Originally posted by: Cogman
Originally posted by: DrPizza
In my experience as a math teacher, the students who have struggled most are the students who never memorized the most basic things: the multiplication table for example. If you can't multiply 6*7 instantly, then of course you're going to be lost when it becomes 6x * 7x = 42x². There's just too much to decipher rather than just one new thing. I can still get students through those things, but in the long run, they typically get the lowest grades in the class (with the exception of the one super-lazy kid I always seem to end up with.)
Another not true. I can do upper level calculus pretty well, but sometimes I mix up numbers when doing multiplication or addition
. The worst is getting a sign wrong (my most common mistake).
<- still got an A in calc.
Note the magic words: "in my experience"
"Sometimes I mix up numbers" is far different from "don't know the majority of the times table from 1x1 to 10x10.
My calculus kids don't have a problem - besides, to put it rather frankly, I prune my students to not care what the numerical answer really is. We work through the hard parts and realize that if what our goal was is to get a numerical answer, well, that's what the computer and calculator are for. We practice calculus as an artform
Besides, it's fun to intimidate the algebra students who come in the next period and see an entire blackboard full of math and equations and funny symbols and a nearly total absence of numbers.
Again, however, I've had many many students in lower level math classes. And, at our school, we provide remediation for all students doing poorly in the lower level math classes. This means that in addition to having them in class, I also have the students who struggle in very small groups (sometimes only 1 or 2 students at a time; never more than 4, for an entire period once every 4 days) where I can really evaluate what their problems are.
The ones who struggle most have difficulty with the simple things that should have been memorized long before they ever got into an algebra class: their times table that they had in 3rd grade. I'm not talking about 13 * 17; I'm talking about 5 times 6. They see "30" and don't have a clue where it came from. Some of them have problems with simple addition skills - if I give them an example on the board "5x + 7 + 8x + 9" and write that it simplifies to "13x + 16", they cannot figure out where the 13 came from or the 16. It's nice to have them in small groups, because it gives me the time to (unfortunately) make them absolutely calculator dependent - not that they aren't already. For the rest of the year, for every such problem, they pick up their calculator and manually add 5 + 8, then they add 7 + 9. Guess which students get involved in so many steps that they sometimes goof and write the answer as 13x² + 16? The rest of the students are simply learning that when you add polynomials, you combine like terms and don't change the exponents. For the students who struggle, they have more steps to learn to be successful.
Another common set of concepts that should have been memorized before algebra (not that I don't go over it extensively) is sign facts. Thanks for pointing out that you have the same problem. Some students don't know & either can't or refuse to memorize that the product of a negative number and a positive number is a negative number. Without that little tidbit of knowledge, factoring something like x² + 10x - 24 becomes a crapshoot. The weaker students will almost always have a 4 and a 6 in there somewhere; I can predict the weather halfway around the globe better than I can predict what signs will be in there.
For the students who have their times tables memorized and understand the sign concepts, I can teach them how to factor trinomials with a leading coefficient of 1 in about 5 minutes. <here are the steps: work on the polynomial from right to left. Write down all the pairs that multiply to 24; go in order so you don't skip any of them on accident. 1 times 24, 2 times 12, 3 times 8, 4 times 6. Then, working right to left, do what the sign tells you to do. This time, it says to subtract. Which pair subtracts to give us - again, working right to left, which subtracts to give us 10? 12 and 2. Good. Now write you pair of binomials, (x 12)(x 2), and keep working right to left. The next thing is that (+) sign. Whatever sign is in that position goes on the bigger number. Now, the thinking step. How do you get that -24? +12 times 2 or +12 times -2?> 2 examples, let them practice 2 more examples, and most students have it mastered by the time they're done with their homework. Meanwhile, there are 3 or 4 kids who can't even come up with the factors of 24. Again, that's from the basic multiplication table! (and why I used this as an example) Of course, in small groups, I get those kids to pick up their calculators and do 24 divided by 1, 24 divided by 2, etc. - by this time, the rest of the class is so good at factoring that they don't write down all the pairs that multiply to 24 (or whatever number) - factoring becomes nearly automatic to them, and the kids who don't know the times table and sign properties are far far behind - and they're the ONLY students who don't master factoring trinomials (with a leading coefficient of 1) rapidly (if ever).
The other students eventually catch on - I work with them and give them extra time. But guess what - now we have to solve quadratic equations by factoring. Now we have to simplify rational expressions with trinomials in the numerator and denominator by factoring and canceling. Now we have to multiply those rational expressions by factoring a shitload of trinomials. Guess what - those students who don't know their multiplication table and their sign facts fall flat on their face by this point. The curriculum I have to teach is this -->|...........|<-- big. The time I have is this -->|........|<-- big. I'll stay after school to help those students, I'll come in early, I'll give up my lunches & my prep periods, I'll work during their extra help period (1 every 4 days), but every single time, the reason they struggle with these concepts is their multiplication table and their sign facts. Unfortunately for them, they end up needing to learn more in the course of the year than their counterparts who already know their basic facts. And, there's often a reason that they start the year behind the other students - whatever that underlying reason it - whatever the reason is that they don't know the times table and sign rules - it usually results in them continuing to do poorly.
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