Two months ago, I asked, “Which graph best represents the importance of teacher knowledge of mathematical content as a function of grade level taught?”

Twenty-six of thirty-five respondents answered C, matching my answer key. Three out of four math teachers agree: content knowledge is important at all grade levels.

For example:

Sure, you need lots of mathematical knowledge in order to be able to guide students to understanding of the advanced mathematical concepts taught at the upper end of school, but it is also vital that for early years teaching, and throughout elementary school, teachers have a strong knowledge of mathematics. Sure, they might only teach basic number skills, but they need to be able to make connections between ideas, understand the deeper significance of these ideas.

Some picked up on my choice of *importance*, rather than* amount*:

You said it’s about the IMPORTANCE of the content knowledge, not the amount they have. For students to develop concepts, they need tasks that help them to engage in and to connect with mathematical big ideas. From the choice or design of tasks, to the good questions that get asked to help students make those connections, the teacher’s content knowledge is critical – in some ways that’s even more important in the early years, but I think an argument could be made that it’s hugely important across the grades.

And again:

I think that teacher knowledge is equally important at every grade level, but a teacher needs to know more mathematics in the higher grades. If the question were about the quantity of knowledge rather than its importance, then I would choose D.

Not all who chose C would buy this amount argument. Not more/less, just *different:*

But the content is different as grade changes. Calc teachers don’t need to know cognitive structures of place value like K-3 teachers do, for example.

My guess is that those who chose E or its poor cousin D (six in all) would cite *complexity*. Tom wrote,

The more I learn about high school math (second year teacher, now teaching Alg I, Alg II, Pre-Calc), the more I realize how nuanced upper level topics are. I sat in on a Calculus class and was blown away at the difficulty of it (coming from a math major!) – we’re not just cranking out derivatives here.Â While TEACHING each grade level requires specific knowledge of HOW students learn each topic, I think the complexity of the math itself increases. Probably not exponentially, but faster than linearly.

Not so fast:

Too frequently it is assumed that elementary teachers don’t need deep knowledge because they’re just teaching kids how to count and add. How hard could it be? But the thing is, elementary teachers are helping very young children build very sophisticated concepts regardless of how easy an algorithm might be to memorize.

Graph A is my take on the complexity question, my response to “Anyone can teach Math 8.” Logarithms in Math 12? Easy peasy lemon squeezy compared to dividing fractions in Math 8. You know the algorithmâ€“just flip it and multiplyâ€“but can you answer the 13-year-old who asks *why*? Then again, maybe I’ve just missed the nuance of logarithms. Thanks for planting that seed, Tom. By the way, nobody chose A.

Only one person chose B. This truly shocked me. I was expecting a much larger number. After all, the role of the teacher has shifted. No longer the primary source of content, no longerâ€¦

But here’s the thing: dispensing knowledge requires only a little bit of content knowledge. That and a chisel tip whiteboard marker/Wacom pen. Posing differentiated tasks that will engage students in and help them develop an understanding of the mathematics to be learned? Now *that* requires content knowledge. It requires that the teacher understands this mathematics deeply. And yes, content is googleable but you need some mad Google-fu skills to get past the procedural.

At the risk of coming across like one of those nutjobs who finds a war on Christmas in “happy holidays,” what importance is placed on content knowledge in “I teach children, not math”? Kids before content. I get *that* part. Given a choice, I’d pick the pedagogue over the mathematician for my kids. Not even close.Â But “not mathematics”? To me, it paints a false dichotomy:

Planning and implementing learning tasks, assessing and supporting students’ learningâ€¦ these must be guided by an understanding of the mathematics at hand (and how this connects to other ideas students see earlier/later).

A better picture:

In fact, some respondents speculated about which graph best matches the importance of PK and PCK across the grades. Most landed on C.

An interesting comment with pro-d implications:

Content knowledge is always important. In the younger grades, teachers need to be able to build and encourage mathematical ability in young students. If they do not have a solid understanding of math, then they themselves can be wary, and students are given Mad Minutes and the likeâ€¦

Here, the mad minute, a *teaching* practice, is seen as a symptom of a lack of *content*, not *pedagogical*, knowledge. This probably goes against conventional wisdom.

A final comment from David Wees:

What I really wanted to choose was a graph that showed that teachers mathematical content knowledge over time should increase, to demonstrate that they are learning. So while I think C would be ideal, teachers could start anywhere on the scale, provided they are willing to put in the same time exploring mathematics as do their students.

What does this mean? First, “this doesn’t mean elementary teachers need to be versed in differential equations.” Content knowledge can grow with experienceâ€¦ if it’s believed to be important.

Note: I’m wondering if responding to the survey implied anonymity. Please let me know if you wish to have your name attached to your comment.

Lovely graphic (Batman).

Also, have you read Ball’s work on PCK, et al.?

I haven’t. Raymond (@MathEdnet) pointed me in the direction of Content knowledge for teaching: What makes it so special? http://blog.mathed.net/2012/09/rysk-ball-thames-phelpss-content.html

I’m okay with the comment being attached to my name. I love the Batman comic too!

An aside, in the final comment (which I think is mine?), I think it should be noted that I believe that regardless of where a teacher’s content knowledge starts, it should show improvement over time. The same is true of their pedagogical knowledge. I’d be concerned if a teacher showed no sign of improvement in either category over any significant length of time (not that I would know how to actually measure either reliably).

I also think that a teacher whose content knowledge is stronger, and has equal understanding of pedagogy is almost certainly better off. The question would be, is low content knowledge and high pedagogical knowledge better than low pedagogical knowledge and high content knowledge? And if so, does that inform where we should be putting our efforts when faced with people who are low in both types of knowledge?

Okay, I didn’t create the Batman comic. It’s a meme. I’d give credit if I could find it again. To answer your last question, in both. I think it’s foolish to address only one. Fair to say student teachers are low in both? I know I was. Yet, in my teacher qualification program, all efforts were put into PK. “A teacher is a teacher” we were told.

Nice, I made the final cut!

I forget whether I chose the linear or exponential function, but remember that I really wanted to choose (C). However, sitting in on that Calculus class made me pick something that increased over time. What I really would have picked would be something like (D) or (E) after a vertical shift up to some magical level. At this magical level, those of us teaching at the elementary levels may not be able to, initially, teach related rates Calculus problems (though maybe I should just speak for myself!). However, we do know Calculus – we know that the idea of average versus instantaneous is an important differentiation that if we can build early on that will make our students’ introduction to Calculus go smoothly. At the algebra level, even if we’d forgotten how to crank out derivatives with the quotient rule, we know the importance of slope and building the differentiation of velocity and acceleration.

Those are the type of connections we all need to know, even if we never teach calculus. And even if we will never teach related rates, we should all be able to understand the connections that can build foundations for a topic such as that.

Agreed. Man, you would have enjoyed @Trianglemancsd’s NCTM session: http://christopherdanielson.wordpress.com/2013/04/21/the-goods-nctmdenver/

And I’d give full marks for the graph that you describe, too.

Great suggestion for discussion:

Just a few of my thoughts on the batman picture:

(1) the icon of batman is likely to hook many students of this time

(2) why does batman have to be violent/slapping the woman for her to get the saying correct, “the sage on the stage”?

Anyways, my initial response to the original question would be C because I believe that all mathematics teachers should have a sound knowledge base of mathematics to at least college level. The idea of them developing this content knowledge on the job is really problematic since there are several other things they have to learn about the workplace. The tendency to act on their immediate environment would take priority over developing their own knowledge base. Actually, many of them would be in survival mode with low efficacy for teaching mathematics. Teachers having high content knowledge and efficacy for teaching mathematics could make the connections across grades and specifically, should be cognizant of the knowledge required below, on, and above the grade level they are teaching. Particularly, i would suggest a combination of C and E for the depth of knowledge over years of teaching. For most, if not all, teachers will continue to enhance their depth of knowledge over the years with Professional Development.

The combination of low content knowledge and high pedagogy is a recipe for an environment where there is minimal to no student learning at the required level taking place. Persons with such combinations should not be placed in the classrooms. They should be recommended for help and support and checked before considering for entry into mathematics classrooms. Should they fail to develop, maybe they are not suited for the classroom. Similarly, if persons demonstrate high content knowledge but poor pedagogy, this is also serious and dangerous for most students. These candidates would not be able to connect to most students. We really need to think through these variations thoroughly in considering candidates and being cognizant of what is needed to prepare these various combinations to an acceptable quality required in both content knowledge and pedagogy for them to have successful mathematics classrooms. However, persons having high content knowledge and the tendency for developing into high pedagogy would be appropriate candidates for supported mathematics classrooms.

.

Uh, I’m confused by your comments re: the Batman image. Hook what students? That’s not a woman. That’s Robin of Batman & Robin fame. He’s not getting her, er, I mean him, to say the phrase. He’s slapping him because it’s a cliche and doesn’t want to hear the end of the sentence. At least that was my take on it.

I didn’t intend for this to turn into a discussion about which candidates should be accepted into teaching. Rather, how to support teachers (and maybe, given your comments re: on the job training, especially student teachers/ST candidates) as they develop both pedagogical & content knowledge. Acknowledging the importance of PCK would be a start.

I think Lee Shulman, the originator of the term PCK, would struggle a bit with the “better picture” venn diagram. In his 1986 AERA address (and the paper that followed), he was trying to describe content knowledge and teaching knowledge as separate, and PCK not as the intersection (things teachers know about math, things mathematicians know about teaching), but as a different set of things that *only* a content area teacher would know. For example, a test of a math teacher’s PCK would contain questions that a math teacher could correctly answer, but a mathematician or a teacher of another content could not. In the overlapping circles of the above venn diagram, it looks like PCK is the stuff *both* teachers and mathematicians would know, which wasn’t Shulman’s original point. But that was the relatively easy part, it seems, as 25+ years later we’re still struggling to pinpoint exactly what all that PCK “stuff” should be!

Thanks, Raymond. This is interesting. I was thinking of PCK as the intersection of a teacher’s own pedagogical knowledge and content knowledgeâ€“ what the teacher knows about teaching and what the teacher knows about mathâ€“ not in terms of the knowledge of teachers vs. mathematicians. I’ll have to check out the article you referenced on Twitter. I’m struggling with PCK as being completely separate from CKâ€“ what a teacher knows about how children learn mathematics being disconnected from his/her own understanding of the mathematics that he/she teaches. For example, when teachers struggle to understand the left to right partial sums method of adding two-digit numbers, this should fall under PCK for it’s the method that most children go to naturally before we teach it out of them. However, I find some teachers believe that addition

musthappen from right to left which falls under CK.Deborah Ball has the best grip on this (see http://blog.mathed.net/2012/09/rysk-ball-thames-phelpss-content.html) but the part I struggle with most is how to sepearate what she calls “Specialized Content Knowledge” from either “Knowledge of Content and Students” and “Knowledge of Content and Teaching.” Even if I could make the theory clear in my head, as a practical matter and teacher educator I’m not sure to what degree it’s useful to focus on specialized mathematical knowledge

withoutconsidering students and teaching.