Making Math Education Even Worse, by Marina Ratner,
It pains me to write but in spite of all of your precollegiate mathematics education knowledge and contributions, Prof. Ratner got it right and you “missed the boat” in response:
The CA Math Content Standards were – and still are – the best in the country. They have problems; e.g., there is too much specialized focus in its thread on Statistics, Data Analysis, and Probability and, even worse, Mathematical Reasoning. No sensible person can be against mathematical reasoning, of course, but that is exactly the point. Sensible people embed it everywhere and, as a standalone item, it becomes almost meaningless – hence the paucity (as in none) of CA Key Standards in that category. The writers included it to help ensure Board of Ed approval because most professional math educators were strongly objecting to the entire Stanford approach. Perhaps the most egregious, is your characterization of California’s problems using poison words: “rote-learning of linear equations by not preparing students for the correct definition of slope.” This is at best misleading and closer to being flat wrong:
From the introduction to Grade 7:
“They graph linear functions and understand the idea of slope and its relation to ratio.”
This is followed specifically with two Key Standards and examples:
3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph.
3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.
In what way(s) do you find the relevant 8th grade standard in the CCSS-M, Expressions and Equations (EE.8 #5,6), to be conceptually superior? (The word is used once in the intro to Grade 7 but it is not mentioned thereafter.) Formally proving that all pairs of distinct points determine similar triangles so that this ratio is well-defined would be mathematically necessary to be completely logical but I doubt if that’s what you meant particularly since traditional proof has been downplayed so badly even in the high school CCSS-M, much less 8th grade, especially in comparison with the CA Math Content Standards.
Regarding the general concept of competent Algebra 1 (not some pretense thereof), it was, it is, and it will remain standard in 8th grade (if not already accomplished in 7th grade) for self-respecting, academically-oriented private schools. As you well know, the Stanford Math group who wrote the CA Standards started with the egalitarian notion that this should be an opportunity for everyone including those who do not have access to such schools. It cannot be and was not intended to be just imposed that traditional Algebra 1 be the math course for all 8th graders but the group worked backwards from that target step-by-step through the grades in order to get there comfortably (such as developing the concept of slope in 7th grade that you appear to have missed). Is every detail spelled out? Of course not, nor should they be, but the key ideas – even set off as Key Standards – are there and presented considerably more clearly than in the CCSS-M.
There is statistical evidence that the goal did improve the state of mathematics competence in California, but we both know the CA Math Content Standards fell well short of the ideal. It was not – as your words could be interpreted to imply – that they reflect an inherent lack of development of student understanding. The primary villain is the overwhelming mandate for chronological grade placement (age-5) for incoming students and almost universal social promotion. Far too many students are not competent with the standards at their grade levels – sometimes years below – yet they move on anyway. Algebra in 8th grade – Algebra in 11th grade or even Algebra in college – is not realistic for all but truly gifted students who lack easily identifiable mathematics antecedents. A less common problem, but damaging to our most talented students, is the reverse situation. Advancement in grade level (as was done with my son at his private school and now chair of Chemistry and Biochemistry at Amherst College) is almost unheard of. Although mandated by many districts, and underscored by the API scoring of schools, mandating that all students be in an honest Algebra class in 8th grade without a reasonable level of competence with the Standards of earlier grades was never the intention. It was to be the opportunity, not the mandate.
“Moreover, Common Core does not place a ceiling on achievement. What the standards do provide are key stepping stones to higher-level math such as trigonometry, calculus and beyond.”
Although these words are regularly repeated, reality is the diametric opposite. Across California, CPM (supposedly, College Preparatory Mathematics) is back with a vengeance. Ironically, it was the very catalyst that spawned the now defunct Mathematically Correct and it pulled its submission to California from the 2001 approval process rather than be rejected by our CRP (Content Review Panel). You’ll recall that it and San Francisco State’s IMP were among the federally blessed “Exemplary” programs for which the only mathematician, UT-SA’s Manuel P. Berriozábal, refused to sign off. Weren’t you among the signatories of David Klein’s full-page letter of objection in the Washington Post? One of CPM’s long-standing goals is to have ALL assessments – even final examinations – done collectively with one’s assigned group. It makes for a wonderful ruse – all students can appear to be meeting the “standards” of the course (even if absent!) – while deeply frustrating those students who are “getting it” (often with direct instruction by some family member who knows the subject). Trigonometry, calculus, and beyond from any of CPM, IMP, Core-Plus (all self-blessed as CCSS-M compatible)? It just doesn’t happen. However, from the homepage of Core-Plus:
“The new Common Core State Standards (CCSS) edition of Core-Plus Mathematics builds on the strengths of previous editions that were cited as Exemplary by the U.S. Department of Education Expert Panel on Mathematics and Science”
What did happen – may already be happening again? Beneath the horizon, schools began to offer a traditional alternative to provide an opportunity for adequate preparation for knowledgeable students with math-based career aspirations. What also happened (but may not be successful this time because of the SBAC or PARCC state examinations?) was that other students and their parents petitioned their Boards of Education for an elective choice and, if unfettered choice was granted, the death knell sounded on the innovative “deeper understanding” curriculum and pedagogy.
Finally, you do acknowledge the ridiculous nature of the 6th grade “picture-drawing frenzy” observed by Prof. Ratner but seem to imply it was an isolated incident instead of her description, “this model-drawing mania went on in my grandson’s class for the entire year.” The fact is that such mis-interpretations of “teaching for deeper understanding” are going on for entire years in classrooms – in entire districts – all across the country; they are even taught by professional math educators as mandated by Common Core. You described her observation as a “failure to properly implement Common Core” and I am sure that you believe that to be the case but your conviction is belied by the fact that one of the three primary writers of the CCSS-M and the head of the SBAC-M is Phil Daro (bachelors degree in English Lit). Phil Daro has been strongly influential in precollegiate mathematics education – curricula and pedagogy – across California for decades, my first working acquaintance with him was in 1988, months prior to the first NCTM Standards. His vision for the “right” way to conduct mathematics classrooms (not “to teach”) helped lead to the 1992 CA Math Framework, MathLand-type curricula, and the ensuing California battles of the Math Wars with our temporary respite beginning in late 1997. Unfortunately, his vision is not only reinvigorated here in California, it is now a huge national problem and Prof. Ratner “nailed it”.