tag:blogger.com,1999:blog-18322056.post80502343820169628..comments2024-03-22T06:46:20.694-04:00Comments on Homeschool Math Blog: Pet peeves with Common Core Standards 1Unknownnoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-18322056.post-84596073403078784222012-02-17T22:41:44.697-04:002012-02-17T22:41:44.697-04:00A couple of random thoughts (mathematician here).
...A couple of random thoughts (mathematician here).<br /><br />It probably isn't -- but I wonder if this is some holdover from or otherwise influenced by the "new math" pedagogical stance of the 1960s. Just because the "official" abstract definition of a function in mathematics is as a set of ordered pairs.....but I can't imagine that has stuck around that long.<br /><br />I also feel like it is more intuitive for students to start with a function like y=x^2 and show that it isn't linear by graphing it, and then see how they could start with ordered pairs or a table and figure out whether or not it is linear. If the teacher restricts the starting rules to adding multiples of numbers from 0, it could work, but otherwise it just seems like it would generate more confusion that exploring the concept from some concrete functions first.quodedhttp://quoded.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-18322056.post-19766768706342318392012-02-17T04:38:34.987-04:002012-02-17T04:38:34.987-04:00Hi Maria,
I read with interest your post about th...Hi Maria,<br /><br />I read with interest your post about the C.C. standards, partly because I love math discussions, but also because I'm studying these standards myself, trying to get a handle on the pedagogical assumptions made by its authors.<br /><br />I understand your concern about the derivation of linear functions through generating sequences by the method of skip-counting. But I think there's a rationale behind the apparent madness: the C.C. standard authors are: a) focused on functions, and b) trying to help students grasp an important aspect of linear functions. In many of the more modern pre-algebraic and algebraic textbooks, there is a strong emphasis on the concept that, for linear functions, a fixed arithmetic change in elements of the x-list necessarily results in a fixed arithmetic change in elements in the y-list. For example, in the function y = 3x, an increase of 1 in two successive x-terms results in an increase of 3 for the two related, successive y-terms.(The same idea you wisely pointed out for the function: y = 2x.) This concept has been gaining in popularity as an important notion for students to grasp. And the people who find this important like to point out that a fixed increase in x terms may also lead to a fixed DECREASE in y-terms. For example, in the function y = – 3x, an increase of 1 in the x-terms leads to a fixed DECREASE of 3 in the y-terms. This aspect of functions is important for students to learn, it seems to me, because it helps students: a) recognize whether or not the values in a T-table are generated by a linear function. And conversely, but just as importantly, once students grasp this concept, they should be able to see that if a fixed increase in the x-values is NOT reflected by a fixed increase or decrease in the y-values, then the relation being represented must NOT be a function. For example, the ordered pairs — (0,0), (1,1), (2,4) — while the first terms of the important y = x^2 function — can not be elements of a linear function, since the same change of 1 in the x-values leads to two different changes in the y-values: 1 and 2. So I think it all boils down to the fact that this concept gives students a quick and easy way to tell whether or not a function is linear. And that's a good thing for students to know as they start working with both linear and non-linear functions. <br /><br />Having looked at both the Common Core and NCTM Standards, I've noticed that both sets of standards place a strong emphasis on the idea of function in general. And so, to my mind, this skip-counting approach to linear functions is just one more lesson that helps students explore functions and grasp a hallmark of the first functions must students learn: linear functions. <br /><br />Thanks for bringing up this question. It's important for all of us to question these new national standards. I have my own concerns about the C.C. standards, but this is not the place or time.<br /><br />Best regards,<br />— Josh Rappaport<br />Publisher, Singing Turtle PressJosh Rappaporthttp://www.algebrawizard.comnoreply@blogger.com