Yesterday I asked which is more important for a math teacher: *teaching ability* or *content knowledge*.

Several commenters reframed the question as *pedagogy* vs *content knowledge. *I found that shift helpful, so let’s run with that.

My original plan was to highlight a few comments here, and then offer a twist (or two) on the original question. But the post received way more than a few comments, and all of them are super thoughtful. Instead of including a few excerpts here, I’m going to push against conventional Internet wisdom and encourage you to go read *all* the comments. (Seriously. It’ll only take a few minutes, and it’ll be worth every second.)

#### New Questions

Alright, welcome back!

This discussion has raised a host of new questions for me. I’ll rattle off three of them here (and possibly more in a future post). Feel free to chime in on one or all. Or just lurk. That’s cool too. 🙂

- Is it possible to separate
*pedagogy*from*content knowledge*? - What are the best ways to develop one’s
*pedagogy*? - What are the best ways to develop one’s
*content knowledge*?

#### Bonus Question

At the risk of damaging the discussion by asking too many questions at once, here’s one more:

- Suppose you’re involved in hiring for your department. What are the top three qualities/skills you look for in a math teacher? (Double bonus: Rank them.)

## Comments 11

When I answered the original question I really had the perspective of if I were hiring a teacher in mind. I might have a slightly different perspective than most being in a middle school. When I have helped with hiring in the past at the middle school level, most applications for math teacher are elementary teacher with a math background, and few/none with secondary education certification:

1) Secondary Education Math Certification – give me assurance they have the content area knowledge to understand the material, prove and explain why it works, understand the sequence and long term goals of where to go with the material.

2) Classroom management skills – a teacher can not give every student an equal opportunity to learn if the classroom is not in control at all times.

3) Comfortable with/have examples of using technology in the classroom – we have so many awesome technology tools available to us today (cough cough DESMOS). Technology has to be used as a way of presenting material to students, developing material and concepts, preparing students for the real world, and engaging students in lesson, I think that in today’s classroom technology is an integral part of pedagogy.

These are my top 3. In addition I would add willingness to work with others, and wanting to improve their craft. The original discussion was about teaching ability VS content knowledge. I think that if a applicant has the above traits, and is willing to learn, in a short amount of time administration could help them develop their ability to teach.

I feel as though my pedagogy development started before I even knew what it was – as I learned math in school. Much of how I taught early in my career was based off of how I was taught. Since that time, I have slowly shifted my practices based on being exposed to other teachers (through PLCs, conferences, etc..).

I don’t think you can separate the two. Being in the role of a student to learn content and see new pedagogy modeled is the best way to improve both, which occurs at the same time.

Unlike some commenters from the original post, when I started teaching math, I was 95% pedagogy and 5% content knowledge. My bachelor’s degree was a double major in literature and creative writing. I’ve spent the better part of the last 10 years developing the content knowledge of mathematical development in children along with the steady progression of my own personal pedagogical perspectives and beliefs.

Why is this anecdote important? Because it is my belief that both, pedagogy and content knowledge, are inseparable and must be developed in tandem as a teacher matures. Robert, John, and Bridget all started with the content and I started with the pedagogy, but we all continued to seek out our own professional development which led us all to where we are now in our careers.

Finally, as a diehard special educator and student-centered advocate, I think that none of this matters because the students will always be different. The pedagogy we learned will change or get redefined. The content focus will shift and the lens will magnify or broaden so our content knowledge will have to along with it.

In other words, my answer is that neither of these things is particularly more important than the other, but give me a teacher candidate who is 100% learner and I’ll hire them every time.

I was just reading a paper (Okita and Schartz on Recursive Feedback) that claims content knowledge can be gained via the teaching process. They lay out three phases for “learning by teaching”. (1) preparation (2) teaching (3) observing. The focus of their paper is the learning that happens in the third phase: seeing how one’s students take up the knowledge you have imparted to them. They called the information the teacher gets via this observation “recursive feedback”. The paper had impressive results showing the recursive feedback is powerful in improving *content* knowledge (separate from pedagogical improvements). (This study focused on a teaching-act done by the learners in the class themselves, not adult teachers)

The implication for teachers is that we need to set up ways in which we carefully observe our students interacting with the mathematical environment, and reflect what that means for the concepts. This is akin to how we reflect for the purpose of formative assessment, but here we’d be focusing our reflection on the concept at hand. Like Robert said in the last thread, he had a B.S. in Mathematics but it was through teaching that the concepts became clear. I have a similar experience.

All the work on Pedagogical Content Knowledge “PCK” from Deborah Ball and others also seems to say the stuff is acquired over time via the continued practice of teaching. PCK is not exactly the content, but it is the type of knowledge that lets you know why someone answered 2x-6=8 with x=10. Carelessness is not the full explanation, right? We have an inkling of what might be going on in the student’s mind, what partial understanding they may have. With more work or by asking the student themselves we may understand a little more.

The short answers to your questions seem to be “keep teaching and you’ll improve in both” but that’d be oversimplifying it. In what ways can we improve the improvement process? Perhaps explicitly reflecting on ideas students brought up in class, and exploring conceptual implications of those. Perhaps reflecting on what we included and what we trimmed in a lesson and how that affected the ideas the students brought to bear on a later problem solving task. Perhaps it is diving into the PCK you already have around a task and taking advantage of that unique position to engage with mathematics from a new angle. What *would* the implications be if we could go from 2x-6=8 to x-6=8. Do we run up against another problem? Can we build a different algebra? These are deep mathematical concepts.

I think maybe one way to speed up the experience process is to literally experience more. Maybe you watch a video of a teacher giving a lesson more, maybe you observe more of your colleagues. I for one would love a database of videos of great teaching to discuss with pre-service and in-service teachers. Maybe there’s a way to generate these without running into any student privacy concerns.

1. Yes, it’s possible to separate them.

2. To develop pedagogy, one needs experience.

3. To develop content knowledge, one needs to read my web comic about personified math! The girls have hairstyles matching their graphs, and the current arc has Versine trying to restore archaic trig by banishing calculators and — okay, you caught me, I’m the 25:75 guy from the earlier post who is bad at marketing. (I really do have a comic.)

Seriously, I think it’s possible to separate them, but it’s like taking seawater and separating the liquid and the salt. First, because at a glance, it’s probably only the pedagogy that you see, or are interested in for hiring purposes. Second, because increasing one or the other can be seen as increasing the salt or liquid content – and still not being sure what the overall mix is. Finally, because what’s the purpose in distilling out the content knowledge? Are you hoping to measure something? (Like ratios?) Or hoping to replace it with sugar? (For French instruction instead of Math?) I suppose you could, but it seems like a lot of effort with little to gain from the exercise.

Regarding the other questions, I think experience is the key in developing pedagogy, because as much as I can tell you that you need to use randomized groupings, and even show you the research, you need more than that to change a habit. (In education, none of us are blank slates.) Observation (or modelling, as Nolan said) is possible as an alternative. For developing content knowledge, curiosity. That’s what leads one to have the motivation to research a student question, or engage in discussion with colleagues, or reflect enough to comment on blogs. Which, granted, can funnel back into pedagogy, linking everything together again.

I’ll pass on Question 4, at least for now. Thanks for reading my randomness.

I think you can have content knowledge without pedagogy, but I’m not sure you can have pedagogy without content knowledge. For example, I don’t know that it is possible to write good curricular materials without having both pedagogy and content knowledge, and good pedagogy is developed through experience.

Something I thought about a lot in college was that I thought all teachers — including elementary ones, should experience math content learning through at least 1 semester of Calculus, because Calculus was where I began to see the importance of the basic skills learned in elementary school, and I better understood the danger of teaching “tricks” without concept. Maybe it’s like the difference between depth of knowledge and breadth knowledge — good pedagogy (for math at least) needs deeper mathematical concept understanding but not necessarily the full breadth of courses that come with a math degree. Pedagogy, as mentioned above — isn’t just explaining a concept — it’s being able to discern good materials and methods, and know how to ask questions to prompt learning and discussion, and what to listen for, and manage multiple learning influences.

Just my humble opinion — and it comes at a frustration at the differences in the requirements for K-12 teachers versus college teachers. College teachers seem to be expected to have more content knowledge, with minimal requirements of pedagogical knowledge, while HS teachers are expected to have both.

What an interesting question! I am a bit surprised that no one has brought up the term mathematical knowledge for teaching. Recently, the Association of Math Teacher Educators released the document, Standards for Preparing Teachers of Mathematics, https://amte.net/sites/default/files/SPTM.pdf

The first standard listed in this document is Standard C.1 Knowledge of Mathematics for Teaching.

I am biased, but I think that our program at South Dakota State University does a good job addressing the Knowledge of Mathematics for Teaching. In our program our students take 37 credits of mathematics courses that ALL math majors take, 13 credits of mathematics education courses taught in the math department, and 31 credits of education courses. I think that the key to the success of our program are the mathematics education courses that include both mathematical content and pedagogy. These courses include Geometry for Teachers, Technology for Math Educators, Assessment for STEM Educators, Mathematics Education Capstone, and Mathematics Methods. These courses include both content and pedagogy and are taught by faculty who model good instructional techniques. I think that these are the type of courses that many of you missed that would have made a difference in your experience.

If I did my math correctly, I would suggest the following: 46% Mathematics, 38% Pedagogy, and 16% Knowledge of Mathematics for Teaching.

First of all, I love Andrew’s 100% learner. That’s actually a question I’ve informally asked at interviews…what books have you read, how do you contribute to a PLC, do you read blogs, on Twitter, etc.

To answer your questions more specifically…I think you develop your pedagogy from listening to kids and talking to your colleagues (and do all of that 100% learner stuff). I think to be able to listen to kids your content knowledge needs to have a depth and breadth (as I mentioned in the previous post’s comments). To develop your content knowledge you need to be able listen to your kids and be open to start from where they are.

So from that…I think maybe it’s all too intertwined to separate. For inservice teachers, I think the best professional development involves doing math and talking about task implementation at the same time. This goes to Sharon’s comment and bringing in mathematical knowledge for teaching. I do think that increased content knowledge (and a learner attitude) makes acquiring mathematical knowledge for teaching much easier.

I agree with the original commenters who said that the two are inseparable in math. If I truly know how math works (not just a trained monkey, which is what I was as a student) then I can explain it. If I can’t explain it then I don’t have content knowledge, all I have is a bag of tricks. I say that for my students and for myself. I’ve learned more math from teaching middle school than I did getting an engineering degree in college.

To develop your pedagogy, observe experienced teachers. See what works. See what doesn’t work. Reflect on your practice. Think about the story you are telling. Where are the gaps? What will they misunderstand? How can you plug those holes in the story?

Want to learn content? Read a book. Go on coursera. Use Khan Academy. Audit a class at the local college. Lots of options. But remember, the lower level of math you are teaching, the deeper your knowledge of the concept needs to be in order to explain it clearly to a young person. I know so much more about multiplication and division of fractions now than I did two years ago, because after 13 years of teaching Algebra 1, I starting teaching sixth grade math. What used to just be an algorithm became a concept I had to explore using multiple concrete representations. That’s real learning.

And that’s why I think you cannot separate the two.

Have you read up on the TPACK model? It adds in a technology components but it covers pedagogical & content knowledge & the interplay between. You’re probably aware of it, but just in case not thought I’d drop it here as it’s related to the question.

Author

Hi Laura. I’m not familiar with TPACK. Time for a little Googling. Thanks for sharing! 🙂