastered in middle school. It is that the only reason this is a legitimate college-level course is that it transcends the concerns of the mathematics classroom.
While I feel strongly that we need to draw a clear distinction between mathematics and QL, I am not saying that mathematics should be taught without regard for the world beyond the classroom. All mathematics should be taught with the goal of promoting student ability to use these tools and ideas in ways that transcend the specific circumstances under which they have been learned. But we also need to recognize how very difficult it is to accomplish this. The research that I have seen suggests that the most effective means of reaching this goal is to lead students through an alternation of theory building and a variety of applications, combined with plentiful opportunities for personal experimentation and reflection. This goal is much more than quantitative literacy. It is the development of mathematical ability.
The Economist recently published an article that is directly relevant to August’s column, The Best Way to Learn. See “The Great Schools Revolution”, The Economist, Sept 14, 2011.
David, I understand how, in a college context, one needs to distinguish QL from "math". But I'm not so sure this is true in K-12. Your last sentence "an alternation of theory building and a variety of applications" captures precisely what I feel ought to be the structure of "math" courses in K-12 (if the applications are genuine real world questions) and what I find missing in much of the CCSSM. Call it what you want but courses in K-12 should show how math helps everyone understand so much in everyday life. Mathematical formalism for its own sake is never going to have an impact on most student's lives.
ReplyDelete(Part 1)
ReplyDeleteHow does someone who thinks mathematics is important object to Quantitative Literacy (QL)? One certainly does not want a quantitatively illiterate society. However, I am made nervous by the idea of quantitative literacy for the same reason I am made nervous by Standards movements. What does one do with people who are taught arithmetical and mathematical material but don't master it or forget it quickly after they were found to have learned it? What does one do with real world people who have troubles with learning what one thinks is important? In particular, with mathematics, I think it is critical that students get to be aware of the great range of mathematical tools that they or mathematical experts can bring to bear on questions which arise in the world, as well as the excitement of aesthetically pleasing mathematics. Good curriculum may not mean that everyone masters what one would like them to master but it may have a staying power on scales other than being able to answer questions about factoring polynomials and percentages on standardized high stakes tests. One metaphor I like to use is that students should learn enough about the nature of mathematics and its applications to know when "to call a mathematician."
While I applaud the quantitative literacy movement's concern with teaching using contexts and the applicability of mathematics, I am nervous that nearly all the domains I see involving QL deal with are very traditional topics in arithmetic. Questions about percentages, fractions, and number size, while important, also seem to me rather "dry." Yes, one sees "bloopers" regularly on TV and in the press but what is far worse is that high school graduates have no inkling of the power of mathematics and its importance in the development of new technologies that are changing our way of life so dramatically. In a general way I would like to see the importance of mathematical modeling stressed, particularly models that use elementary mathematics. Shouldn't this be in our curriculum, too?
(Part 2)
ReplyDeleteFor example, despite QL how many graduates of high school understand the role mathematics has played in making compact cell phones possible? When asked, lay people invariably see cell phones as gifts from physics and engineering, and while this is true, there is wonderful mathematics that is no more complicated than many arithmetic, algebra, and trigonometry topics taught in K-11 but have been "banished" by the CCSS-M. More specifically, your cell phone works because of:
a. Error correction codes (Richard Hamming)
b. Data compression codes (David Huffman)
c. Frequency assignment algorithms (graph coloring algorithms)
d. Global positioning technology
Mathematics has played a very significant role here. Another way to think of the issues is to look at specific modeling domains that many students find highly interesting as sources of questions: operations research and fairness questions. How can we improve voting and election processes including redistricting? Is the method used to assign each state a number of representatives in the House of Representatives fair? Is the weighted voting system used by the European Union fair? In the domain of OR there are problems like the Chinese Postman Problem (things to be done on the edges of a graph or weighted graph) and the TSP (things to be done at the vertices of a graph or weighted graph). There are also lots of wonderful applications of mathematics using elementary tools to studying genomics.
Why aren't data compression codes and error correction code ideas taught in K - 11? It seems to me the reason is that the mathematics community has tied itself to a curriculum which views Calculus as an entry point to STEM disciplines, thereby writing off large numbers of students whose mathematical talents and interests lie with more discrete and geometrical mathematical thinking. Furthermore, the current curriculum (here I mean the CCSS-M) poorly serves future parents and those who will pursue careers that stray from STEM majors.
Best,
Joe Malkevitch
email:
malkevitch@york.cuny.edu
David, perhaps you'll write in a future column about the so-called ELA Standards with respect to QL. I say "so-called ELA Standards" because their full name is “Common Core State Standards for English Language Arts & Literacy in History/Social Studies, Science, and Technical Subjects.”
ReplyDeleteThey say:
"When reading scientific and technical texts, students need to be able to gain knowledge from challenging texts that often make extensive use of elaborate diagrams and data to convey information and illustrate concepts. Students must be able to read complex informational texts in these fields with independence and confidence because the vast majority of reading in college and workforce training programs will be sophisticated nonfiction. It is important to note that these Reading standards are meant to complement the specific content demands of the disciplines, not replace them." (p. 60, CCSS-ELA)
Among other things, students in grades 9–10 are expected to:
"7. Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.
8. Assess the extent to which the reasoning and evidence in a text support the author's claim or a recommendation for solving a scientific or technical problem." (p. 62, CCSS-ELA)
Somewhat off the topic of the blog post: I disagree with some of the assertions made about the CCSS-M in the New York Times editorial. These are briefly noted, together with relevant sections of the CCSS-M, on my blog.