kitchen table math, the sequel: conceptual knowledge
Showing posts with label conceptual knowledge. Show all posts
Showing posts with label conceptual knowledge. Show all posts

Wednesday, December 31, 2014

Frederick Hess: Common Core tests were the fix for NCLB

Frederick Hess writing in National Affairs:
In 2001, the No Child Left Behind Act marked a dramatic win for standards-based reform — but at the price of abandoning the push for "national" standards. NCLB required states to adopt standards in reading and math, administer annual tests geared to those standards, use tests to determine which students were proficient, and analyze the outcomes to determine which schools and systems were making "adequate yearly progress" — including the absurd requirement that 100% of students be proficient by 2014. Schools and systems that didn't perform adequately were subject to federally mandated sanctions. The crucial compromise was that states could set their own standards and tests. In fact, NCLB specifically prohibited national testing or a federally controlled curriculum.

What followed was not difficult to anticipate. The possibility of sanctions gave more than a few state leaders reason to adopt easy tests and lower the scores required for proficiency. A "race to the bottom" was soon underway, prompting an effort to combat the gamesmanship.

[snip]

The real power of standards lies in their ability to change what is tested, and thus to change how curricula and textbooks are written, how teachers teach, and how students learn. As Finn and Petrilli put it, the standards are ignored, and "[e]ducators instead obsess about what's on the high-stakes test." This is why advocates are so impassioned and why critics are justified in fretting about the implications of the Common Core. When coupled with tests, accountability systems, and teacher evaluation, the Common Core becomes the invisible but omnipresent foundation of American education.

[snip]

[T]he Common Core is neither necessary nor sufficient for fixing the problem it was designed to solve. The critical rationale for the Common Core was concern that states had gamed and manipulated testing under NCLB. But a more modest solution was already available. Every state has long participated in the National Assessment of Educational Progress, which tests students in reading and math (and sometimes in other subjects) in grades four, eight, and twelve under carefully controlled conditions and provides a rock-solid means for comparing performance. In fact, NAEP results were already being used to flag states that appeared to be gaming their NCLB tests. Common Core advocates, however, thought that relying on NAEP was an unsatisfactory, makeshift solution. Instead, they embraced the Common Core standards.

Solving the "race to the bottom" problem would have required the Common Core tests to replicate NAEP's careful protocols. However, perhaps recognizing that states might not have signed on if they were subject to transparent coercion, Common Core advocates were remarkably laid back about what states would actually be required to do when it came to policing test conditions, accepting mandatory passing scores, or establishing strong oversight boards. Thus, advocates failed to build in controls to prevent states from manipulating outcomes. States can administer the Common Core-aligned tests much later in the school year than is recommended (thus inflating measures of student learning), ignore guidelines on testing conditions, and set their own proficiency scores. The only "safeguard" against any of this is state officials' inclination to do the right thing — which is precisely what it was before the Common Core. Meanwhile, many Common Core states have decided not to use the program's new tests at all; as a result, barely 40% of students are currently slated to be tested with one of the two new Common Core tests, and at least 19 different tests will be used nationwide next spring. Given the critical role of the tests for maintaining standards, this undermines the purpose of the Common Core — and in a fashion that seems unlikely to lead to purposeful experimentation or rethinking. Within a few years, testing may be only slightly less fragmented than before the Common Core, and many established tests will have been jettisoned for slapped-together replacements.

How the Common Core Went Wrong
Common Core was all about the tests.

NCLB had failed -- so the thinking went -- because (some) states gamed the system by writing easy tests (or writing hard tests but setting easy cut scores).

So Common Core would create common standards and, hence, common tests. Hard tests.

Like Jason Zimba, I might have thought, a few years back, that changing the tests would do the trick. Create good tests, let the schools take it from there.

But having sent three children through public schools, and having read some of the literature on foreign aid and its many debacles, I now have a much greater appreciation for the slipperiness of reality.

Or the slipperiness of culture, more like.

I'm extremely tardy getting Barbara Oakley's op ed & new book posted, and it looks like I won't get to it today, either.

But I do want to quote her passage on culture:
Today's Common Core approach to teaching STEM is at least superficially appealing. The goal of placing equal emphasis on conceptual understanding, procedural skills and fluency, and application is laudable. But as with any new approach to teaching, the Common Core builds on the culture that's already there. And the culture that has long reigned in STEM education is that conceptual understanding trumps everything. So bewildered math teachers who are now struggling to teach the Common Core are leaning on the old thinking, which has it that if a student doesn't understand—in the "ah-ha," light-bulb sense of understanding—there's no way she or he can truly become expert in the material.

True experts have a profound conceptual understanding of their field. But the expertise built the profound conceptual understanding, not the other way around. There's a big difference between the "ah-ha" light bulb, as understanding begins to glimmer, and real mastery.

How We Should Be Teaching Math
Like NGOs disbursing foreign aid, the Common Core had to build on the culture that was already there.

The culture that's already there inside public schools pits "knowledge" against "thinking," "problem solving," and "understanding," with knowledge the loser. Here in my district, in fact, our curriculum director has produced a new Powerpoint, titled "Teaching for Understanding," which poses a rhetorical question:
Is it possible to have a great deal of knowledge but limited understanding?
Harder tests aren't going to raise student achievement inside a culture whose denizens believe that knowledge is an impediment to understanding.

Tuesday, September 4, 2012

teaching math facts: equivalent sums or iterative grouping?

ABSTRACT

This experiment tested the hypothesis that organizing arithmetic fact practice by equivalent values facilitates children's understanding of math equivalence. Children (M age = 8 years 6 months, N = 104) were randomly assigned to 1 of 3 practice conditions: (a) equivalent values, in which problems were grouped by equivalent sums (e.g., 3 + 4 = 7, 2 + 5 = 7, etc.), (b) iterative, in which problems were grouped iteratively by shared addend (e.g., 3 + 1 = 4, 3 + 2 = 5, etc.), or (c) no extra practice, in which children did not receive any practice over and above what they ordinarily receive at school and home. Children then completed measures to assess their understanding of math equivalence. Children who practiced facts organized by equivalent values demonstrated a better understanding of math equivalence than children in the other 2 conditions. Results suggest that organizing arithmetic facts into conceptually related groupings may help children improve their understanding of math equivalence.
It Pays to be Organized: Organizing Arithmetic Practice Around Equivalent Values Facilitates Understanding of Math Equivalence.[Article]
McNeil, Nicole M. 1; Chesney, Dana L. 1; Matthews, Percival G. 1; Fyfe, Emily R. 1; Petersen, Lori A. 1; Dunwiddie, April E. 1; Wheeler, Mary C. 1
Journal of Educational PsychologY | Publish Ahead of Print, POST AUTHOR CORRECTIONS, 25 June 2012

Saturday, March 17, 2012

the 4 Cs at the Celebration

Judging by the program content at this year's Celebration of Teaching and Learning, "21st century skills" have won, and knowledge has lost. Although there was ceaseless talk of the "new Common Core," the Common Core standards were in all cases that I witnessed* assumed to be a synonym for 21st century skills. So game over, at least as far as the two unions are concerned.

The only good news (I guess) is that "the Partnership"** has finally decided what the 21st century skills actually are, just 10 years after making them up:
  • Critical Thinking and Problem Solving
  • Communication
  • Collaboration
  • Creativity
The four Cs.

There was universal agreement amongst the participants that the 4 Cs cannot be tested.


* I'm guessing this workshop was different. I'll see if I can get my hands on the Powerpoint.
**tech companies, the NEA, and the Department of Education

Monday, February 7, 2011

A view from computer science college professors

I hear from a friend who teaches college computer science that many CS educators agree on the following:
  1. There is too much emphasis on calculus, and more focus should be placed on discrete mathematics (graph theory, logic, automata theory, etc) and statistics.
  2. K-12 education places too much emphasis on memorization at the expense of conceptual understanding.  This leaves college students ill-prepared for their computer science course work.
Interesting.

Cross posted at Education Quick Takes

Wednesday, July 8, 2009

Facts, Facts, Facts!

I've just written Amazon.com review of Dan Willingham's book "Why Children Don't Like School." Here it is:

Every once in a while, an empirical study comes along that provides solid evidence against one of those Constructivist practices that some of us whose thoughts on education come more from actual practice than from education theory have often been skeptical about. There is, for example, Jennifer Kaminski’s Ohio State study, which suggests that too much of a focus on “real-world” math obscures the underlying mathematics, such that students are unable to transfer concepts to new problems.

Dan Willingham's book Why Don't Students Like School presents a whole bunch of these experimental results. Together, they challenge the notions that:

1. Students need to learn inquiry, argumentation, and higher-level thinking rather than tons of facts.

2. Integrating art into other subjects enhances learning; so does integrating computer technology.

3. Children learn best through self-guided discovery.

4. Drill is kill. Multiple strategies in a given lesson are better than a single strategy practiced multiple times.

5. Students learn best when constructing their own knowledge.

6. The best way to prepare students to become scientists and mathematicians is to teach them to solve problems the way scientists and mathematicians do.

The empirical data that Willingham cites show that, in fact:

1. Factual knowledge, lots of it, is a prerequisite to higher-level thinking.

2. Students are most likely to remember those aspects of a lesson that they end up thinking about the most. Corollary: Incorporating art or computer technology into another subject may sometimes cause students to think about the art or the technology more than the lesson content, such that they don’t retain the latter.

3. Discovery learning should be reserved for environments where feedback about faulty strategies is immediate: "If students are left to explore ideas on their own,” Willingham writes, they may “remember incorrect 'discoveries' as much as they will remember the correct ones."

4. In Willingham’s words, "it is virtually impossible to become proficient at a mental task,” or transfer ones learning to new environments, “without extended practice."

5. Unlike experts in a field, "students are ready to comprehend but not create knowledge."

6. Novices don’t become experts by behaving like experts do. "Cognition early in training,” Willingham writes, “is fundamentally different from cognition late in training."

Of course, Willingham could be making all this up. But consider just one of his empirical claims:

"Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts... The very processes that teachers care about the most--critical thinking processes such as reasoning and problem solving--are intimately intertwined with factual knowledge that is stored in long term memory..."

This is a strong statement that could easily be debunked by anyone who knows the empirical literature. There are plenty of highly articulate, outspoken people out there who don’t like what Willingham has to say, but I haven’t seen a single critical review that contradicts his empirical claims.

Of course, if all that matter in life are inquiry, argumentation, and “higher-level” thinking rather than lots and lots of facts, one can say whatever one wants to about Why Children Don’t Like School.

Thursday, March 13, 2008

National Mathematics Advisory Panel Releases Final Report

On March 13, 2008, the National Mathematics Advisory Panel presented its Final Report to the President of the United States and the Secretary of Education. Copies of these ground-breaking reports, rich with information for parents, teachers, policy makers, the research community, and others, are provided below.

Foundations for Success: Report of the National Mathematics Advisory Panel

Final Report PDF (851 KB) Word (1 MB)

I ranted and raved about the Math Panel's report over at Mindless Math Mutterings.

Thursday, December 13, 2007

why is there 1?

Cleared away one of my floor stacks today. I'm glad I did because one of the objects in the stack was the Algebra 2 notebook in which I'd written an anecdote about C. & math. The notebook went missing a few months ago and now I've found it.

This is C. last summer at the picnic table outside our kitchen. We were probably working on percent (scroll down).

I don’t like math.

Why is there 1!?

1 + 1 equals 2!

Why is that?

I don't like math.

Words make sense. If I read “the dog,” I know what “the” means.

Math doesn't make sense.

- CHB summer 2007, 12 years old

I know you will all be impressed by the fact that I did not say, "You know what 'the' means?"

Myrtle on procedural and conceptual knowledge

Maybe I can make a convincing argument that a student who only thinks of multiplication as iterative addition and can't multiply 24X86 has neither procedural nor conceptual knowledge.

There is more to the "concept" of multiplication than iterative addition. (Try applying iterative addition to 1/8 x 2/5.) Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?) But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well). The multiplication algorithm invisibly makes use of the distributive "concept," and does not employ an iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.

A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers without the multiplication algorithm:

24 X 86 means that
(20 + 4)(80 + 6) which means/implies that...

Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.

Notwithstanding operations on super hairy numbers, he is capable of doing the algorithm on paper when he needs to and can resort to "concepts" when he needs to do mental calculations.


the multiplication algorithm invisibly makes use of the distributive "concept"

I love that!

I love the whole Comment, in fact. People like me -- people who value liberal arts education in general and mathematics education in particular but who aren't expert in mathematics and probably never will be, have no way to get at these things.

I intuitively grasp the notion that there is some kind of "starter understanding" a person can have without being fluent in procedures. Seeing that 6x4 is the repeated addition of 6 4s or 4 6s as the case may be (I've spent quite a bit of time muddled over that one!) strikes me as superior to not seeing it. (I had no idea multiplication could be called repeated addition until I started reteaching myself math, and then I noticed it on my own.)

But at the same time I am gripped -- and gripped is the correct word -- by the conviction that a starter understanding is not a real understanding.

And yet because I lack a real understanding I have no way to express this and thus no means of combating the forces of reform math when they threaten to overrun my son's education.

I'm logging this post under Greatest Hits so I'll know where it is when I need it.