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What the NY Times Doesn't Know About Math Instruction

12/9/2013

 
A New York Times editorial on December 6 called for improved math instruction, calling the current system “broken.” Although I agree we could be doing a better job of teaching math, the suggestions in the editorial showed a striking naiveté about what it will take to improve.

The editors of the Times offered four suggestions:

  1. A more flexible curriculum (especially better integration of engineering)
  2. Very early exposure to numbers
  3. Better teacher preparation
  4. Experience in the real world (perhaps via collaborations with industry)

The editorial ignores the fact that 1 and 4 will be meaningless without 2 and 3. And it grossly underestimates the difficulty of implementing 2 and 3.

The editors’ idea regarding how to implement “early exposure to numbers” is to ensure better access to preschool. But that won’t do it because “exposure” won’t do it. Math is not learned like a language. Children can learn vocabulary and more complex syntax by mere exposure. They can’t learn math that way.

Still, the editorial is right to argue that very early learning is important. I’d argue it’s the key to math reform.

That National Math Panel concluded that three things need to be in place for students to be proficient in math: (1) you need to have memorized a small number of math facts to automaticity; (2) you need to know standard algorithms that apply to standard problems and; (3) you need a conceptual understanding of number, and of what algorithms do.

American kids are okay (not great) on math facts and okay (not great) on algorithms. On conceptual understanding, they are terrible. A student can get by for a time by memorizing algorithms and when they are to be applied, but eventually, not understanding what you’re doing catches up with you, meaning it affects your success in math.

But lacking conceptual understanding makes people hate math long before that. What could be more boring than executing algorithms you don’t understand? And if you don't understand what you're doing, why wouldn't you conclude "I'm not really good at math?"

This conceptual understanding ought to start in preschool with ideas like cardinality and equality. “Very early exposure to numbers” is not going to do it. That doesn’t mean taking what we had been doing in first grade and asking kids in pre-K to do it. That means putting activities into pre-K (e.g., games and puzzles that emphasize the use of space) that will provide a foundation for conceptual understanding so that first-graders will be in a better position to understand what they are doing. (Though first grade math will also have to change for that happen.)

The other recommendation that the editors of the Times get wrong is “Better teacher preparation.” They focus on high school, noting that many teachers of physical sciences did not major in these subjects. That may be a problem, but it ignores a much more serious teacher problem.

Most American teachers—like most American adults, including me--don’t have a deep conceptual understanding of math. They are a product of the system we are trying to change. You cannot teach what you don’t know.

So what’s my recommendation for American mathematics?

We need to pay much closer attention to preschool and to early elementary grades. That will entail developing methods of helping children understand the conceptual side of math—methods we now lack. It will also entail professional development to train teachers in the conceptual side of math.

The size of this undertaking is massive. But it directly addresses the issue encapsulated in the editorial’s title: “Who Says Math Has to Be Boring?” The editorial focuses on the idea that it’s boring to do things without knowing why you’re doing them. So the proffered solution is real-world application. But I think a worse problem is not understanding how math works, being asked to execute algorithms with no understanding of what is really happening. That’s a heavier lift but will ultimately be more rewarding.

Dave Marain link
12/9/2013 12:38:34 am

Some excellent points made here but sadly one could read editorials and letters on this topic from every decade for the past 40 years and I have and little has changed. Experienced math educators have forever recognized the problems identified here but what substantive change hhs occurred other than more testing and expecting more accountability from teachers who are expected to change water into wine.

Helping young children develop conceptual understanding of numbers, operations, relationships (spatial as well) requires specialized training that is not currently the norm in teacher preparation. We are very good at appointing commissions to draft world class standards and creating more ambitious testing but not very good at providing prospective and current teachers with the training necessary to implement these ambitious changes. THERE ARE NO SHORTCUTS HERE. It requires a sea change in teacher preparation and an investment of time and money that no one up to now has been willing to make. The money is out there to make this happen but saying we want to be the best is very different from preparing to be the best. There will always be fads and theories about how to improve the education of our children. But it's not rocket science to figure out that changing standards and assessments is putting the cart before the horse. Teaching children HOW to think isn't easy but it is doable. A young NFL player who happens to have majored in math was asked how math could help him with football. He replied, "It's all about problem-solving." Perhaps we should be listening more to young people like this...

John Miller link
12/11/2013 07:24:57 am

I saw the (a) story about the football player who said that. Wasn't NFL but was collegiate. He is a graduate student, publishing, and teaching math at a university.

David Wees link
12/9/2013 01:01:27 am

This is sort of true:

"Math is not learned like a language. Children can learn vocabulary and more complex syntax by mere exposure. They can’t learn math that way."

It depends on how you define mathematics for this to be true. If you define it as "the set of procedures in order to solve problems in an area of thinking labelled 'mathematics' that have been previously developed by humans" then the only procedures people will learn are the ones that are present in their environment. My son, who through me, is exposed to much rich mathematics and a LOT of number talk, has developed some of his own procedures (none of them is original). However, this environment is not present for every child. In fact, it is present for very, very few children.

Early number talk and exposure to mathematical reasoning (assume a gradient scale of mathematical reasoning, rather than the binary logic/not logical so many people use) helps a lot. It helps students so that when they are exposed to the earliest number algorithms, they are able to understand, even without someone trying to explain it to them, why the relationships between the operations and what they know about numbers hold.

In terms of the recommendations from the national panel of experts, I would re-order them, not because they aren't all important but because #3 is the one that is the most important to get right. You know how lists work, people check them off, starting at the top of the list. "Did #1? Yep. Did #2? Yep. Did #3? No, not yet, but we'll get to it."

It also means that we would need:
1. More emphasis on professional development in mathematics and mathematics education for teachers, particularly early childhood educators.
2. More time set aside on an ongoing basis to continue that professional development as teachers continue through their profession.
3. We must not forget the role parents play in this. Maybe we could use some of that broadcast media we are so fond of and use it to teach parents (via some entertaining TV shows?) how to talk math with their kids. "Boy, I sure like how Homer talks to his son about thinking ideas."

Jeremy Wang
12/9/2013 06:17:10 am

Yes, this comment stood out to me as well ("Math is not learned like a language...").

I know that Dr. Dan has done some work in the area of implicit learning, an area of research that looks at how people learn without intention or awareness. Arthur Reber did seminal work in this area using an artificial grammar task to show how people can learn (at least above chance) whether a string of letters is grammatical or not.

Are you saying that math can't be learned implicitly? To what degree?

As David argues, I believe exposure early and often to a lot of mathematics can have a positive effect on students' later learning. That is, mere exposure to math concepts can help build conceptual understanding. I think Dan would agree, but the statement about math learning not being like language learning makes me wonder...

Dan Willingham
12/9/2013 07:05:27 am

Point taken, I probably stated it too strongly. One can learn incidentally. That's actually pretty much what I think would make sense at PreK--it's experiences that lead to understanding these concepts, not direct study/instruction. Still, it will need to be thoughtfully implemented, planned--that's where I think it goes beyond exposure. . .and language need not. Data consistent with that (but notconclusive) are that socioeconomic status differences are sizable for language, but not mathematics.
Jeremy--we def. don't want this knowledge to be implicit! :)

Jeremy Wang
12/10/2013 05:11:10 am

Agreed - we want math knowledge to be explicit!

Mark Chuoke
12/9/2013 09:27:19 am

I think the "why" in math equals understanding the "how." If you see that a simple algebraic equation can be used to solve a problem of proportionality, or a more complex equation can solve the problem of predicting change over time, you will also see the how. Also, I've heard from math teachers that spiraling math instruction -- alternating algebra with geometry with probability with factoring, etc, -- and doing so within the course of an hour and twenty minute block lesson is more effective than hammering away at algebra all class long, every day. It is what is recommended by Doug Rohrer, a researcher who's worked with the Spacing Effect and other concepts of Robert Bjork. The spiraling creates the "desirable difficulties" of shifting topics and "successful forgetting," and at the same time it teaches the ways in which math concepts are applied across a rich field of various expressions.

Heike Larson link
12/9/2013 09:56:13 am

A large part of the challenge of math instruction is how math is taught--i.e., the actual math curriculum. In many cases, math is taught in a paper-and-pencil only fashion--to children who are too young to understand math at a purely conceptual level.

Montessori mathematics, instead, use a carefully sequenced series of math materials to illustrate and concretize concepts--and to help children to slowly transition from concretes to abstraction.

We've put together a very brief description of Montessori math on our web site:

Preschool: http://www.leportschools.com/preschool/math/what-we-teach/
Early elementary: http://www.leportschools.com/grades-1-3/mathematics/how-we-teach/

Montessori teachers often comment that the first time they really understood math was when they took the Montessori training! These materials really concretize mathematical concepts, and make learning a great experience for children.

Hung-Hsi Wu
12/9/2013 03:25:15 pm

Dan,

Congratulations on setting the record straight. People should read your piece instead of the NYT editorial.

Wu

Dave Marwin link
12/10/2013 06:13:40 am

(a) 9+9+9+9+9+9+9+9+9+9 vs
(b) 10+10+10+10+10+10+10+10+10
Is their equality a coincidence?

On your grid paper, make 10 rows of
* * * * * * * * *
Which addition problem does this represent?

Should how you could represent the other addition problem without drawing any more stars. [Rotate paper 90°]

Now write both as multiplication sentences...

We can pontificate about all of this ad nauseam but in the end teachers have to be trained to provide an environment which BLENDS explicit and implicit instruction. I learned much about arithmetic and number sense from playing Monopoly but I didn't learn everything that way! Some concepts/skills/procedures had to be clearly demonstrated to me. I was observing my precocious 6-yr old grandson learning to play Monopoly. From playing a couple of times he decided to buy every property he landed on. When he ran out of money I told him he'd have to wait until her could collect $200. "No problem PopPop. Just let me be the banker!" Will he improve his understanding of the game without formal instruction? Of course. Will he also develop some misconceptions if not corrected and given a clear explicit explanation? Of course. INFORMAL LEARNING CAN GO ONLY SO FAR IN MATHEMATICS. This must be balanced with the child developing proficiency with skills/algorithms, attention to detail and recognizing the appropriateness of approximate vs exact results. I'm only scratching the surface here. But I do know that none of this happens by accident. CCSS are necessary to raise the bar but without the"heavy lifting" required to train/prepare teachers, it will be futile. But nothing substantive will occur until the education of our children is genuinely seen as an investment instead of an expense. When we truly put our money where our mouths are...

Teacher_P
12/11/2013 09:03:27 pm

"Of course. INFORMAL LEARNING CAN GO ONLY SO FAR IN MATHEMATICS. This must be balanced with the child developing proficiency with skills/algorithms, attention to detail and recognizing the appropriateness of approximate vs exact results"

Exactly.

In England, we've gone the other way on this having decided in the 1980s that calculators made learning basic algorithms obsolete and everything has been loaded towards "conceptual understanding" ever since then.

We now have an entire generation of Primary School teachers who can't reliably multiply 3 digit numbers without a calculator (because the method they use which illustrates the concept very clearly becomes unwieldy beyond 2 digit x 3 digit problems). We also have many kids confused about multiply decimal numbers by 10 because Primary teachers reinforce the conceptually important "moving the number" to the left instead of the algorithmically simpler "moving the decimal point" one place right.

Emphasis on the concepts in the above leads to very inefficient algorithms and that means the amount of practice a child can do in a lesson is very limited, perhaps no more than 3 examples. And when you add that to our inspection regime constantly demanding "evidence of progress" which, until only a month or so ago meant "this teacher will be regarded as failing and subject to capability processes if an inspector sees children spending more than a minute or so in practice" and finally to high stakes testing which rewards slight knowledge of many topics instead of deep proficiency in few you have a near perfect storm.

Things may now be changing. The current government is keen to see a knowledge based curriculum and the current education minister is a "fan" of Professor Willingham, but governments come and go so there's every chance the next one will reverse both the good and bad parts of the reforms put in place. (And there are quite a few bad parts in among the good, not least giving more power for head teachers to bully classroom teachers when it is quite clear many of our problems in England are due to head teachers focusing on pleasing the inspectorate instead of what will actually educate the children!)

Dave Marain link
12/11/2013 08:47:27 am

I think his name was Verner, a defensive back for the Tennessee Titans. He decided to go back to UCLA to finish his degree in math. I'm entering this on my S4 phone and it's hard to post with the Captcha. That's why my previous post was duplicated 4 times! Sorry about that. And it may happen again...
Would be nice to get a comment on the other 500 words I posted re math ed of course!

johnymilton001 link
12/12/2013 01:25:42 am

hamm good that mean its provide good maths education.I am really happy to heard about it. So than for provide info like this and plz frnd comment little small. You have comment like an article. :-p

Tanveer link
12/12/2013 04:10:35 pm

Conceptual understanding is a must for subjects like maths. Maths cannot be mugged up like any other subject. The author has rightly pointed out the need to explain the concept in preschool. It needs to be seen if early elementary grades will be introduced in schools.

Sheryl Morris
12/14/2013 05:01:09 am

I really appreciated Heike Larson's comment:
"Montessori teachers often comment that the first time they really understood math was when they took the Montessori training! These materials really concretize mathematical concepts, and make learning a great experience for children." I know this personally.

I value comments made by David Wees, also. I feel we underestimate what a child learns by mere exposure in a 'thoughtfully prepared environment' (fundamental to Montessori.)

Thank you for including this informative article.
http://www.aft.org/pdfs/americaneducator/summer2010/Newcombe.pdf

Iowa math teacher
12/15/2013 08:25:54 pm

One problem is with American education in general – we do not encourage thinking. My students come to me in high school with the goal of regurgitating whatever I tell them onto a worksheet or a test, earning an A for this surface memorization, and then promptly forgetting everything they have “learned”. The grade or test score is the goal, not the learning. Unfortunately, this strategy has served them well in the past. Students have to be allowed; no, encouraged, to think if we are looking for true understanding in any subject matter.

What is really missing in mathematics education is the wonder and the joy; the interlocking pieces and the student-developed patterns that encourages understanding of basic and not-so-basic mathematical concepts. But in order to model this for students the teachers need to actually LIKE math and many do not. They take refuge in the safe and boring repetitions, wicking away any personal input by stressing the one “right way” regimentation.

Unfortunately, math these days is merely a Common Core checklist with no place for such conversations. Where is the time allowed for trying to draw a hypercube after talking about “Flatland”? Can’t we gather data to graph and discuss linear equations instead of stressing standard form, slope-intercept form, and point-slope form of the equation? Why not talk about other methods of calculating pi that do not include the diameter and circumference of a circle? But there is only so much time and all students must meet all standards. So we are still teaching “a mile wide and an inch deep”.

Math and all of its patterns is not some dry, unchanging study; it is really just art, and therefore extremely personal. It is a vibrant subject with a long history and a future that contains as-of-yet unimagined mathematics. Who knows what else may be discovered? I do not know, but it will not be uncovered by someone who stays safe and repeats only what he or she has been told.

Priscilla Bremser link
12/15/2013 11:48:09 pm

Well done -- thank you. Unfortunately, the December 6 NY Times editorial was just the first of three. Yesterday's was similarly misguided (you've nailed it with "naiveté"), including an endorsement of higher AP participation, as if that's done any good over the past two decades. (My rant on that topic is here: http://horizonsaftermath.blogspot.com/ .) I hope you'll write a response to the Dec. 15 editorial as well.

Heike Larson link
12/31/2013 02:04:45 pm

This post got me to work on putting up a new blog post on our school's blog. In the post, we explain how we teach math conceptually, from preschool through eight grade. Linking to it here, as I thought people in this thread might find it interesting:

http://www.leportschools.com/blog/teaching-math-conceptually/

Joe Morin
1/5/2014 03:07:58 am

The NY Times hockey stick curves showing expected growth in STEM jobs are simply not credible; unless they mean the number of STEM jobs needed at overseas companies where most of the USA venture capital is flowing.


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