Why Teaching Math Is So Hard

1. Why Teaching Math Is So Hard Dan Kennedy Baylor School

2. Teaching math is so hard because… Mathematics is not something you learn. Mathematics is something you do. This makes mathematics different from virtually every other subject in the traditional school curriculum.

3. Here are some things that some people can do: Juggle. Waterski. Play the violin. Dunk a basketball. Run a marathon. Mathematics. Other people cannot.

4. People never say, “I never could do English,” or “I never could do History.” These are subjects you learn. People do not readily excuse themselves for not learning.

5. And who defines what it means to “do math?” Math teachers. This is a big difference between the ability to do mathematics and the ability to read!

6. Someone who can read this sentence knows how to read. How about this sentence: Ontogeny recapitulates phylogeny.

7. What does it mean to do mathematics?

8. Do we teach students to do mathematics or to understand mathematics? + + + – – – 0 + 2 + 1 – (–4) – (–4) – 0 = 11

9. Compare this to: So how do we justify teaching a meaningless computational trick that is ONLY good for computing 3-by-3 determinants? It does not generalize to higher orders. It does not even suggest anything important about how determinants work!

10. Teaching math is so hard because… The people who must identify which students cannot do mathematics are the same people entrusted with teaching them. Their math teachers!

11. A sobering thought: There are people walking the streets of Buffalo right now who became convinced years ago that they could not “do math” -- because they could not “do” some things that we no longer even teach today!

12. For example, here is what we were doing 45 years ago:

13. Teaching mathematics is so hard because: This is America. “The problem is, this is such an unusual country.” -- Jan de Lange, Director, Freeudenthal Institute for Teaching and Learning,Utrecht, The Netherlands

14. “My Ph.D. was in mathematics; by most standards, I was very 'well trained.' Nonetheless, the education that I received was in many ways impoverished.” -- Dr. Alan Schoenfeld, Reflections on an Impoverished Education, from Mathematics and Democracy: The Case for Quantitative Literacy, NCED 2001

15. Teaching math is so hard because… Our professional leaders cannot agree on what mathematics we ought to be teaching!

16. In 1983, the College Board published a small booklet called Academic Preparation for College. It listed the following “basic academic competencies” for mathematics…

17. The ability to perform, with reasonable accuracy, the computations of addition, subtraction, multiplication, and division using natural numbers, fractions, decimals, and integers. • The ability to make and use measurements in both traditional and metric units. • The ability to use effectively the mathematics of:− integers, fractions, and decimals;− ratios, proportions, and percentages;− roots and powers;− algebra;− geometry • The ability to make estimates and approximations, and to judge the reasonableness of a result. • The ability to formulate and solve a problem in mathematical terms. • The ability to select and use appropriate approaches and tools in solving problems (mental computation, trial and error, paper-and-pencil techniques, calculator, and computer • The ability to use elementary concepts of probability and statistics.

18. Ironically, it was that very same year, 1983, that another document was published, destined to change the rules for high school academic preparation for years to come… A Nation at Risk: The Imperative for Educational reform

19. From A Nation at Risk: “If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war.”

20. The Math Wars: Quantitative Literacy vs. Calculus Preparation Theory vs. Applications Rote vs. Constructivism Tracking vs. Mainstreaming Etc.

21. No matter what mathematics we choose to teach or how we choose to teach it, some people will believe that we have made the wrong choice. Some of them will say so. And they might be right.

22. Teaching mathematics is so hard because: Everybody recognizes the importance of mathematics, even if they do not understand it.

23. There are only three R's: Reading ‘Riting ‘Rithmetic God help us; we are one of them.

24. If Johnny can read and Johnny can write, then the fate of Johnny will be determined by whether Johnny can do 'rithmetic. The mathematics teacher is the Gatekeeper. Whether Johnny becomes a wealthy CEO or a penniless beggar is entirely up to you. Have a nice day.

25. Teaching mathematics is so hard because: Most people believe mathematics is constant over time. Unlike any other science. Sort of like religion. Thou shalt factor.

26. Parents realize that physics, chemistry, biology, history, and geography are different today from when they were in school. So they allow those subjects to change. Not so mathematics.

27. The following things upset people: Inability to do their child's homework. Any appearance of the words NEW and MATH in the same sentence. Calculators. The assertion that mathematics is for everyone.

28. Some people seem to think that pre-college mathematics is timeless. If it was important for our parents, how can it be unimportant today? But technology has been rendering our parents’ mathematics obsolete for decades. For example, consider log tables.

29. Here is a 1928 College Board mathematics achievement exam. It looks a lot like today’s college placement tests. But that is another talk.

30. And speaking of logarithms…

31. NYSRegents Teaching mathematics is so hard because: Assessment is out of control.

32. What should we assess? • What we value • Learning

33. I learned an important fact about classroom assessment when I began teaching AP courses: It changes the entire classroom dynamic when the teacher honestly does not know what will be on the test. The teacher has no other option but to teach the students how to think for themselves!

34. Why students don’t think on tests: • Thinking takes time. • Thinking is only necessary when you cannot do something “without thinking.” • If you can do something without thinking, you can do it very well. • Students who can do something very well have been well-prepared. • Therefore, if you prepare them well, your students will proceed through your tests without thinking!

35. This is how the educational game is played: • We show the students how to do math. • We let them practice at it for a while. • Then we give them a test to see how well they can mimic what we did. • The game is won and lost for BOTH of us on test day.

36. We must value correct mathematics more than we value correct answers. We must let our students know it. What we don't see can hurt our students.

37. Good algebra:

38. Bad algebra:

39. Teaching mathematics is so hard because: Articulation is out of control. High School College

40. Once upon a time there were 11 AP courses. One of them was in mathematics. Today there are 37 AP exams in 20 subject areas. Three of them are in mathematics.

41. “Currently, the greatest growth in the high school curriculum is in courses that have traditionally been taught in colleges. “The greatest growth in the college curriculum is in courses that have traditionally been taught in high schools. “It is not clear that either institution is serving its clients very well.” --Dr. Bernard Madison, Chair of the MAA Task Force on Articulation, 2002

42. Unofficial 2009 point u u 2008276,004 exams 2003212,794 exams u 1993101,945 exams u 198651,273 exams u 196710,703 exams 1955285 exams u u

43. What once was designed to be a program for validating college-level study of a few courses by a small percentage of exceptional students for the purposes of college placement has gradually morphed into a program for validating a rigorous high-school curriculum in a wide variety of courses for the purposes of college admissions.

44. What effect is this AP scramble having on the students? On the one hand, they are condensing or skipping foundational courses, so they are less prepared for advanced courses. On the other hand, they are taking more advanced courses, assuring that their lack of preparation will be exposed!

45. The Race to Begin Algebra I 8th Grade… 7th Grade… 6th Grade…

46. College Mathematicians want our best high school students to have: Deeper understanding More proofs Harder problems Challenges like the AMC Competitions No rush through foundational courses No courses in high school beyond freshman calculus

47. What some colleges are doing now: Demanding 4’s or 5’s for credit Changing freshman calculus to a course that assumes students have already studied calculus

48. How students are reacting: They are less likely to seek credit. They are more likely to play it safe. They re-take calculus in college… …unless they opt out of college mathematics and “take the credit and run.”

49. Other secondary-to-college articulation problems: Technology. Pedagogy. Placement tests. Math wars continued. Graduate students. Etc.

50. Teaching mathematics is so hard because: Technology is out of control.