## Thursday, February 12, 2015

### Math problems of the week: Common Core-inspired sequence problems vs. sequence problems from the 1920s

From ixl.com, where "All of IXL's dynamic math and language arts skills are aligned to your standards, including: The Common Core State Standards All 50 states and D.C."

Accessed from ixl's high school math page, here are three problems from the "Find terms of a sequence (Precalculus - U.1)" page:

The arithmetic series problem set in Wentworth's New School Algebra:

ixl bases its pre-calculus sequence problems on this standard from the Core Standards for High School:
High School: Functions » Interpreting Functions » Understand the concept of a function and use function notation. » 3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Extra Credit:

Arguably, ixl's problems meet the above standard: arguably, doing ixl's problems encourages students to have the various recognitions that the standard specifies. Recognizing "that sequences are functions," etc.,  doesn't necessarily involve finding particular functions for particular sequences, or, for that matter, actually doing anything with sequences. Recognizing, after all, is not the same as doing.

So, in choosing words like "recognize" and "understand," have the Common Core authors chosen their words wisely?

Auntie Ann said...

Reminds me of this, by Ze'ev Wurman on a stab at science standards:

---

[...] I was excited when the National Research Council recently published its new Framework for K-12 Science Education, in which it outlines its vision for improving teaching science in America in the 21st century.

[...] This certainly looks promising, particularly because the framework for the first time introduces engineering as a subject of study for our K-12 students. Yet as I kept reading the document's 280 pages of lofty prose, I noticed something odd: The framework does not expect students to use any kind of analytical mathematics while studying science.

For example, the framework promotes a practice called Using Mathematics, Information and Computer Technology, and Computational Thinking (p. 3-13). Yet one observes that after singing paeans to the importance of mathematics, it only expects students by grade 12 to be competent in "recognizing," "expressing," and "using simple 'mathematical expressions' to see if they make sense," but not in actually solving science problems using mathematics.

Anonymous said...

ixl.com was there before Common Core, and these problems are the same as they had before, just now the standard they might sort of fit attached to them. Common Core aligned is not the same as Common Core inspired. It is just a bunch of online problems, and then they tried to figure out what standard to label them with when those came along. I think.

kcab said...

Those IXL problems belong in the, I dunno, maybe fourth grade section. I could see them as warm-ups in 6th grade, but they in no way belong in pre-calc.

At least Wentworth explicitly identifies them as arithmetic progressions. Without that label, I could choose any suitable answer in The On-Line Encyclopedia of Integer Sequences and justify it.

For example:

1,5,25,25,625,3125,...
https://oeis.org/A121007
"Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/5."

Similar to https://oeis.org/A121003, "Denominators of partial sums of Catalan numbers scaled by powers of 1/5."

Or I could make up a polynomial with the sequence numbers as roots, and add one more root chosen arbitrarily, and declare that the sequence names the roots of the polynomial in ascending (or descending, for the third problem) order.

Because, technically, a sequence is merely "an ordered set of mathematical objects". The definition of sequence says nothing about any relationship that may exist between successive members of the sequence.

Auntie Ann said...

FYI: Google has scanned Wentworth. It's available here: New School Algebra: Wentworth

and the answers here (although it's missing pages 34, 35, 42 and 43):