Yeah, that must be it, Dept.


An article in EdSource talks about how California is doing in its implementation of Common Core. The big message (wait for it): Teachers need more training. (Theme music, audience applause, and fade to the smarmy host drinking coffee.)

I read as far as this passage before I realized I’d heard all this before:

“Teachers from elementary school to the college level need additional training in Common Core standards because, under previous standards, teachers focused on the memorization of procedures and rules instead of understanding the concepts behind math problems, Dell told EdSource.

“Similarly, he said many administrators are still hanging onto “old school thinking” that kids should be practicing the same kinds of problems over and over again instead of learning how math works and applying it in real world situations.”

Right; and for schools that insist on students not practicing the same kinds of problems, but instead are required every day to solve different ones for which they have little or no prior knowledge of how to solve: How’s that been working out for everyone? And the drawing of pictures and inefficient methods used in lieu of standard algorithms for two to three years before teaching the standard algorithm? Everyone on board with that?

Well, everything’s OK then. I just worry over nothing, I guess.


Old Legends Never Die, Dept.

I had read that USC has an interactive guide to Common Core, so I decided to look at it. It does a nice job describing what CC is about, and its history. And of course it includes the “shifts” in math instruction. These are the same shifts described at the CC website, except that they take the three shifts so described and split them up into six. In either case, they still bother me.

What’s bothersome about them is the inherent assumption that all math that came before CC was faulty. And while there may be some truth to that as far as the progressivist ideology that has been permeating the lower grades for the past 25+ years, that is not where their subtle criticism lies. The shift in “Focus” for example states:

More time is to be devoted to important concepts. Rather than covering many topics quickly, the standards stress the need to deepen instruction around pivotal ideas.”

And part and parcel to this, there is also this particular side-bar which appears in the interactive guide to CC:

“The majority of today’s parents learned math by memorizing algorithms, so learning about conceptual-based number sense may be a difficult transition.”

As I’ve shown in articles about textbooks from the era that supposedly “failed thousands of students in math”, the reformers of those eras (which included the notable William A. Brownell who is still revered by organizations such as the National Council of Teachers of Mathematics and by present-day reformers) explained the concepts underlying various mathematical procedures clearly. It was not always and universally true that students were simply shown the algorithm with no context or conceptual explanation.

A retired school teacher from Ontario wrote to me recently stating:

“I was always given thorough explanations throughout my schooling, though they may well have missed the explanation re division of fractions. Yet, that is the example that is always pulled out. Unfortunately,all the other explanations that have always been routinely given are forgotten and the extremely good math instruction that has taken place is dismissed. And all this was done with much less money and much less fuss. And thank God, I now understand division of fractions. Having understanding come at a somewhat later date is not an inferior type of learning. Probably part of normal human development.”

Nevertheless, the legends continue and the new methods of teaching (alternative methods first, and algorithms last, to ensure deep understanding lest the use of the algorithm eclipses and precludes it) continue on, unabated by complaints of parents and/or teachers brave enough to speak out.

We Need to Do Something, Dept.

The usual ploy of politicians, it has been joked, is to say “We have a problem” (description of problem then follows). Then: “We need to do something about this problem.” (Some strategy, usually ineffective, then follows). “There,” they say proudly. “We did something. Re-elect me.”

I was reminded of this joke when I read about Alabama’s State Senator Del Marsh, talking about the problem of education in Alabama and how something needs to be done.


“The state of Alabama needs to adopt and agree on a comprehensive education plan going forward.That encompasses all of our entities of education. Because unless you have that, how do we know where we’re spending our money is the best place to spend it?”

Sounds good (except for the “going forward” phrase which I wish would die a long-called for death, but I digress.)

” I can’t find anybody in the education community that I’ve talked to that does not agree that we should have a comprehensive education plan,” Marsh said. “But we don’t have one.” Directors of two groups representing education leaders said Marsh’s idea has merit. “

Ah, there we go. Everyone agrees there is a problem and a plan is needed.

Another Senator, Dick Brewbaker, then joined in with this familiar refrain:

Sen. Dick Brewbaker, R-Montgomery, chairman of the Senate’s Education and Youth Affairs Committee, said Alabama needs a plan tailored specifically for the state. Brewbaker has been a critic of the state’s adoption of Common Core standards back in 2010, saying that was driven by the desire for federal grant money. “The only plan we have so far is to try to pull down every federal dollar we could get our hands on,” Brewbaker said. 

OK, it’s shaping up. Let me guess what happens next! Alabama decides to revise the state (Common Core) standards. And the revision is just a few tweaks here and there, a change of the name so it is rebranded and VOILA!! We did something. Re-elect us!

Stay tuned for further developments.



Articles I Never Finished Reading, Dept.

This one is about project-based learning and has the usual disparagement of traditional teaching:

“Wetherington described learning through STEM as moving away from teachers lecturing and students completing worksheets, and more toward hands-on integrated lessons.

“We are providing the context for students to learn math and science and integrate them together and apply their knowledge in new and powerful ways,” he said. “It’s a very exciting thing to see students take knowledge that they’ve traditionally seen on a whiteboard or handouts, and really transform that into hands-on work.”

What isn’t mentioned is that such approach has been the purpose of science labs for years. The difference now, is turning much of instruction–including math classes– into one big science lab. Sure it was exciting when I took high school physics and saw how the trig I was learning in pre-calc was used to solve actual problems. But the textbook problems (disparagingly referred to as the dreaded “whiteboard” or “handouts”) prepared me for doing that.

One of the differences of PBL implementation now is using the lab or “project” as the means as well as motivation to learn what should have been prior knowledge. “Just in time” learning has its place when done right–I’ve seen it done right and have done it myself– but I’ve also seen it implemented in a fashion akin to throwing a kid in the deep end of a swimming pool and shouting out instructions to swim from the side of the pool. If by some miracle the kid makes it to the other side, the kid will say “I don’t know how I did that but I sure don’t want to do that again.”

Which, I don’t think, is the result these people are after, but there you are.

Nothing New Under the Sun, Dept.

Bar modeling, the technique of solving math problems using a strip or bar, has become synonymous with Singapore Math. The math program in Singapore uses such technique to solve a large variety of math problems.

But the technique has been around for many years as evidenced by this arithmetic textbook that I recently found, published in 1919.  (Hamilton’s Essentials of Arithmetic)


Ignore These Messages, Dept.

One of the many popular tropes of the internet is referring to someone who offers criticism of a piece as a “troll”.  Not that there aren’t bellicose and baiting-type comments, but the word is often applied to any type of criticism as in : “If you don’t agree with this, you’re a troll.”

So I thought I’d provide some comments I received in private regarding the discussion following Keith Devlin’s piece in the Huffington Post, as well as what I wrote about Devlin’s piece in the post below this one.  Chances are pretty good that these commenters would be viewed as trolls by many, so if what they have to say is disturbing, I won’t hold it against you if you want to call them that. If it helps you to ignore what they have to say, go for it!  They won’t mind.

Troll No. 1:

With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really. Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.

Troll No. 2

A question for such consultants that I sometimes ask: ‘Would you consider yourself a competent, satisfied person? Like, professionally and/or personally?’ [After short-circuiting for a sec, they almost always say yes.] ‘Well, you reached that point–gainfully and satisfactorily employed, passionate about your subject, personally satisfied, etc., etc.–all without having to like the way you were taught very much. So…it worked for you, all without you having to ENJOY it. Can you reasonably say you’d have done better? That you’d have turned out differently? And if so, how would you have rather turned out? I  hate the argument from successful people that, ‘I could’ve been so much MORE!’ It’s not something they can confirm, and the ideas probably didn’t cross their minds a single bit as they were being instructed on the ways to their success. Still, they want to distract from (and, with their methods, most likely hinder) today’s students’ progress, all based on something–a personal memory, a phantom–that absolutely cannot be validated.

Troll No. 3

Does what they do work? Can they even define what success means? I’ve seen no explanation. They don’t show how any sort of discovery for one problem translates into other areas. Discovery in basic fractions will help very little with factoring quadratics, let alone rational expressions. It also takes a lot of class time and few topics can be covered. What “deep” concepts helps students solve all of the variations of Distance/Rate/Time problems? Draw a picture? Think backwards?  How about studying the governing equation, seeing how the variables relate, and doing a whole lot of individual homework problems of all sorts of variations?

In preparing for the AMC math competition, you don’t just study concepts or some general top-down process. You prepare by going to their web site and doing ALL of their past problems. There are tricks and understanding subtleties that will never be discovered with some general process. By working on all of these problems, you start at a much closer solution point when you encounter a new variations. This is how success for any highly-valued math test is created. You don’t have to have any conceptual knowledge if you can define the variables and equations. Just let the math give you the understanding.

The people pushing these new ideas need to show how a full curriculum based on their ideas works, not just some sort of small delta success for kids already damaged by fuzzy K-8 math curricula. They define their own version of success without regard to the needs of where these kids are going after high school. A local highly-regarded vo-tech school in our area requires the AccuPlacer test. Do the techniques of these reformers help those students do better on that test than a traditional process – done well?

Not This Again, Dept.

In a recent article in Huffington Post, Keith Devlin (a mathematician from Stanford who writes about math education) says that math as it was taught in the past focused largely on computational skills. But it should be about “number sense” since computations can largely be done by computers, calculators, “the cloud”, etc, these days.

He waits until the end to tell us what “number sense” means–which in his case means “conceptual understanding” as opposed to procedural fluency.  As if he has heard the arguments before (and he has), he takes a pre-emptive strike against potential criticism from people like me by paying lip service to the need for learning procedures and then goes into his case for “understanding”.

Make no mistake about it, acquiring that modern-day mathematical skillset definitely requires spending time carrying out the various procedures. Your child or children will still spend time “doing math” in the way you remember. But whereas the focus used to be on mastering the skills with the goal of carrying out the procedures accurately — something that, thanks to the learning capacity of the human brain, could be achieved without deep, conceptual understanding — the focus today is on that conceptual understanding. That is a very different goal, and quite frankly a much more difficult one to reach.

This, by the way, is the new state of affairs that the mathematics Common Core was created to address. Outsiders, including politicians in search of populist issues to incite voters and others with an axe to grind, have derided caricatured portrayals of this important new educational goal, by describing it as “woolly” and “fuzzy”. But I disposed of that uninformed red herring already. The fact is, number sense is (rightly, and importantly) the primary focus of 21st Century K-12 mathematics education that millions of children around the world are receiving today. Children who are not getting such an education are going to be severely handicapped in the world they are being educated to inhabit.

While he may have disposed of the “red herring” of the educational goal of understanding, I’d like to revisit it.  As I’ve written about in numerous articles (thus disposing of the “red herring” of traditional math done poorly being the definition of traditional math), a glance at textbooks from the past shows that in fact the conceptual underpinning of various procedures has in fact been included and was a part of math education. Granted, the explanation of why the invert and multiply rule for fractional division works was not provided. I knew how to use it as well as what fractional division is and how to solve word problems with it, but as I said in my last missive, I didn’t learn how the rule is derived until 10 years ago. Nevertheless, I did manage to major in math despite the omission of something that brings gasps  of astonishment and “tsk tsks” from the education establishment–and largely from people who themselves have benefitted from traditional modes of math education.

As Prof. Devlin asserts, in many US classrooms today,students must  demonstrate an “understanding” of computational procedures.  How this has played out in the last 25+ years, and now even more so under Common Core, is that before they are allowed to use standard algorithms, they must learn alternative procedures to provide the conceptual underpinning in the belief that teaching the standard algorithms first obscures what Devlin calls “number sense” or “understanding”.  The teaching of mathematics has thus been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

In so doing, students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding. But this “stop and explain” approach to understanding undermines what the reformers want to achieve in the first place. It is “rote understanding”: an out-loud articulation of meaning in every stage that is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.

The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education. Their views and philosophies are taken as faith by school administrations, school districts and many teachers – teachers who have been indoctrinated in schools of education that teach these methods.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. While students undergoing instruction under the prevalent interpretations of Common Core may be able to recite an explanation they have been told–um, that is, “discovered” with some “guide on the side” guidance from the teacher on how best to phrase such explanation–that is not the test of effectiveness of a math program. In many cases, what passes for “understanding” or “number sense”  is “rote understanding”.   In so doing, the math reformers have unwittingly created a body of students in which “understanding” foundational math is not even “doing” math.