Monday, April 11, 2011

Bloom's Taxonomy in Math

First, I should say that I graduated with a math degree and then went on for my masters in math before ever taking any education courses. By the time I went back to take some education courses, I just went to a local school to get certified and move on. That is to say, I don't know that I got the best understanding of how education courses were meant to go.

That being said, maybe I was taught Bloom's poorly, but we were often asked to write various questions that coincide with the different levels of Bloom's. The subjects were often accompanied by various verbs that would help us get started. The person in charge usually took whatever suggestions were made, but that just confused me more.

For example, the lowest level (now "Remembering" but formerly "Knowledge") was often accompanied by the typical verb "define." I can see how that might be a low level recall question. In class I write on the board, "A rational number is any real number that can be written as a fraction a/b where a and b are integers." On the quiz I ask the students to "Define a rational number." This only requires them to spit back out what they were told.

On the other hand, I often like to kick off my math classes by showing the students how it's important to define things precisely in math. I ask students to try to define "table" in such a way that anything that fits the definition must be a table and anything that does not fit the definition is not included. The question could be phrased, "Define the word 'table.'" This is certainly much higher level thinking. Maybe it fits the verb list because the question could be phrased as "Create a definition of the word 'table.'"

At the same time, I can see how it shouldn't be that high level of a question if I've taught the students how to solve the equation "x + 2 = 5" and then give them the question "x + 1 = 8." But where should it go on the taxonomy? What if the context is slightly larger? For example, I teach my students a number of integration methods (substitution, integration by parts, partial fractions, etc.) and then just post an integral question. Students must determine which method works in order to work the problem. This seems like some sort of evaluation must be going on.

Certainly the open-ended WCYDWT problems are high level, but what about more traditional word problems? They are often labeled as typical "application" problems, but are more often taught as simple modifications of problems that have already been worked in class which seems to lower it to more of a recall level.

I think writing Bloom's questions out of context of what's been taught is deceiving. If I show my students a proof of something and then ask them about it later, it's recall, but if they are asked to produce the proof on their own, it moves up in level. Right?

Anyways, in math classes we often teach the methods and then work a number of examples. Quizzes are usually composed of variations of the examples we've presented. While I can see how it might be argued that almost all of high school (and higher) math could be synthesis of all the algebra, addition facts, etc. students have learned along the way, it seems to often be low-level questioning.

Another issue I have with this is the perceived level of difficulty in the questions on the different levels. Maybe it's just me, but coming up with an explanation of why the substitution method is better than elimination for a system of equations is much simpler than actually working the problem in many cases. According to my understanding, though, the explanation is relatively high level while solving the problem should be fairly low-level.

The method of questioning also seems to get wrapped up in the level. I've been told that a multiple choice question cannot go above a certain level in Bloom's and that essay questions are inherently higher level. I think I can see how that tendency may work, but I also think that multiple choice questions could be carefully crafted to ask some deeper concepts while an essay could just be regurgitating something the teacher has already said.

At the heart of it, I feel like I know what are "deep" questions and what are not. I realize that in a field like education the "science" of taxonomy is a little more fuzzy than in the more physical sciences. So, I think I get that Bloom's is meant to be more of a "guide" than a rule-book. That being said, I've been to many professional development seminars or various school meetings where we are asked to classify parts of a quiz we've written or to write new questions that fit this taxonomy and I still feel confused about where to classify things.

Where do you think typical math quiz questions fit in with Bloom's?

1. Definitely agree with the guidelines rather than hard boundaries interpretation of Bloom's. And we have to be wary of dual use words like definition.

Literacy teachers have a simpler framework for questions that I feel like fits more naturally in math. Literal, Application, Inferential and Analytical. I wrote about it a bit here http://mathhombre.blogspot.com/2010/05/glyphs-to-data-to-display.html

I feel like frameworks are meant to be supportive. When they stop working for you, let go or revise.

2. Bloom's is stupid. Ok. It's not stupid but it's one of those things that's misused and so has become stupid.

1. It was intended to be domain specific. That is, math should have its own. Which is one of those "so obvious I don't know why I didn't realize it" kinds of things. I didn't realize it either until I ran into it in some paper. There may be a math-specific one floating around somewhere. I don't know. If not, I nominate you.

2. It's not really a hierarchy. Stupid pyramid. It's just a classification system. You don't work your way up it and students definitely don't learn one and are ready to move up to the next step. I've seen it redrawn before where recall was the one at the bottom and everything else is parallel at the top. Or just sideways so its all equal. The pyramid throws people off. The top isn't better than the bottom and I don't really know why the bottom is wider. That doesn't even make sense.

3. Verb changes don't guarantee content depth. So you're definitely right there.

Mainly the strength is it helps force variety. If you look at your test and it's all in one Bloom's skill you're probably doing something wrong. Or at least you're doing something boring. But yeah, don't kill yourself trying to get to the top of the pyramid, especially if it's forced.

Oh and there's also Webb's Depth of Knowledge and the Marzano/Kendall taxonomy if you're interested. They're all useful in their own way.