# why did common core remove so many topics from Algebra II?

I don't understand why common core: High School Algebra II (NYC, NY, USA) removed so many topics. I think these ideas are pretty important and useful (especially since a handful of these topics come up in SAT). These also help with critical thinking, thinking abstractly and mathematical reasoning (proofs and problem solving/algorithms).

• Rationalizing binomial denominators
• Dividing complex numbers (incl. rationalizing binomial denominators, ex: 10/(3 – i)
• Solving absolute value equations algebraically
• Solving absolute value inequalities
• Given a real-life scenario, write an absolute value inequality that models it
• Solve problems involving direct and inverse variation
• Simplifying complex fractions
• Using L. of Sines and L. of Cosines to solve triangles
• Finding area of a triangle using (1/2)abSinC
• Ambiguous case (SSA, using L. of Sines)
• Binomial Probabilities/Bernoulli experiments
• Finding probabilities based on comparing areas
• Finding probabilities using permutations and combinations
• Composition of functions: writing an algebraic rule for f(g(x)) given f(x) and g(x) and finding the domain of f(g(x)) given f(x) and g(x)
• Co-functions (applying the idea that cos(A) = sin(90 – A) in various ways)
• Angle Sum, Angle Difference and Double Angle identities
• Solving Trig equations (linear, quadratic, equations requiring use of the identities above)
• Where did this list of cut topics come from? – Daniel R. Collins Oct 8 '18 at 14:35
• Nothing is stopping you from continuing to teach these topics. especially since a handful of these topics come up in SAT The right reason to teach a topic is because it's important or useful, not because it's on the SAT. – Ben Crowell Oct 9 '18 at 21:51
• This question would benefit from some contextualization. In addition to explaining what the common core is and where to find information about its implementation in New York City, it would be useful to explain the specific context (when? who? why?) of the supposed removal of topics from these standards. This is all background that is largely opaque to someone not operating within the New York City school system, or at least the US educational environment. – Dan Fox Oct 10 '18 at 8:54
• That is a long list. It seems to me that just those topics could make up a whole course. U.S. education (especially in math) has been faulted for being a mile wide and an inch deep. Taking out topics is an important part of improving the situation. If you think those topics are more important than what is left in, that would be a more interesting question to me. – Sue VanHattum Aug 16 '19 at 16:24
• @Sue VanHattum: To continue your comment, many of these topics (e.g. the trig. topics, binomial probabilities/Bernoulli experiments, finding probabilities using permutations and combinations) seem to me to belong to the next level course, precalculus, and not to algebra 2 (unless an honors algebra 2 course, but then we'd be out of the realm of standardized topics). Indeed, until 2016 trigonometry was not even tested on the SAT, and even now most of the trig. topics the OP listed are not tested. – Dave L Renfro Aug 16 '19 at 18:36

The Common Core State Standards for Mathematics (CCSSM) provides a list of standards that should be covered; the inference that a topic not mentioned in the algebra strand should, therefore, be omitted does not necessarily follow, although time constraints no doubt ensure that will be the case for many courses.

Rather than going into the process for developing CCSSM, I believe it may be helpful to wonder about the '2' in the course title: What exactly is meant by Algebra 2?

• It follows a course called Algebra 1

• It precedes a course called Algebra 3? (No, unless you mean Linear Algebra; followed by Abstract Algebra; then Abstract Algebra 2 = Galois Theory; then Algebraic Geometry or Algebraic Topology or Algebraic Number Theory or...)

• It precedes pre-Calculus, but doesn't that make it - for all those who eventually take Calculus - fall under the umbrella of (literally) pre-Calc?

• It is pre-Calculus, and the next course is to be Calculus

The last two points are especially important: Is this course leading in to a follow-up course that examines further function families (e.g., rational functions), trigonometry, and other topics? Or is it headed straight for secant lines that segue into tangent lines and eventually limits in a first course on Calculus? Alternatively still, is Algebra 2 supposed to position students to make a decision around whether they should continue on a path to Calculus or Statistics/Probability? This could have an effect on topics for inclusion.

In fact, could Algebra 2 be expected as a terminal course? For some US colleges/universities, yes, this is the expectation; in fact, some do not require even Algebra 2. Consider this piece from Inside Higher Ed:

Michigan State University has revised its general-education math requirement so that algebra is no longer required of all students. The revision reflects an increasing view on college campuses that there is no one-size-fits-all math curriculum -- and that math is often best studied in connection with everyday life.

(I note with no irony or sarcasm -- only interest -- that MSU truly has one of the best programs in the country/world for mathematics education.)

Returning to your list of topics, it would probably take me a full year to cover a proper subset of those topics in ways that are mathematically meaningful. If you will permit a mixed-metaphor, then I note that one can "cookbook-blitz" these topics by asserting facts and having students carry out various procedures (e.g., transforming trig identities into statements about just sine and cosine, and then using a very select few axioms to prove the desired consequence). Personally, if I were to teach about trig identities in a way that felt meaningful (in some ways) to me, then I might opt to use them as a way to introduce more formal proof writing (cf. my answer to MSE 2905643) but this would diverge from certain expectations around "mathematical maturity" in an Algebra 2 course, and it would take far too long for me to get to all the "omitted" topics in your post - as well as those that remain!

For a fun mathematical closer, here is one example of a cookbook-blitz approach to trigonometry: Accept the fact (whatever it means) that $$e^{ix} = \cos(x) + i\sin(x)$$. Consider next two ways to change the LHS to $$e^{i2x}$$:

• Squaring both sides

• Replacing $$x$$ with $$2x$$

Since each of these results in the same LHS, the corresponding RHS's must be equal. By accepting that fact, we arrive, then, at the following identity:

$$[\cos^2(x) - \sin^2(x)] + i[2\sin(x)\cos(x)] = \cos(2x) + i[\sin(2x)]$$

Observe that matching real and imaginary parts immediately yields the double-angle formulae for cosine and sine. Now, I happen to believe this is an interesting derivation (and you can use this to derive many other identities provided, e.g., you know a special value or two of sine and cosine from the unit circle) but I do not think that a one day lesson on this fact is an ideal way to build meaningful mathematics with (most) students. In particular, I am concerned that the given is too strong, and that the techniques (observing two ways of getting to the same LHS; matching real parts and matching imaginary parts) are too novel to be sprung all at once. Imagine instead a course that developed these ideas in multiple ways, and in which student discovery was centered rather than speed or initial efficacy. Under such conditions, I would not, personally, be able to do justice to these ideas along with all the many other ones that could fall under as broad a name as Algebra 2.

With apologies to those who gave us the CCSS-M and who were well meaning, in my view the CCSS-M have been a "disaster" for mathematics, computer science, mathematics education, and most importantly American public school students.

In the years since World War II the world has been transformed by new technologies, in particular those that were driven by the development of the digital computer. The digital computer has also dramatically changed the practice of mathematics and the content of mathematics, as well as helping create the academic discipline of computer science. Mathematics curriculum in K-12 should reflect the tremendous growth in mathematics in the last 75 years as well as the growing number of domains where mathematics is proving itself useful in providing significant insight. There was a time when biology was viewed as being too complex to be amenable to mathematical investigation, but I don't think many feel this way anymore.

As you read what I write below, be aware that without my writing a book length treatment many issues are over vastly simplified.

First, remember some "boundary conditions." In America, unlike most European democracies, education is not "directly" under the control of the United States government. Since education is not mentioned in the US Constitution, educational policy is left to the states, and more precisely, local school boards. Whatever benefits this arrangement may have, it does not include efficiency in the delivery of mathematics education to American students. Second, America was the lucky beneficiary of the NCTM's decision to develop what have come to be called the "NCTM Standards," without external funding. Part of the idea was to develop a nation-wide way of looking at the mathematics curriculum and the teaching of mathematics that could serve as the basis for what individual states did with regard to mathematics education. In a general way one goal of the NCTM Standards was to get rid of tracking of students by race, gender, and income that "encouraged" some to pursue STEM careers (more generally, college careers) and dissuaded others. The development of the CCSS-M was not primarily a grass roots movement of those involved in K-12 teaching. In some ways it was the "negative" response of the mathematics "research" community about how influential the NCTM Standards had become. The issue of the high stakes testing that accompanied the CCSS-M is in part independent of the curriculum issues involved.

There were many differences between the NCTM Standards and CCSS-M. One can compare and contrast the NCTM Standards and the CCSS-M regarding continuous mathematics over discrete mathematics, algebraically rooted mathematics over "geometrically" rooted mathematics, the importance of "theory" over the "applicability" of mathematics, and mathematical tools that would be useful for students planning STEM careers vs. other careers. The CCSS-M implicitly assumes that all public school students should be "prepared" to take Calculus even when data show that perhaps 1/3 of high school students who go on to college major in STEM fields or ones that are "artificially" required to take Calculus. (Lots of research in biology takes advantage of Calculus-based ideas but lots of biology research also benefits from knowing about mathematics used for phylogeny and genomics which is more or less Calculus-free. One commonly stated aim of the CCSS-M was to change K-12 mathematics curriculum from one that was a mile wide to a curriculum which was less broad and to give students a better "conceptual" understanding of mathematics by downplaying some procedural aspects of the prior curriculum. This, despite the fact that for many students procedural fluidity was a path to conceptual understanding, and, for me, emphasizing a conceptual understanding, say, of complex numbers for all students is misguided. For most students having more breadth in the mathematics they see (graph theory, recursion, mathematical modeling, social choice (voting, apportionment, game theory) ideas, etc.) is a wiser choice. Thus, that certain algebra topics treated in the past in K-12 are no longer present does not bother me. In fact, my feeling is that K-11 mathematics education should be designed to serve the needs of students independent of their career plans and that the 12th grade be used to treat more technical parts of mathematics for those who know in high school that they want to major in STEM fields when they get to college. Having students have their first exposure to Calculus in college has much to recommend it.

Hopefully, NCTM will return to the curriculum leadership role it had when it developed the Standards, and America can return to having a K-12 curriculum that serves all American students better than the patchwork of modified CCSS-M that the states have turned to when the "national" version of the CCSS-M "collapsed," in many cases not for the "right" reasons.

• "Lots of biology research also benefits from knowing about mathematics used for phylogeny and genomics which is more or less Calculus-free" — When biology students, nay, professors have no knowledge of calculus, they invent it themselves: fliptomato.wordpress.com/2007/03/19/… Yay for individual research and project-based learning I guess. Grants well spent. – Rusty Core Sep 1 '19 at 18:39
• NCTM was the single biggest force in the destruction of American school math education in the late 1980s - early 1990s. Its so-called "standards" were rejected by most states, and the math programs authored according to these standards retreated. But don't you fret, as the failed 25-year old programs have been rebadged as "aligned with Common Core", and now you can have the same unbridled calculator-heavy, geometry-free project-based groupwork back, courtesy of TI, HP and Microsoft. Why learning about discriminant when you can punch in solve() and get your roots? Schweet! – Rusty Core Sep 1 '19 at 18:41