My husband moved us out to NY so that he could get a physics PhD (I know, I couldn't bring him over to the much more beautiful and elegant world of math.) He has an Iranian classmate that we've started hanging out with. The other night he invited us over for dinner with his roommates and friends all of whom are Iranian and all of whom are either studying physics, mechanical engineering or computer science. Because most of them have TA-ships and are teaching undergraduate courses, when they found out I was a math teacher they all turned to me and asked a ton of questions along the vein of "why don't American undergrads know any math?! What DO they learn in high school?" My husband's friend had been struggling with his undergrads in a physics lab because they couldn't make a simple algebraic substitution (I can't remember what the problem was, but something like if a=b/c and d=2a, then d=2(b/c). Of course instead of a, b, c, and d they had maybe q with subscripts.) I asked him if maybe the subscripts had confused them, and he said he went back to simple a, b, c and d variables and they were still stumped. It took him 2 hours to explain this substitution to these students. He said they had no sense of variable at all. They could solve equations by rote, and they had bits and pieces of algebraic techniques, but no logical understanding of what algebra is and why they need to know it for physics. The other Iranian PhD students chimed in with their own anecdotes of students who have come to college to study the hard sciences with very little mathematical aptitude. They spent a while discussing how the Iranian education system is much more rigorous compared to what we have in the US.
This is not a low ranked college. The students who come to Stony Brook University should know their algebra, especially those who want to study the hard sciences and math because it has very competitive science and math departments. And New York has the Regents. How can students who passed the grueling Algebra 1, Geometry and Algebra 2/trig Regents exams not know simple substitutions (and not be able to grasp them even when a physics TA comes over and personally explains the process for over an hour?) With such a small sample and only anecdotes from overworked TAs who aren't trained to teach math this is not a fair base from which to judge the New York high school math curriculum, but I'm feeling a little judgy at the moment especially after wrestling with the New York math standards and the regents for the first time this year.
Pre-algebra was a sacred class for me at my old school because it creates the base the rest of students' algebra understandings must rest on. For this reason I went really slowly and carefully in my pre-algebra class and made sure students were really understanding the jump from concrete to abstract mathematics. I strongly believe that pre-algebra should spend as much time as possible on cementing the ideas of what variables are, how to write expressions, and how equations and formulas are linked to variables and expressions. These are DIFFICULT ideas. Students need time to process them. They need the freedom to explore them in their own ways. They need to see how variables aren't just unknown numbers- that they're so much richer and more flexible than number- that's why they're so useful in algebra. Students should spend time observing patterns in variables (specifically, combining like terms and the exponent rules are a great way to do this) and how we can generalize number patterns using variables in simple and elegant ways. I believe this is what pre-algebra is for. It's NOT for statistics! It's NOT for quadratics and FOIL. It's NOT for re-drilling fractions, decimals, ratios and percents again for the 50th time. The New York (and Oregon for that matter) standards cram so much into each school year that students don't cement their knowledge or have time to make meaningful connections. This means that each topic appears in the math standards for at least four years in a row because students have to constantly review stuff they should have learned last year but only "covered" because there wasn't time to go into it in depth. (i.e. adding and subtracting fractions appears from 5th-9th grades.) Each topic gets "covered" each year but not taught each year. So quick students have to relearn the same content year after year, while students who struggle never properly learn it at all.
I know this argument doesn't necessarily have traction. Students need to review no matter how deeply you taught the material the year before, but I do know that I spent a month on developing variable sense and then another month showing students the usefulness of variables and expressions in writing out general number patterns placing specific emphasis on exponent rules and geometric patterns at my previous school and when the students needed the exponent rules again in algebra 1, we only needed a half-hour review and ALL my students were fluid with using them in very complex situations. I'm getting algebra 2, pre-calc, and calc students now who don't understand their exponent rules and their eyes glaze over every time I try to show them the logic behind the rules because to them, they're just a random assortment of letters to be memorized when needed and forgotten the rest of the time. You can't learn differentiation in calc without being able to turn roots and rationals into exponential expressions instead. This inability to understand that this one seemingly random technique (exponent rules) is rooted deeply in mathematical logic and needs to be understood logically because it is a foundational piece of the structure of algebra I believe is a symptom of the standards push for breadth over depth. Students have memorized math techniques as a history student memorizes dates. They may sort of have a sense of order, but no sense of significance.
Variable sense is important and deserves time. If given time in pre-algebra, students will be much more successful in their higher math classes. It does not deserve a week a year spread over 4 years. So to answer the question posed by our Iranian friends on what is wrong with American education, I think it's the standards. And more specifically, that no one seems to know what should be shoved into pre-algebra so they make it a hodgepodge of random techniques they think will be useful for algebra 1 rather than spending that year to really develop variable sense. And I am a part of the problem too because I'm correlating my lesson plans to NY state standards so that my students will be able to pass the Regents. I'm scared of going off in the direction I feel is right because it doesn't cover the "standards" I'm supposed to cover. I think pre-algebra is the problem and I wish I could go shake the people who put "determine if a relation is a function" and "describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations)" on the PRE-ALGEBRA standards. There's a reason we have a whole year of highschool geometry and two years of algebra. Give them time to get used to the idea of variable BEFORE rushing them into function translations!
