Setting aside the autistic genius category, most students come to understand their own thinking better as they struggle to put it into words.

We adults need to recognize that putting ideas into words is often a struggle. We need to give kids time, to allow for intuitive explanations, to recognize the kernel of truth in their flailing attempts and help them draw it out.

Math explanations, IMO, should be oral and informal, at least in the elementary and middle school years. Written explanations are much harder than oral ones, just as essay writing is much more difficult than relating a story to your friends.

And most of all, written explanations should NEVER be required on elementary-level high-stakes standardized tests.

I do ask students to explain their mathematical reasoning because it helps me assess what they actually know, and because clearly explaining their reasoning in writing is a skill worthy of development, and one that students should be developing across disciplines.

That said, I also make the point that writing out “I multiplied 3 times 8 to get 24 feet in 8 yards because there are 3 feet in a yard” is in no way superior than writing: “3 feet per yard x 8 yards = 32 feet.” Some students find thinking symbolically simpler; others don’t. I am relatively indifferent.

However, I do think clarity matters, and explanations can tell you a great about how much a student really understands. Consider these responses to the following question from a preassessment I recently gave my sixth grade students.

When dividing a whole number by a fraction, would you expect the quotient to be greater than or less than the whole number?
Explain your answer. You may use any pictures, numbers and/or words to explain your thinking.

Responses many from my students (many who could accurately compute a quotient on the same assessment) included:

“greater, because dividing a whole # by a frac it is the equivalent of whole x whole”
“what’s quotient? can’t figure out”
“less, a fraction means ‘a part’ also multiplication can mean ‘of’ so when you multiply a whole by a fraction it is part of a whole number”
“less because if you want to separate something you need to take away!”
“less than, I think it would be less than because if you are dividing a whole number by a fraction you would half [sic] to make the whole number a fraction too”
“greater because you are reversing the order of doing it with the whole numbers”
“lower because when you mult a number by one it stays the same number because it is”

Incidentally, not one student chose to draw a picture. Given the cloudy thinking and clumsy communication, my students definitely have some work to do around understanding fraction division, even if they already know how to find the answer.

When did we decide that maths needs to be explained in words? I am quite insistent on my students providing explanations; a call them ‘workings’ and they are the series of mathematical steps that they have followed to arrive at their answer. This is how things are explained in mathematics.

However, for some reason this does not show understanding. In order to understand mathematics, we need to be able to waffle on about it in English. And yet mathematics was invented in order to make it easier to express notions that are cumbersome to express with words. That’s part of the beauty of it.

It is as if we were to insist that the only way to understand German is to translate it into English; that reasoning in German alone does not show an understanding of German. Clearly, translation is a useful device for novice learners and a key component of teaching, but being able to work entirely within the target language is a sign of sophistication rather than of a lack of understanding.

Every year, I teach my senior physicists about wave particle duality. Light, I suggest, can be thought of as a wave or as a particle, depending on the situation. “But what,” they ask, “actually is it?”

“Ah,” I say, “If you really want to understand what light is, you need to understand it through the maths. It doesn’t translate well into English.”

I’m with Greg here. I find, especially in younger students, that some mathematical concepts are understood with absolutely no hope of them explaining it verbally (until older). The language part is necessary for us teachers who want to see the maths translated as one demonstration of understanding and probably a pretty poor one. And at the other end of the spectrum, where language itself it incapable of expressing concepts (Gödel?) you’d be asking an impossible task.

Lisa, if by “crunching numbers” you mean following an algorithm, then I’d partially agree with you but following an algorithm isn’t necessarily blindly following. If students can competently get an answer with zero understanding then the teacher has started from the wrong place or failed to construct any understanding.

This post resonates with something I’ve implemented recently in my class. I’ve started providing the final answers to everything question on exams. I want students to prove to me why the answer is what it is. I want them to be consumed with the question and how to arrive at the answer, not the answer itself. So far, so good. I’ve seen more proof (i.e. work) this year than I ever have.

I feel like a lot of the clarity desired in showing your work comes from using units in the math to promote and communicate understanding of relationships. When I have high school chemistry students unsure about whether to take density divided by mass or times mass or mass divided by density to calculate volume, there is a fundamental problem with their understanding of the relationship between the types of measurements and their units. From grade school upward, instead of multiplying 5 people x 3 crayons/person = 15 crayons, we just accept 5×3=15 as adequate to show their work. Do they know what they are multiplying, or did they just deduce that this problem gave them 2 numbers, and it’s a multiplication worksheet, so they should multiply? Most teachers at least require units on the answers, but writing units within the working math that is put down on the paper helps show students how the units guide their work. When given a novel situation involving a relationship ratio, their understanding of unit relationships, not of rote mathematical operations, will get them to the correct answer. Yet, in my experience (which may be the exception, or maybe kids just forget things), this skill isn’t taught until they are in high school science classes and can’t perform elementary math operations.

Just as a thought experiment, what is

a.) a problem that could be justified via only algebra notation just fine?

b.) a problem where in all cases algebra steps wouldn’t be enough would _require_ words to have a correct justification?

Hi Dan

I think that this is where we differ. You look at expert performance — in this case, the writing of academic mathematics papers – and you infer something about the teaching of high school maths. On the other hand, I am sceptical that novices attain expertise by simply emulating the behaviour of experts. Paul Kirschner describes this as confusing epistemology — how the field of knowledge advances — with pedagogy — how we teach novices what is already well established. Essentially, I see this as fallacious and one of the ideas that drives constructivist educators towards less effective teaching methods.

It is quite clear that experts and novices differ in many significant ways. “How People Learn” — of which I am not a fan — actually contains good descriptions of some of the key experiments that demonstrate these differences. I would personally recommend Chapter 6 from Dan Willingham’s “Why don’t students like school” which addresses the expert/novice confusion issue directly.

In academic maths papers, an author is often setting out to explore new and sometimes contested domains. They will need to explain their reasoning and spend time outlining definitions. Sometimes, new notation and formalisms will be required.

On the other hand, I can think of no standard high school maths problem to which the solution cannot be fully ‘explained’ using well-known, often centuries-old formal steps. A teacher will recognise these steps and be able to deduce whether the reasoning is correct. Different ways of representing such a solution only become necessary if you value students deriving their own suboptimal methods i.e. it is a logical consequence of a discovery learning approach. And discovery learning is ineffective.

Dan, Please state the unsubstantiated claims that I have made about expert performance. Thanks.

I have not claimed that problems are useless. I have simply claimed that the solutions can be written using well-known, formal steps.

For “standard” think of typical textbook questions.

Greg,

There is an interesting philosophy concept called ‘Philosophical Zombies’ which seems to be relevant here: https://en.wikipedia.org/wiki/Philosophical_zombie

I think far too many of us have encountered ‘mathematical zombies’- students who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work- to have a ton of faith in the claim that correct procedures is necessarily equivalent to correct understanding. David Wees gives an excellent example of one such instance above.

Asking for full explanations from students, in addition to accurate calculations, has the double advantage of catching many conceptual misunderstandings early so that they can be fully addressed and encouraging students to consider math as a meaning-full subject as opposed to a long list of procedures to apply in various contexts. Constructivist teaching techniques may be less efficient in producing procedural fluency on a given schedule, but they are often more effective at producing students who have developed a solid conceptual grounding in mathematics.

Ze’ev,

“I thought the purpose of a problem in a classroom is to check whether a student knows sufficient math to solve it, rather than learn bout the nature of human thinking processes.”

I can not disagree with this enough. The purpose of a problem in my classroom is almost always to understand the nature of that human’s thinking processes. This allows for amplification, further investigation into how the student is able to navigate similar problems with subtle variations and complications, and attempts to draw student mental models into internal conflict to create pressure for remediation and revision of said mental models.

I suspect that I am not alone in this view, which is why you see many of us arguing that explaining one’s thinking is important while simultaneously arguing that strict adherence to formal structures often hinders this process instead of helping.

These conversations always bewilder me. They take place waaaay up in the stratosphere of abstraction. They remind me most of dorm room debates about morality and religion. That is, very, very abstract and unconstrained by the rigors of any discipline that might provide structure, rigor or checks on the arguments.

Here’s what I wish for, whenever one of these arguments bust out between traditionalists and progressives.*

1) Can the traditionalist and progressives find a lesson/activity/short video that they both agree is lovely teaching.
2) Same thing, but they both agree it’s lousy teaching.
3) Can each identify a whatever that they like, but their sparring partner doesn’t. Can they explain why.
4) If the disagreement persists, can they explain why they think it does?

* I refuse to use the term educationist. It sounds stupid and doesn’t describe anything.

Katherine Beals , according to the description on her website, has three “left-brain” children (her terminology). She’s written a book on RAISING A LEFT-BRAIN CHILD IN A RIGHT-BRAIN WORLD. I think a perusal of that web site or a look at her book would be useful in ascertaining where her views about mathematics education are coming from.

Many people argue, incorrectly in my view, that student who can compute correctly (or at somewhat higher grade levels, solve algebra and other secondary math problems correctly) do not need to explain anything about their thought process. Further, there is a point of view that it is unfairly punitive to ask or expect these students to put anything in writing about how they arrived at their answers.

But for my money, at the K-5 level especially (but not exclusively), some degree of explanation beyond writing math symbols is necessary for the vast majority of students to gain the sort of mastery. Focusing on only a special group and then expecting that everyone else MUST follow suit is unreasonable. Basing a national approach to mathematics education on such foundations is almost certainly continuing practices that have not served a majority of our students at all well. I don’t disagree that some teachers, particularly those who don’t understand the point of getting students to explain or whose own grasp of mathematics may be such that they don’t feel comfortable explaining their own process, may undermine the explanatory aspects of math pedagogy. And others, well-meaning but also ultimately harmful, may for various reasons wind up being overly-rigid in how they address the issue of explanations with some or all of their students. In this regard, it’s important for teachers to recognize that there are MANY ways to explain mathematics other than purely symbolically or mostly in words. And for students at various developmental levels, demanding/expecting written or oral explanations that are predominantly or exclusively verbal can be inappropriate or unreasonably frustrating for a given student.

Just like using manipulatives, assigning “real-world” problems, involving technology in lessons and assignments, and much else that has been debated in mathematics education over the last 25-30 years, this issue has been unnecessarily obscured by ways in which some people utterly reject the idea of mandatory explanations out of hand, and others apply the idea insensitively and inflexibly.

No amount of reasonable discourse will satisfy everyone. It is, however, necessary to try to offer sound arguments and explanations for this practice. At the same time, teachers and those who train, employ, supervise, coach, or otherwise work with them professionally cannot wait until there is universal agreement on a practice before educating instructors about it and helping them effectively implement it with their students.

I know the “traditionalists” discount the idea of turning to mathematicians to see how they communicate, but I don’t. I find it quite instructive. A few favorites.

Ian Stewart:
“A proof, they tell us, is a finite sequence of logical deductions that begins with either axioms or previously proved results and leads to a conclusion, known as a theorem….This definition of ‘proof’ is all very well, but it is rather like defining a symphony as ‘a sequence of notes of varying pitch and duration, beginning with the first note and ending with the last.’ Something is missing. Moreover, hardly anybody ever writes a proof the way the logic books describe….A proof is a story. It is a story told by mathematicians to mathematicians, expressed in their common language….If a proof is a story, then a memorable proof must tell a ripping yarn….When I can really feel the power of a mathematical storyline, something happens in my mind that I can never forget” (2006, 89-94).

Marcus du Sautoy:
“A successful proof is like a set of signposts that allow all subsequent mathematicians to make the same journey. Readers of the proof will experience the same exciting realization as its author that this path allows them to reach the distant peak. Very often a proof will not seek to dot every i and cross every t, just as a story does not present every detail of a character’s life. It is a description of the journey and not necessarily the re-enactment of every step. The arguments that mathematicians provide as proofs are designed to create a rush in the mind of the reader” (2015).

Paul Lockhart:
“A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light—it should refresh the spirit and illuminate the mind. And it should be /charming/” (2009, 68).

“A proof should be an epiphany from the gods, not a coded message from the Pentagon” (75).

Paul Halmos:
“The best seminar I ever belonged to consisted of Allen Shields and me. We met one afternoon a week for about two hours. We did not prepare for the meetings and we certainly did not lecture at each other. We were interested in similar things, we got along well, and each of us liked to explain his thoughts and found the other a sympathetic and intelligent listener. We would exchange the elementary puzzles we heard during the week, the crazy questions we were asked in class, the half-baked problems that popped into our heads, the vague ideas for solving last week’s problems that occurred to us, the illuminating problems we heard at other seminars—we would shout excitedly, or stare together at the blackboard in bewildered silence—and, whatever we did we both learned a lot from each other during the year the seminar lasted, and we both enjoyed it.” (1985, 72-73).

Ian Stewart:
“When two members of the Arts Faculty argue, they may find it impossible to reach a resolution. When two mathematicians argue—and they do, often in a highly emotional and aggressive way—suddenly one will stop, and say, ‘I’m sorry, you’re quite right, now I see my mistake.’ And they will go off and have lunch together, the best of friends” (2006, 28).

If only this crowd had been there to tell these mathematicians to cease and desist with all this needless talking, arguing, explaining, listening, and charming each other! How much time could have been saved! And how many “needless” ripping yarns, poems, well-written, well-crafted arguments, and rushes in the minds we could have skipped!

Ze’ev,

“I suspect that Gillands employer, and certainly the parents of his students, would also disagree. Some quite strongly. The primary purpose of school is to educate the kids at hand, not to train the teacher. This doesn’t mean that teachers do not learn from experience, but if gaining experience and insight is the primary reason for what the teacher does, he’d better get approval from an IRB and a waiver from each individual student or parent that attend his class.”

Funny thing, that. My employer, my parents, my students, my district, the state evaluator for my school, etc. all support my teaching. Most quite strongly. This might be due to the fact that when most people hear “I work really hard to understand your child’s thought processes so that I can better guide their thinking and draw out subtleties and conflicting mental models,” they don’t think “Oh my God, that man is performing experiments on my child to improve his educational practice.” Instead, they think “Oh my God, that man really cares about what is going on inside my child’s head and is attempting to tailor instruction to what he finds there. Thank goodness he isn’t stuck with a teacher who believes that teaching is just lectures interspersed with quizzes to determine if my child gets it or needs to be droned at more with another utterly useless generic explanation that takes no account of what my particular child is thinking!”

Honestly, if you weren’t such a well known advocate for traditionalist instruction, I would assume that you were trolling me with this comment, as it displays either an utter failure at reading comprehension, or a self-parodying failure to incorporate even basic pedagogical techniques (such as asking questions to investigate what a student is thinking). If I was attempting to straw man traditionalist instruction, I would trot out this line, but I would worry that it was a little over-the-top to be believable.

I can think of some cases where the symbols alone don’t give sufficient insight into the thinking that went into forming them.
Eg. Reproduction of the symbols does not result in the child in question actually being able to connect the mathematical ideas later.

I was working with a pair of children and I learned from my work with them that they understood that there are ten thousandths in a hundredth and ten hundredths in a tenth. They were able to say that 1.8 and 1.80 are equal because 8 tenths and 80 one hundredths are equal. They knew that 1.207 is less than 1.40 because 207 thousandths is smaller than 400 thousandths. They seemed to have a very clear understanding of decimals at least up until the thousandths position.

I then asked the child how much money $1.207 is worth. They told me that 207 is two hundred and seven cents and that 100 cents make up a dollar and that means $1.207 is the same as three dollars and seven cents. I asked them to write that down, which resulted in the statement $1.207 = $3.07.

If I don’t push for at least the occasional explanation, then this way of thinking doesn’t emerge, and this child is likely to wander off with an incomplete understanding of decimals.

As for statements like this,

“There is nothing wrong with having a student occasionally explain his thinking when s/he is called on in the class. There is a lot of wrong with having students explain their thinking in a voluminous verbal/written manner on every test item and every home assignment.”

This is an appeal to extremes. Of course we don’t want kids write volumes of text for every assignment they do but from this you are concluding that only occasional writing or verbal explanation is acceptable?

As for discovery learning being ineffective, this is actually another example of taking an extreme position and using it to argue against more moderate positions.

Kirschner, Sweller, and Clark (2006) reported that there is little evidence to support pure discovery learning, and then grouped together a bunch of things that no pure discovery ideologue would consider pure discovery learning and said that none of these approaches work. Here’s a critique of their paper which took 5 minutes to find and actually cites some studies that suggest that constructivist approaches are not as terrible as is claimed repeatedly.

I don’t advocate for pure discovery learning. I’m in agreement that it is probably not effective at meeting the goal of children learning specific facts and skills. I argue for a more moderate position that there are portions of teaching (or teaching practices) in which mathematical ideas should be made explicit to children but that much more attention needs to be paid to the internal models children actually develop as they understand mathematics. It fascinates me that cognitive load theorists offer schema as a way that knowledge is stored and retrieved but seem to pay no attention at all to what those schema actually are and how they interact with each other.

I ask my students to tell me the story of how they solved the problem. This prompt invites reflection and detail. I want them to explain their thinking, show me their understanding, and tell me how they solved it. They can use words, symbols, pictures, tables, poetry, videos, audio clips, the ShowMe app. It takes effort and practice, but the result is worth it. I get to know my students in a different way. I know how they communicate and I can push them deeper into mathematical communication and understanding. I learn who they are as learners which helps me in my planning. “Tell me the story of how you solved” is a powerful assessment tool. Try it!

I don’t ask kids to explain their work. Too much hassle and most kids confuse “I put down words” with “I explained”.

“I will argue that if you arrive to your destinations time after time without ever being lost it may not tell people much much about the city or its traffic patterns, but it does clearly tell people that know your way around the city. ”

Well, no. It used to. But then you start running into people who, when given a destination, find a map and painstakingly commit the specific route to memory. Then, once they’ve found their destination, they immediately forget that route.

This used to be an absurd thing to do, and most people would never take the time and trouble, so it was a safe assumption that everyone who got to their destination knew their way around the city. No more.

And “math zombies” is a great term.

Examples I see constantly:

1) Kid can find a derivative using the chain rule flawlessly, but has no idea why the derivative of a line is a horizontal line, or why the derivative of a quadratic is a line. None. Zip. Zilch. They don’t even understand the question. “Well, what line? Give me a line, I’ll find the derivative.”

2) Kid can graph all trig functions from memory but has no idea why the tangent function has an asymptote, much less why the tangent and cotangent functions have asymptotes at different values. “Because that’s what the graph does.”

3) Kid is given a graphed quadratic with zeros clearly identified, vertex specifically on a point (but not identified) and a (stretch) clearly visible and, asked to put it in vertex form. Uses the zeros, sets a=1, distributes, uses standard form to complete the square. Gets it wrong because, of course, he set a=1. But also, in map terms, reveals he only knows how to get to the London Bridge using one specific path, incorrectly, rather than the obvious short cut that people who “know their way around the city” would instantly take. When told of his error, is upset because he did the best he could “without knowing a”.

4) Kid working coat problem has no idea what 100% represents, much less 80%. Doesn’t know why he goes from 80% to .8 other than “that’s what I was told to do”. Can’t explain why he doesn’t just increase 160 by 20% or why that’s the wrong thing to do. And, given a problem in another form, might just do the equivalent of increase 60 by 20% because he doesn’t understand why it wouldn’t work.

Like I said, I don’t ask kids to explain their work. But I do ask test questions that ferret out zombies. And more than one zombie has failed my class despite a perfect record of As from teachers who consider accurately worked math a proxy for understanding.

It should be. I understand this. But it often isn’t these days.

And I agree with Michael Pershan; I really think most traditionalists would not be able to find fault with all but the most determinedly reform class.

” I wonder how you would explain in words, when you are 6, why 1 plus 1 equals 2? What is a satisfactory answer and what isn’t?”

First, I am skeptical of the “explain your thinking” approach.

That said, it’s pretty obvious that the questions are designed to get kids to understand the number line.

So a simple, correct answer for “explain your thinking” for 1+1=2 is the number line, with 1 marked, 1 added, and 2 identified. Or the words “because 2 is exactly 1 away from 1 on the numberline.”

Likewise, the weird grouping stuff is an attempt to get kids to understand place value.

The underlying objective is to focus all math around the numberline, and to instill in all kids an understanding of the meaning of place value. This is something math professors have moaned about for decades, that kids don’t understand what “=” means and that they don’t grok place value.

I’m not saying I agree with it. And certainly, if that’s what they are teaching, then it’s easy enough to test without open-ended “explain your thinking”. Or maybe it isn’t, because zombies.

I dunno. Not a fan. But it’s not hard to understand the objective.

Most of my experience comes from dealing with university students, but I have worked both with math majors and students in Quantitative Literacy (an alternative to college algebra), both in groups of students 20-30, where I feel I can see individuals as well as the group. In both cases I see issues with the “just get the right answer approach” from school. For the strong high school students (which makes up most of the math major community) their use of formal symbols in their working is itself very poor, and does not respect the meaning of those symbols. A classic example is using the = sign to mean “and the next step is”. Worse when I need to break the rules they have learnt (and every mathematical rule can get broken for some system) the result is confusion and resentment, rather than the joy at new understanding (and breaking rules) that students who go beyond the methods show.

For the weaker students many do not seem to have arrived at the first step. Many years of math instruction have left them with some ability to move symbols around but no consistency in the meaning from line to line. This is understandable as they don’t seem to feel that mathematical concepts have meaning.

A competing issue is the fact that the goal of mathematics is to process ideas reliably without constantly checking the meaning. As a result a computer can do mathematics. This is the wonder of the subject, but does create problems!

I do believe that a lot of understanding does come from mechanical manipulation, and that fluency there is a very important goal. Yet if it is the only goal then only a lucky few will gain understanding that they are also doing more.

Metacognition! Thank you, Tim Hartman and David Coffey. I can’t imagine anyone not seeing the value of that, but you never know.

Let’s look at other countries that are outperforming us in math. Are they requiring the sort of cumbersome, verbose explanations described in the article? No: while American students are busy writing up verbal explanations and diagramming their thought processes and belaboring multiple solutions with problems that are comparatively easy in terms of actual math, students in other countries are moving ahead with conceptually challenging mathematics. These students–from Finland to Japan–are doing the sorts of problems that you can’t do unless you understand the underlying math. Compare the level of actual mathematics required in the mostly multiple choice test questions on our Common Core inspired high school exit exams (many of which, yes, can be done without deep mathematical understanding) with the sorts of problems that high school seniors are tested on in those countries that outperform us in math.

Here are is a link to a page with sample problems from the National Matriculation Examination of Finland: http://oilf.blogspot.com/2014/… . And here is a link to a page with sample
problems from the Chinese Gao Kao: http://oilf.blogspot.com/2015/…

The American approach is to build conceptual understanding
through time-consuming student-centered discovery of multiple solutions and explanations of relatively simple problems. An internationally more successful approach is to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems.

It would be interesting to see how American students would do on the Finnish and Chinese exams as compared with their Finnish and Chinese peers.

Katherine,

As you point out in your discussion of the Finland model (http://oilf.blogspot.com/2014/10/the-finnish-fallacy-drawing-wrong.html), people often see what they want to see when they evaluate other countries to see what they do correctly. You apparently feel so confident in your analysis of what Finland does right that you are willing to overrule the former director of the Finnish Ministry of Education and Culture.

This is a problem for both sides of these debates, and comes from failing to analyze the data set as a whole and instead simple trying to cherry pick conclusions from those who do well. This introduces survivorship bias and allows us to project whatever positive traits we wish to emphasize without acknowledging all of the counterexamples extant in the original data set.

Also, while I would be interested to see American education move in this direction, this is still far from being an accurate description of most of American mathematics education:

‘The American approach is to build conceptual understanding
through time-consuming student-centered discovery of multiple solutions and explanations of relatively simple problems.’

This is not true.

There are too many elementary teachers who have no solid conceptual grasp of mathematics to actually lead exploratory lessons. Most still teach lessons via lecture and using only one technique, because that is the only one they understand/trust to get reliable results. And most HS teachers (and college teachers) are still massively lecture based with large problem sets, because that is how they were taught and what they know.

I know that we all like to pretend that progressives have taken over mathematics education in the united states- progressives because it makes us feel less hopeless and traditionalists because it gives them something to rail against), but it just isn’t true and has never really been true. So comparing America’s ‘constructivist’ mathematics education to the systems in other countries is about as effective as comparing American unicorns to other countries’ horses.

At a recent PD for grade 3 teachers, I lamented the fact that PARCC assessments do not have a means for students to convey understanding other than using words and clumsy equation editors. This has the unintended consequence of pushing teachers towards delegating increasing amounts of class time to helping students to develop written explanations. It is difficult as a district, to communicate the significance of encouraging various representations of understanding and process, when the high stakes test that is the basket into which all of the eggs have been placed, is getting it all wrong in such a glaringly obvious way.

” However, both these quotes seem to suggest a ‘they are bad at it so I don’t bother teaching it’ approach that seems a bit backwards.”

It’s pretty clear that I do bother teaching the “why”, in my previous comment. However, I don’t say anything like “show your thinking”. What I do instead is ask questions that the kids will only understand if they understand the concepts.

Brett is completely correct in the last two paragraphs of #49. Reform math, as it’s called, is very popular in elementary school because teachers prefer it. But it’s made no headway in high school, or does for only a limited time, because high school teachers mostly despise it. Overwhelmingly, math teachers in high school teach exactly the way Barry and Katherine want it done–problem sets and lectures. And they lose millions of kids completely, while giving As to a lot of kids who don’t understand a thing.

I’m not a fan of reform. I know kids who can do math in their heads, understanding concepts perfectly, and I’m fine with that. I don’t much care for discovery. But I’ve spent too much time getting squicked out by automatons who don’t understand a thing.

And no thoughtful, reflective mathematics educator advocates the sort of rigid application of “explain your thinking” to every problem, particularly those which are trivial for students at a particular developmental and grade level.

IF you’re teaching a classroom of students where you have adequate evidence that for those students the problems you’re asking for explanations on are trivial. If not, then good teachers are obligated to try to probe where students are having difficulties, confusion, etc. And one way that proves effective is asking for explanations.

But of course, there can be difficulties brought on by “The System.” If I want information that allows me to do formative assessment – to the benefit of my students and my own ability to more effectively teach them – I need both the time to write the individual comments that most experts in formative assessment believe are essential, and the sort of school culture that allows non-graded formative assessments to be taken seriously by all stakeholders: students, parents, administrators, and faculty. Without that, formative assessment doesn’t work. So then I am forced to GRADE the explanations students offer – can’t get the kids to take them seriously if they’re not graded in that case. And then I have very upset students and parents who don’t see why a kid who gets that “right answer” should have to explain anything at all.

There are many excellent reasons for teachers to ask for students to explain their thinking, and many benefits for students who learn how to reflect on their own problem-solving strategies and practices. But none of this will work in places where everything has to be graded and that which isn’t graded is trivialized. When numerical/letter grades are king, real learning is kicked to the curb, along with meaningful assessment.

We can focus solely on those teachers who mechanically demand explanations on everything. We can focus on the Common Core and high stakes testing which drive many teachers to do things mechanically because they’ve been told – directly and indirectly – that their professional judgment doesn’t matter. We can focus on publishers and textbook authors who reinforce some of the worst tendencies of teachers prone to teach mechanically.

Or we can look at the larger culture of “school” as an institution in which learning is the booby prize instead of the ONLY prize. Where getting ‘right answers’ suffices for earning the Mickey Mouse stamps of grades, and where parents – and hence students – care about nothing else. Everything is a means to an end, but there’s no actual end, and the process is utterly devalued and trivialized. And then we may well conclude that we get what we ask for from our schools, though it might just not be what we or our children actually need.

To assume that the only students who are incapable adequately explaining their answers must have Aspergers is quite naive. I think that you may not realize that these explanations are required of children who are not developmentally ready to explain their answers. You are asking students in the Concrete Operational stage (7 to 11) to
1)Have counter factual thinking by explaining why an incorrect calculation has been observed and what can be done to fix it.
2)Have abstract logic and reasoning where they need to explain more complicated topics on a level appropriate for an older child.

These are all things that the majority of the students will not be able to do until they become 12. These are concrete thinkers here. Depending on the question, asking them to explain their answer may be hindered by their development. Its a shame to penalize a student because they are on track developmentally.

Remember, children are not small adults.

@Michael and @Brett, thank you for your thoughtful comments. Great contributions.

I am constantly amazed by the amount of stress, angst, ink, and fury over reforms that, by and large, HAVE NOT HAPPENED.

Katherine,

It is interesting to me that you chose to double down on your initial argument without actually addressing the critiques given. Pointing to the top 20 performing countries doesn’t actually support a given system unless you demonstrate that this system is distinct from the lower scoring nations. Moreover, it would help if you demonstrated that those countries with the best scores were more in line with this system than those who did worse. So, for instance, it would help to know if Sweden’s mathematical education system was significantly different than Finland’s. Since they are culturally quite similar, that might eliminate some confounding variables. Without this, you are cherry picking and not establishing significant impact from differences in pedagogical methods.

Similarly, though it has been pointed out that NCTM/CCSS methods (as I take you to be using the phrase) are incredibly rare in USA schools, you simply reassert that this is what you mean by “an American Approach”. The problem here is that this is not actually the approach taken in most American schools. So that can be what you mean by an American approach. However, no statistical analysis of American success (on the PISA, for example) will actually be available, because this isn’t actually the approach to mathematics education that is common in American schools! Both of the above are fairly basic ideas from statistics and research design methodology. Please account for them.

The Canadian study, I am more interested in, since we seem to at least have some attempt at quality research design. However, it will take some time to dig through it and I am quite frankly quite suspicious of research from clearly biased sources which has not yet been vetted by the larger research community, especially when it comes on the heels of such shoddy PISA based analysis. Not saying that it isn’t valid. Not saying it is. Just stating my priors.

@Katherine Beals wrote in part: “What are your sources for “NCTM/CCSS methods” being “incredibly rare” in USA schools? (Keep in mind that NCTM-inspired curricula include Everyday Math and Investigations).”

Keep in mind that textbooks are not teaching methods. I’ve seen many teachers using the textbooks you mention, as well as philosophically-similar books in middle- and high schools who barely follow the approaches suggested in the teachers’ manuals, for one thing. I’ve seen teachers where such textbooks are the official curricular materials put the books away and eschew their use. I’ve seen teachers teach from those texts in ways that would be anathema to their authors, turning what we can call “NCTM-style” mathematics teaching into traditional teaching with ostensibly progressive reform curricula.

In brief, teachers who do not accept teacher-centered instruction, discovery learning, what is rather oddly called “constructivist teaching” (odd because no matter what gets said, constructivism is a theory of learning; none of its theorists ever wrote a constructivist theory of teaching), etc., can easily stay light years away from it regardless of the official textbooks through any number of ways and do so routinely.

Add to that the teachers who may be neutral or even positive about such books and, ostensibly, the teaching methods espoused by their authors who either don’t understand how to effectively implement the methods or who find that it simply doesn’t suit them to do so (and hence they quickly revert to the traditional approaches with which they were taught), and it’s quite easy to explain how Brett, I, Jim Hiebert, or anyone else who might be looking at classrooms in various parts of the country could assert that what I’ll lump into the term “progressive math teaching” is rare in this country, more’s the pity.

I’m sorry, but the 1989 NCTM Standards never called for what you describe, though many of its most vocal critics claimed that it demanded an end to paper-and-pencil arithmetic, to “standard” algorithms, etc., and also that math must be discovered by each individual student.

No matter how often those claims are debunked, they rise again like the undead in a horror move to stalk living and stoke their fears. The facts, however, are quite different. The words that appeared throughout that volume were ‘less emphasis’ on X, ‘more emphasis’ on Y. That’s a pretty mild request (and that’s all NCTM could ever do: request shifts in emphasis).

What individual teachers, districts, states, publishers, and textbook authors chose to do with those suggestions, how they interpreted and tried to implement them, was a highly diverse enterprise. But since I first arrived at the U of Michigan as a graduate student in mathematics education in 1992, I’ve read so much from people who claim that NCTM called for eliminating everything and anything “traditional” from US mathematics classrooms (and how similar organizations did the same in other countries, including England, Canada, and Australia). Never mind that the people I’ve studied under, worked with, gotten to know through their writing online, listened to at professional conferences, etc., seem pretty consistent in making modest suggestions and calling for shifts, not revolutions, in how teaching is done and classroom time is spent (and I’m speaking about people from many countries besides the United States): the claims just will not cease that supporters of progressive mathematics education are all mad professors experimenting on innocent children, always wrongheadedly and to the utter detriment of the nation’s and world’s youngsters.

How, exactly, so many thoughtful, highly educated, dedicated professionals manage to get EVERYTHING wrong is never explained. It seems to suffice to make the claim that our mathematics classrooms are in a shambles, all due to progressive reform ideas and thinkers, while in those “other countries,” all is marvelous, all kids are above average, all teachers teach just the way that those who oppose NCTM, et al. were taught once upon a time in the good old days.

Well, maybe that’s true. But it pretty well defies credulity, as well as runs counter to my experience over the last quarter century. I think it runs counter to the experience of a lot of very smart, creative, thoughtful mathematics teachers who contribute to this blog and other practitioner blogs I’ve been following for going on a decade.

So you must excuse me if I read the description of Australian math education with some skepticism. It’s too one-sided, too different from the work Australian colleagues over my career have provided, too much like similarly slanted descriptions of US classrooms I’ve read that go far afield from my own observations in schools throughout southeastern Michigan and in New York City. As the cheap headline writers are wont to put it: It Doesn’t Add Up.

Katherine,

I want to address your last comment with the care it deserves so I am breaking this into several responses to draw out different threads in your last post.

“Well-controlled experiments on this topic are few and far between. The Canadian study is one of them. What makes you think the source is “clearly biased”?”

Initially, the red flag was that it was only really addressed on clearly biased sites and had very few further citations from other published paper. That is a big red flag. Now that I have had more time to look over what data I can find (and looked at the earlier draft of their paper), I find their discussion of the PISA to be particularly problematic. To hear them tell it, Quebec either held steady (which would not support their findings) or declined. However, when you look at the data (http://cmec.ca/Publications/Lists/Publications/Attachments/318/PISA2012_CanadianReport_EN_Web.pdf), specifically Table 1.6 in the link, you notice that PISA scores grew pretty steadily (though by small amounts) over that time, while other provinces declined. That does not build confidence in their case.

In fact, the fact that they fail to consider the performance of other Canadian provinces over the same time is a warning signal on its own. The lack of a control comparison when one is available suggests an unwillingness (or at least a lack of interest) to account for possible historical threats to validity.

This might explain the fact that while many Canadians readily acknowledge a poor roll out of the change in curriculum (http://www.cea-ace.ca/education-canada/article/15-years-after-quebec-education-reform-critical-reflections), no one in Canada really questions whether it was effective. More recent results have supported that claim. (http://www.theglobeandmail.com/news/national/education/quebec-students-place-sixth-in-international-math-rankings/article15815420/). And by 2006, people weren’t questioning what Quebec was thinking, but instead were trying to figure out why their provinces had failed so miserably to follow their lead (https://www.umanitoba.ca/publications/cjeap/pdf_files/raptis.pdf)

“In the absence of controlled studies, we can compare curricula, high school exit exams, PISA results (I said “20 countries” earlier when I should have said “about 30 countries”), and the relative representation of American vs. foreign students in math/science BA, BS, MA, MS and PhD programs and in jobs that require high-level (or even non-trivial) math skills. We can interview foreign exchange students about how much they learned in their American math classes vs. their math classes in their home countries. We can spend time in foreign classrooms or read eye witness accounts. In my experience, all of this points in a troubling direction. Widespread, systematic studies and controlled experiments, of course, would be nice.”

We can do all of these things, but we must do so in a way that is rigorous and with an eye to challenging our biases, not to confirming them. Given that my experience with all of these considerations point in the opposite direction from yours, I suspect we may be at a stalemate of priors. And while I hate to beat this into the ground, half (or more) of your comparisons only hold if you can show that American mathematics education is significantly progressive at any level. I will take that question up again in my next comment.

@Tempe: It is impossible for me, at least, to meaningfully engage in that conversation if you’re going to insist upon such an extremely one-sided take on what you describe. I don’t think putting things as you do is conducive to exploring. You have a strongly-held point of view (all this stuff is bad) and you want agreement from me and others. You have a villain (university teacher educators). You appear to have a hero (those who support the good old days).

That’s not how things look to me from inside of classrooms, particularly not in the low-income, high-needs schools and districts where I coach secondary teachers. I’ve only been to Australia twice, and was not in a position to visit schools on those trips. But I’ve known mathematics educators from Australia. They were without exception a bright, hard-working, insightful bunch of people. I believe they would take exception with how you’re characterizing things there.

But all that notwithstanding, if you’re after a productive conversation, I can’t offer you one given the stance you’re taking. Maybe others can do better.

(Continuing my conversation with Katherine)

“What are your sources for “NCTM/CCSS methods” being “incredibly rare” in USA schools? (Keep in mind that NCTM-inspired curricula include Everyday Math and Investigations).”

I really don’t mean to be rude, but I have a question. Have you ever worked in K-12 mathematics education? I don’t mean that sarcastically or as a rhetorical question. I just noticed that we seem to be talking past each other from time to time and it seems to be linked to either a lack of exposure to American mathematics classrooms or an exposure to classrooms that are wildly different from those I have experienced in the three states in which I have worked.

Thus, the short answer to your question is “over a decade of experience teaching and collaborating with educators in three different states (and collaborating with others who work as trainers in many more states than that).

The long version… well, Michael addressed a lot of that above. Assessing what is actually happening in the mathematics classrooms of America by the adopted curriculum would be as silly as trying to assess schools of education by simply looking at their syllabi. No one would ever take such a process seriously. (And yes, my tongue is firmly in cheek right now.) If you aren’t going into mathematics classrooms on a regular basis, preferably unannounced and with no possibility of it being taken as evaluation, then you have no basis on which to ground your claims.

I should add that Michael’s explanation covers a large amount of what I see at the HS and MS levels. But there is another factor that massively affects mathematics education at the elementary level- mathematical incompetence. The simple truth is that many, if not most, elementary school teachers are very open about their weakness in mathematics. As such, they tend toward an algorithmic approach to mathematics instruction and are unable to actually implement any curriculum particularly well. However, that does make them particularly unable to teach reform curriculum. Now, this may leave you convinced that reform curriculum should just be abandoned. I know others at both elementary and HS levels that were very successful teaching reformed mathematics who maintained that it shouldn’t be implemented broadly for exactly this reason. I respect that opinion, but believe that the better response is math specialists at all levels that possess a deep and nuanced understanding of mathematics.

“I agree that high school math is still generally more “traditional”; the problem here as that the mathematical material taught in U.S. high schools (and found in U.S. high school math texts) is, from everything I’ve seen, read, and heard, significantly below that found in most other developed countries, both in terms of mathematical challenge, and in terms of mathematical depth. It’s what Barry calls “traditional math done badly.””

The (very small) data set I have to work from doesn’t support this claim. However, I am quite aware of how limited my knowledge of foreign mathematics education is and won’t try to extend that to an argument that this can’t be true. That said, your argument from test difficulty really doesn’t hold water, as I will address in my next comment.

Hi,

I recently responded to both Dan’s post and the original article with a piece of my student’s thinking.

Would those who think only arriving at the “right” answer conveys mathematical proficiency be satisfied with this student’s understanding of place value?

https://thelearningkaleidoscope.wordpress.com/2015/11/13/is-explaining-your-mathematical-thinking-really-necessary/

On reading Brett’s response, I remembered belatedly that the folks here talk teaching, not policy. I’m a policy nut, and Katharine made a policy case, so I responded on those terms. I’m not going to pursue this save to respond to Greg’s criticism.

I don’t think it’s possible to “attack experience”. You can attack validity of experience, conclusions from experience, or restriction of experience.

Barry is, from everything I can see, an expert in elementary math pedagogy. He’s been reading and thinking about it for a long time, and has worked with other experts. I think Barry would readily agree that he’s not an experienced high school math teacher. It’s a shame he wasn’t given the opportunity to expand his experience, most likely due to age discrimination, which is a real problem in school hiring practices. I found his memoir credible. It also revealed good teaching instincts. He described his method well, and also revealed that the results weren’t as effective as he expected (and in that, he’s welcome to a very large club!). I should also make it clear that I often agree with Barry’s reasoning, I just think his conclusions (that we’d get better results) are unfounded. But if Jo Boaler were to show up in the comments thread, I suspect I’d side much more with Barry than Jo.

Greg says arguing from experience is fallacious. In teaching, given the lack of data, experience is what we have. Rest assured, convincing *teachers* to adopt a desired policy requires experience or data that is a) demographic specific and b) leads to an improvement so significant that the effort to change is worth it.

When Katharine and Barry (and the other adherents) argue that forcing first graders to explain their thinking is inappropriate given the complex nature of the underlying math, they are on reasonably solid ground. They have a case supported by both data and practice.

When they moved to the middle school example and percentages, their argument lost both. First, high school math teachers typically think students lack all conceptual understanding of percentages. Saying “they need procedural fluency over conceptual understanding” is far less compelling. At the college level, too, math professors complain that remedial and lower level math students lack all conceptual understanding of percentages. Second, the argument lacks any research supporting the advantages of procedural fluency.

Finally, K&B ask: “Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers–from multi-digit arithmetic through to multi-variable calculus–doesn’t understand the underlying math?”

and on that point, clearly, experience matters and the answer is, from many high school teachers, yes, it can be the case. Dan’s opening post raised that possibility, and many teachers chimed in with knowledge of “zombies”.

Arguing from experience isn’t fallacious in teaching, although there are of course limitations. It’s particularly appropriate when the other side is trying to argue from logic and clearly lack experience.

Katharine then makes three unsupported assertions, comparing us time and again to other countries. As no other country has America’s considerable heterogeneity in income, race, and ability, I consider all comparisons of this nature utterly pointless.

But China, in particular, is egregious. Here I didn’t argue from experience, but data. China’s corrupt academic practices and culture are well-established, ,and indeed shared throughout most of Asia. We have no idea what is typical for a Chinese student, not even a suburban kid from Beijing, because we don’t know the degree to which the tests are made more difficult as an attempt to thwart cheating. It goes without saying, but I’ll say so anyway, that China’s corruption doesn’t lead to the conclusion that all Chinese people cheat.

On an experience basis I, too, have extensive experience working with Asians from the mid, south, and east of the continent. As America has a much more permissive immigration policy than Australia, I’ve probably seen a much broader range of ability:

a) outstanding procedural and conceptual fluency
b) excellent procedural fluency, acceptable conceptual fluency within limited parameters.
c) rote zombies
d) utter incompetence, unable even to reach zombie status.

b is the mode. Next c, then a, then d.

Americans (regardless of race) don’t fall into those categories as neatly. Many of our students are incredibly uncomfortable with c, and less determined to reach b. So we get a and d, and then a lot of flailers.

@Brett

I don’t think the answer to 6 is 0. It ought to be in terms of the a’s. Both x and any a_i can be negative, meaning x is merely a translation any direction along the number line before you sum squares. There’s no reason to believe the optimal translation is 0. If a_1 is 0 and a_2 is 1, x=-0.5. I like this problem. It’s clear that it has an answer but not clear what that answer is.

Lola. You are correct. I am just going to hang my head in shame and blame it on fatigue. Now I just want to play with the problem and see what comes out.

Brett,

I appreciate your discussion of the Finnish problems. Can you share with us some problems from the PARCC/SBAC that you consider to be of similar mathematical difficulty?

And can you share with us some problems from NCTM/CCSS-informed textbooks that you consider to be of similar mathematical difficulty?

(FWIW, when I talk about traditional American textbooks, I’m referring to textbooks that date back to the 1960s and earlier, which contain more proofs and more conceptually challenging math than most contemporary American textbooks, and more closely resemble textbooks still used in other developed countries.)

I understand your concern here, Michael. However, I also believe that textbooks really do shape the world we work in. One of the reasons that I am really intrigued by the work of the MTBOS as of late is that they are starting to formalize and piece together curriculums instead of just offering modules or questions. I have great respect for those that generate their own curriculum from scratch. I used to do it myself. However, I also don’t believe that this can be the basis for high quality mathematics education at a national or even state level. This is primarily because it falls afoul of the “reforms that ask for teachers to work harder are doomed to failure” maxim. Maintaining work/life balance and high quality classroom interactions (along with grading and various other requirements of the job) is tricky enough without adding “autonomous curriculum development” to the list.

I can’t wait for Dan, Christopher and the rest of the team at Desmos to release the first great textbook of the new century. Until then, I content myself with another teacher led effort from the last one, a program that I KNOW makes it easier to be a good teacher instead of making it harder.

So, Katherine, I will bite. I will pick a few at random from my textbook’s homework help section (help for every assigned problem, scaffolded heavily at first and with reduced support in later lessons when the concept has been addressed a few times). I should also add that I am not picking for problems I love in this instance, but looking for problems that seem to meet the parameters of what you consider high quality critical thinking questions. I am using the HW help section because it is freely available to all.

First section chosen at random from Algebra 2 included a gem in problem 4-26 (http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CCA2/chapter/Ch4/lesson/4.1.2) Click the problem number on the left.

I liked all of the questions in this assignment from the first Geometry section I chose. (http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CCG/chapter/Ch7/lesson/7.3.1) The expected value work in 7-121 is nice because it forces kids to come at the problem from multiple angles. 7-121c is the sort of thing I think you like, in partiular. The proof work on 7-122 is typical, but do note that most requests for proofs in our text include the possibility of a proof not being possible, in which case we explain why. 7-125 is interesting because it causes students to brainstorm about what is necessary to describe a shape and what is sufficient. Class conversation about that one the next day should be excellent. The question is almost certainly is a preview of the next lesson.

First randomly chosen set from Algebra 1 (http://homework.cpm.org/cpm-homework/homework/category/CC/textbook/CCA/chapter/Ch3/lesson/3.2.4) 3-71 is a nice problem that mixes multiple representations to generate a rule for an arithmetic sequence. Atypical in that it requires connecting the representations to make sense of the problem. 3-74 is a straightforward question about closure of sets under a given operation, but seems like the sort of thing you would like for its ‘proofy’ aspects.

I can go on, but honestly you can go look for yourself. Our adopted curriculum is saturated with stuff like this. It is also not very widely adopted because kids freak out when they encounter problems like this without being specifically told how to work them (your complaint, as well) and so they scream bloody murder and turn to articles like yours to trash progressive textbooks and get them thrown out of their districts. The irony is hopefully not lost on you.

PARCC Algebra 2. PBA Test Items from #13 on all seem to hit what you are looking for. I think you would like #14 in particular. http://parcc.pearson.com/resources/practice-tests/math/algebra-2/pba/PC194854-001_AlgIIOPTB_PT.pdf
Algebra 2 EOY Item #1 is a nice look at polynomial divisibility (or it can turn into a slog of guess and check. # 2 is, IMO, harder than the polynomial expressions question on the Finnish test, as it requires the student to deal with extraneous solutions. #7 is some pretty nice conceptual work on exponents. Etc.

I should reiterate here that I am very mixed on the PARCC test. The sample questions last year had issues with combining multiple concepts within one question, making the data gained from such tests problematic. In addition, switching it from paper based to computerized is a nightmare. That isn’t a natural medium for much of what our students do and made explanations (especially symbolic explanations) damned near impossible. I also have concerns with what level of proficiency should be expected for HS graduation, but that would be true of any standardized test.

The examples above were just to give you some evidence for the claim that US kids weren’t being deprived of these sorts of questions. Nor, if they use halfway decent textbooks (of which there are far too few and too sparsely distributed), are they deprived of them in their daily math work.

While understanding may be the ultimate goal, it is confusing the issue because communication is the immediate goal (not explaining or justifying or giving reasons either). The language of mathematics is often arithmetic mixed with algebra. I have been emphasizing (to my M2 HS students) that it is always the student’s job to communicate their solution (to formative assessments) to me. When it is tough, like with systems, it is still their job. The teacher must be demanding. For example (with linears): Using y=mx+b without stating it is poor communication. Not explaining or showing why m=4 (for example) or why b=2. Never mentioning any math words like slope, or y-int, or delta y, or delta x, or making a poor graph or a sloppy incomplete table. I call this ‘poor communication’ (and deduct as much as 25 percent) and the students are getting it. I like what they are showing me and I think they are enjoying the opportunity to treat their work as a conversation with me.

This last turn in the conversation is interesting, because it doesn’t actually seem to have anything to do with the original debate. At least not to my reading.

I would love it if we could get students to the level of mathematical ability that that Finnish exam seems to require. Of course, the link stated that they only need to complete 10 of the 15 questions. I’m going to estimate that if a student can correctly answer, say, 7 of those, then they will pass. This is not a low bar, but it’s not as high as might be implied. Also, according to the Wikipedia entry on Education in Finland, only 42% of the population, approximately, completes that matriculation examination – 50% of the population never takes the academic track of the last two years of high school at all, and this is in a population that by and large is not having quite as many cultural battles about the value and importance of education as we are having. This in a country whose median net household worth is around $120,000 (110000 euros) compared to $69,000 in the U.S, despite a 60% (on average) tax burden.

Still and obviously, America is not even doing well enough to get 42% of the population, or even 42% of the non-poverty-stricken population, to that level, but blaming that on any sort of traditionalist vs reform curricular choice seems a bit bizarre to me.

First of all, a point of historical argument: in the 1960s, textbooks had harder problems in them because the vast majority of Americans never came close to completing high school, so those books were never seen. In 1960, around 45% of urban Americans and 32% of rural Americans graduated high school. By 1970, it had gone up about 10 percentage points in each region. My father, who has worked as an outrageously successful medical doctor for 40 years, graduated in 1970 at the top of his high school class and earned a full scholarship to college and admission to a top tier medical school having never taken a calculus course, so he DEFINITELY would not have been able to pass the Finnish exit exam based on his exposure to those halcyon days. To pretend the American textbooks and education of that era are superior to now is simply to ignore the fact that very few students ever used them, and those that did were a select, privileged, and talented few.

Interestingly, I teach at a high-end independent school, and the select, privileged, and talented are my purview. I truly believe that there is not as much mathematical understanding or knowledge in their heads as their should be, but I certainly can’t blame any sort of reform curriculum for that: my students pretty much entirely come through traditional-as-possible curricula, filled with memorization, mnemonics, and mathematical mimicry. Our AP Calculus students – AB and BC both – do 30+ exercises a night, exactly in the AP format, memorize every possible way to integrate or derive, and generally ace the hell out of the AP Exams, which may not be quite difficult enough for your taste but are certainly as difficult as we get in any standardized way. That Finnish exam would eat them for breakfast, though, because you are right that they don’t really understand math in that way and, perhaps more importantly, they’ve never seen an exam like that.

But here’s the thing: there are two reasons Finnish students might be passing that test. Option one: they have seen problems similar to those enough that they can reproduce them. This is traditional mathematics education in its strength, and is how the students at my school so regularly dominate the AP Calculus exam. Option two: they are solid mathematical thinkers who can approach problems they don’t know how to solve without freaking the hell out, calmly apply knowledge they know is in there somewhere, and come up with a solution to a difficult problem they HAVEN’T seen before.

If option one is the method you want to encourage, then, sure, make the PARCC harder so it looks like that. Pay people who actually know math to grade it and you might be able to make that happen (ask College Board how they do it on the AP Statistics exam, if you don’t mind paying that high a price for each student). Publish enough sample tests that it can be taught to. Engage in some high quality, high stakes chalk and talk. Make sure that you find a way to track the ones who can’t “hack it” to a lower track that doesn’t need it so that you don’t have to be embarrassed by their scores. Make sure to blame their elementary teachers when they forget how to divide fractions in the middle of a calculus problem.

But if option two sounds like the better choice, then you believe in the tenets of reform education. The entire point of reform curricula, as I see it, is to encourage and require students to engage in mathematical reasoning ALL THE TIME. To become used to problems they are uncomfortable with. To get good at taking a deep breath and trying something. Most reform curricula attempt to include at least some of the research techniques that have been shown to improve long-term retention, such as mixed homework and test questions, which would definitely improve student ability on a multi-year exam like the Finnish one.

Of course, you can get this sort of thinking without using a reform curriculum. You can assign the hard problems in any Larson book, have students work them and discuss them with limited scaffolding, and get the type of mathematical engagement and thinking that this exam requires. But if you do that you are teaching a reform class with a traditional textbook. And probably requiring a LOT of homework in the process.

My favorite reform curriculum is the one used at Exeter, found here: https://www.exeter.edu/academics/72_6539.aspx . I’m sure many would be shocked to hear that Exeter, with all the New England Prep School implications, uses a reform curriculum, but there it is. Problem-based approach, problems worked in teams and discussed as a class, with the teacher serving as a moderator and facilitator. Is there direct instruction? Surely, as there is in any decent reform classroom. These students are certainly as or more prepared for an exam like the Finnish ones as anybody, and a reform-style curriculum will get them there.

If we had a graduation exam similar to the one in Finland, perhaps a push toward Exeter-style curriculum would be possible everywhere. With my current realities (in which I can only assign half as much homework as them, if that) I have to choose: an efficient chalk-and-talk that teaches them to do lots of problems but not argue why they can do them, or a less efficient exploratory curriculum that encourages them to explore, discover, and debate mathematics at the expense of content. And when it comes down to it, I think THAT is the heart of the debate here.