I just came across this old post by Ken DeRosa!
Apropos in this summer of SAT math prep.
Maybe PWN will tell us what level of difficulty this problem would be rated on the SAT. I'm thinking 3 or possibly 4, and it would be a 4 only because a lot of students haven't taken algebra 2.
Stumbled across this website while meditating on how to enhance mathematical reasoning skills. I'm giving a workshop to K-12 teachers in about a month where we will be using computational thinking and computers to improve student's modeling and abstraction skills.
ReplyDeleteIn my region of the planet, beginning college students simply aren't prepared to do college level work. They have no idea that the level of expected performance in K-12 is leagues below college level. Students and parents don't know their K-12 school system set them up to fail.
I believe two root causes must be addressed before the situation will get any better:
1. Align the incentives for K-12 teachers, staff, and administrators to what students really need to be successful,
2. Create a culture that values education, knowledge, and teaching.
Currently, the incentives encourage the K-12 systems to pass students on to the next grade in order to avoid punishment. For example, teachers are fired if the failure rate in their class is too high. This is the direct result of a system that is dominated by "high stakes testing" as the sole form of school system assessment. It is just one example of the public war on teaching. My state legislature is dishing out punishments to all of education while proclaiming "education is vital to the future of this state".
Over the July 4th holiday, I listened to several of my cousins who recently finished their undergraduate degrees discuss how "getting a degree" is just a hurdle to be conquered. They clearly articulated the current values of western culture about education; "no need to learn anything, just get the paper."
My consulting customers are unhappy with the educational attainment of their new hires, study after study confirms the US is falling behind in education, and we can't even get enough students in STEM fields to satisfy the available jobs in a recession.
I believe it will take a significant shift of the public attitude about learning and the value of knowledge. You may never "need" to know how to compose a function. But, it is basic knowledge that every high school graduate should know without a moment's hesitation.
The SAT does do some nested function questions, but they're not all that common, and they usually involve actual values. This would be more "SAT-ish" if it said something like "If f(3) = n, what is g(n)?"
ReplyDeleteI'd say it's difficulty 4, as it is, but it's tough to say for sure since I can't think of an instance of something like this appearing on the real thing.
The greatest danger is the temptation to multiply it out and collect like terms, increasing the chance for arithmetic errors.
ReplyDeleteBut the question only asks, "what is an expression for f(g(x))...
f(g(x)=4x²+30x+56 (ahhh, that feels better)
Well, calculus isn't on the SAT, but if you can't handle composition of functions, you're sunk in calculus. Being able to substitute one function for another is vital (and so is being able to recognize that's what's going on when you're faced with a expression like (x^2 + 1)^5), even in a non-rigorous situation like business calc (which I happen to be teaching right now in summer school ;) ). How many kids figure that they don't have to worry about algebra, etc. because they have no plans to major in math, and then get weeded out because they can't pass the biz calc hurdle?
ReplyDeleteThe other really big math deficiency most kids face is their near complete inability to simply and manipulate fractions. They are so used to punching things out on a calculator that when they start seeing rational expressions that they have to manipulate and simplify, they bomb. Again, a vital skill if they want to get through weeder classes like finite math and business calc, and they struggle with something that is in essence fourth grade math (at least working with fractions was when I was in fourth grade).
I'd say it's difficulty 4, as it is, but it's tough to say for sure since I can't think of an instance of something like this appearing on the real thing.
ReplyDeleteInteresting.
It's true: I've only seen nested function questions involving numbers.
This is the direct result of a system that is dominated by "high stakes testing" as the sole form of school system assessment. It is just one example of the public war on teaching.
ReplyDeleteAnonymous - thanks for posting!
I see the public war on teaching a little differently.
As I see it, public schools have for many years been focused on the needs of the adults. I'm not saying that as a 'punishment' to teachers but as an observation of reality. When kids' needs coincided with adults' needs both needs are served, but when adult needs are in conflict with kids' needs, the adults' needs prevail. (Keeping incompetent teachers in the classroom is an example of an adult need for employment security trumping a child's need for an effective teacher.)
For my money, education reformers have simply taken this focus on the grownups and stood it on its head: with high-stakes testing, we're **still** focusing on the grownups, only now we're firing them (or trying to).
Speaking as a parent, I need testing to be about the kids.
If my child does poorly on a **good** test, I need the school to a) know that he's done poorly and b) remediate his learning at once so he's good to go next year.
Richard DuFour is great on this subject.
As a young principal he was determined to raise student achievement in his school, and he spent hours and hours and hours observing teachers and conferencing with teachers, etc., etc.
Student achievement didn't budge.
He finally had a revelation: focus on the students, not the teachers.
The rest is history.
Adlai Stevenson High School has had continuously improving student achievement since the 1980s; mode score on AP exams is 5.
It's a truism that we need good teachers, but the way to get good teachers is to focus on the kids.
Down in the basement of Functions 'R' Us, Gus, the function mechanic, gets an order for a function H(c) that will map the number of cups of flour into the number of happy kids.
ReplyDeleteGus scans the function shelves for ideas...
There is a function H(g) that takes the number of glazed donuts and outputs the number of happy kids.
"Well, that would work, but I don't have glazed donuts; I just have cups of flour."
Soon his eyes light on a recipe function, G(c) that maps cups of flour into glazed donuts. Gus smiles...
"HoG(c), aka H(G(c)) is just what I need."
As I see it, public schools have for many years been focused on the needs of the adults.
ReplyDeleteI absolutely agree, the focus is all wrong. The present system's incentives don't support the goal of providing a meaningful and useful education for students. Focusing on the adults is one (excellent) example of how the incentives are *NOT* aligned to promote student success (point #1).
To clarify my remark about the public war on teaching, I mean that I believe the US has lost sight of the real purpose of learning and obtaining an education. When I hear students say things like "D stands for degree" and "I just need the piece of paper", it tells me that the message of what an education is about has been perverted. I must assume that this new message is emanating from our culture, including parents, government, institutions, and popular media. We need a cultural shift that reinforces the intrinsic value of education. As long as completing K-12 or college is treated as nothing more than a check box to be ticked off, I don't believe we will be able to make the required adjustments to the educational infrastructure in the US.
Thanks for the insightful response and I'll "Google" Richard DuFour.