Thanks, Brett, for these various links, and for your suggestions about which problems to look at. I hope to blog about a number of these in upcoming posts on my Math Problems of the Week series. My analysis will undoubtedly provide support for one of your earlier statements: people often see what they want to see.

Happiest ending, as far as I’m concerned.

Thanks, Katharine & Barry, for seeding the conversation, and everybody else for disagreeing with each other seriously and respectfully. Out of 90 comments, I had to moderate only the smallest handful. I’ll look forward to following the conversation on your respective blogs, Twitters, newsletters, zines, etc. Comments closed.

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.

While I write about elementary math pedagogy, my teaching credential is secondary math. I have not taught elementary school math, and my subbing experiences have been in high schools and middle schools.

My book describes two situations; a six week stint at a high school and a semester long stint at a middle school. The algebra classes at the high school did not go well for a variety of reasons, the major one being that there were enormous math skill deficits among many of the students. For the record, the majority of students were white, and many had very bad family situations.

The middle school algebra class went well. I may have mentioned at one point a panic when the class did poorly on a test, but said that generally class averages were in the high 70’s, low 80’s. I didn’t go into detail about grades, but the majority of students got A’s and B’s in my algebra classes; there was 1 F and 3 or 4 D’s given to students who shouldn’t have been placed in algebra.

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.

By the way, I’ve been meaning to mention that mathematical proofs are in part about someone sharing his/her mathematical reasoning with others. Of course, at the professional level, the discourse demands a level of abstraction and sophistication (“mathematical maturity”) that is far from what we expect in K-5 or even K-14. College students do not for the most part begin to learn how to do real mathematical thinking (if they ever do) until they finish the basic calculus sequence. Nonetheless, it is commonplace in mathematics for people to be expected to explain their reasoning and no one will get very far at all if all she can do is produce the correct numerical answers to computational problems. As Keith Devlin already suggested, we have loads of tools for doing that faster and more accurately.

There is absolutely no reason not to expect students to be able to demonstrate their reasoning on a regular basis (though not for every problem or for trivial cases where at their grade level and personal development, it’s silly to ask someone to “explain” a result).

I seriously doubt that there is a single person in this discussion advocating the sort of absurdity that we can all find examples of. Now, will we continue to beat against the patently obvious, or will we move towards a deeper and more productive level of investigation into how to go forward with explanation in various grades and contexts?

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.)

But as knowledgeable mathematics educators know, what comprises “explanation” in classrooms where teachers aren’t operating either with some sort of high-stakes testing gun at their heads or where they developed their practice slowly over time, grounded in increasing pedagogical content knowledge so that they can more easily resist being bullied into mechanically applying the very idea of asking for explanations, what comprises appropriate explanation in various grades and at various ages is not some monolithic idea that can readily be put onto a standardized test.

Look at the classic 3rd-grade lesson from Deborah Ball in which students are asked to explain their understanding of odd v. even numbers. Shockingly, no one offers an algebraic explanation. But much of great value is revealed that simply would not have emerged in traditionally-taught classrooms.

I don’t expect everyone to agree. Indeed, I don’t believe there is an amount of evidence that some people in this conversation would admit makes it reasonable to ask students to provide explanations of their thinking on a regular basis. That’s just the nature of these debates, I’m afraid: a great deal of entrenchment.

I do have to wonder, however, how many teachers and parents who had the opportunity to view the teaching of people like Ball, Magdalene Lampert, or some of the practitioners here would be able to honestly reject what they see as wrong-headed, harmful to children, inferior to traditional mathematics teaching in K-12 (and particularly in K-5). I suspect that number is a good deal smaller than some math warriors would like to believe. And it is clear that progressive, student-centered teaching isn’t going to vanish by fiat. So maybe some of those who seem to abhor it out of hand should reconsider 100% opposition. Or not.

All that said, the idea that we have an alignment between the Common Core Initiative (and its concomitant assessments) and teaching that seeks to put more emphasis on student understanding, explored in part by expecting students in various ways to examine and discuss their thinking, strategies, problem-solving efforts, etc., is specious. I see no fundamental agreement between progressive teachers here or elsewhere and the folks who’ve given us the Common Core or the assessments thus far produced by the two testing consortia about what comprises good assessment. I have absolutely zero investment in or commitment to the Common Core as a political and economic instrument for further undermining free public education. I have none, either, to any instance of curricular materials that Big Publishing is selling to fit their notions of what the Common Core calls for.

In other words, the Common Core is an enormous red herring in this conversation. The issue of probing student thinking in mathematics exists for teachers regardless of the existence of CCSSI or any particular books. In my work as a math coach, one of my constant refrains about textbooks is that they are resources, not bibles. That some teachers, administrators, students, parents, or politicians fail to grasp that is unfortunate but ultimately something to be critiqued and, it is to be hoped, dispelled as another artifact of 19th century thinking about pedagogy (at best).

Bad assessment, particularly when forced through the narrow portal of high-stakes, standardized testing, is another vestige of weak thinking about education that we must repeatedly struggle against, with or without some gigantic edifice like the Common Core driving it. But one of the biggest mistakes we can make is to claim that getting students to reflect on their own thinking about mathematics and mathematical problem-solving and sharing that thinking with teachers and peers is a bad idea because such and such a textbook, teacher, or test gets that badly wrong. The answer is to improve and broaden the base of understanding among teachers and other stakeholders about how this can be done effectively and used to the advantage of everyone, not to drop the idea.