I love the Khan Academy videos. I love them because they have opened up a discussion about education, learning, theory, and bringing all of it together in meaningful ways to support students in their mathematical development. Relatively new to the education scene, Khan Academy receives a lot of criticism. In the most recent critique of Khan Academy, we have been amused by the "Mystery Teacher Theater 3000" videos.

First, I think that Khan Academy gets a bit of a bad wrap. I'm pretty sure it was not Sal Khan's intent to "revolutionize" or "reinvent" education by posting his videos online. In my opinion, a public misconception about mathematics, learning, and education has allowed KA to rise to it's current position. The public views mathematics as a body of factual information, one that is passed from the "knower" to the "student." Learning, then, is perceived as memorizing/understanding factual information. This public perception applies to much of education in general, but especially to mathematics education. 

Recently, the MTT3K video exploited some mistakes in Khan's explanation of multiplying positive and negative numbers. Among other criticisms, the makers of the video point out that Khan mistakenly refers to the "transitive property" and explains "-4 x 3" as "negative fours times itself three times." In addition, the video points out some questionable pedagogical decisions by Khan which might be confusing or difficult for students first learning the topic and that fail to explain why the multiplication facts are true.

I welcome the dialogue that is happening surrounding these videos, but want to take the discussion of why Khan Academy is an ineffective learning tool one step further. In doing so, I want to reference Constance Kamii and her application of Piagetian principles to teaching:

"Piaget's theory of memory is very different from the empiricist belief that 'facts' are 'stored' and 'retrieved.' According to Piaget, a fact is 'read' differently from reality by children at different levels of development because each child interprets it by assimilating it into the knowledge he has already constructed."

In this next quote, she refers to facts about addition. I think the same could also easily be said for multiplication:

"In Piaget's theory, there is no such thing as an 'addition fact.' A fact is empirically observable. Physical and social knowledge involves facts but not logico-mathematical knowledge. The fact that a ball bounces when it is dropped is observable (physical knowledge). The fact that a ball is not appreciated in the living room is also observable (social knowledge). But logico-mathematical knowledge consists of relationships, which are not observable. Although four balls are observable, the 'four-ness' is not. When we add 4 to 2, we are putting into an additive relationship two numerical quantities that each of us constructed, by reflective abstraction; 4 +2 equals 6 is a relationship, not a fact."

At first, these quotes seem hard to swallow. However, Piaget (and others) have conducted many experiments to show that even children who "knew" facts about addition lacked the ability to perform on different tasks involving those same operations.

We construct in a way that is personal to us and relative to our current ways of understanding. My point in all of this is that, video or no video, you can't make someone learn. The 'knowing' that is so prized in education comes from every person's natural ability to think and from the human tendency to maintain internal/cognitive equilibrium. My suggestion is that we don't tell students what they should know or how they should think. Give them a task that pushes on their way of knowing, let them do mathematics, and watch and listen closely as they sort things out together.
I am nearing the end of my first year in grad school. Even more than I anticipated, blogging and getting feedback from readers has been exceptionally helpful in pinpointing the things in mathematics education that really matter to me the most and in finalizing my action research proposal. With that said, I wanted to post my preliminary research proposal in case anyone is interested in reading it. It is rather long and still relatively "un-polished" but I owe much of it's creation to readers who regularly push my thinking. Thanks!

How does a mathematics classroom centered in habits of mind support students in realizing mathematical agency?

I hear people talk a lot about the importance of context in mathematics....usually, I think, for all the wrong reasons. There is a common misconception that if we show students how they can "use math" that they will find it more enjoyable and see the value in learning it in school. I'm pretty sure the problem isn't that students really want to know how this stuff is valuable in usage. The problem, I think, is that when the subject is relegated to procedures/facts/tricks it often disrupts the way a student naturally thinks.

Applied Math

By it's very name, this approach implies that we are "applying" something. Usually, there is a mathematical topic presented and then problems are selected in which students have to apply what they have learned to solve these problems. Sometimes, the problem might come first and then topics are presented (as facts) to provide students with the "tools" necessary to solve a problem ("necessitating" content). The focus here is on math as a tool to solve a problem. The assumption is that the mathematics exists before, and inspires, the problem.

Doing Math

Doing math is an act of creation by students. Usually, there is a problem/task/situation that requires students to think in the form of reflective abstraction. Mathematical "facts" arise as generalizations students make by looking for patterns and consistencies. These facts might, then, be explored in the abstract. The focus here is on problems as an entry to creating mathematics. The assumption is that the problem exists before, and inspires, some mathematics (I say "some" because although that may be anticipated by the teacher, it is ultimately defined by the students).

As I see it, yes, context is important but not to show students how they can use math. The context is important to show students how their mathematics is a natural extension of how they think and live in the world. Constance Kamii says it best:

"Most math educators think about verbal problems (word problems) as applications of computational 'skills,' rather than as the beginning point that eventually leads to generalized computation, without content, context, or practical purpose."

Kamii, C. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York, NY: Teachers College Press.
Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem.  I would love some blogger critique so please let me know what you think...

HORIZON: How far away is the horizon?

This is a pretty famous problem in mathematics (I think). I remember seeing it and being intrigued. I think there is an opportunity for pretty rich math here. Students would have to create an abstraction to represent the situation, develop rules about their abstraction, and use them to approximate an answer to the unit question.

PROBLEMATIC PACKAGING: How can you optimize this packaging?

This one arose out of a curiosity I had about optimizing parking lot design. I thought it might be fun as a puzzle where students get a package and different "items" (blocks of two or three sizes and shapes and worth various points) and they have to figure out how to maximize their point value. Could be extended by looking at different point values or different size boxes.

GET ME OUTTA HERE: How do you know if a game is solvable?

I love these puzzles. I'm not exactly sure how to turn this into a full-blown unit, but I'm pretty sure it can be done. This might not be the best unit question. I have also considered giving students a specific puzzle and asking "What is the fewest number of moves?" Maybe we can extend it from there and move into "solvable" setups?

7 CLICKS FROM KEVIN BACON: What is the minimum number of clicks to get to Kevin Bacon from ANY person on Facebook?

This one I'm REALLY not sure about. Obviously, it comes from the popular game about social connection (although it might be a good idea to use a celebrity my student will have actually hear of). I thought there might be some connections to graph theory here and it might be a nice extension of a combinatorics unit that I will do again next year ("How many combinations are there at Chipotle?"). Feedback please.

WIN, LOSE, or DRAW: Can you draw this without lifting your pencil or going over the same line twice?

The Bridges of Konigsberg is such a GREAT problem that I want to turn it into a whole unit next year. I'm not sure I love the context of the original problem because, to students, I think it feels like a pseudocontext. I thought about giving them a crazy network diagram and asking the unit question about that (maybe with "bridges" showing up along the way?). However, I usually prefer to start with a concrete situation and abstract from there....not sure.

...something more complicated but this was the best picture I could find for now

GOING COASTAL!: How long is the California coastline?

I did a similar project this year where students figured out the area of the Koch Snowflake. I'm thinking about trying to start concrete and move abstract next year with this one. I'm sure the Koch Snowflake will still rear it's ugly head somewhere in our investigation.