Eureka Math, Dept.

An article about a school using Eureka Math. It is the usual puff piece about a math program, but I guess in the interest of a “balanced story” (which is rare in math education reporting) they included this sentence:

“She acknowledged that with Eureka, fluency with math facts is not a daily practice, however teachers are finding other ways to introduce math facts into the middle school curriculum.”

And parents are probably drilling them at home.

I’ve heard mixed reviews about Eureka Math, and had some words with one of the people in charge of the program.

I also made reference to Eureka in this article.


New Boss = Old Boss? Or Is CC really going away in FL?

Florida is one of a growing number of states claiming to do away with Common Core.  We’ll see whether Florida’s new standards are really different or just a tweaked version of CC as has been the case in other states that claim they’ve jettisoned CC. But the quote below from Commissioner of Education Richard Corcoran does sound promising. (From Miami Herald.)

One of the biggest changes could be the way students learn math. The previous, Common Core-based standards, emphasized understanding the logic behind math equations. Corcoran on Friday said math should go back to the basics of arriving at the answer to form a stronger foundation.

“When you’re trying to remember what’s four times four, and you have to think about it and it’s not automatic, you’re never going to be able to conquer algebra and all those other courses,” he said.


Articles I Never Finished Reading, Dept.

From an article about “how math might have changed since you were in school.”

“With Common Core, students are learning more complicated and in-depth methods of doing math that focus, in part, on a simple idea: Teach students to think critically and to do so at an earlier age compared with previous standards so they can pursue career training or college, Mooney said.”

And how’s that been working out for the nation?


“Mooney said an obstacle in implementing the standards comes from figuring out how to overcome conventional wisdom about how people learn and talk and think about math.”

Right. Because in past eras which progressives deem to have “failed thousands of students” math textbooks taught standard algorithms first, and gave many practice problems (including word problems). After mastery of the standard methods, students were then shown alternative strategies–but they had as an anchor the standard algorithms from which they could then venture out. In that way, the alternative strategies helped spotlight why the algorithms worked as they did.

The strategies taught in those bygone days were the same ones that are now taught first under the Common Core standards to ensure understanding. Now, students don’t get to learn the standard algorithms until they master the strategies that are thought to provide the conceptual understanding.  And for the most part, in those days that supposedly failed all those students, there were explanations as to why the standard algorithms worked. That is, the conceptual understanding accompanied the explanation of how to use the standard algorithms. (See this article for further rants and perambulations on this topic).


Much Ado About Distribution, Dept.

A common complaint–as in “see, math is being taught wrong”– is that students fail to see that equations like 3(x-5)=60 can be solved by dividing both sides by 3 first. Progressives seem to make a big deal about this to the tune of “If students are doing this, they lack ‘deeper understanding’ about equations.

Textbooks that claim alignment to the Common Core now make it a practice to show this.  The problem is that if you have 7(x-5)=60, the process isn’t so neat.

In my experience students ignore the lesson and go ahead and distribute.  I point out that they can do it the short-cut way (since all the problems in this particular lesson are structured so that the short cut can be used), but they still do the “long-way” distribution method. At first I was worried when students were not “getting” it, until I realized I was succumbing to the embedded “deeper understanding” aspect of teaching it this way. I caught myself so now I don’t make a big deal out of it. I tell them to use whichever way they find easiest for them.

Students just learning algebraic rules do not readily see (x-5) as a single entity, and so they also fail to see 3(x-5) as something like 3A, where the three can be divided to undo the multiplication.

Says Robert Craigen, Math Professor at U of Manitoba:

” When students distribute first, is that really a sign of lack of understanding?  Why not instead see it as “showing understanding of the operations”?  Why isn’t cancelling the 3 being a math zombie?  These guys aren’t consistent at all.  Either approach might signal understanding, and either one might be the result of mindless application of mechanical rules.  I would rather focus on whether students are correctly performing the steps and can deal with variant problems. With experience comes fuller understanding.  Initially they only need enough understanding to avoid making obvious errors.  The early goals should be fluency and functional (not deep) levels of understanding.”

To which I say, “Amen”.


5,000 viewers, Dept.

The YouTube video of a talk I gave on math education in the US (encompassing comments on Common Core) has reached 5,000 viewers.

For those who want the bottom line, the last bullet of my conclusions is “Mistakes should not be clung to because of the time spent making them.”

For those with greater patience who only wish to hear my comments on Common Core, go to minute 19:25.

And for those with time on their hands, the whole talk is about 30 minutes long.  I originally gave it at a researchED conference held at Oxford in 2016.  I gave the talk locally in 2018 at the San Luis Obispo IHOP, where it was videoed.  The local chapter of the Sons of the American Revolution kindly sponsored the talk.

Some people told me I should not associate with such groups.  Well, I suppose if the KKK had asked me to give a talk on math education I would refuse, but I didn’t think I was jeopardizing my role as math education advocate by accepting their sponsorship. It came about when someone in my town read my book “Math Education in the US” which he bought at the local bookstore. He asked the bookstore owners if he could get hold of me, and they gave me a note with his phone number. The rest is history. Draw your own conclusions!

Some thoughts on Devlin and Boaler

For those who have read and heard Keith Devlin, he is pretty close with Jo Boaler who you may also have heard about. Keith Devlin, you will recall, writes a column called Devlin’s Angle in MAA and also is known as “that math guy” at NPR.

He made a big name for himself some years ago when he claimed that multiplication  “Ain’t no repeated addition”.

Well, yes, in formal, higher level mathematics, there is a general definition of multiplication that must meet several conditions. Technically, it is a function which maps two objects (numbers, functions, even shapes) from a set, into one object, (e.g., f(2,4) = 8 ) and the function is commutative, associative, distributive, and has an identify function called “1” in which a*1 = a.

What he seems to miss is that this general function does in fact include repeated addition as a means of informing the particular relations between numbers. For example, the times table chart showing the various facts is formally taken as an axiomatic definition; i.e., 4 x 2 = 8. That is, it equals 8 because we define the function that way. The inconvenient fact that Devlin likes to dodge is that the definition is informed by repeated addition.

Devlin and others of like mind think that teaching multiplication as repeated addition results in confusion when we teach fractional multiplication. Actually it isn’t that confusing, and using an area model incorporates the ‘repeated addition’ form of multiplication to get the end result.

Devlin and others who repeat his refrain got to their higher understanding by starting with repeated addition, which they now disdain. Remove the ladder much, Keith?

It is a throwback to the 60’s new math in which multiplication was defined formally. The joke was that kids knew that 5 x 4 = 4 x 5, but didn’t know that it also equals 20.

Funny that the top performing nations like Singapore manage to teach multiplication as repeated addition.

Devlin came under a lot of criticism for his series of articles, yet he stood by them and defended his stance vigorously in subsequent articles.

Boaler doesn’t go quite so far, but she is more for “fluently deriving” the math facts than straight memorization. So 9 x 8, should be looked at as 9×9 – 9. That way, kids who don’t know all their facts can derive them. And also satisfy her idea of what “deeper understanding” is. Really, folks, multiplication isn’t that hard. Nothing against the strategy she talks about, but the disdain for memorization (because it supposedly eclipses understanding) is just more money-making nonsense for Boaler and her ilk.

In my opinion, of course.

Don’t Tell Jo Boaler, Dept.


A group of students in Lake Charles, Louisiana is promoting knowing the multiplication facts.

Those who think that the traditional ways of teaching mathematics have been shown to be harmful for all students, will find this quote from the article to be heresy:

“The ability of an individual of any age to be able to multiply consistently and effectively can build confidence in other areas of life,” Nonnette said. “We as an organization will succeed in our mission to enhance the awareness of math education through one multiplication chart at a time.”

Now, if only we can find some way to make the progressives think they came up with this!

Reactions to Barbara Oakley’s op-ed: Revisited

I’ve noticed a spike in traffic at this site, looking at a post I wrote over a year ago. The piece I wrote addressed a blog post that criticized an op-ed on math education written by Barbara Oakley.

The blog post is here but comments have long been closed. I recall there was some flap with the blogger who wrote it, when I said that comments from Barbara Oakley and myself hadn’t been published.  She did publish Barbara’s comment, but I notice that my comment is still “awaiting moderation”.  Meaning she forgot about it, or didn’t want to publish it. I have no idea which may be true, but if you’re curious, here is the comment:

In your post you state: “It is true that traditional ways of teaching mathematics have been shown to be harmful for all students, and even more harmful for non-dominant populations, including girls. This phenomenon has been widely documented by professors of mathematics education such as Rochelle Gutierrez, Jo Boaler, and others.”

What research has shown this to be true for ALL students as you state? You cite Boaler who has gone on record as saying that learning times tables can be injurious to students. She believes that memorization does harm and undermines “understanding”. You also you cite Gutierrez who claims that minorities and girls don’t do well in math because of the way it’s taught. Even if we agree with Gutierrez’s claim, that would say that not all are injured, but just minorities and women.

I wrote an article some time ago about traditionally taught math and showed data from the Iowa Tests of Basic Skills (for grades 3 through 8 ) and the ITED (high school grades) from the early 40’s through the 80’s for the State of Iowa. (See

The scores (in all subject areas, not just math) show a steady increase from the 40’s to about 1965, and then a dramatic decline from 1965 to the mid-70’s. One conclusion that can be drawn from these test scores is that the method of education in effect during that period appeared to be working. And by definition, whatever was working during that time period was not failing. And this was at a time when traditional math teaching was the mainstay.

It is true that traditionally taught math can be done poorly. It can also be done well, and there have been many people who have benefitted from such instruction. Regarding memorization, I suggest you read “Memorable Teaching” by Peps McCrea which is an exploration of how memorization is an essential part of the learning process. (

Ch 13 of “Out on Good Behavior”

For those following the continuing series Ch. 13 is now up at Truth in American Education.


In planning my future classes during the summer before the upcoming school year I proceed from an undying faith in my expectations of how things will be. During the actual school year, I then deal with the reality. In the end, it is always astounding to me how some intuitions turn out surprisingly well.

My Math 7 class at Cypress my second year was the non-accelerated version. I had taught accelerated Math 7 the year before, but was now faced with a challenging group of students who I knew were disheartened about math and likely dreading the next year. While planning my lessons during the summer using the JUMP Math teacher’s manual, I had a vision that the students would upon succeeding and getting good grades on tests and quizzes, eventually discover that the math was actually interesting and that they could manage it.

The reality was slightly different as I was finding out and as I’ve written about in preceding chapters. I knew that something was happening. Just not in the manner I had envisioned.

Read the rest here.

Extended metaphor, Dept.


We frequently hear about how in math education we should engage students in “productive struggle”. While there is some value in having students synthesize prior knowledge from worked examples and scaffolded problems, this is generally not what is meant by “productive struggle”. Generally it means having students solve problems that are usually one-off types that do not generalize. What prior knowledge students may have to draw upon is in most cases very small and lacking in sufficient practice for students to be able to apply it efficiently. And if prior knowledge is absent, students are expected to obtain it via “just in time” learning, which would arrive without sufficient practice and mastery.

Students are expected to collaborate with fellow students, and dissuaded from asking the teacher for help. If the teacher is asked to help, the teacher is usually instructed to not give answers to students questions but to facilitate the student to answer their own questions.

The result is like throwing someone who lacks swimming skills into the deep end of a pool and asking him/her to swim to the other side. The result is generally a struggle to keep from drowning–which is not the same as learning how to swim.