What Level of Teaching is Right for Me?

Just as you get songs stuck in your head, sometimes I get graphs stuck in mine. Like this one:

Now, like any catchy song, this graph runs its risks. The astute and thoughtful Tracy Zager objects to the entire premise of this graph, making three key points:

  1. Teachers of young, fresh-faced kids need deep content knowledge, too! Try explaining what it means to divide or multiply fractions. It’s shockingly hard! Most high school teachers can’t do it. And yet deep understanding of core concepts like is crucial for elementary teachers.
  2. Teachers of old, decrepit kids need pedagogical skills, too! It doesn’t matter how expert you are if you only know how to deliver dry-as-chalk-dust lectures and pass out worksheets. You’ve got to meaningfully and creatively engage with your students, whatever their age.
  3. This zero-sum approach is probably not healthy for teachers!

(For what it’s worth: I totally agree. Check back for a follow-up post soon.)

Still, I hear this graph echoing my experience. It’s not a photorealistic portrait, but it’s a recognizable caricature. The basic ingredients of teaching don’t change, but the recipe does.

The question is: Does this graph show us the world as it should be, or merely the disappointing reality we’ve got now?

For preschool, I think this is a fair model. The content demands are low: you’ve gotta know the colors, the farm animals, and how to say “please” and “thank you.”

But the job demands exquisitely specialized knowledge about human development, which I’m here calling “pedagogy.” (Can you spot how motor development interacts with language acquisition? Are you all read up on your Piaget?)

Elementary school shifts the balance a bit. To teach reading, yes, you need to be a good reader yourself. But even more, you need to unpack and inspect the challenges of early literacy in a way that most adults never have.

Is that “pedagogy” or “content”? It’s hard to say. The binary breaks down a bit.

A crossover happens at middle school, when you go from teaching a single group many subjects to teaching many groups a single subject.

Expertise now really matters, but pedagogy remains vital. It’s no good knowing cool, sophisticated stuff if you can’t make it accessible and appealing, too. Emphasizing one over the other is a losing battle.


High school tips the balance, for the first time, decidedly towards content. It’s damn hard to teach trig if you don’t know it yourself.

But it’s also hard to teach it if you can’t climb inside the minds of your students and design lessons that will inspire deep thinking.


When it comes to teaching undergraduates, I don’t want to undersell pedagogy. Clearly it matters, and many of the best undergraduate educators are keenly aware of new research and fresh methodologies.

But at the same time, good undergrad pedagogy can be as simple as “explaining things well,” “listening sensitively to feedback,” and “offering time and energy to your students.” Tactics like these aren’t the specialized provenance of teachers. They’re just good communication.


And finally, when teaching graduate students, human warmth and social skills still matter, but content expertise is so rare and precious that it necessarily dominates.

For example, I consider myself a caring, thoughtful person and a pretty decent pedagogue. Those are invaluable skills for a PhD advisor in any subject – say, anthropology. But don’t pick me to advise yours, because I only know about 90 seconds’ worth of anthropology.


Why has this graph been occupying my thoughts (at least, the half that’s not already occupied with Hello or Hamilton)?

I think it’s because, Monday mornings this year, I welcome my 6th-grade homeroom then immediately turn around and teach 12th-grade “higher-level” mathematics.

I’m only shuffle-stepping around the middle part of that graph, but already, the differences are stark.

Some days, I prefer teaching the older guys—fully formed scholars, actively specializing, seeking content expertise.

But other days, I prefer working with the little fellows—budding humans, still maturing, just beginning to know themselves.

POSTSCRIPT: Suffice it to say that my thinking is ever evolving on this stuff! (And so is this post; sorry for those who have commented on bits that I’ve now edited out.)

I agree with a lot of what Tracy says in the comments below; but I also think there are good reasons that our educational system is structured the way it is: with primary school taught by generalists, and middle/high school by subject specialists.

Anyway, check back soon for a follow-up!

46 thoughts on “What Level of Teaching is Right for Me?

  1. This is great. This got me thinking about whether the SUM of the content knowledge and the pedagogy is constant. You drew the two as straight lines in the plot, so the sum must be linear, but I suspect having it be U-shaped is more accurate. That is, for middle school, the amount of pedagogy drops substantially, while the amount of content knowledge does not increase that much either.


  2. I strongly disagree and feel really sad right now. This may be the picture of how it is, but not at all how it should be. Elementary teachers need much stronger content than everyone assumes. I mean, we teach huge concepts (equality, reasoning, pattern-sniffing, the foundations of all the algebraic properties, etc.). Meanwhile, high-school teachers have long dismissed pedagogy, thinking content is all that matters. I had high-school teachers I’d only recognize from the back, one arm raised with chalk. Know what? Their pedagogy sucked and I didn’t learn much, whether their “explanations were clear” or not. So, I’ll grant that your graph is how a lot of people think about K-12, but if we have any hope, we have to upend the assumptions. Every single teacher needs strong content and strong pedagogy, no matter the age of the kids they teach. I’d vote for identical, horizontal lines.

    1. The references for elementary teachers’ content knowledge are Liping Ma or Deborah Ball. Both argue that, whatever piece of mathematics we teach, we need to know it deeply.

      One more thought. This weekend at PCMI, I was sitting with HS teachers who freaked out when asked to write a story problem for something like 4 divided by 2/3. They were also stuck with numeric, formulaic area of triangle solutions for this problem and were stunned I was able to solve them with visual arguments. https://twitter.com/heather_kohn/status/673231944189132804 They haven’t thought about fractions the way I have though, because they only teach fraction manipulations, not fraction concepts. Of course, they knew much more about quadratics and modular arithmetic than I did. How lovely that we were able to learn from each other. That’s what I want. I normally love your work, Ben, but I think this post is really destructive.

      1. I think this is a totally fair critique!

        I’ve got some questions and clarifications, but first I’m going to tweak the body of your post a bit, because I’d like people to read your thoughts, too.

        1. Indeed! Or, in my case, some kind of sugary cider drink. (I have the alcohol tastes of a 17-year-old.)

          I’m glad this post struck a chord, even if it’s a “oh my gosh that’s a grating and frightening sound” kind of chord.

          I totally see the danger in this framing: underselling both the content demands of elementary school teaching (which do run deep) and the pedagogical needs of high school teaching (with are equally profound).

          Still, I find myself shackled by this framing a bit. I see where it’s dangerous. But it also feels fairly true to my experience.

          One thought: At the elementary level, is math perhaps a special case? It seems plausible to me that a well-educated college grad knows enough “content” to teach 3rd grade science, social studies, and reading; what they’d need is deeper knowledge about how kids learn the subject, not about the subject itself. But on reflection, this feels less true for math. The typical college grad doesn’t know enough math to teach it comfortably.

          Another thought: Is the problem not so much with where I’ve drawn my lines, but with the idea that they are separate lines at all?

          I’ve always experienced them as fairly separate, feeling like I could tell when I was struggling with content (hello, that year I taught earth science) vs. pedagogy (hello, first year of teaching middle school math). But maybe they aren’t so separable, and are unified in what people like to call “pedagogical content knowledge.”

    2. @tjzager Thanks for sharing this response! I think that the broad assumption made by this graph is instructive for beginning a conversation about the way “a lot of people think about” education. From there, it would be a productive, hope-building exercise to bridge some gaps and upend those assumptions. I appreciate your framing of this professional challenge!

    3. “Elementary teachers need much stronger content than everyone assumes.”

      While I agree with this, it SEEMS like you’re suggesting that “elementary teachers need to know the high school content” while implicitly assuming that’s also all the high school teachers need to know. In my opinion (ignoring the semantic debate between content and pedagogy) the high school teachers need to know all the content that elementary teachers need to know, plus more. Yes, the elementary teachers need to understand algebra and geometry so they can properly teach arithmetic, but I don’t want anyone teaching high school unless they have learned ring theory, linear algebra, and hopefully some real analysis if they are teaching anything close to calculus.

      In other words, ALL teachers need much stronger content than everyone assumes.

      1. Sure, all teachers need stronger content than everyone assumes. But I don’t hear anyone arguing that HS teachers don’t need content. I do hear people assuming that HS teachers don’t need pedagogy and ES teachers don’t need content, so that’s what I’m pushing back on.

        And I’m not suggesting ES teachers need to know the HS content. Nope. This is where Liping Ma is fascinating. In her study, she showed how deep Chinese teachers’ content knowledge was in K-8 mathematics–much deeper than US ES teachers–even though many of the Chinese teachers had no higher education themselves. I don’t mean their knowledge was FURTHER. It wasn’t. It was a DEEPER understanding of elementary mathematics. This is a concept we struggle with in the U.S.

        1. I think the deep/far distinction is really powerful. I find it super meaningful in math, though a little trickier in other subjects, like science.

          If I understand Bohr’s model of the atom, but not quantum models, then am I lacking depth or reach?

          Or, if I understand the transcription/translation process, but almost nothing about other mechanisms of gene regulation (e.g., histone wrapping or whatever), am I lacking depth or reach?

          I’m genuinely not sure about examples like that!

        2. Great question. I’m not either! But the example that springs to mind for me in science came from a video I just googled and couldn’t find quickly. What I remember is teachers gave students an acorn and a branch. They asked, “Where’d the stuff come from? Where does the tree get its mass?”

          The students could describe photosynthesis with textbook representations and equations. But when asked this question about where the stuff comes from, they’d say, “From minerals that the tree sucks up through its roots.”

          That’s my clear-cut lack of depth example. Yes, it’s mind-blowing to say OUT OF THIN AIR! But they’d been using that equation for a while and never really thought about what “fixing carbon” really meant. What that CO2 –> O2 transaction leaves behind.

          So, if a student had depth and really understood photosynthesis, as an example, what’s farther for her? Application? Farther into the chemistry and physics that underlie photosynthesis? Or is that depth again? I dunno.

          I’m grateful the distinction is clearer in math! I’m not sure why it is, but it is for me too.

        3. That one popped up on my search too. I look forward to watching it. The one I’m thinking of was a classroom-based one, though. The main focus was how persistent misconceptions are. After good formative assessment, teachers tried to teach the kids where the mass really does come from. Good teaching, too. It went nowhere. Kids still said it came through the roots.

        4. I think that the deeper vs further distinction needs more clarification. For example, deeper understanding of arithmetic IS algebra, whereas mere procedural understanding of algebraic symbol manipulation is not necessarily algebra.

          So I think that “high school content” will be a source of semantic disagreement between us. Instead, I will offer that what is needed at every level is an understanding that is not merely procedural and leave it at that.

  3. One middle school principal in my school encouraged us. “Don’t smile before Christmas.” I hope the principal you describe did not actually ascribe to that standard.

    As to the main theme of the graph, it might be more accurate to display the X by a decline in pedagogy with a rise in andragogy. While I’m not a fan of buzzwords, I see the methods of teaching as gradually shifting from nanny and charge to master and child to an eventual peer relationship with appropriate challenges offered along the way by the professional.

    The “content delivery” should be a straight line. The content needs always to be appropriate in volume and type for the individuals (even if delivered to a group of same-age learners). That tailoring for each learner is the most difficult component of a teacher’s skillset. No plan endures the first question from the third row (to paraphrase the military adage).

    It is also humbling and valuable to remember that from infant to child to adult, the human buckets we want to “fill with content” are able to teach us a thing or two along the way.

  4. Math is a special case, but for a different reason than the one you suggest here:

    “One thought: At the elementary level, is math perhaps a special case? It seems plausible to me that a well-educated college grad knows enough “content” to teach 3rd grade science, social studies, and reading; what they’d need is deeper knowledge about how kids learn the subject, not about the subject itself. But on reflection, this feels less true for math.”

    Not true. Elementary teachers spend a great deal of time learning and re-learning all their content areas because you have to know it at a much deeper level to teach it. For example, as a college grad, do you know what phonemes, onsets, and rimes are? Do you know the difference between phonemic awareness and phonics? Do you know how continuous sounds differ from stop sounds, and the world of r-controlled vowels? If not, you’re not ready to teach reading. That’s one part of the content knowledge. The content pedagogical knowledge comes in thinking about how to teach and sequence those skills, while also marinating children in a text-rich environment, teaching comprehension and the habits of great readers, and, in ways big and small, creating conditions for kids to fall in love with reading. I really miss all of that, to tell you the truth!

    Same goes for all the other subjects. For example, I was a science major, did original research at MIT, worked at NASA, studied Physics at Oxford. I had to re-learn E&M at a much deeper level to teach it to 4th graders.

    I’m a writer and an editor. With all the instruction I ever had, teaching kids the 6-traits writing model (Ruth Culham) is what really taught me how to write. And so on.

    But math IS a special case in a different sense, because elementary teachers have weaker content knowledge in math than they do in the other subjects. That’s one of the major challenges we face in preparing and supporting elementary school teachers. That’s one of the reasons why your blog freaked me out. They think just the way you do: the younger the kids I teach, the less math I need to know. So the most math anxious teachers often teach preK-2. They think like you thought here, with contempt for the content, dismissing it with, “I think I can handle 2 + 2.”

    Except, no. NO! PreK-2 includes the foundations of cardinality, ordinality, counting, operations, parts, wholes, commutativity, patterns, place value, equality, sameness, difference, attributes, etc., as well as the bigger essential idea of sensemaking in school mathematics. These are giant concepts, worthy of deep study.

    On the flip side, I’m spending quite a bit of time these days wondering how to get secondary math teachers to recognize they need to work on their *teaching* and support them as they do. I feel like you just told them they don’t have to, as long as they know the math. That’s a comfortable and unproductive belief for them.

    And for you, I think.

    1. Ah, okay, now I can see where we’re parting ways!

      First of all: Where you seem to be seeing a lot of disagreement on the nature of teaching, I’m mostly seeing agreement. For example:

      ME: “To teach reading… you need to unpack and inspect the challenges of early literacy in a way that most adults never have.”
      YOU: “Do you know how continuous sounds differ from stop sounds, and the world of r-controlled vowels? If not, you’re not ready to teach reading.”

      (Same thing, right?! You’re calling this “content”, because it’s stuff you need to know about reading. I’m calling it “pedagogy,” because it’s stuff you need to know to teach reading, but not to read.)

      ME: “When it comes to teaching undergraduates, I don’t want to undersell pedagogy. Clearly it matters… but [it] can be as simple as… ‘listening sensitively to feedback’ and ‘offering time and energy to your students.'”
      YOU: “I feel like you just told them they don’t have to [work on their teaching], as long as they know the math.”

      (I see where they’d get that from the graph alone, taken from context; but the paragraphs below specifically say the opposite!)

      So I think we’re agreeing on substance, but disagreeing sharply on framing and presentation (which are also important issues).

      It seems your approach is, “This graph shows the conventional wisdom, which is wrong for reasons X, Y, and Z.”

      Mine is more, “This graph shows the conventional wisdom, which has a lot of truth, though it has caveats X, Y, and Z.”

      Looking at the post again, I do see what you’re saying. My rhetorical strategy is often, “Affirm the truth in what people already believe, and then try to help them expand a bit.” But you’re probably right that this post doesn’t do enough on the “expand” front.

      1. Really interesting. Thanks for this reply. So if we go back to Shulman, who defined pedagogical content knowledge, that may help our communication gap. I think of it like this:

        Content Knowledge: The WHAT of teaching.
        Pedagogical Knowledge: The HOW of teaching.
        Pedagogical Content Knowledge: The best HOWs for this particular WHAT, in this particular context, with these particular students.

        I hope he’ll forgive me for the licenses I’m taking in the interest of brevity.

        So, if content knowledge is that a + b = b + a and pedagogical knowledge is how to manage a classroom discussion, the PCK is how to facilitate a rich discussion about a + b = b + a. What problem should we choose, how should we represent it, what questions do I ask, what do I anticipate students will notice, what connections do I want them to make, etc.?

        All successful teachers of students at any age need to have deep and rich content knowledge and pedagogical knowledge if they’re even going to begin to develop pedagogical content knowledge. And PCK is what’s needed for effective teaching.

        That’s why I’m still going to push back at the entire premise of the graphs (linear and bar).

        If I had to make generalizations, this is what I’d say (with caveats and exceptions):

        Elementary teachers tend to have stronger pedagogical knowledge and need to work on their content knowledge. That will enable them to develop strong pedagogical content knowledge, which is what’s needed for good teaching.

        Secondary teachers tend to have stronger content knowledge and need to work on their pedagogical knowledge. That will enable them to develop strong pedagogical content knowledge, which is what’s needed for good teaching.

        So, in this post, I see a real difference between describing the world as it is and imagining the world as it should be. That’s the expansion I’m missing.

        1. That’s fair. Thanks for the conversation.

          I think I’m gonna give this piece a bit of an overhaul tonight and scan new drawings tomorrow. I’ll archive the original (so anyone reading your comments can know what they refer to!).

  5. @tjzager I totally agree with your first critical point. Fraction teaching and the dumb-ass approach to algebra are the two things that confuse and alienate the kids. Calculus comes a close third.

  6. I hear a lot of comments crying, “high school teachers need pedagogy too!!! content isn’t the only thing that’s important!!!”

    High school teachers need good pedagogy, of course. I agree. But I personally find that pedagogy isn’t usually the main thing lacking for poor high school teachers–it’s content. And lack of content-knowledge deserves more “crying out” than lack of pedagogy in high school.

    Like was said in another comment, we all need a little more content than what we’re actually teaching–elementary school teachers need at least a middle-school (algebra) level of math content knowledge, and high school teachers should have at least a college-level of math content knowledge. However, I find that most high school math teachers don’t have college-level math knowledge.

    How often do we receive “content” training as math teachers? Rarely, I would guess. As a high school teacher, I go to countless trainings on pedagogy and discourse and depth of knowledge and literacy and and and.. When was the last time school districts issued a *required* training on an advanced mathematical topic? Rarely, in my experience.

    If I was in charge of professional development for high school math teachers in my district, I would require that teachers, every five years (say), take an advanced math course from a menu of options. What do you think of that proposal?

    1. “As a high school teacher, I go to countless trainings on pedagogy and discourse and depth of knowledge and literacy and and and..”

      Seriously???? I have never heard of such a thing. Most HS teachers I know have had incredibly limited exposure to ideas around discourse, DOK, etc. I am amazed! I hope at least some of them were effective.

      FWIW, I’m all for content growth for everyone. This zero-sum approach doesn’t suit my fancy. We all need to work on it all.

      1. Yes, I agree, we all need to work on it all. Well said.

        It goes without saying that our experiences and perspectives are all different as we come to this conversation.

        But I’m still putting my money on what I said about required trainings for high school math teachers. I don’t think that I’m projecting my own experience too much when I say that most of these trainings are around pedagogy, not content.

        One thing that you said earlier that I really liked was about “Pedagogical Content Knowledge” that represents a hybrid middle ground. This is some of the best teacher education that we can receive–especially when we’re given the opportunity to get together with other teachers and share our methods for conveying particular ideas. The pedagogical training I was speaking of, that I’ve received, (DOK, discourse, etc) is often too general to be useful “on the ground,” in my opinion.

        Great conversation here. In the end, I think we’re all on big fans of excellent teaching.

        Many thanks to you, Ben, for inspiring the thread.

  7. I found both the premise of the article and many of the comments very interesting. I guess a major stumbling block for me was deciding what ‘pedagogy’ really meant, because no dictionary could really cover the range of topics under discussion. I would like to propose a definition of pedagogy as ‘investment of thought into how to bridge the gap between your content understanding and your students’ content understanding’. [which after re-reading is pretty similar in flavour to tzjager above, but I might as well say it anyway.] I don’t think this immediately makes the graph objectively accurate in the contexts people have been mentioning, but it seems to fit with my perception of how teaching works. And it would then seem reasonable that to bridge a smaller content gap (because the students’ content understanding is greater) requires less pedagogy, and maybe more something else?

    Two related points:
    1) In an ideal world, one would have infinite quantities of both, but I was thinking: what I would be nervous about if under-prepared for teaching various ages? And then I think the graph fits kind of. I would be nervous about not having a good idea why dividing fractions is confusing to a 10-year-old and not having a good idea how to de-confuse it. But if I was teaching undergrad linear algebra, my primary concern would probably about whether I’d describe the Gram-Schmidt algorithm correctly or whatever, on the grounds that if I did it correctly then sharing my thought process about how I suspected that what I was doing what the right thing would be pedagogically useful for the students.

    2) I think a lot of the commenters rightly draw on the fact that everything depends on the students, but it also depends on you the teacher. I’m aware this analogy may not generalise brilliantly, but try teaching 1st year undergrad classes in two topics: one well-suited to your research, the other not even closely related. The only difference then is *you*. In the latter case, you feel much more invested in the struggle to understand the content, because it’s a struggle that you yourself feel (hopefully to a slightly lesser extent than your students). So you can cover all the “here’s why I think you find this difficult – why not try this?” angles without really thinking too much, whereas being fully on top of the more intricate content details may be the deal-breaker to giving a good class. Whereas in my preferred topic, knowledge won’t be the problem – really the first step is trying to remember what it was like to *not know* this stuff, which under any definition feels like it should count as pedagogy?

    1. *upvote*

      Great comment, especially your #2 point. A lot depends on where we are, personally, as a teacher. And our teaching practice has to be characterized by self-reflection, whether we’re in year 1 or year 30 of our teaching career (early on you might focus on gaining more content knowledge, and later you might focus on gaining more pedagogical knowledge).

      1. Thanks for the link. The article is quite dense but also interesting, and I’m sure the definition of PCK makes sense to us on lots of levels. Though I think it’s still meaningful to say that what might hold one back from having good PCK on a particular topic might be *either* content knowledge itself, or *pedagogical experience* within that area. Or more normally some complicated mixture of both, all of which I’m advance with experience and planning.

  8. As far as not laughing (or was it smiling) before Christmas, am I the only one who looked at the cartoons and laughed, especially the comment ” I think you have just answered your question”!

  9. I teach remedial math in a community college. I think the x-axis should be relabeled “mathematical sophistication of student.”

    My remedial math students, many of which are my parents’ age, require more pedagogy and less content than the high-school calculus students I work with.

  10. Where do leadership or management skills fit into the picture?

    Watching the old movie “Up the Down Staircase” recently, I was struck by how many of the problems the idealistic new teacher faced arose from both her misdirected upper management (Dilbert’s pointy haired boss seemed mild by comparison) and her own inadequate in-class management. Teachers have little influence on higher managers, of course, but a good teacher might better his/her own performance. For example, from the movie, one might make better use of time by doing repair work-orders or supply requisitions for the classroom before or after the students are present, rather than immediately upon discovery of a need.

    Whether a sergeant leading a squad, a foreman leading a work crew, or a parent settling all family members down for a shared meal, each seem to require similar essential soft skills directed at getting everybody informed, motivated, attired, assembled, in-position, equipped, readied, and on task. Are teachers taught such skills, with emphasis on the age of the assembly and task to be accomplished? Does that collection fit into the Pedagogy box?

  11. As someone who is a grad student currently transitioning from working with preschoolers to working with high schoolers, this post resonates with me in so many ways.

    To answer your question, I think the graph shows how things currently are, which is not how they should be. While I agree that the need for content knowledge grows as the students age, I think that pedagogy should really stay quite high over time. It may not need to be as high as a preschool or elementary school teacher’s level in grad school, but definitely at a middle school teacher’s level. I’ve been through four years of undergrad and am in my third (and last for the time being!) year of grad school, and, in my experience, pedagogy makes all the difference. My first grad school class (before I switched into education) was taught by a very nice, very smart, and very experienced person who was a former attorney at the Securities and Exchanges Commission. He was an awful teacher. While I managed to be successful in the class, I know many of my classmates weren’t so lucky, and I know it’s because of his lack of ability to teach. While the pedagogical skills of fostering a love for learning and developing good study habits aren’t present, it is replaced by the need to engage with fellow adults, who are likely tired and poor because they are grad students. I am so grateful that my current program has so many amazing professors who have the pedagogical skills to keep me engaged for a three-hour class even though I’m burnt out from student teaching and keeping up with my other coursework.

    To me, the pedagogy shouldn’t be decreasing for older students; it should just be changing to meet the needs of older students. Especially in P-12, we are always teaching kids first and content second.

    1. Your last paragraph – actually your entire comment – is spot on in my book. Teaching isn’t telling – as evidenced by your SEC prof. It seems this is one of the great cultural myths regarding American education – let’s just get really smart people to teach math and science. Content knowledge is key, but the ability to convey that content so kids can do what the teacher is able to do is equally if not more important in my mind. Michael Jordan may be the greatest basketball player of all time (content knowledge and skill), but I’m not sure he’d be a very good AAU coach.

  12. Since I love Ben’s stuff so much, I have to pile onto this comment flurry (albeit a couple days late) with something irrelevant/irreverent:

    The linguistics of animal sounds is really interesting (comic version. As I see it, there are two dimensions:
    (1) what animals does each culture bother to teach its children about, why, and how?
    For example, American kids learn about cows, Thai kids learn about elephants. I’m guessing that similar proportions have actual, frequent, personal experience with the respective animals (i.e., not much).
    (2) what sounds do the individual animals make? Associated sub-questions are why they make those particular sounds, what degree of variation there is within a language, why they vary across languages, whether they change over time, etc.

    Anyway, my actual contribution to the main points:
    I think one reason that math is different from other elementary school subjects is because the vast majority of the kids’ experience with math is coming from their teacher. I don’t think that is the case for any other subject. For example, the vast majority of their language experience is not coming from the teacher.

    Also, of course, math is high stakes. Animal sounds aren’t.

    1. A correction about my math-oriented claims came via twitter via Tracy Zager and Christopher Danielson. I am posting here for reference and general enlightenment. This is my summary/paraphrase, so any (further) errors are mine:

      No, kids are not just getting their math education and experiences from teachers in school. Instead, they are seeing math all around them, like language, thus arrive in the classroom with a lot of background. Reference: Talking Math with Your Kids (which links to further references).

      Part of the TMWYK revolution is (a) helping parents recognize these experiences so that they are explicitly considered “mathematical,” (b) increasing their frequency, (c) increasing their quality. To explain (a), comparison with language is helpful. When parents read to their kids, they know they are reading to their kids and (mostly) everyone considers it part of teaching kids to read. When parents naturally do math with their kids, neither may recognize and label it as such. This can lead to the perception that math learning just happens in school and an overly narrow understanding of what “math” is.

  13. I think the graph would be more natural as inequalities (like, it wouldn’t be percentages anymore, but that could be relabelled.)

  14. Thanks for your great work – you have a big fan base among college math profs! This thread contains an amazing array of resources and I love the discussion your post has started.

    One perspective from higher ed about this bit:

    YOU: But at the same time, good undergrad pedagogy can be as simple as “explaining things well,” “listening sensitively to feedback,” and “offering time and energy to your students.” Tactics like these aren’t the specialized provenance of teachers. They’re just good communication.

    This is probably true of the view of higher ed. math circa 1990. It’s certainly not now. There’s just too much evidence that lecturing (i.e. explaining things well) is ineffective – and worse for some groups than others (women, minorities). There was a especially convincing report last year (called the Freeman Report) out of the National Academy of Sciences: passive lecturing classes had a 55% higher drop/fail rate than anything more interactive. It’s probably true that the majority of college STEM classes are still passive lectures – but that’s not the way it should be, and STEM professors increasingly know that.

  15. I went to a military academy (uncollege). All the courses were like high school: 25-35 students in the room, no grad student teachers, no poor English, had to go to the board and do problems, homework collected and graded, lots of quizzes and tests, not just an end of the term final. Teachers were available after school, for set office hours, for arranged office hours, and by phone until 11 PM!

    I irked my classmate’s dad who was a professor when I bragged about my college was just like high school. But I meant it POSITIVELY. Superior pedagogy. S..orry. Military is the same way in service schools after graduation. They actually care about the learning process and the end result.

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