This is just a quick post to reflect on something that went really well this past week. We have been working on a modification of the Pig Unit from IMP. In the process of investigating, we were trying to wrap our heads around this problem:

If you roll a die two times in a row, what are the chances that you will get at least one 1?
What if you roll 3 times? 4 times? "n" times?

We usually do these problems in small groups (3-6 students) but I decided to mix it up for this one. I split the class into two large groups (each of about 12) and we formed an "inner circle" and an "outer circle." The inner circle got 5 minutes to form some initial thoughts about the problem individually and then we opened it up for a whole group, collaborative sense-making. I told them they were "done" when every person in the group both agreed with and could explain their collective solution. The other half of the students (the outer circle) were partnered with someone on the inner circle and they were to be silent observers, recording the ways in which their partner was positively contributing to the group discussion. I also removed myself from contributing to their efforts in any way.

Afterwards, we talked about ways the group worked well together, ways they didn't work well together, and we identified the habits/strategies that were useful for each group. There were a lot of amazing things that happened and some really great discussion that followed. Mostly, I loved that the students had to (and did) push, challenge, and support each other in an attempt to justify their solution and prove to each other that it made sense.

I was left wondering:
1. Why doesn't this work as well with a WHOLE class discussion/collaboration? What would support that? I think this post and research might help....haven't read it yet.
2. Does this collaboration happen with small groups also? It's harder to tell when you are monitoring multiple groups.
Constructivism and "discovery" learning are two popular methodologies in progressive mathematics education that are easily misinterpreted and, sometimes, get confused for being the same thing. I have been thinking a lot about both lately and would like to 1. try to outline what I believe to be a definition (comparison) of both and 2. propose some possible implications and questions as they pertain to mathematics education.


Constructivists suggest that we cannot be certain of any absolute truth but that people construct, or create, "knowledge" based on their experiential reality (or their interactions). "Truth" or "knowledge" is a social construct attained when people agree on particular mental models that appear to be consistent with our collective experiential reality. However, this social agreement does not necessarily imply "universal truth." For instance, people once believed that the world was flat (collective social agreement) but, as they discovered, this turned out not to be a "truth."


Discovery learning is based in a different belief about knowledge and truth. This theory posits that there is a certain body of knowledge available and that teachers can help students "come to know" (or discover) this knowledge by implementing well-designed tasks in the classroom.

As it pertains to mathematics, I believe constructivists would suggest that the collective set of rules, procedures, and beliefs we call "mathematics" are not, in fact, universal truths but rather mutually agreed upon constructions from the mathematics community. I have some questions about how the implications of this theory for the teaching of mathematics:

1. Social constructivism posits that we rely on other people to both challenge and confirm our ways of knowing. When they have been mutually agreed upon, they become our ontological reality (our truth). What happens when students agree upon an ontological reality that is different from that of the teacher or that of the mathematics community at large? Whose reality is deemed correct in this instance?

2. Constructivist learning theory suggests that "learning" happens when an existing mental model is confronted with an experience that does not conform. The learner must then revise their mental model to accommodate this cognitive conflict. This suggests that the role of the teacher is to present students with situations that create this cognitive conflict (or disequilibrium) in students. Doesn't this imply that the teacher is pressing on students to rewrite their mathematical ways of knowing in a way that is more consistent with that of the mathematical community? Isn't this, in effect, discovery learning?

3. The questions above seem to indicate that it is impossible to create a socially just form of mathematics education that involves topics about which there is already a socially agreed upon "reality" (content standards). Because there is already an agreed upon reality, the students' interaction with them will always be measured against, and subject to, this greater authority. What, then, is the purpose of mathematics education and how can it be socially just?

Please, help me "sort this out..."
I have been thinking a lot recently about the subtleties of problem-based approaches to math education. The following is a gross oversimplification, but I think it will illustrate the essence of what I am interested in. Let's compare two well-know approaches, Interactive Mathematics Program and Exeter Math:

Interactive Mathematics Program (IMP)
The Game of Pig - Year 1

MODEL: the unit starts with a "unit question/problem" and then smaller sub-questions (sometimes out of context of the unit question) are explored to deepen understanding before returning to the unit question again at the end of the unit.

Exeter Math
Year 1

MODEL: students are given a set of problems that are, more or less, completely unrelated. Each smaller problem stands on its own; it does not tie in to a larger context.

EXAMPLE: This unit follows the following progression:

1. Students are introduced to the unit question by playing games of Pig and thinking about strategy.

2. The first section detours from the unit question to help students "define" probability. There are investigations about gambler's fallacy, experimental versus theoretical probability, and measuring probability between 0 and 1.

3. The second section introduces "rug diagrams" as a way to represent probability. There are some investigations about this and the end of the section ties rug diagrams back to coin and dice games.

4. The third section looks at how things play out "in the long run." It involves investigations about the law of large numbers and expected value.

5. The last section looks at a simplified version of the game of Pig before returning to the original unit question.

EXAMPLE: Here are some problems, in sequence, from the Year 1 Exeter problem set:

Clearly, there are advantages and disadvantages to both approaches. What are they? Do you prefer one over the other? Why? In short, "What's the Difference?!?"
What does it mean to "educate?" Here is's definition of education:
The next section was alarming:
I'm just (very) concerned with the perception that education is all about the transfer of "facts." I know this isn't a new argument, but I can't help but feel this is a pretty critical time in education (particularly for math). National and state budgets are in bad shape. People perceive math as a fact-based discipline. Computers provide fact-based instruction effectively. I find all of this really alarming.

There was a great post by Grant Wiggins recently in which he suggests:

"I propose that for the sake of better results we need to turn conventional wisdom on it is head:  let’s see what results if we think of action, not knowledge, as the essence of an education; let’s see what results from thinking of future ability, not knowledge of the past, as the core; let’s see what follows, therefore, from thinking of content knowledge as neither the aim of curriculum nor the key building blocks of it but as the offshoot of learning to do things now and for the future."

And another section from the same post in which he references Ralph Tyler:

"According to Tyler, the general aim is "to bring about significant changes in students’ patterns of behavior.” In other words, though we often lose sight of this basic fact, the point of learning is not just to know things but to be a different person – more mature, more wise, more self-disciplined, more effective, and more productive in the broadest sense."

To me, these quotes speak to what it means to "educate." I think we would be well suited to redefine math education as one that nurtures and cultivates habits of mind. These are the habits that influence future action, that inspire innovation, and that, in my opinion, are what it means to be "educated." More to come...
I have been writing a bit about the problem with "right" answers in the mathematics classroom lately. I think it is a big concern. Upon further reflection, I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:

"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."

There is some great reading about how this relates to "mathematics for all" and teaching for social justice. Building this community is extremely difficult. Students have developed expectations about what learning and teaching mathematics "should" look like. As a result, if we are to promote this type of student discourse it becomes necessary to renegotiate the "didactic contract" (Brousseau, 1986). This "contract" both explicitly and implicitly outlines the role of both teacher and student, the expectation for classroom discourse, and, as a result, the locus of authority.

This type of discourse simply isn't going to take place if "mathematics" is practiced as "solving" a bunch of related problems (read: 1-30 odd). Instead, find the big idea, pick one rich question to lead an investigation of that idea, and then let the students sort it out. You are likely to see students doing mathematics. To return to Schoenfeld,

"This <is> their mathematics. They <have> ownership of it, not only in the motivational sense, but in the deep epistemological sense that characterizes the true mathematical knowing and understanding possessed by mathematicians."

You always hear people say, "kids don't like math!" don't like feeling dumb. People don't like feeling dumb. Feeling dumb comes from being told you're "wrong" over and over. Math education effectively does this better than just about any other subject in school. It's no wonder that people are sending out tweets like this:
Things get even worse when we start throwing grades in the mix. Now, all of a sudden, not only are we telling kids they're wrong...we're punishing them for it. It shouldn't be surprising that students are reluctant to take risks, persist with difficult problems, and trust their own thinking; they don't want to get it wrong. Today, I saw this tweet:
To which, I replied:
On second thought, I should have written: "It CAN'T exist independent of a specific group of kids." I mean, of course we know it can, and it does, but isn't this putting the cart before the horse?!? When your curriculum is decided in advance, you've already told students what the measure for "knowing" is and if they don't meet that then something is wrong with them. I've written about other curriculum/lesson structures that respond more directly to students but my concern goes beyond that.

I have had the pleasure of "mentoring" (which is a total misnomer...we really just learn from each other) one of our new teachers this year. She's amazing. Yesterday we were planning a probability unit together and we were trying to figure out an answer to the unit question that would drive our investigation of probability. In the process, her and I were collaborating, conjecturing and testing, investigating smaller problems, drawing diagrams, and listening closely to each other. Honestly, I don't really care if, as a class, we are successful in calculating an exact answer to this problem. I want students to come up with an answer that makes sense to them, that responds to their ways of knowing, and that is reflective of their deepening understanding of chance and probability. To quote a brilliant mentor of mine, "I guess my point is, 'solving' the unit problem will certainly be in the discussion but we'll be 'successful' moreso because students invented the solution, rather than being told." Mostly, I just want the students to have the same experience we had; the experience of playing with and doing mathematics.

I know there are "realities" in a lot of schools that make this difficult. Benchmarks, AYP, API, etc, etc. There are all sorts of measures of success and progress out there (most of which, I would argue, are false indicators of "learning"). Well, here's my vote for success/progress:
I want people to know that we are all mathematical in our thinking...maybe just not in the ways that school has defined mathematics. I want fewer people to hate math.