I don’t believe more representations necessarily means more understanding. As I said in my comment below, understanding only comes from linking symbolic representations (graphs, tables, expressions, equations) to concrete, visceral representations. If a student can accurately convert an equation into a graph, this implies no understanding (unless the graphical representation helps them translate to the concrete, visceral one and they are using the graphical representation as a proxy for that.) Concrete, visceral representations of ideas are the only ones that make sense to humans. We’ve invented symbolic representations to make it easier and more efficient to think and communicate about real, tangible things, but unless you are imagining real objects when you are looking at symbolic ones, you don’t understand what you are looking at.

Also, “explaining” the connections between representations isn’t what we should look for to know whether or not someone has made a proper connection between representations. Translating one representation into the other, not explaining the connections between the two, should be the skill we look for. This is an easier thing to measure and a more fundamental behavior. Explaining the connection between two representations, let’s say a table and an equation, can be thought of as translating the table representation into a verbal representation, then translating the verbal representation into elements of the equation representation. This is messier for two reasons: 1) the imperfection of verbal representations muddy the translation; 2) it brings three representations into play instead of the two that were intended…

Yes, good call! Karen’s example comes up more often, and gives a clearer illustration of the “procedure w/o concept” phenomenon.

I have a book called The Math Book. If we ever have extra time a the end of the day, I open it to a random page and do a little “story time” for lack of a better term. Today I showed students some cool stuff with the Fibonacci sequence. Last week I talked about Gauss and him summing an arithmetic sequence quickly–to his teacher’s surprise. A few weeks ago we cut a mobius strip in half and showed them that it remained one piece. Sometimes its more interactive than others–usually depending on the time we have–but there is always a lot of theatrics and suspense built into the discussions. The students love them. I don’t know that anyone would chose to watch if they weren’t a captive audience anyways, but that is probably the closest thing I can think of to a spectator friendly aspect of math, which is really the antithesis of a spectator sport–I’d rather be playing the game!

I’m intrigued. You’ll have to explain the rules of this game in more detail.

I think kids who are good at memorizing unrelated ideas can be taught procedures pretty easily. I think it is a big reason why the “math gene” myth exists. It IS true that some can more easily commit unrelated information to memory than others and thus I believe those who struggle, also believe they “can’t” do math.

If mathematics is being taught as if it is just unrelated rules, steps and procedures, then it would make sense that so many believe they fall in this “bad at math” group.

You might be interested in the research regarding the CMI (comprehensive mathematics instruction) framework from researchers at BYU and Utah’s K-12 public schools. Just google it.

I was surprised when I read the quote from Dan’s blog because it is *exactly* the definition of conceptual understanding that I use in my classroom. Yes, procedural fluency is important, but if we never make the explicit connections between say factoring a quadratic to find its roots (procedure) and the understanding that those roots are the x intercepts of the graph of that quadratic (conceptual), why are we even bothering to teach them the procedure?
I agree with Dan that the implication of most articles that argue for more drill is that those of us on “the other side” are just trying to make our classes fun so students will like math. I’m not trying to make my classes fun at all; rather, I am trying to engage students with the math because an understanding of how the concepts connect and are interwoven is what makes the subject of mathematics inherently fun. Math is cool, not because I can put on a dog and pony show, but because it is full of patterns that we can learn to recognize and connect to other patterns in other places where we might not expect to find them. The more we find those patterns, the more we connect them in interesting and unexpected ways, the cooler and more engaging and more fun math becomes for us.
And, yes, sometimes it is necessary to develop a certain level of procedural fluency to make connections, but most times, you can use the drill time strategically to say “do these problems and then let’s look for the pattern that the answers make.” Once you establish the pattern you can say, “let’s do a few more problems and see if the pattern continues.” Just making students perform problems over and over again without making the connections is what turns them off to mathematics.

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Does Will mention procedural fluency there? It seems to be an analysis of why mastering number skills is important, not procedural fluency. He does mention it elsewhere on his blog. e.g. here http://bit.ly/2wPzW21
“I’ve never understood the ‘drill-and-kill versus teach-for-conceptual understanding’ argument. In that, I mean it’s not a dichotomy. Both reinforce each other. Students who try to problem solve with fluency but no conceptual understanding can’t apply their knowledge to new contexts. Students who try to problem solve with conceptual understanding but no fluency fall into working-memory overload in doing the basics and lose sight of, or can’t form, the strategy for solving the problem.”

How so? The blog is about pinning down what is meant by conceptual understanding and the exercise drives that home (as is the intent). Nowhere has anyone stated that conceptual understanding devoid of procedural fluency was preferable …its just far less common since procedural fluency is the easy part of the conceptual-understanding/procedural-fluency/application trifecta that is rigorous math.

Also, sometimes to enact change you have to push the pendulum. While i don’t agree with any argument that conceptual understanding should be a greater focus in math classrooms, I do agree that, generally, it still needs more focus on classrooms than it is getting (as does application). That said, I do value your piece and its effort to keep that pendulum from swinging too far in the other direction.

Thank you :)

Dan writes that our study “attempts to discredit conceptual understanding by linking it to ‘movement and music’ (p. 186) in math class.”

This is incorrect. Instead, as we stated in the study’s introduction, movement and music have been suggested as ways to teach students math, including concepts. It is easy to find suggestions of exactly this. For example, see this NEA link (http://www.nea.org/tools/lessons/music-and-math.html). NCTM is currently offering teachers grants to use music to teach math (https://www.nctm.org/Grants-and-Awards/Grants/Using-Music-to-Teach-Mathematics-Grants).

Dan incorrectly attributes to us a claim made by others that movement and music might help students learn math. Instead, we empirically evaluated this claim.

We empirically evaluated whether instructional practices–that teachers themselves reported using–were associated with math achievement gains by 1st grade students. One type of instructional practice was using movement and music. Others were teacher-directed and student-centered instructional practices.

We examined for achievement gains by 1st grade students who had struggled in math during kindergarten as well as by those 1st grade students who had not. We adjusted the estimates for many other explanatory factors.

We found no evidence to indicate that use of movement and music were associated with achievement gains (Table 5). In contrast, teacher-directed instruction was consistently associated with achievement gains including both by students who had previously struggled and those who had not. Student-centered instruction was associated with achievement gains, but only by those who had not previously struggled. Our finding that student-centered instruction may not benefit students who struggle is consistent with other work (https://www.tandfonline.com/doi/full/10.1080/01425692.2015.1093409).

Yet we found that 1st grade teachers were more likely to be using movement and music as well as manipulatives and calculators when leading classrooms with greater shares of students who had struggled (Table 3).

We should be using instructional practices to teach math that are empirically supported. This is especially true for students who are already struggling. Mischaracterizing a peer-reviewed study examining which instructional practices look to have more empirical support only detracts from our shared efforts to help all students learn math.

SteveH,
I couldn’t disagree more.

First of all, The traditional vs. integrated is about content sequencing, not delivery methods. Both are CCSM and both prescribe a core focused on rigor: the pursuit with *equal* intensity of conceptual understanding, procedural skills and fluency, and application. To say any of these three aspects is less important than the others is to demonstrate a misconception of the nature of mathematics. Too many people in this country think that math is nothing more than a set of procedural tools. They don’t realize that those tools are used for something and that an appropriate application of those tools (or better ones) requires some understanding of where those tools come from and why they do what they do. The “zombie” argument is certainly not fallacious. There are many students who perform memorized procedures without any thought as to weather their use is appropriate. A good mathematician is one who decides which tools at their disposal are best suited for the task. Part of this “conceptual understanding” is knowing when, where, and why we are using a particular tool/method. “Mastery” is not synonymous with procedural fluency. Mastery goes beyond that. A student who has truly mastered a mathematical concept understands the concept and the procedures it encompasses (which, by the way, are only a subset of the concept–there are many other aspects of mathematical concepts than just the procedures) and are able to apply the concept in a useful way.

As for AP/IB, ACT, ect. One issue we have right now is that it is hard if not impossible to assess any aspect of rigor other than procedural fluency without expending a lot of time and resources. To use Dan’s example, any mathematician would agree that a student who uses a discriminant to determine if x^2+2 has real or complex roots has not yet obtained mastery. But without seeing how a student goes about answering the question “Does x^2+2=0 have real or complex solutions?” we do not know if they are at that mastery level or not.

…And I’d hardly say that a curriculum devoid of conceptual understanding has been very successful at pumping out STEM students that are at a mastery level anyway. On the other hand, our University offered a summer high school program were high school students were invited to participate in a course on number theory (basically Math 4100). The course was derived from the Arnold E. Ross Mathematics Program (which originated in 1957), and is very conceptual. Here is an example question from day 1:

Consider the following number systems: Z, Q, R, 2Z, Z/3, Z/6, Z/8, Z/11. Each of
these systems has two operations, addition and multiplication. Consider various algebraic
properties and determine for which of these systems those properties hold. Some properties
to consider are: if a^2 = 1, then a = 1 or a ≠ 1; if b*c = 0 then either b = 0 or c = 0; if
d ≠ 0, then the system also contains d^-1; etc.
Make a list of several such basic properties, noting which systems have which properties.
Which of these systems would you consider to be the “most similar” algebraically? Why?

You can see that from day one, the students started building their own conceptual understanding. Granted, we didn’t leave them out to hang–eventually as a class we pooled our lists and sorted out what defining characteristics each system has and from that built the axioms from the ground up. We did not give them a list, they made the list and we, with students as mathematicians, sorted it out until we settled on what an “Intro to Number Theory” course would introduce for axioms of the integers. THIS IS WHAT MATHEMATICIANS DO!! To say these concept-first methods were “time-wasting” is truly “failing to see the forest through the trees”.
The only true rote procedures in the entire course were division with remainder and Euclid’s Algorithm. Everything else in the course is either conceptual, or application of E.A or the division algorithm. To say that mastery meant being able to do division with remainder and E.A. is to sell the entire 4100 course–and number theory itself–short. A mathematician uses tools at their disposal to discover concepts… P-E-R-I-O-D! To say conceptual understanding is over emphasized in today’s math classes is to say that mathematics is over emphasized in today’s math classes. You can’t make “traditional” math better without bringing about a balance of conceptual understanding, procedural fluency, and application. both research and experience demomnstrates this.

In the sense of a graduate level math class, the problems students DO *are* conceptual problems. They are not practicing rote procedures throughout on homework sets. The last class that I can remember where there were more than a mere handful of algorithms to learn was maybe PDE’s. Sure there’s an occasional algorithm that shows up almost all courses (like Page-Rank) but the *procedure* of the algorithm is hardly the meat of the course. It is the concepts that surround it that are of interest. Anyone who thinks that algorithms are the end-all-be-all in mathematics has not delved deep enough into what mathematics is. They are simply a means to an end, the end being understanding mathematical concepts or applying the procedure *correctly* to some application.

Take Green’s Theorem. If a future engineer/mathematician knows how to use it. Great! Do they understand the necessary conditions on the boundary curve? Do they realize that it is a generalization of the FTC and a special case of Stokes’? Can they apply it to answers questions in electromagnetism? Can they apply the theorem to build a platometer or antenna? If the answer to all of these questions is “no”, then I would ask, why the h – – – do they know Green’s Theorem anyway?! In fact, I would say their mastery of the theorem is minimal …hardly more than a computer program’s mastery–perhaps less since the program is far more infallible.

As for your argument that ‘traditional’ math [education] is the only thing that works, research–even plain observation–would demonstrate otherwise. 90% of the people I meet admit to me right off, and quit emphatically “I can’t do math!” These aren’t millennials who are participating in this progressive trend (which by the way isn’t that progressive, nor is it a trend since conceptual understanding and application have long had equal standing with procedural fluency in nearly every other countries education system), these people are products of what you would call a “traditional” curriculum. And the 10% that claim to be good at math, turns out they know just a bunch of random facts/procedures. They think because they know some algorithms/formulas/theorems that they are good at math. After a few minutes of conversation, it becomes very apparent that they have never done anything beyond regurgitating what a teacher spoon fed them or what they read in a book or a blog. I have yet to meet someone outside of my circle of mathematician friends who is able to carry on a conversation about *real* mathematics. I’m getting rather lonely. It’s not that we want the process “slowed down to spend more time on in-class concepts and engagement” We want more *real* mathematicians in this country than we currently have.

Finally, I can see why your comment would have been deleted. If it was anything like this one it was full of subjective opinions about what you think math ed should look like, not research based evidence. Opinions do not determine fact and they do not determine best practice, no matter how strongly we believe them. If you truly believe in math, then you should believe the math behind the studies. If I want to hear anecdotal dribble about how an engineers son struggled with real mathematics, I’ll go talk to the 10%!

“The traditional vs. integrated is about content sequencing, not delivery methods. …the pursuit with *equal* intensity of conceptual understanding, procedural skills and fluency, and application.”

Equal? That’s not what it is in practice.

“Too many people in this country think that math is nothing more than a set of procedural tools.”

“People” can think what they want, but that’s not how AP/IB math is taught.

“There are many students who perform memorized procedures without any thought as to weather their use is appropriate.”

Appropriate? Were they able to solve the problem? What additional litmus test are you applying? All mastery includes understanding and there are always more levels and nuances to learn from P-sets.

“A student who has truly mastered a mathematical concept understands the concept and the procedures it encompasses (which, by the way, are only a subset of the concept—there are many other aspects of mathematical concepts than just the procedures) and are able to apply the concept in a useful way.”

“Useful?” Truly mastering a concept is the end result of mastery of the skills from individual homework, becase introductory concepts are only words and any in-class group oriented project wastes too much time stretching thought in only one direction. There is no proof of transference.

“As for AP/IB, ACT, ect. One issue we have right now is that it is hard if not impossible to assess any aspect of rigor other than procedural fluency without expending a lot of time and resources.”

That’s because you don’t appreciate all of the layers of understanding that are required to do homework and pass tests. Perhaps they are layers you now take for granted. If you have another method of testing rigor in mind, then you best share it. Then you can compare your students with those in AP/IB curicula. Right now I don’t see any alternative curriculum.

“…And I’d hardly say that a curriculum devoid of conceptual understanding has been very successful at pumping out STEM students that are at a mastery level anyway.”

Really? What do you propose that will work better? Where are your curriculum examples/results? I might be your biggest fan – honestly!

“The course was derived from the Arnold E. Ross Mathematics Program (which originated in 1957), and is very conceptual.”

Go ahead and create a full curriculum based on this. Just make sure it’s opt-in.

“Anyone who thinks that algorithms are the end-all-be-all in mathematics has not delved deep enough into what mathematics is.”

Strawman.

“As for your argument that ‘traditional’ math [education] is the only thing that works, research—even plain observation—would demonstrate otherwise. 90% of the people I meet admit to me right off, and quit emphatically “I can’t do math!” ”

Where is your research on a full curriculum? Does “works” mean getting “people” to say that they can DO math? That’s wasn’t my definition of works. I was clearly talking about STEM preparation. AP/IB sequences are the benchmarks for producing these students. Where is your specific alternative?

“procedural fluency is still overwhelmingly emphasized.”

In procedural learning lies all of the subtle understandings. Are P-sets only for speed?

“Getting an answer by brute force methods when another method is clearly more efficiante shows a lack of mastery.”

Strawman. That’s not what AP/IB do.

“.. there is research that shows students who had a curriculum that equally valued conceptual understanding outperformed students who had a curriculum focused solely on procedural fluency on standardized tests—and these tests where heavily procedural!”

Which specific multi-year curriculum is this? What specific curriculum was it compared to? Was it compared to AP/IB math or was it for K-8 at a non-STEM level? Did it slow down coverage? Anyone can slow down and do better. Anyone can have properly-trained teachers and do better with any curriculum.

“I don’t think you appreciate how easy it is to answer those types of questions.”

I don’t think you appreciate all of the subtle variations in problems that cannot be solved by concepts – general notions. When I taught college math and CS, nobody could pass my courses with rote knowledge or brute force techniques. Change a problem slightly and that requires a lot more than rote brute force.

“…research, perhaps most famously by Jo Boaler”

You forgot the quotes around research. I have to stop giggling. Prove me wrong?!?

“The bottom line is this: no one here is saying conceptual understanding trumps procedural fluency.”

Really!?! Where is the proper grade-by-grade enforcement of mastery in K-6 that gives students a chance of getting ready for a STEM degree without help at home? Math education has to be more than a “trust the spiral” process that puts the entire onus of mastery (above low slope CCSS) on students.

“…you have yet to offer any that concludes a balance of both is detrimental, or even that emphasis on procedure is beneficial.”

Strawman. Educators never do that research. They wouldn’t get the funding even if they were so inclined.

“I don’t value anyone’s opinion over research and my own personal observation.”

How about the reality of the low slope of CCSS to no remediation in College Algebra? How about the reality of the dominance of AP/IB? How do you think students get ready for those tracks in high school? Ask us parents of the best students. We focus on mastery of unit skills. I did get my son to use GeoGebra in middle school where he studied trig functions to model saw-tooth functions, but that was neither necessary or sufficient.

I don’t see anyone here talking about improving AP/IB – and I could make many comments on that. This is all about replacing them with something that is never explained in longitudinal curriculum form. It’s easy to cherry-pick reasearch, but quite another thing to prove it in reality. Where is your reality? Again, I might be your biggest fan, but don’t give me K-6 examples because traditional math has been gone from there for decades. Our lower schools improved results when they moved from MathLand to Everyday Math. I tutor some of those students to help them make the non-linear transition to the high school AP math track.

It certainly exists. Its just much less common a problem. the reason being that procedural fluency is usually easier for students to obtain and for teachers to impart. That’s not to say it doesn’t exist. when it does happen, its usually from lack of continuing down the path uncovered by the conceptual understanding.

My first example: I observed a class of 10th graders writing recursive equations for linear, exponential, and quadratic discrete functions. Their teacher had taught them (and I had the impression that most understood) that quadratics have a linear rate of change and therefore the recursive function of a quadratic would be something of the form f(n)=(n-1)+mn+b
However, though most students understood this concept, very few were actually able to write down the formulas.

My second example: This one is personal. Steve H.’s comment about Greens Theorem made me realize that, although I conceptually understand what Green’s Theorem is all about and were it comes from and what its applications are, my Calc III days are so far behind me and my integration skills so deteriorated, I doubt I would actually be able to apply Green’s Theorem in any meaningful way in my present state of abilities.

…which shows the equal footing that procedural fluency has with conceptual understanding and application. Any of the three without the others is pretty pointless. One is not more important than the other, but traditionally conceptual understanding and application have had to take a back seat when in a rigorous math course all three should be up front driving.

Great point Christopher Danielson,
the art of finding a [irrational] square root by hand is hardly mentioned today (possibly for good reason) though I think conceptual understanding isn’t diminished because of it.

Trig ratios are another realm. I once had a student ask how the calculator knew that sin(20 DEG) was 0.342020143326…

I was impressed that their conceptual understanding was deep enough to realize that this was even an issue. They understood how one could geometrically find things like sin(45 DEG), sin(30 DEG) etc, but they were able to pick up on the fact that the tools at hand were not enough to obtain what the calculator had shown them. They even understood that simply drawing a 20 degree right triangle and measuring couldn’t possibly lead to that kind of accuracy. After class I pulled this sophomore aside and we had a brief conversation of the angle sum and angle difference identities that revealed conceptually how one could arrive at the result.

Dan’s original question was the right one. How should conceptual understanding be defined?

Cognitive science (the science that defines and measures conceptual understanding) is in consensus on its definition. Conceptual understanding is made up of of observable, measurable cellular material. It consists of neurons in the brain, their stored information, and the wiring between them.

When you solve a math problem, science can SEE conceptual understanding occurring — in real time — using new technologies such as fNMI, PET, and MEG.
In 2018, this science is fundamentally well understood.

When you read a problem, your vision moves the problem data into “working memory.” Working memory then searches the neurons of your long term memory (LTM) for matching data elements. If it finds matches, the matching neurons “activate” and “fire” — send out a measurable electric current — to up to hundreds of other neurons they are “wired” to (connected to via synapses).

If the current is strong enough, those connected neurons “activate” and “fire” as well. The relationships stored in the neurons of memory that are activated can then be recalled into working memory, where they are applied duing the steps of solving the problem. That’s how mathematical problem solving takes place.

Those activated “conceptual frameworks” — the neurons and wires among them — are the source and substance of your conceptual understanding. The key question is: How do we help students construct those conceptual frameworks?

That process is also well understood by 2018 science. First, fundamental facts must be moved into long-term memory and made recallable — they must be “memorized.”

AFTER fundamental facts are recallable, they must be used in procedures that are memorized and used to solve problems. As recallable facts are used in procedures, the facts and procedures in memory are wired together (“neurons that fire together, wire together”) , and implicit (tacit, intuitive) conceptual frameworks grow in number and strength.

But neurons cannot grow conceptual connections until AFTER facts and procedures are stored in them AND those facts and procedures are made recallable for processing to at least some extent.

And according to science, making math facts and procedures recallable takes considerable time and effort (over-learning/ retrieval practice).

So, in answer to Thaslam’s comment: For the initial (non-expert) learning in a topic, “memorization” (storing fundamentals in neurons and making them recallable) must physiologically occur BEFORE “learning” — which is gaining an intuitive sense of WHEN to recall a relationship by the brain relying on the strength of neural electrical impulses.

For more on conceptual understanding, I recommend the short book “Learning How to Learn” by leading US neuroscientist Dr. Terry Sejnowski (and co-authors). It is jargon-free — yet clearly explains the neuroscience of how students construct conceptual understanding.

Thanks, Dan, for asking for the scientific definition of “conceptual understanding.”

— Eric (rick) Nelson

Thaslam —

Much of what you have quoted above (not all, but most) involves people with no expertise in how the brain works claiming that the brain can do what cognitive experts say the brain of students cannot do. That does represent how U.S. math standards have been set since 1989.

And what has been the outcome? By the international standards, at or near the end of their schooling, how are U.S. students doing?

In 2012, the OECD tested skills in 22 developed-world nations including the United States. In “numeracy” (solving problems with mathematical content), U.S. 16-24 year olds ranked 22nd out of 22: Dead last. See http://bit.ly/2tMzJ08 .

In a world where a nation’s technical and engineering skills are essential for self-defense, that outcome devastating for our children and grandchildren. Your math standards that deny science have been tried for 30 years. It is time to ask cognitive scientists to join in making decisions on what our math standards should be.

— rick nelson

Thaslam –

Could you (or any readers) possibly provide the filename of the post with the typos? I don’t doubt you are seeing them, but the file I am downloading at http://bit.ly/2tMzJ08 does not have those errors. That said, with tech, I do expect disasters.

I believe readers looking at the sentences you cite as “without references” will see plenty of them.
On the need for memorizing facts as a necessary foundation for building conceptual frameworks, permit me one additional reference.

In the theory of learning described by Jean Piaget, moving facts and procedures into long-term memory (LTM) is termed “assimilation.” Resulting changes to the brain’s conceptual frameworks in LTM are “accommodation.” Herbert Simon and his colleagues write that a

“careful understanding of Piaget (shows) that assimilation of knowledge also plays a critical role in setting the stage for accommodation–that the accommodation cannot proceed without assimilation.”
— from Anderson, J.; Reder, L.; Simon, H. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer (emphasis added)

The article, downloadable from https://files.eric.ed.gov/fulltext/ED439007.pdf , takes some hard shots at the 1989 NCTM standards which I suspect you will take issue with.

But how math can be applied to a variety of problems is research for which Dr. Simon was awarded a Nobel Prize. Readers will find the article to be especially interesting.

For the record, I think everyone was highly disappointed to hear from science, 10-20 years after the 1989 NCTM standards were written and adopted in most states, that the standards asked students to do what their brains could not. Absent clear science at the time, the NCTM made an optimistic guess about how the brain worked. It worked out the way optimistic guesses about science usually do.

Everyone hoped the CCMS would fix what was wrong. It was disappointing to find the CCMS, though better aligned with science than the NCTM’s standards, are not well aligned.

If we must have “state” standards, they must work for students both mathematically and cognitively. But for kids and the nation, it would be better if standards in education were set by front-line practitioners who learn from and respect scientific research, in the manner doctors set the standards of medicine.

— Eric (rick) Nelson

In response to Ms. Harrison and many of the comments so far:

If K-14 students need to explicitly (verbally or in writing) explain WHY they take some of the steps they take to solve math and science problems, shouldn’t math and science teachers be required to correctly explain the rules of grammar they apply for at least some of the sentences they write? If you can’t “diagram a sentence,” are you incompetent in communication?

Actually, science says you don’t need to be able to recite the rules of English linguistics to speak, write, or teach. Your brain knows the rules implicitly or you could not talk or write, but like most of our knowledge in memory, it is unconscious. If we can correctly solve the problems of speech, or math, or riding a bicycle, explaining verbally why we do what we do is a waste of our limited cognitive resources.

Science says the goal of most learning is to be able to do the right thing intuitively, without having to stop and think about why. Members of our species who had to stop and think during confrontations at the watering hole were weeded out by natural selection.

NSF figures tell us that of the 1.8 million US BA graduates in 2015, only 1% (about 18,000) were math majors. Less than 1% (about 15,000) were chemistry and physics majors.

Majors need to explain the why of their major. But every K-12 student takes math, and all engineers and health care workers take chemistry and physics — because they will likely need what is taught in those courses as a tool to solve problems.

Experts on the brain agree that solving problems requires intuitive, often subconscious understanding. They say your brain simply does not have time to gain explicit (explaining why) understanding for all of the fields in which you need to solve problems in life and “on the job.”

Isn’t that consistent with our real life experience?

— Eric (rick) Nelson

Here’s something scary, I was helping a Pre-calc student with some homework on solving rational adding and multiplying rational expressions and solving rational equations. Their teacher told them to they have to limit the domain when they multiply, but not when they add or solve. In fact, one of the examples the teacher did for them showed the solution to be 3, but the denominator of one of the terms was x-3. Talk about fluency without understanding!