Why Teaching Mathematics is So Hard The Realities of Teaching Mathematics in the Modern World (Distilled from overhead transparencies) Dan Kennedy Baylor School Chattanooga, TN Dkennedy@chattanooga.net Teaching mathematics is so hard because: Mathematics is not something you learn. Mathematics is something you do. People never say "I could never do English" or "I could never do history." These are subjects you learn. People do not readily excuse themselves for not learning. Here are some things that some people can do: Juggle. Waterski. Play the violin. Dunk a basketball. Run a marathon. Mathematics. Other people can not. Do we teach mathematics, or how to do mathematics?                         –   –   –  +  +  + – 105 – 48 – 72 + 45 + 84 + 96 = 0 FOIL SOHCAHTOA ETC. Teaching mathematics is so hard because: The people who must identify which students can not do mathematics are the same people who are entrusted with teaching them. Their math teachers. 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! Someone who can read this sentence knows how to read. How about this sentence: Ontogeny recapitulates phylogeny. What does it mean to do mathematics? 5 + 2 = 24 ÷ 6 = Solve for x: (x + 2)(2x – 3) = 49. Find cos(ð/3). Find the product: 874539  374958. Find . What is  ? Which of these groups can do mathematics the best? Men. Women. Caucasians. Blacks. Hispanics. Asians. Math teachers. Teaching mathematics is so hard because: People can not agree on what mathematics we ought to be teaching! "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 Mathematics is like swimming: Young children can do it. People who teach swimming are not teaching you how to swim. You can appreciate and enjoy swimming without ever winning a race. Not many swimmers can swim the 200 Individual Medley. Nor will many ever need to. The Math Wars: Quantitative Literacy vs. Calculus Preparation Theory vs. Applications Rote vs. Constructivism Tracking vs. Mainstreaming Etc. 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. Teaching mathematics is so hard because: Everybody recognizes the importance of mathematics, even if they do not understand it. There are only three R's: Readin' Ritin' Rithmetic. God help us; we are one of them.   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. Teaching mathematics is so hard because: Most people believe mathematics is constant over time. Unlike any other science. Sort of like religion. 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. 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. Teaching mathematics is so hard because: Assessment is out of control. What should we assess? What we value Learning RULES OF THE EDUCATIONAL GAME: We tell students what they need to know. They practice at it for a while. We give them a test to see how closely they can match what we did. Thinking takes time. Thinking comes into play when you can't do something "without thinking." You can do something without thinking if you really know how to do it well. If your students can do something really well, then they have been well-prepared. Therefore, if you and your students both do your jobs perfectly, they will proceed through your test without thinking. 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. Good algebra: Bad algebra: Teaching mathematics is so hard because: Articulation is out of control. "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 Once upon a time there were 11 Advanced Placement courses. Today there are 35. Once upon a time we spoke of the possibility of a rare AP student being admitted with sophomore standing. Today there are nearly 2,000 AP National Scholars. These students must have grades of 4 or above in at least 8 AP courses. AP Statistics = 1/2 course. Other secondary-to-college articulation problems: Technology. Pedagogy. Placement tests. Math wars continued. Graduate students. Teaching mathematics is so hard because: Technology is out of control. Teachers can not keep up. The curriculum can not even come close to keeping up. But remember: Mathematics is not something you learn; mathematics is something you do. So, now that you have a computer algebra system in your hand.... What does it mean to do mathematics? 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 WHAT MATHEMATICS TEACHERS HAD TO DO FOR HIGH SCHOOL STUDENTS IN THE PAST Prepare the college-bound for calculus. Prepare the non-college-bound for employment. Identify which were which, usually by predicting the likelihood of (1) in individual cases. WHAT MATHEMATICS TEACHERS HAVE TO DO FOR HIGH SCHOOL STUDENTS TODAY Prepare them all for calculus. Prepare them all for employment. Prepare them all for a life dominated by computer technology. Prepare them all to pass state-mandated competency tests. Teaching mathematics is so hard because: There is so much more to teaching mathematics than just teaching mathematics. WHAT HIGH SCHOOL MATH TEACHERS HAD TO KNOW IN ORDER TO TEACH PROBLEM-SOLVING IN THE PAST 1.      Algebra. 2.      Geometry. 3.      Probability. 4.      Trigonometry. WHAT HIGH SCHOOL MATH TEACHERS HAVE TO KNOW IN ORDER TO TEACH PROBLEM-SOLVING TODAY 1.      Mathematics. 2.      Astronomy. 3.      Economics. 4.      Statistics. 5.      Physics 6.      Biology. 7.      Etc. This is such an unusual country. There is too much in your textbook for your students to learn. If you understand every problem in your textbook, it probably isn't much of a textbook. Learning is enjoyable. So is teaching. Any other philosophy of teaching and learning is counterproductive. Even in mathematics.