Gee, That’s the First Time I’ve Heard That, Dept.

At a math blog, I came across the following paragraph:

“Word problems in math textbooks often give much of the information needed to solve them. There is no mystery. Students walk away from a math course with the only skill acquired being the ability to decode the textbook. They are just swapping numbers and plugging in different information. As a result, the so-called problems are no longer problems. They are routine and predictable. The problems are too scaffolded and the students realize that it’s an exercise in futility. An insult to their intelligence. While practice is indeed a fundamental part of math, when problems are variations of the same one, the motivation to complete them is lost. They don’t see the point of it all.”

This is the standard complaint levied against the typical word problems one sees in math textbooks. Well sorry to disagree with this grand master, but I’ve been using Dolciani’s algebra book for my 8th grade algebra class and believe me, the students are not finding the problems predictable. Dolciani, like other good math book authors, does vary the problems so that one is not working the same problem over and over, which is the standard complaint–i.e., it’s just plug and chug.

But this blogger doesn’t even like such variations and scaffolding. Sorry, but that’s how you understand and master the basic skills in math–by getting students to see the structure of problems and how to solve them through the initial worked examples, and then stretch their capability and extend the problem solving principle to situations that are just a bit different, and more difficult.

No, what this blooger likes are what are called “Fermi problems”

Look to Enrico Fermi. The Italian physicist had a gift for making accurate estimates of seemingly unsolvable questions. Given little information, he was able to provide educated guesses that came very close to the actual answer. His most famous question, “how many piano tuners are in Chicago?” seems to make no sense, but through a series of questions, estimations and assumptions, he arrived at a reasonable answer. Legend says that Fermi calculated the power of an atomic explosion by looking at the distance his handkerchief travelled when he dropped it as the shockwave passed. He determined it within a factor of 2. For a discipline that is always looking for realistic applications, math class would do well to use Fermi problems. It doesn’t get more real-life than that!

While there’s nothing wrong with such problems per se, they should not be used as a starting point or replacement for learning math nor as the fundamental definition of what math is used for. Fermi problems are percent/scale-up problems. How many golf courses are there in the US? How many molecules are in a mountain? These are classic IQ and job interview types of questions. The fallacy is that one cannot memorize (!) a lot of facts to make these estimates easier. You can practice these problems to get better at them. They can convince others that you are a genius. A big fact to memorize is how many people are in the US. A second one is how many people are in your state. Then estimate the number of golf courses (or whatever) in your state and scale it up. Often, when someone asks you one of these questions, they are happy if you can come up with some reasonable process for estimation. 

A steady diet of these things does not teach students general and transferable problem solving skills that they will need in other math courses. The belief seems to be “Give them top down type problems that force them to learn things on a ‘just in time’ basis, as if there is a problem solving schema that will emerge, given enough time and enough off the wall problems. Solving Fermi problems depends on memorizing simple facts and using simple math. It is not what math is all about.

Talk to the parents of the students who are on the track to AP calculus and on to STEM majors. They solved lots of the traditional word problems people like this blogger hold in disdain. Dolciani’s algebra books didn’t skimp on problems. Every chapter had word problems tailored to the particular math skill that was the focus of that chapter. If the chapter was on algebraic fractions, then the rate/distance problems and mixture problems given in that chapter relied on knowledge of algebraic fractions to set them up and solve.

My students are finding the problems challenging. It took some time before we were at the point where a certain type of problem was familiar, and for me to then up the ante. But unfortunately, the beliefs espoused by this blogger are very typical and people who teach the traditional problems are viewed as doing their students a disservice.

A math professor I know has this to say about teaching students how to solve problems using things such as Fermi problems:

I WANT my students to look at the sophisticated work I give them and say “Huh, this is no problem I just do such and such and so and so, and this will get me to the answer”. It is the students who have to struggle and fret over straightforward stuff that I worry about. Why do they insist on making easy things hard and putting roadblocks in students’ way?”

But what does he know? He’s just a mathematician who happens to think like one.


Shut the Hell Up, Dept.

This article focuses on the changes a school district is making in its classrooms to encourage collaboration. The unstated and unproven assumption is that working in groups is a 21st century thing, that it prepares students for getting along in the work-force, and is superior to a traditional type arrangement.

The sub-headline reads “Only ‘matter of time’ before similar spaces pop up across Iowa, expert says”.

At least they didn’t call them “maker spaces” but that, too, is probably a matter of time.

“Laura Wood, a 21st Century Learning specialist for the area education agencies that oversee much of southern Iowa, has helped advocate for those spaces.”

I find it interesting that one can now have a job title of “21st Century Learning specialist”. I might try calling myself that too and see if people find me more credible.

The school district’s vision of this brave new world in which students can move desks around at will and “collaborate” is summed up in these statements:

“While students can collaborate in traditional classrooms, 21st century spaces treat group work as the default, said Cindy Green, Cardinal’s director of curriculum and instruction. “Our focus and our push is to get kids collaborating,” she said, “because when they leave and go out into the workforce, they’re going to need to be cooperating with their peers.” “

Well, you may not know this but I worked in the real world for 40 years, up to and including 2010 when the modern day workplace was in existence. Collaboration consisted mostly of having some meetings in the beginning of a project to hash out ideas. But once things got going, we worked pretty much on an individual basis. People brought various levels of expertise to the table. Engineers worked on engineering aspects; lawyers worked on legal aspects and so on. In school, students are still novices and do not have levels of expertise to offer other students. It’s pretty much the blind leading the blind, unless there is some guidance being provided by “teacher as facilitator”.

In this brave new world of collaborative, student-centered, inquiry-based learning, instruction from the teacher is generally kept to a minimum. Direct instruction is considered inferior; getting instruction from a fellow student, however, is considered a good thing. Why direct instruction from a teacher is considered inferior to direct instruction from a student is a mystery to me. I did have someone invested in this theory explain it to me. “Students have more faith in something they think they came up with than something the teacher tells them.”

Not mentioned is to what degree such “collaborative instruction” is coming from expertise gained from Kumon, Sylvan and other sources of outside tutoring. Might be good if reporters asked such questions, but I don’t see that happening anytime soon.

Articles I Never Should Have Started Reading, Dept.

Once in a while I visit Edutopia’s website to find out what the current educationist trends are. If you’re looking for bad ideas, Edutopia will never let you down. To wit and for example, this article: Brain-Based Strategies to Reduce Test Stress in which “A neurologist shares ideas for beating stress before and during test time.”

The article has no shortage of platitudes and trendy edu-thoughts, including this gem of a paragraph. It starts off well enough:

“When the goal of learning is only test preparation, students will not be prepared to apply their learning to novel questions or problems.”

Yes, very true. But is anyone saying that the goal of learning is “only” test preparation? The author continues, undaunted with what is REALLY needed:

“But engaging students in authentic performance tasks and project-based learning helps deepen their understanding on both the factual and conceptual levels. In addition, when students experience their learning as personally meaningful, their intrinsic motivation strengthens long-term, durable memory networks. These are far more accessible for test retrieval (and longer term access) than rote memory.”

In other words only if students engage in “authentic” activities and PBL, will learning really take hold. Otherwise nothing is “personally meaningful” and is therefore relegated to the catch-all category reformers love to use: “Rote memory”.

Apparently, the author has written books on education. Well at least the article didn’t mention “maker spaces”.

If You Repeat Something Long Enough, Dept.

This article in The Windsor Times (which serves Windsor, California) explores the changes in math education going on in that school district. There is a discussion of how the “integrated math” option of Common Core is superior to the “old ways” of doing things:

“Part of the issue is the substantive changes to the curriculum itself, which everyone agrees is better in the long run and significantly more rigorous that previous versions but is also significantly different. Gone are the old paradigms of Algebra I, Geometry, Algebra II/Trigonometry, followed by AP Calculus for the elite few. In its place now exists Integrated Math 1, which incorporates early concepts of both Algebra and Geometry, with slightly more emphasis on Algebra, Integrated Math 2, which introduces more advanced concepts of Algebra and Geometry, with slightly more focus on Geometry, and Integrated Math 3, which takes both sets to the next level and along with some Trigonometry.”

Excuse me, did you say “everyone agrees” the new curriculum is better? Everyone, you say? Sorry. Go on.

” “The belief is that students who have grown up in the integrated model will be much better equipped and have a much more extensive math education that those without. However, students at all levels are struggling with the staggered roll out from the old model to the new.

” “We are no longer asking them just to compute,” Director of Educational Services Lisa Saxon said. “They have to think about numbers in a different way, explain, defend and justify their answers, and undertake a completely different level of rigor.”

Excuse me again. I teach algebra and believe me, I don’t just ask students to “compute”. Where are you getting that? As for defending and justifying their answers, and this completely different level of rigor: I’m aware some people think there are such things as “math zombies” who operate from a “rote” level of executing procedures. I would suggest that there is also what I call a “rote” level of understanding in which students mouth the explanations the teachers want to hear. Is that what you meant?

Shut the Hell Up, Dept.

I was going to file this under “Articles I Never Finished Reading, Dept” but I felt that “Shut the Hell Up” was more apt, considering this quote:

Often what’s lacking for U.S. students, and Massachusetts students as well, is a conceptual understanding,” Chester said. “They may learn the mechanics, but without that conceptual understanding it’s not as clear to those students how that math gets applied.” Mitchell agrees. It’s the skills to apply math that are lacking. She says students need to be taught how to think with math and not just memorize.”

Given that this has been the complaint for the past 28 years or so, don’t you think the excuse that “we’re just not teaching reform math right” sounds a little lame?

On the issue of “understanding” vs “procedures”, a math teacher I communicate with in New Zealand has this to say:

“I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.

“Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.

“Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding. Most kids really don’t want to understand very much. Every now and then I have a student who refuses to learn a new skill until they “understand” it — and that causes problems, largely because they learn so unnecessarily slowly as a result, which I find difficult as a teacher.”

I recognize that this quote will cause cognitive dissonance. That’s OK.  Productive struggle is good for you.


A question worth considering

From one of our commenters, a retired math teacher of 30 years who wonders what a lot of us have been thinking. The context is that much traditional-math bashing relies on the mischaracterized memories of people who claim that they were taught by rote with no understanding.  This person raises the following question:

“As an interested parent, I am always curious about something. When parents bring up the issue of their kids lacking the basics of arithmetic and the need for tutoring, they are often ignored because their stories are anecdotal. Aren’t the memories that parents have about their own math education,10 -30+ years, ago also anecdotal? I don’t understand why faded memories are believed and present-day ‘anecdotes’ ignored? Could someone explain! Anyone!”

Where is the Evidence, Dept.

Another comment came in that is worth putting out there.  This one was in response to my missive called Chicken Little’s Rebuttal.  For those who don’t wish to read that and get to the bottom line now, I was responding to a serial tweet (i.e., a tweet split into two to get past the 140 character barrier) that came in response to my piece on The Dog Whistles of Math Reform.

“I’m also a bit confused. So CCSS contains dog whistles which signals to reformers that they should focus on the sorts of things they were inclined to focus on? So then what’s the effect?”

Here’s the comment:

I see no Chicken Little here. My children’s experience echoes what Barry has been saying all these years.

In a way, I think of my children’s education as Before Common Core and After Common Core. Before Common Core, my children had teachers who would get on their knees and beg parents to do math facts with their children at back-to-school-night. (Unfortunately, even then math facts weren’t covered enough in the classroom.) After Common Core, the teachers just tell parents to work on math strategies with their kids. Before Common Core, my children would have timed math fact tests. After Common Core, nope. Before Common Core, the teacher would provide direct instruction. After Common Core, most of my children’s time in public school was spent in student-centered groups, where the kids would discuss the math problems with each other. My child would come home from these student discussions with the completely wrong idea of how to do basic, fundamental math.

One of my kids started Common Core about two years before the 7th grade split, so we have had a test of sorts to see how everybody is doing. Some kids got through this type of teaching just fine, and got into advanced math and are doing fine. Other kids used to be strong at math before Common Core but now…. aren’t. It’s interesting, too – I’ll look at those smart kids who didn’t make the split, those kids who used to be “good” at math before Common Core started, and automatically think that the teaching style of the last two years had something to do with their missing the cut-off. Their parents, though? Their parents blame the kids, in a way. The parents just tell the kids that math isn’t their strong subject. Two years ago, it was.

I’m grateful that I came across Barry’s writing shortly after Common Core started, and I’m grateful that I read the comments on his articles. I would have worked with my kid anyway when she had problems. But reading the comments helped convince me to take her help to the next level – to buy additional curricula that the school wasn’t using and to go through the math (all of it) with her myself (I used Art of Problem Solving’s series). That technique helped keep my child’s fundamentals strong, and I would recommend it highly to all parents who have children learning with reform math in public elementary school.

An Important Word from One of Our Commenters

SteveH is a commenter on many of the missives on this blog.  I’m running a comment that he recently made on “Chicken Little Rebuttal” because I thought it hit on a lot of important points. I especially like his characterization of one particularly bad educational practice which you’ll read about below: “This is ed school fairlyland thought.”

Enjoy it–and comment on it as you wish!  Here it is:


CCSS institutionalizes low expectation, NO-STEM math. The highest goal at the end of high school is a 75% likelihood of passing a college algebra course (no remediation), and this track officially starts in Kindergarten. Traditional math allowed me to get to calculus in high school without ANY help from my parents. This is now institutionally almost impossible. The issue is not just about HOW math is taught, but about providing proper curriculum tracks. Low expectations and fuzzy math have been going on for 20+ years with reform math (MathLand, TERC, Everyday Math, etc.), but CCSS now institutionalizes low expectations. K-6 is now officially a NO-STEM zone.

How are students able to make the transition from a barely-algebra-end-of-high-school-no-college-remediation-slope to a STEM level AP calculus math track in high school? They need to have a proper algebra course in 8th grade. CCSS pushes the idea that 8th grade algebra is not really necessary and that students can make the transition by doubling up on math in high school or taking a summer course. This is ed school fairlyland thought. Their pedagogy is just cover for low expectations and some sort of natural curiosity/engagement process that apparently works by definition. If you aren’t successful, don’t blame them.

So, how do students do this now? Ask us parents. Really. Ask us. It’s not about showing a love of learning or taking our kids to science museums, or even asking them real world questions in the grocery store. We had to ensure that Everyday Math was more than repeated partial circling or letting our kids choose the lattice method. We had to “Practice math facts at home” all of the time. We had to ensure that basic skills (and understanding) were mastered. We had to push even our math brain kids. Their fundamental flaw is to assume that education is natural. This issue is far more than just how math is taught.

The problem is that with full inclusion, K-6 doesn’t push to get kids to learn anything more than the CCSS basics. They know that proficiency in CCSS will never reach their high math track split in 7th grade. They just assume that something like the Everyday Math spiral will get the job done naturally. It does no such thing. Been there, done that. Talked to other STEM-kid parents. Nope. It doesn’t happen. We parents hide the skill tracking at home. CCSS increases the academic gap.

Many schools seem to understand the problem with differentiated instruction, but the solution often becomes differentiated self-learning when they group equal level kids together in the same classroom and give them only a fraction of the teacher’s time. (Note that I use “level” rather than “ability.”) It’s made worse when the work focuses on enrichment rather than acceleration. Besides, this ends up being a form of hidden academic tracking in class – hidden to parents, not the students.

Many educators seem to understand the need for more math tracks in K-6 because of the wider range of abilitites, but they don’t know how to do it properly. They think it can be a natural “trust the spiral” technique, but fail to see that those kids in the higher level groups in class are the ones getting the basic skill help/push at home. They are not necessarily better ability kids! Their lack of understanding of this need for mastery condemns all of the other kids with no help at home to a NO-STEM career path by 7th grade.

As I said, I was able to get there without any help from my parents long ago. This is virtually impossible today. This issue is NOT just about teaching pedagogy, but about expectations and what is really needed for students to have a chance at living up to their potential. Reaching this level does not have to be a multi-generational process, but for many, it’s all over by 7th grade. They can’t make the non-linear transition. Educators are just happy if students become the first in their family to get to the community college. Maybe their kids will have a chance at reaching their potential with parents who know enough to push and ensure basic skills and content knowledge at home.