The California Mathematics Framework has taken another step, with the latest draft released last week (many Word documents, many idiots). Critics who wish to comment have been given about three minutes to digest the thousand pages and then do so. For those who care to try, Jelani Nelson has posted an anonymous person’s monster work in compiling and PDF-ing the CMF documents, and tracking the (mostly lack of) changes. It is pretty clear that the CMF Powers don’t give a stuff what anyone thinks, but opponents gotta do what they gotta do.
There’s not much point in us working hard on this (our earlier snipes are here and here and here). Undoubtedly, Brian Conrad will soon update his site with comments on the new draft, and Greg Ashman has written on the CMF gang’s love of making things up. We’ll just comment a little on one aspect of the CMF madness: memory.
The CMF acknowledges, albeit barely, that memory plays a role in students’ doing mathematics. The term is mentioned ten times in the draft CMF, all but once in at least a minimally accepting tone. Typical is the note in the introductory Overview:
These [mathematical] concepts are studied with a goal to develop both understanding and fluency, which requires knowing, efficiently retrieving, and appropriately using facts, procedures, and strategies, including from memory.
The wording is weird and unsettling, since the required “knowing” of facts et al make their appearance including from memory. One wonders about the location of knowledge other than in memory, and the more one wonders the more unsettling is the phrasing. Still, the proper role of memory is at least intimated in the CMF.
If there is a hint of the proper role of memory, however, there is none such for the role of memorisation. The terms “memorize/memorization” are mentioned twenty-four times in the draft CMF, and in every single instance the connotation is negative: the terms are invariably used as a straw man or a bogeyman or both.
The CMF’s denigration of memorization begins in Chapter 1, on “Mathematics for All”:
Absent tasks or projects that enable [students] to experience … connection and purpose, they end up seeing mathematics as an exercise in memorized procedures that match different problem types.
…
Students also self-select out when mathematics is experienced as the memorization of meaningless formulas—perhaps because they see no relevance for their learning and no longer recognize the inherent value or purpose in learning mathematics.
In Chapter 2, on “Teaching for Equity and Engagement”, they really go to town:
[Planning teaching around big ideas] also helps teachers move beyond the unproductive notions that mathematical ideas and understandings should be sequentially organized in the same manner for all students or that algorithms that must be memorized.
…
| Unproductive beliefs | Productive beliefs |
| Mathematics learning should focus primarily on practicing procedures and memorizing basic number combinations. | Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse. |
…
Rather than focusing on specific procedures and memorization, instruction is more effective when teachers aim to develop understanding of bigger ideas and procedures.
…
The math task analysis framework from Stein and colleagues (2000) shown in [the table below] offers helpful descriptions of two types of narrow, low cognitive demand tasks—those that require only memorization or procedures without connections—and two types of open, high cognitive demand tasks—those in which students employ mathematical procedures with connections or do mathematics tasks.
…
| Lower-Level Demands | Higher-Level Demands |
| Memorization Tasks | Procedures with Connections Tasks |
| involve either reproducing previously learned facts, rules, formulae or definitions OR committing facts, rules, formulae or definitions to memory. | focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas. |
| cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure. | suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts. |
| are not ambiguous. Such tasks involve exact reproduction of previously-seen material and what is to be reproduced is clearly and directly stated. | usually are represented in multiple ways (e.g., visual diagrams, , symbols, problem situations). Making connections among multiple representations helps to develop meaning. |
| have no connection to the concepts or meaning that underlie the facts, rules, formulae or definitions being learned or reproduced. | require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding. |
…
| Unproductive beliefs | Productive beliefs |
| The role of the student is to memorize information that is presented and then use it to solve routine problems on homework, quizzes, and tests. | The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge or familiar contexts and experiences, and considering the reasoning of others. |
And on and on and on they whine. There’s not a single reference to memorisation without it being characterised as low level, as a drudge, as other than a pain in the butt. And yet somehow the memory of things, which the CMF gang kinda sorta acknowledges as necessary, is supposed to magically appear.
The denigration of memorisation is just one tiny aspect of the CMF, which appears to be solid, unceasing nonsense. To get a sense of this, readers with the stomach can view the full tables from which we excerpted the memorisation parts. Or, just pick a page. Then pick another. If anyone can find a page without nonsense, let us know.
We admire the amazing hard word of Brian Conrad and Jelani Nelson and the others who are fighting the CMF nonsense. The work is obviously worthy and necessary. But it is not necessary for us. For us, the denigration of memorisation is enough. The CMF people are plain nuts.
California Sunset (on mathematics education)
Whenever I see an article/blog post on math education on the West Coast, I more or less know what to expect and yet, each time, I find myself surprised. The state of Washington isn’t different to California. They all lost the plot a long time ago.
Is it just the West Coast? The NCTM seems to be a bunch of loons.
That could be the case, Marty. I don’t claim to be a specialist, but I encounter more articles about West Coast for some reason. In any case, they don’t stop to surprise.
Here is another exhibit.
Click to access Math%20SDS%20ES%20Framework.pdf
Yeah, I know. I think I meant to write about it at some point, but my stomach rebelled.
I understand you 100%. Frankly speaking, I don’t know how to write about something like this without profanity.
Australians will now see some action with a new national approach to reforming Initial Teacher Education known as “Strong Beginnings”.
You’re better at deadpan than Buster Keaton.