*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.Common Core was all about the tests.

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

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.Like NGOs disbursing foreign aid, the Common Core had to build on the culture that was already there.

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

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.

## 19 comments:

The question "Is it possible to have a great deal of knowledge but limited understanding?" is a good one to ask. The answer is an obvious yes.

Of course, you also want to ask the other question, "Is it possible to have a great deal of understanding but limited knowledge?" That is a more difficult question to answer well, but I think that the readers of this blog will agree that the answer is "no".

And the other, other question is: Do we currently have a problem of copious knowledge paired with insufficient understanding (such that fixing this would improve matters somehow)? I think the answer to that one is also "no."

If they had decided to make everyone use the NAEP instead, trust me, people would be screaming just as loudly.

As for knowledge vs understanding in math, I don't think there is a difference. If you don't understand what you are doing in math, you don't know it. In the simplest case, a kid who can tell you the times tables, but has no idea that 3x3 = 3 + 3 + 3 certainly does not know multiplication. Similarly, a kid who memorized the rules for dividing fractions, but who doesn't understand why it works (usually because he or she doesn't really understand fractions in general) does not KNOW how to divide fractions. In math, you have to understand the concepts before you move on to something else.

"But as with any new approach to teaching, the Common Core builds on the culture that's already there."

Yes!

That's why we should be using the Iowa Tests (ITBS)! They are tried and true and offer everything we need.

Has anyone made a coherent argument against using the ITBS as our common school accountability test?

Similarly, a kid who memorized the rules for dividing fractions, but who doesn't understand why it works (usually because he or she doesn't really understand fractions in general) does not KNOW how to divide fractions. In math, you have to understand the concepts before you move on to something else.So if a kid can solve the problem of how many 2/3 oz servings are in 1 3/4 oz of yogurt by knowing that you divide 1 3/4 by 2/3 and does the procedure correctly, but cannot explain why the invert and multiply rule works (as was the case for me in 6th grade), do you conclude the kid lacks understanding? How are you defining this holy grail of "understanding"? Sometimes procedures lend themselves to understanding, sometimes they do not.

This is going to be a "it depends" answer. If the kid can solve that problem after having be shown how to solve that same problem explicitly, and maybe having been shown a bunch of examples of that problem, then I would say "no" - if he or she can't explain, most likely it is just a memorized procedure which will disappear once no longer practiced. Unfortunately, this is the way a lot of math is taught.

All procedures lend themselves to understanding. Remember, someone had to INVENT those procedures - I sure hope that person understood them. Also, if we want to be able to solve these problems by computer, developing algortihms to do this, we better understand the procedures!

I don’t think procedures “lend themselves to understanding” or not. We have to actively take steps to understand them and to communicate that understanding. Invert-and-multiply is a concise rule with a lot of math behind it. But it’s not the only one like that. Math education is filled with such things for many years to come.

In general, we have these two tasks: teach the algorithm and teach the reasoning. We want students to be able to use the algorithms with ease so that they can apply them to problems and also use them as building blocks for the next algorithm. But we should also teach where the algorithm comes from. I know that students are not going to hold that reasoning in their heads every time they apply the algorithm. But if we don’t teach this part, then we are turning math into what it was never intended to be: a collection of unconnected factoids (that some would call “knowledge”).

Also, don’t we hope that at least some of our students will reach the point where they can create their own algorithms, come up with their own mathematical ideas? That is not likely if you have not been teaching the reasoning as well as the algorithms. It can be done in either order, but I agree that complete understanding includes the reasoning behind the algorithm. Like many people, I don’t think I was ever taught why invert-and-multiply works.

Phil

I was shown why invert and multiply works; it was interesting, and I think the teacher wanted all of us to be able to follow why it worked. But that is a FAR cry from having each student be required to be able to call up that reasoning on demand, and describe how and why it works as part of their graded seat work or homework, or on a test.

And there, I fundamentally disagree. yes, students should be graded on whether they understand the reasoning. That kind of mathematical thinking is important.

Students who memorize the procedure without understanding it will lose the procedure unless they are doing it over and over. Generally, that doesn't happen. This is also a reason why many bright kids hate math so much. They learned it as a set of memorized procedures that made no sense to them. As soon as those kids no longer have to do invert-and-multiply, they quickly forget. That is how we get so many students in college who seemed to have had good grades in math, and who seem bright, but who can't do basic math. They learned a class of problems, like the one posed above, by memorizing a bunch of similar examples, then aced the test by just doing the same thing. They quickly forget, though they may have to rememorize in their SAT prep course. But as soon as that SAT is over, bye-bye.

That is also why so many of my CS students insist on memorizing examples of programs rather than understanding them. They get upset because we don't teach CS the same way they learned math. They want us to give them a program, which they will memorize, and then give them an assignment which is exactly the same with different numbers. It is very upsetting to many students that we don't do this in CS, and that we want them to understand the reasoning behind the algorithms.

But, froggiemama, it's entirely possible to understand why an algorithm works without being able to express is efficiently in words. I agree that students (at least those who expect to go to higher levels in math) need to understand why/how; I just think it's unrealistic to think they can begin to explain that, in words, and especially at an early age. So then the question becomes, how do we provide chances to demonstrate that understanding?

I agree with Anon @ 1:38 that it's not necessary for young kids to be able to explain in words but that the older kids who would have been on the 60s-version of the college prep track should understand the whys and hows. I don't think it's necessary for all HS kids. Many, if not most, would be better off if they could actually DO what used to be k-8 arithmetic, without calculators; basic facts, operations, std algorithms, fractions, decimals, percentages, mean, median, mode, std deviation etc. The whys and hows are important for higher-level math, but not everyone is able or willing to operate at that level of abstraction. The one-size-fits-all approach doesn't work for many kids.

"Today's Common Core approach to teaching STEM is at least superficially appealing."

PARCC does NOT do STEM. We can't get very far in a national debate when people get the basics wrong.

NCLB and CC do NOT deal with how to get to honors or AP or to STEM. With NCLB, nobody expected that. I'm not sure why CC leaves people with that impression. Is it because they talk of "college readiness"? Do they just gloss over the fact that the highest level in PARCC only means that one is likely to pass a college algebra class?

So now we get backtracking from all sorts of people to say that it's not "college readiness" - that CC's goal is really much more modest. But we still don't have a national test. CC actually defines very little. There are several players in the test mix. ACT is appealing because they claim that they will use the same sort of college-ACT numbers (up to 36) so that students and parents can track their progress from the earliest grades. Unfortunately, this is where the funny business of nonlinear calibration happens. How does an ACT of 30 in fifth grade translate to an ACT of 30 at a junior in high school? Is it done via help at home or with tutors?

With PARCC, your child might get top-level "distinguished" ratings all through K-8 and then find out that he/she is completely not prepared for a STEM program in college no matter how many summer courses or doubling up is done.

CC is a one-size-fits-all curriculum where there is no common national test or calibration.

This is going to be a "it depends" answer. If the kid can solve that problem after having be shown how to solve that same problem explicitly, and maybe having been shown a bunch of examples of that problem, then I would say "no" - if he or she can't explain, most likely it is just a memorized procedure which will disappear once no longer practiced. Unfortunately, this is the way a lot of math is taught.No. If a kid understands what's going on with the problem so that he/she knows that division is the way to solve it, but cannot explain why the invert and multiply algorithm works, so what? And yes, many things disappear if you don't practice them; again, so what?

Students are usually shown fractional division via a whole divided by a fraction. Then gradually to fraction divided by fraction.

The invert and multiply action becomes more obvious in 7th or 8th grade when they've had algebraic equations and can also know what reciprocals are and that a number multiplied by its reciprocal equals 1.

And there is no shame in being shown how to solve a problem via examples. If the problems then become harder by variants of the same problem, then they are properly scaffolded. Sweller, Kirschner et al talk about how the worked example effect can and should be used in teaching math.

Understanding is only shown by being able to do all sorts of variations of problems. Do we expect students to do a proof to justify how to divide fractions? Does a proof mean that the student will be able to divide 2.34 by 7/16? How about 9/13 divided by 2 1/2? What words prove understanding in a way that would show that they can solve any problem? What about the transition to dividing 5/(x-3) by (x+3)^2/22? This requires a good understanding of factors and basic identities? Everyone can spit back the "understanding" that a/1 = a, but do they understand what that means in a situation where you have to divide 5 by 3/16?

When I tutor high school students and back when I taught college math, it's easy to get them to say things in words and I can tell that they "understand" when we go over something like synthetic division. However, the proof is when they do the homework assignment all by themselves. In a properly taught math class, it is not possible to pass with just rote understandings. They might have partial understandings and gaps, but there are no understanding words that will fix the missing understanding that comes from a lack of practice.

One thing I've seen repeatedly is that it is often more efficient to go back and deepen understanding of a previous topic than to require a deep understanding at first encounter. Learn things to a limited depth, memorize what you don't understand (and what you do), and come back later to deepen understanding. You must not omit that last step.

The key is the proper sequencing of topics. Do enough explanation, memorization, and application of a topic to push it over a certain threshold of usability, then move on to topics that require its ongoing use in ever-expanding ways. As you proceed through the sequence of topics, you both move forward and methodically revisit and deepen the understanding of previous topics using your new skills. (You don't just "spiral" over unusable skills in a random sequence trusting that some of them will eventually "stick".)

For example, I take a kid whose operations on algebra I equations have become habits, and have him look at his old "invert and multiply rule" in terms of his new algebra. I show him: A/B = A(1/B). Okay, that's now obvious to him. And what does it mean? Since he now knows that the reciprocal of X is 1/X, it means: "A divided by B equals A times the reciprocal of B". And if B is a fraction, N/P, what would 1/(N/P) be? We write (N/P)(P/N) =1, divide both sides by (N/P), and 1/(N/P) is obviously (P/N). Whatever N and P are, the reciprocal of the fraction is the inverted fraction. So A divided by (N/P) equals A times (P/N). Invert and multiply.

When we look at an old idea through the lens of the new skills, it becomes an "Oh, I see, of course!" experience replacing the original, arbitrary "rule". Some teachers seem to think that if you don't fully understand invert and multiply at first, you will be handicapped, but trying to gain this degree of understanding prior to learning the later ideas is a waste of time. Other teachers seem to assume that a kid in algebra who has thoroughly memorized invert and multiple, uses it correctly, and is doing well in algebra must understand invert and multiply algebraically, but that's usually not the case. Without explicitly revisiting it and examining it using algebra, it will probably remain fossilized as a memorized procedure. You have to explicitly revisit old topics with your new tools.

The sequence is what matters. Get enough understanding to aid memorization, get enough memorization to make it usable, keep using it until habits form, keep adding new skills, then go back with those skills and deepen the foundation of understanding. Eventually the memorization will become just a convenient summary of the deep understanding applied to common problems.

What is understanding? Is it a formal proof? Clearly, that's not possible or necessary for students just learning math. Here is a "proof" of "invert and multiply", which really isn't a proof, but it's what I used when I taught math.

http://www.moveitmaththesource.com/realfractions/mrnovaksproof.html

However, does this explanation tell you what to do when you see 3/(7/5)? Do you need a different "proof" for that? Understanding is not one thing, and there are different levels of understanding.

How about 3/4/5? One of my pet peeves is order of operation. There is no such thing. It's only a construction needed for equations when they are defined in a linear text format. I only learned about it when I got to programming. Proper math is a beautiful two-dimensional construction. There is no order of operation. There is only what you can do and what you can't do. This leads to understanding issues when students are constantly told to "simplify" as if that form is correct and any other form is incorrect. I liked to mess students up by moving factors up and down by changing the signs of their exponents. I would also write things as X2 instead of 2X. Understanding is a on-going process at all levels.

Is understanding a process whereby you can solve problems with some sort of vague Polya technique - draw pictures, label variables, think backwards? In those terms, I see understanding as the ability to know which governing equation applies to a problem and solving it using developed skills with that equation on a number of basic variations. In most cases, one will not have to "think". If the problem is a new variation you haven't seen, then you are starting at a base camp, not some blank Polya slate at sea level.

When I was in 8th grade algebra, we were required to explain what rule (identity) we used for each step. We found it annoying for really simple things. We wanted to do two or three steps at one time. So each of our problems had to be "proved" as in the link above. There is understanding to be had in each new problem you do. Understanding comes from mastering homework sets, bottom-up, not top-down with a vague thinking process.

However, one might want to have some sort of big picture understanding like when to use explicit, implicit, and parametric equations, but that's not the sort of vague trivial understandings I normally hear about in educational pedagogy where mastering skills is only rote. There is no way to pass a math course with rote skills unless the teacher is incompetent in testing and grading. Doing problem variations proves understanding. There is no understanding that is compatible with lack of skills. Skills (understandings) might be limited, but the solution is still bottom-up, not top-down.

I think Mr. Novak's proof is completely appropriate for pre-algebra students. And you only have to replace the numbers with letters to have a general proof.

I also would like to see some exercises that increase the intuitive feel for the result. For example, 20 cookies are used to provide a bunch of students with servings. How many students can you serve if each gets 5 cookies? 4 cookies? 1 cookies? 1/2 a cookie? 1/4 a cookie? 1/5 ? 2/5?

But also, I agree with Glen's post about adding the reasoning at a later look -- as long as that actually occurs!

Phil

I agree with Glen that sometimes, showing the procedure and then going back and showing how it works later is valuable.

I audited calculus in high school and took it for real as a freshman in college. With that experience, I have always thought that teaching students the quick-and-dirty way of taking a derivative, spending a little time telling them the power of it and what you can do with it, and then going back and showing how and why it works would be a better sequence.

Good morning,

I ended up posting a little more about "invert and multiply" on my blog, here:

http://advancedmathyoungstudents.com/blog/

Then I got curious. I realize that I don't actually know how elementary schools teach this topic. (I taught a lot of math to my children in an informal way, but forgot to closely monitor how they were taught in school.) So what I am curious about is: is this kind of step-by-step intuition building part of the process?

One reason I am asking is that some of the anti-common-core material that I have seen seems unfair. I am talking about the kind where they point at a series of exercises designed to build understanding and then mock it by showing how much quicker the standard algorithm is. I am in favor of the algorithms. But mindless application does nothing to build number sense or understanding.

Phil

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