Last weekend my wife and I had the privilege of sharing a meal with Scott Miller, David Sladkey, and each of their wives. Awesome food, excellent conversation. It was the highlight of a great weekend at the DuPage Valley Math Conference.

At one point, David asked an intriguing question that sparked a lengthy discussion. I’ll share the question here in the selfish hope that I’ll be able to hear a few more opinions, and continue my own pondering (and learning) in the process.

Here’s the question:

*Suppose a teacher gets to divide 100% between two categories: teaching ability and content knowledge. What’s the ideal breakdown?
*

**Update:** Several commenters reframed the question as *pedagogy* vs *content knowledge*. I find that shift in language helpful. If you prefer that formulation—or if you’d like to push back and offer your own related question—go for it.

**Update:** I want to learn more about this, shared by elsdunbar: “Deborah Ball describes a horizontal content knowledge as ‘[A]n awareness of how mathematical topics are related over the span of mathematics included in the curriculum.'”

**Related post:** These thought-provoking comments have sparked a new set of questions for me.

**Update:** Mark Chubb addresses this question—and a great deal more—in his recent post Professional Development: What Should It Look Like? It’s fantastic. Go check it out.

## Comments 47

I’d love to have a little more info on the question. Does a 50-50 split mean the teacher was only mediocre at both teaching and content knowledge? Or that the teacher focused their attention equally between their teaching craft and their content?

If it’s the former, I’m afraid no percentage combination would be a winning one.

If it’s the latter, I would say that it’s pretty important for teachers to remain balanced. I am in the camp that believes teachers teach best when they know their content very well, and continue to seek out ways to learn more. But at the same time, just because you know your stuff doesn’t mean you can teach it. A focus on the art of guiding learners is just as important.

Author

Thanks for weighing in, Suzanne! I’m intentionally leaving the question and terms open (vague?) because I think it’ll lead to more interesting discussion. :fingers crossed:

i’ve seen amazing teachers teach from places of content weakness and do it well and have students learn. i’ve also seen really smart people be terrible teachers. i’d go with pareto’s law. 80-20. instruction > content.

plus, isn’t the content bucket easier to add to?

The most idealistic, non-mathematical answer is 100% pedagogy and 100% content knowledge, but I know that’s a cop out to your question! 🙂

My real answer is that there is no breakdown, both are vital and must be achieved simultaneously which is why teaching is so hard, but also so rewarding!

It doesn’t matter, unless the teacher is 100% reflective.

I agree that content knowledge without excellent teaching ability is useless in the classroom. But how many of our students, particularly math students, suffer because their teacher’s content knowledge does not allow them to teach math as a narrative, connecting it properly to prior knowledge and structuring concepts to allow students to transfer their understanding to later math?

Case-in-point: aren’t there many (but nowhere near all) teachers in elementary school who are a bit math-phobic themselves? Does the fact that they are excellent teachers overcome this content shortcoming? I would argue no. In fact, if I had it my way, teachers of elementary students would specialize by content at least to some degree, just as high school teachers do.

Is instruction really greater than content? Or do the two need to have more of a symbiotic relationship? Can excellent instruction really exist without excellent content knowledge?

I wonder if there is a relationship between teaching ability and learning ability. If I have a high percentage in teaching ability, am I better able to learn content I don’t know?

My favourite lessons are the ones in which I learn alongside my students rather than transferring my knowledge directly to them.

If I know I lack some content knowledge, but am confident I have the ability to learn that knowledge, am I more effective in the classroom?

I love this question and have been considering it seriously for a couple hours. I think teaching ability far out weighs content simply because content can be learned. Reflecting back, the topics I was less secure on, I taught the best because I boned up on my content knowledge. I also was much closer to the learning process so that made me a better teacher and understander of mistakes and misconceptions. I know I can learn more math. Much of teaching is instinctual. It’s like you can teach a tall person to play basketball but you can teach a basketball player to be tall. Number wise I say 70% teaching ability and 30% content.

It’s a really tough one. Teaching would be an easy job if it was just about transferring context. But without content knowledge, how will you be able to help your students acquire a connected view of a discipline you teach if you are unable to see connections yourself and recognize when your students are making those connections.

I hear a lot about the shift that occurs elementary and secondary, from pedagogy to content. I have no experience in secondary, but I think elementary teachers education programs need more content courses in addition to pedagogy focused ones (my 2 years BEd on top of 4 year BA degree was all pedagogy).

In my opinion, teachers should have a good content knowledge to enter the classroom and good mentors to develop pedagogical strategies.

Back to your question, 50/50.

I’m going to go big or go home here. I’ll go 100% pedagogy and 0% content knowledge. Here’s my case against content knowledge: I have a B.S. in mathematics from UCLA and I didn’t realize how little I actually knew about mathematics until after I graduated. I was a high functioning math robot. I could do the math algorithmically but had very little understanding about what it all meant.

I started out as a full time teacher with an emergency credential having never taken an education class or done a minute of student teaching. So at that point, I was roughly 100% content knowledge (a B.S. in math) and no pedagogy. . My “content knowledge” was not valuable because I could do but didn’t know how it worked. I have since learned so much about mathematics that I never knew. I think pedagogy is much harder to teach (sort of like the ease of teaching computation versus teaching critical thinking).

So, give me an armada of pedagogically sound teachers who know nothing about math and I can mold them into masterpieces. Give me an armada of content knowledge experts without pedagogy and I’ll take you to my upper division math classes at UCLA.

This is an interesting question. Without the ability to make relational connections with students and communicate effectively (teaching ability), content knowledge is nearly irrelevant. I also think that teaching ability drives the desire to become more knowledgeable about the content so that you can better meet the needs of students. My vote: 80-20 in favor of teaching ability.

Similar story to Robert above. Pure math undergrad with no classroom experience. Tons of content knowledge, but I needed something else. I would heavily weight toward teaching ability, maybe 90-10.

When I think back on my favorite teachers from elementary on up the ladder, I don’t remember the teachers who knew the material better than everyone else. I remember the ones who knew how to teach. Those were the ones who created a better learning experience for me and eventually put me where I am today.

I’m going to have to agree with Daniel and lean on Robert on this one. For me, it’s somewhere in the range of 90-10, so I’ll go with 88.8-11.2, teaching ability:content knowledge.

Similarly to Robert, my degree is in mathematics and I was kicking cabinets, ripping up papers, and wondering why the hell I got into the profession during my first two years. It wasn’t until I found my stride with a sound pedagogical perspective that I was able to start *teaching* effectively. I knew the math, but didn’t know how to write the narrative.

I think there’s some content knowledge that needs to happen, to speak to Suzanne’s point. There are not enough teachers of mathematics who can build out the story beyond the unit, and there’s no denying that. It’s great when I can build up to a college-level concept in an Integrated I class, but it’s more important that I create an environment in which students want to be there, expect a challenge, and are constantly curious.

So yep, mathematically speaking, 88.2 to 11.2

Final answer.

88.8 to 11.2

Dang it. Go to bed, Stevens.

I agree with Robert that 100% content- 0% pedagogy is an ineffective mix. However I don’t think that the inverse of that statement is necessarily true. I think 100% pedagogy with 0% content leads to math “tricks.”

Pedagogy > content but not by much. I think 60% pedagogy: 40% content.

(Which is why I always feel bad for my students the first two years I teach something. )

It seems to me that some are arguing that it is easier to acquire content than pedagogy. I would counter that both are equally challenging to acquire, yet crucial to successfully teaching students. As an instructor of both preservice and inservice teachers, I have seen both groups show natural talent for one or the other. I have also seen them both work hard to acquire what did not naturally come to them. I may know all sorts of mathematical concepts, but if I don’t know how to teach others these ideas the there will be little to no success. On the flip side, if I am a pedagogy ninja, but don’t have the content knowledge to teach with any sort of depth, this too is a losing equation. To slight content as the easier to learn is to slight the teaching profession, in my opinion. A true teacher constantly steives to improve in both domains and never settles for a fraction of their best self.

I don’t buy that the two can be separated. Maybe we should do better measuring content knowledge by charting how well you can teach it. My goal for a student is that she will speak with authority on a topic to which she was just introduced. The definition of art is creativity within the language and constraints of a medium; what good is knowing something if you can’t express it in the language of others?

What I have noticed in reading these comments is that the issue might not be what a teacher brings to the table (knowing more content or being more skilled in pedagogy), but how that teacher approaches refining his/her craft. What I hear Daniel, John, and Robert saying is that they have changed and grown since they started teaching. Which matters more? That they are now more skilled w/ pedagogy then when they started or that they were driven by reflection to change and learn.

I would cheat on the original question and say that we have less than 100 percent when we start teaching. I started as a volunteer in adult literacy. I had been a pretty mediocre math student on high school and avoided math like the plague in college, so let’s say I had 10% content knowledge. In terms of pedagogy, I had no preparation other than three volunteer orientation workshops which equates to 0% pedagogical knowledge. I’m not sure what the balance is now, but let’s say it’s 40/40. I feel like I know a lot about pedagogy, but there’s is so much to know. And I’ve learned a lot of basic math in the last 18 years, but mostly enough again to realize how little I know.

The most important element for my success (such as it is, since I know I’ve done some crappy teaching) had been my good fortune to be in schools where teachers were learning. I learned to teach by watching colleagues. I learned content from them as well, informally and in math circles and workshops. The balance between pedagogy and content is abstract unless we are given time and structures to meet with other teachers and talk about the content and how we teach it in a grounded way. We could have 50/50 balance between minimal pedagogical knowledge and minimal content knowledge…

Mathematical question for those who lean heavily towards pedagogy: if you increase the level of math content knowledge, aren’t you skewing the ratio?

I believe that strong teaching ability must be defined as an ability to foster learning. One who excels at fostering learning, is also an inquisitive learner. Therefore in my experience, really good teachers are strong learners and thus develop content knowledge. The opposite is not necessarily true for those with content knowledge.

I’m going opposite of some others that I highly respect. I’m leaning towards content knowledge…but not just more…deep and wide. Deborah Ball describes a horizontal content knowledge as “‘[A]n awareness of how mathematical topics are related

over the span of mathematics included in the curriculum.’ Example: Knowing how the algorithm to multiply together two numbers is related to multiplying together two polynomials.”

I have a degree in mathematics and began my career teaching math to students with disabilities. This experience was invaluable because it forced me to find connections between topics and across grade levels in order to meet each student where they were. Additionally, I took an opportunity to teach a math for elementary teachers course in order to further dig down into math across the grade levels.

I’ve found that the more math you have at the upper level can limit your ability to see approaches to topics at a lower level. This makes it hard to meet students where they are. I think this awareness is harder to teach and grow than pedagogical awareness.

This is a really good question, and leaving in “vague” open it up for a lot of discussions. I have taught at the middle school level my whole teaching career. Over the almost 20 years I have been a teacher, it seem as if more challenging material has been pushed down to the middle school level. As that has happened I have notice that some excellent teachers (ability) struggle to teach some of these new and challenging concepts, I believe that is true because they don’t have the background (content knowledge) to do so. At the high school level almost all teacher have a degree in mathematics and usually a minor in education (at least that’s how it is in my home state of NJ). Which means at the HS level most teachers should have the content area knowledge necessary to teach the material. At the middle school level people need to be “highly qualified” in math to teach it, which has many ways of being accomplished, but not necessarily checking that the person has the content area knowledge necessary to present the material in the correct manner, or to see the development of the material to the higher levels of mathematics. My perspective on the question lead me to actually wonder which “short coming” is easier to work on. If a teacher has high ability and lacks content area knowledge the journey to acquire more content area knowledge is a long and arduous one. If a teacher has strong content area knowledge and lacks some ability to pass that knowledge onto their students, I think there are easy ways to overcome this problem. So taking this into considerate I guess I would say something like 60% content area knowledge and 40% ability, mainly because if the teacher wants to become a better teacher I think it is easier to work on their instructional techniques.

Content Knowledge versus Teacher Skill. I know it’s not part of the given dichotomy, but teacher beliefs about what teaching math are foundational here, see the Math Ed podcast 1505 Patricia Campbell linked below: https://www.podomatic.com/podcasts/mathed/episodes/2015-03-07T05_39_44-08_00

In this podcast, Pat found that for middle school teachers who have traditional beliefs about what good math teaching is, their content knowledge was absolutely critical to student achievement. Teachers with high content knowledge had high student achievement, but students with low content knowledge had very poor student achievement. I’m guessing that most teachers who are responding on this site are not of this mindset, but I would argue that nationwide a majority of teachers are. She also found a similar relationship with pedagogical knowledge. (Interestingly, in the lower grades the relationship between both content and teacher skill and student learning is lower). What she found most important in the lower grades (and still quite important in middle school) was that teachers were aware of the importance of students having positive mathematical dispositions and that they reported doing things with their students to bolster their mathematical dispositions, particularly when paired with higher teacher content knowledge.

Personally, I concur with Pat, based on my own experiences in education. I was always really strong at math in school. I had a good conceptual understanding of math, so even though I didn’t major or minor in math or education in school, when I decided to change careers and become a math teacher, I always had students who showed high growth. That was true in spite of the fact that early in my career I had a very traditional view of math teaching (that was how I had been taught after all). NOTE: When I did do a 1 year alt teaching certification program my mentor from the university told me I was a natural teacher because I knew the right questions to ask to help make sure students really understood the concept, and not just what is the answer type questions, so I guess I had some teacher instructional skill as well, even then.

Over the years, my mindset about teaching and learning has changed, I like to think that I’ve moved in a direction of understanding why it is important for students to take a more active role in their own mathematical sense making, and that I’ve begun to understand the importance of a positive math disposition (see Mark Chubb’s post about Growth Mindset among others). Really the two go hand in hand, the more students can make sense of mathematical contexts on their own or with the support of their fellow students in discussions, the more postive their mathematical disposition.

It was this change in mindset about what math teaching ought to be that has pushed me to develop different teaching practices/and content understandings than I would have developed otherwise. I think that once you have a meaning making mindset about math instruction, you realize that you have to develop your ability to teach in that way (why I’ve chosen to independently read Intentional Talk and Beyond Answers and want to carry out their ideas) as well as your content ability (why I watch webinars on different ways to teach/understand a concept, even though I already have a good conceptual base, and why now that I’m transitioning into being a coach I’m learning about the MQI rubric, and how to implement it with teachers.)

So to loop back to the question as I read it, should teachers focus their time developing their pedagogical skill or their content knowledge? You can’t really answer that question without knowing the teacher and their beliefs about what good teaching even is. I would actually argue that teachers need to develop their understandings of the importance of student dispositions and develop their understandings of what the mathematical practices really mean, before they will even take advantage of the most useful types of ways to improve either content knowledge or instructional capabilities. I would say that once a teacher has a solid grasp on how teaching math should be about students making meaning together in different sized groups, and how the teacher should scaffold this meaning making (see the Appendices in Intentional Talk for how to lead productive discussions) they will start to see their own PD in similar terms and be able to determine from their own reflection on their practice which of the two areas they need to improve. Every teacher I’ve ever met with this type of mindset recognizes the need to understand the content even more conceptually, so you can understand the different misconceptions students have. In addition, they know they need to increase their pedagogical skills so that they better understand how to move the student from a misconception to a corrected conception in a way that makes sense to that student. This type of teaching is definitely more challenging to me, but also much more rewarding.

At the risk of overthinking here, it seems that both content knowledge and pedagogical knowledge should be growing on a pretty regular basis. I see Michael’s question as a fluid one of balance between the two. If I grow in content knowledge for a while without adding any deeper sense of pedagogy the balance starts skewing in that direction. So, given this vision of the question I would definitely fall toward the 80-20 camp (or Stevens’ brilliant 88.8 – 11.2) My early teaching life story is different from Kaplinsky’s. I have three ed degrees now and no math degree but I have picked up enough math content AND teaching skill over the years that I am confident that I am not doing any harm to my students even when I teach AP Calc or AP Stats. I do know that my energy, especially over the last 6 – 8 years, has been spent largely on expanding my pedagogy repertoire.

One of the things I love most in life is a sense of synchronicity. On Tuesday night I spend the evening at a great local pub debating this issue with a colleague who had sent me some terrific research on this very issue.

I wonder about his often. In my personal experience, I was always comfortable with math as a student and typically excelled. I have also been comfortable teaching since I was 15. I learned the moment I started teaching in elementary that what I learned/memorized in math classes was far from sufficient, but my pedagogy was on point. If I hadn’t studied math content deeply on my own, I do not think I would have fully recovered from my experience as a math student.

I feel like my definition of content knowledge varies from Robert’s in that I don’t feel like either of us actually came into teaching with much content knowledge. To me, if you don’t understand conceptually, you lack content knowledge.

It is not always easy for teachers who have always been good at the teaching part to accept that they need some help with the content part. I work with amazing teachers; inspiring mentors, but I am not sure that they have the content knowledge to help students gain a deep, conceptual understanding of math. You cannot use collaborative learning strategies or behavior management your way into quality math teaching without content knowledge.

There must be a mix, but I am not sure what it is. Maybe 65-35 pedagogy?

So much fun and depth in this discussion. I’ll throw in Einstein’s ” You don’t really understand something until you can explain (teach?) it to your grandmother.” This speaks to the interrelation of content and pedagogy but there has been little discussion of the paradigms and metaphors that form the basis of your understanding on pedagogy. We could use the word ‘teach’ but isn’t great teaching helping students to ‘learn’ effectively with transfer and deeper understanding? For this to happen, you would need a depth of understanding in your content knowledge. You would have to know the big ideas and how they fit. You would need this to keep students in the flow of the learning, catching/correcting growing misconceptions and for encouraging and extending their learning. It also involves how confident you are in murky areas. I have seen many cases of teachers telling students not to do certain questions (because they didn’t know how to do them) or they alter the questions so they are back in their wheelhouse of ‘tricks’. Instead, these questions are crafted to deepen knowledge and should be embraced together. It was so freeing early in my career to realize that I didn’t have to be an expert at everything to teach. I could learn along side of my students, collectively they are great teachers.

Pedagogy and content knowledge are not discrete parts making a whole. As one increases, the other doesn’t decrease. As I effectively utilize good pedagogy, the lessons crafted should increase my own awareness and depth of content. I have been taught a lot of content from my students (and other teachers) and this increasing depth allows me to craft/utilize pedagogy to better create lessons for accessing and extending all of our learning community’s understanding. As I add pedagogical understanding, it needs to be supported upon a scaffold of experience/knowledge built during my growth in content knowledge.

My first thought is I would be absolutely useless teaching a French class and the content mountain would be very difficult to ascend. I think I would also be mostly useless in a kindergarten class. How would I help a group of 5 year olds learn how to write or read or even stand in a straight line? I am sure I would be clueless and have great respect for anyone brave enough to enter that arena. Does that mean it is 50/50 with a drive to continually grow in both?

What a great conversation going on. I’d like to offer two anecdotes. When I was a senior in college, I had built a reputation as a good tutor for calculus, multivariable calculus, linear algebra, and other “intro” math classes. My roommate was an economics major who struggled with some of the math that he had to deal with, and asked me to help him out. I didn’t know the first thing about basic microeconomics, and honestly still don’t. Even so, I found that through some good lines of questioning, I was able to help him understand things much better. At the end of the semester, he aced his final, credited me profusely, and I still didn’t understand a thing.

Fast forward to my first year of teaching, in a tiny private school where there were 3 high school teachers (English, Science, and Math) for 19 students. Each of us had to take on a history class as well, and I ended up teaching U.S. History. I enjoyed history, but hadn’t studied it since I was in high school, and was a chapter ahead of the class the whole year. Even so, I found myself gaining essential skills as a teacher, partially because I hadn’t mastered the content. The class ended up learning a lot that year about U.S. History, and so did I, but I am pretty sure I grew more as a teacher that year in teaching the content that I was less familiar with.

Now, tutoring is far different than teaching, and I don’t think it’s a good idea to have teachers teaching outside of their area of expertise, but both of these experiences made me realize that building a relationship, finding the right entry points, meeting a student where they are at – these are the things that make a great teacher. The content is absolutely essential, and to a degree my love and enthusiasm for math is a big part of my pedagogical approach, but I’d have to definitely put the bulk of the importance in successful teaching on the pedagogy. 70-30? 80-20? Maybe it depends on exactly how those two things are measured, and I don’t have a good way to quantify either of those.

Agree with Claire’s 60-40 and that the meaningless tricks come from not knowing the math, and with jstevens009 about needing to know where the content leads to be able to weave the story.

@serratore4

25% teaching ability and 75% content knowledge. 25:75. Yes, you’re reading that correctly. I see three key flaws in the arguments people are making.

1. Anecdotal evidence. I was not ready for teaching, ergo, having that ability (at the outset) would be more important? No! No one is ready for teaching. But one can know content. Reaching for an analogy, a museum tour guide better know where most of the items are located in the building, to not get people lost – and oh yes, they likely need to know enough to keep track of everyone in the tour group. That analogy is flawed, most visitors to a museum choose to be there, but there IS no good analogy because no one is ready for teaching! So yes, 0:100 is inadequate, but with the 25:75, let the passion for the content (it’s NOT dry and unemotional) shine through, even if you’re not sure how to set up your classroom desks.

2. Intrinsic motivation. Content can be learned more easily? CAN, perhaps, but not unless you have some desire! (I won’t delve into the existing arguments about both aspects being equally hard, which also have merit.) Humans are, I’m sorry to say, creatures of habit, and inertia. If you have a lesson that works, you will tend to go back to it. Particularly since there isn’t sufficient time in a day to browse the latest on quantum computing. Even MissMathTeacher314 mused “It’s not always easy for teachers who have always been good at the teaching part to accept that they need some help with the content part.” (Granted, she went 65:35.) I’m arguing 25% is enough to establish a rapport with the students who aren’t engaged by the material itself.

3. Style over substance. To pull another analogy, this emphasis on ability over content is (to me) like increasing our marketing, at the expense of what we’re actually “selling”. Because you want the latest piece of special shiny math right here, yeah? It hooks onto the vintage interface! What’s it all do? I dunno, that’s not my job – I’m in marketing! Buy my stuff! (Frankly, I think this is why society is in dire straits these days, running for the shiny thing without caring to do the research, but that’s another argument.) At least 50% content is necessary to know what the heck is going on “inside the black box”, and I’m ramping that up more, to 75%.

In conclusion… my argument is frankly as flawed as all the rest, because if EVERY teacher was 25:75, we’d probably be in trouble. There are classes where that balance won’t work. (But is it a majority of classes? I don’t know.) But likewise, the 80:20 (or 88.8:11.2) that I’ve seen multiple times is equally flawed (maybe MORE flawed, because it’s so pervasive?). You need someone in your department higher than 20%, and there’s added bias here, because so many people are saying they ALREADY had that (and needed better marketing). I know the question is vague, but if you’re going 80:20, I’m stealing all your knowledge of polynomials, and good luck. Plus as Dylan implied, a majority of teachers out there may not be of our mindset – and hey, that’s a good thing.

Thanks for reading.

Knowing my own limitations, I’d like to start by saying that my reply is only from the perspective when the content = mathematics. (I would like to imagine that this pertains to all content, but I have no basis for making such an assumption).

I think what is 100% necessary is mathematical pedagogy.

First and foremost — I don’t think there is something as pure as the pedagogy being described in many of the responses above. I think pedagogy only exists within content, and I distinctly believe that there is a pedagogy that surrounds mathematics learning that is unique to how students learn mathematics. For the sake of argument, I will momentarily abandon this and pretend the two are discrete…

So, I believe I’m ready to start a math fight…I rarely disagree with Robert Kaplinsky, and as I admire him a great deal have never thought to do so publicly — but I will in this case. I think an “an armada of pedagogically sound teachers who know nothing about math” would be just as dangerous to mathematics education and student learning as “an armada of content knowledge experts without pedagogy”, and equally difficult (or easy) to transform.

There are two pieces to this —

If we first consider just the comparison between the two camps — I hope you can recognize that student learning would be suffering. If you don’t know math, you can’t teach it. If you don’t know how to teach, you aren’t teaching anything, math included. Either case — I’m thinking very little student learning of mathematics would be happening.

Secondly, I think we all agree that both mathematics content and mathematics pedagogy (if you believe me that it is a thing, otherwise just ‘pedagogy’) can be taught. Consider your own progression of learning — I imagine you can points to a time when you ‘learned’ each of these, and likely can even point to the tasks, experiences, teaching moves, lessons, etc that made it possible for you to have learned. If not, at least you can agree that they are attempted to be taught in all kinds of institutions all over the world.

So then, is this a discussion about which is easier to learn? Which is more comfortable for us (perhaps individually/personally) to teach someone else? Or which can students survive if the teacher is ‘faking it till we make it’? Because we might feel that those with strong pedagogy and no mathematics content are just easier to teach because we feel knowledgeable about the mathematics content to be learned, while the pedagogy feels complex. Perhaps we are sensitive to our own limited understanding in one or the other, and so we look for teachers to have that which we ourselves don’t have.

And this is where I’ll go back to my original point — I believe that there is a mathematical pedagogy which really is the complex relationship between mathematically understanding and pedagogical thinking. I truly think this is what we’ve been after in all of the shifts of recent times — not just a realignment of what we teach, but how we teach it. Consider the CA Mathematics framework… how much of it is about what to teach, and how much of it is about how to teach? Why the imbalance, and why isn’t there just a framework on ‘pedagogy’ — why is it wrapped up inside content?

So here’s where I need help — if you really feel pedagogy can be separated from content, that you can have pedagogically understanding absent from content understanding, can you describe to me what pedagogical knowledge is without using content or referencing content?*

*I’ve learned a lot through ‘math fights’, so I’m hoping I can learn a lot here, too. Would love to continue the conversation here, or on twitter @Audrey_Mendivil

Y’all are a bunch of cheaters! It seems to be answering a different question if your response is: I would pick X% and Y% today because tomorrow both will be higher!

My answer is 85:15 content over pedagogy. My reasoning is that you can only truly teach as far as your content knowledge and a little pedagogy can go a long way. For those answering near 100% pedagogy (0% content knowledge?!), I challenge you to test that by asking a primary teacher who identifies as “not a math person” to teach you a bit of math about which they know nothing, and see how effective that lesson is. Psshh… 0% content.

Of course, in the spirit of all the qualifications and disclaimers, I would completely change my answer if I were interpreting the question in a slightly different way…

Great thread to read all the response. I agree with a lot of the 80:20 pedagogy to content. I’m going to go with 70:30. My first year I had awful pedagogy therefore weak classroom management. I do. We do. You do. No time to practice really. Of course I now do the 5 practices as often as possible. Selecting a good task, anticipating responses and misconceptions, selecting student work, and sequencing their presentation of said work in closure.

I think that process need to be an essential part of any teaching credentialing program. I felt the need to rush and go chapter 1 through 10 in order at the start. I still use a book now but it’s supplemented w resources from #mtbos.

Now there’s not much to get tricked by content wise from middle school curriculum but wrangling their attention and putting them in a position to be eager to share their thoughts can only happen with sound pedagogy.

As someone else mentioned it takes a special kind of pedagogy and content knowledge to see how to respond to students misconceptions with counting and one to one correspondence in the lower grades.

On the flip side I wouldn’t feel comfortable teaching AP calc. I would feel comfortable teaching algebra 1, because i taught it to an accelerated group once. But, it would be a different animal being taught to a heterogenously mixed group of 9th graders with varying levels of motivation.

Also if I were to teach a geometry course for the first time, I’d be more confident doing so with a Cpm book because I’m familiar with the structure of the text and of cooperative learning groups.

I see teachers who have both high pedagogy and high content knowledge in a different light. Specifically ones prepping kids for an AP calc or stats exam. In my head that’s the pinnacle where you can make it interesting, answer their questions, facilitate good discussions, and possibly help them achieve college credits in a high school setting. Clearly something I’ve never done but definitely aspire to do later in my career. It’s a lofty goal.

Wow this has been really fun to follow.

I opened class with the same question to my students yesterday. It was unbelievable. The majority went straight for the teaching ability. The students were eloquent in their opinions. I would sum up many of their responses with this one: “It won’t matter how good the content is unless we are ENGAGED in the class.” The students also listened and responded to other students answers. It was really a lot of fun. One student asked, “Why do these have to be mutually exclusive?” Whoa. I would encourage you ask your students the same question when you get back to school. I know you will have an engaging 10 minute discussion with your students.

I have a few questions that have been rolling around in my brain.

1. What percentage of professional development time does your school/district devote to improving teaching ability compared to knowledge of the content?

2. Which category does “Passion of Content” belong? (this question came from Heather Fenton at the original dinner)

3. Does a first year teacher have the same percentage as a veteran teacher?

4. Would the percentages be different for an AP Calculus course compared to an Algebra 1 course? (this was a question my wife asked)

With all that said, I’m finally leveling off at 70% teaching ability and 30% content knowledge. My best learning experiences have been with teachers that knew me and could motivate me to get better in a variety of ways.

If there is one thing that I have learned from tutoring and teaching math over the years, it is that most students can rest when they understand. [The most common reaction is: “That’s it???!!!”] Most often feel unsatisfied, or just plain out relieved, that they were able to mimic their teacher’s doings even though they had no idea what they were really doing or why they were doing it.

As a math major in college, I hit a conceptual wall that made me question my education (and in particular math education) up to that point in my life. That’s when I came upon “The Mathematical Experience,” by Philip J. Davis and Reuben Hersh. Looking back now, I realize how much that book has shaped my approach to learning and teaching math. If we can agree that the overarching goal in the mathematical experience is to understand, then for sure pedagogy trumps content knowledge, because the methods that you use with your students are the ones that allow both the teacher and the students to acquire content knowledge (sometimes simultaneously!), but most importantly, to understand. Interestingly enough, I never stop finding new ways of understanding the same topic I have taught many times before because different groups of students demand different explanations and experiences in order to meet their educational needs.

I’m going to depart from our modern algebraic decimal preference of numbers and go with the Greek standard: 1 whole part pedagogy, no content knowledge necessary. That sounds horrible, so let me rephrase: The more I learn, and the more I want to transfer that knowledge to my students, the more I realize that my job as a teacher depends not on what I know, but on my ability to create a developmentally appropriate learning experience for my students to learn (part of) what I already know. And the more I do that, the more I am reminded that we should never underestimate any student, because they just might end up teaching you, the so-called expert, a thing or two.

Two things, if you have time. One, check out the conversation we had around Ben Orlin’s blog on this: https://mathwithbaddrawings.com/2015/12/09/what-level-of-teaching-is-right-for-me/. The comments are where it’s at. Two, I replied to this idea of Ben’s at TMC last year. No need to watch the whole thing, but the set-up might get you thinking. https://youtu.be/T25YN-bTrmk (Can’t seem to get it to fire from the beginning, but just reset it at the beginning.) Hope it’s helpful!

Tracy

Love this discussion. I’m more on the teaching side. Like if someone read Tracy’s book and “got it”, I’d be happy to let them have a go. Getting it for me is understanding what math understanding looks and feels like. I’d be much happier with a 2nd grade teacher who knew what place value understanding looks like than if the same teacher was procedural with place value and an A calculus student. Know the practices or process standards for a good start. The key if content is an issue is a willingness to learn along with the students.

I’m curious what you meant by teaching ability. I think people have been interpreting it like pedagogy or pedagogical content knowledge, but I’m not sure that’s what you meant.

This has been an amazing read!

Ignoring the 5 bazillion variables here 🙂 I’m going to say 50% teaching ability and 50% content knowledge.

Ponderings:

Does the original question here imply an assumption that all teachers are “equal”… or that the teachers to whom these percentages apply all start at some common “baseline”? Of course, we teachers are NOT all the same… and thank goodness! How boring would THAT be!?!?

Teachers vary the same way our students vary, in our strengths and weaknesses, both with teaching ability and content knowledge. I don’t see these percentages as fixed in the “now”. Rather, I see these percentages as teacher learning goals. If we strive to improve in each of these two areas with equal emphasis over time, we aim to be our best selves… because the best teachers never stop being learners.

Sure, we all show up to our first classroom experiences with some sort of foundational pedagogical skill set and some content knowledge, but these understandings don’t remain fixed for long. Teaching is such an on-the-job-training profession! And again… we’re all different.

None of us will ever arrive at some ideal understanding of our content or of pedagogy.

There is always room to grow, and neither of these two teaching areas is less important than or more important than the other.

I read this post a few days ago and let the question sit in the back of my mind. And then this afternoon, as I was working on a Desmos activity, I had a thought I usually have – ‘I should run this by my colleague and see what insights she can offer. She’s really good with Desmos activities and has stronger content knowledge with this particular topic. I’m sure she’ll have some good suggestions for me.’ And then this post floated back to the forefront of my mind. The great thing about teaching is that we don’t have to be experts (i.e. 100%) at either pedagogy or content knowledge. If we’re lucky, we can surround ourselves with teachers who can balance out our areas of weakness to become stronger together. So regardless of what percentages we bring to the table, I think the important thing is that we support each other to strive to be 100% in both areas.

I don’t think you can have one without the other. I also think this goes for many subjects. Deep content knowledge is vital. Knowing the structure of mathematics, the connections and the relationships between concepts is vital. I was a math major and was good at math until college, then I just survived. I could do math. But, my content knowledge is extremely strong now because I have learned the structures and progressions of mathematics. That could be considered having a deep pedagogical understanding or a deep content understanding. I have seen so many elementary teachers who have structures in place for teaching math: 3-acts, 5 practices, number talks, claim-support-question, complex instruction, but without the deep structural understandings; they don’t notice the intricacies of student thinking and know how to connect and help students build on that. I also see the same thing in high school. Teachers can do the math, but don’t necessarily think deeply about the progression of learning and relationships and connections and also have fewer “teaching strategies”. I don’t know where it falls: pedagogy or content, but to me it is a mix, lets call it a deep “contentogy” or “pedatent” that is the most vital: structure, connections, relationships, and progressions. To be effective you need a deep understanding of both.

This came up in a dept heads meeting today. My initial thought was they go hand in hand. But one teacher said that he thinks content knowledge must come first. After all, how can you develop classroom practice if you have nothing to teach. Being confident of content knowledge then frees one up to work on classroom practice, develop relationships with students, etc. So now I see it like mortgage payments. You pay a lot of interest at the beginning (content knowledge) and then you get to pay more and more of the principle (classroom practice).

I am digging the terrific discussion. Thank you to all who have contributed their thoughts and reasoning. I am at 65% teaching ability and 35% content knowledge. I am continually searching, learning, collaborating, and working to improve my teaching ability so that I can better design learning experiences for my students. My content knowledge has increased by teaching and learning from colleagues. This increase in content knowledge is a means to an end to better engage my students and improve teaching ability. Like Dave posted, I encourage you to raise the question with your students and other teachers. The discussion is facinating to listen to.

Great question and discussion. I’m not sure this is answerable. Teaching is so fluid. We constantly bounce back an forth between content and pedagogy throughout a given day. Taking one over the other is a difficult pill to swallow. That said, I’m chewing on this more and more and I’ve come to the conclusion that It’s hard to think of pedagogy without content, but I can work on math content without thinking about pedagogy. So, I’d give an edge to pedagogy (not sure how to quantify it). Still chewing…

Math content knowledge is important… but Math Knowledge for Teaching is what is the most important! Debra Ball has been explaining this for quite a long time, but basically this is where content knowledge meets pedagogy. David Wees shared this image explaining the features of her work:

https://davidwees.com/sites/default/files/screen_shot_2014-09-27_at_8.26.33_pm.png

Let’s take multiplication for a minute. Knowledge of multiplication facts or the process of multiplying efficiently without the ability to help others isn’t helpful. On the other hand, having great classroom control, knowledge of having great classroom discourse, knowing what to say to those who struggle so they still believe in themselves… without understanding how multiplication develops over time or which models to share, or what contexts and numbers might be appropriate… that might actually be just as bad.

In my first scenario, students that are already good at math might do well because they might be able to pick up on what the teacher is talking about. In the second scenario, I think again only those who are already good at math will be helped.

Hopefully we will continue to learn more about the developmental nature of mathematics (deepen our content knowledge) and discuss the pedagogical moves we make as we do so.

While saying this… I think far more teachers are willing to delve into pedagogical conversations than content ones. Yet, deepening our math knowledge for teaching requires us to go really deep into our content. So, if I were to place a stance on one side or the other, I would have to say that the best way to deepen BOTH is to talk more about the content we are teaching!

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