I've been debating on whether or not to write about this for the past week, but I keep thinking about it so I figure I might as well record it somewhere. My class is in the middle of a unit on probability and last week we were working on this problem from the IMP Year 1 "Pig" unit:
Students played around with it for a bit and we even did some experimental trials before trying to tackle the theoretical probabilities...which is when things got tricky. They started listing all of the possible ways to get a sum of 2, 3, 4, etc. when rolling a pair of dice (pair of die?? not sure? well, you would say "pair of shoes" and not "pair of shoe" so I'm sticking with dice). Anyways, there was a big controversy about whether or not we should count 1+2 AND 2+1 as two different options or if we should just count them together as one option (which, in all honestly, I tried to intentionally bring out by giving students pairs of dice that were different colors).

The group that said we should count them as two different options ended up with 21 total possible outcomes and the following theoretical probabilities:

2            3            4            5            6            7            8            9            10            11            12
4.8%    4.8%      9.5%      9.5%     14.3%     14.3%    14.3%      9.5%       9.5%       4.3%        4.3%

The group that said we should count them together as one option ended up with 36 total possible outcomes and the following theoretical probabilities:

2            3            4            5            6            7            8            9            10            11            12
2.8%    5.6%      8.3%      11.1%    13.9%    16.7%    13.9%     11.1%      8.3%        5.6%        2.8%

Neither side was willing to budge, so I suggested we conduct a HUGE experiment with LOTS of trials across all three of my classes so that we could put the results together and see what conclusions we could draw. We did something like 3,500 trials...and here is what we found (experimental probabilities in brown on the far right):
In the end, there were a few people from the "21 camp" that decided to change their mind and join the "36 camp" but most people stuck with their original idea.

I was reminded of a quote from Les Steffe's work that I read a while back:
"A particular modification of a mathematical concept cannot be caused by a teacher any more than nutriments can cause plants to grow. Nutriments are used by the plants for growth but they do not cause plant growth."
I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think.
Something happened in my class last week that really got me thinking and reflecting. My class was working on this problem...
...when one of the students came up to the board and drew this...
If you haven't already, it might be helpful to think about the prompt and the student's proposal for a bit. Essentially, the question they were considering was "if we started with a perfect square, are all five of the divisions also perfect squares?" As you might have noticed, it is possible to draw it so that it in a way that makes you stop and think. There are a host of interesting things to discuss here including measurement, drawing versus abstraction, and others, but here is what interests me most at the moment...

Another student in the class responded to this student's drawing with a friendly counterargument. In essence, he provided the student with a logical argument about why these five divisions were not squares. The student who had drawn the proposal on the board seemed to recognize the fault in his argument and we kept discussing other cases. Then, on his way out when class was over, I overheard him say, "I wonder what size starting square would make my five case possible?" It seemed to me that this student was still searching for ways to preserve his mental equilibrium; hesitant to make a (perhaps?) necessary accommodation.

All of this got me thinking about the ways our minds work and the role of the teacher in working with students. It's fascinating to me how people work with their ideas. I'm convinced that any lasting change must come from a student. I use the example to make a larger point about our goals in working with students. In the context of this puzzle-like problem, it's pretty easy to suggest that we just keep letting this student play with other size starting squares; to keep letting him play with his own ideas. I know that in my own teaching, it isn't always as easy to preserve this play-like quality with things that we are "supposed" to teach. But to hold students accountable to those things (or what we perceive to be their mastery of that Mathematics) is an oversimplification of the workings of the human mind and a coercive, unjust position to hold in education. Because if we were to force this student to abandon his ideas about that square, to insist that his thinking is wrong, he would inevitably start to reject his own thinking as mathematical. He would inevitably start to surrender his own agency.

I think it brings up some interesting questions about the goals of education. I've been reading The Having of Wonderful Ideas recently (which I highly recommend) and I think Eleanor Duckworth puts it quite nicely:
"Many people subscribe to the goals of the it's-fun (interest) and I-can (confidence) types, but when it comes to detail, almost never does one see a concern with anything other than the-way-things-are beliefs. Lesson-by-lesson objectives are almost without exception of this type, despite the fact that general goals very often mention things like interest, confidence, and resourcefulness. This is, of course, because it is difficult to produce a noticeable change in any of these in the course of one 50-minute lesson. But notice that as a result all the effort is put into attaining the objectives stated for the lesson."
I'm not sure where all of this leaves me other than the realization that when we listen to kids and their ideas, working with students is challenging, perplexing, and a ton of fun.
I recently read 'The Child and the Curriculum' by John Dewey. The essay was written in 1902 and is still relevant today. The article is packed with passages that will make you think...
...but I have tried my best to summarize my reading and interpretation of it through direct quotes from the essay:

The fundamental factors in the educative process are an immature, undeveloped being; and certain social aims, meanings, values incarnate in the matured experience of the adult. The educative process is the due interaction of these two forces.
Instead of seeing these two as an interactive whole, we often view them as conflicting parts, leading to what Dewey views as the "child vs. the curriculum" or "individual nature vs. social culture." Often, he argues, educational movements side with one or the other which leads us to polarized extremism. The two camps, as Dewey describes them:
One school fixes its attention upon the importance of the subject-matter of the curriculum as compared with the contents of the child's own experience...studies introduce a world arranged on the basis of eternal and general truth.
Not so, says the other sect. The child is the starting point, the center, and the end. His development, his growth, is the ideal...Not knowledge, but self-realization is the goal...subject-matter never can be got into the child from without. Learning is active.
Dewey's position is that these two extremes set up a fundamental opposition, left for the theorists, while any settlement on a solution will vibrate back and forth in perpetual compromise. His proposal is that we must refrain from seeing the experience of the child and the subject matter of the curriculum as opposing forces:
From the side of the child, it is a question of seeing how his experience already contains within itself elements - facts and truths - of just the same sort as those entering into the formulated study...
From the side of the studies, it is a question of interpreting them as outgrowths of forces operating in the child's life...
Abandon the notion of subject-matter as something fixed and ready-made in itself, outside the child's experience; cease thinking of the child's experience as also something hard and fast; see it as something fluent, embryonic, vital; and we realize that the child and the curriculum are simply two limits which define a single process. Just as two points define a straight line, so the present standpoint of the child and the facts and truths of studies define instruction. It is continuous reconstruction, moving from the child's present experience out into that represented by the organized bodies of truth that we call studies.
Throughout the essay, Dewey refers to the psychological (of experience and process) vs. the logical (of finality and fulfillment). The two forces are similar to that of the child vs. the curriculum and he argues for "psychologizing" the subject-matter ("restoring it to the experience from which it has been abstracted"):
If the subject-matter of the lessons be such as to have an appropriate place within the expanding consciousness of the child, if it grows out of his own past doings, thinkings, and sufferings, and grows into application in further achievements and receptivities, then no device or trick of method has to be resorted to in order to enlist "interest." The psychologized is of interest - that is, it is placed in the whole conscious life so that it shares the worth of that life. But the externally presented material, conceived and generated in standpoints and attitudes remote from the child, and developed in motives alien to him, has no such place of its own. Hence the recourse to adventitous leverage to push it in, to factitious drill to drive it in, to artificial bribe to lure it in.
And his, perhaps, more action oriented response:
There is no such thing as sheer self-activity possible - because all activity takes place in a medium, in a situation, and with reference to its conditions. But, again, no such thing as imposition of truth from without, as insertion of truth from without, is possible. All depends upon the activity which the mind itself undergoes in responding to what is presented from without. Now, the value of the formulated wealth of knowledge that makes up the course of study is that it may enable the educator to determine the environment of the child, and thus by indirection to direct. Its primary value, its primary indication, is for the teacher, not for the child.
And Dewey's final message to the reader:
The case is of the Child. It is his present powers which are to assert themselves; his present capacities which are to be exercised; his present attitudes which are to be realized. But save as the teacher knows, knows wisely and thoroughly, the race-expression which is embodied in that thing we call the Curriculum, the teacher knows neither what the present power, capacity, or attitude is, nor yet how it is to be asserted, exercised, and realized.
I can't quite decide what that final passage means to me. At present, I take Dewey's words to be a reminder of the subjectivity and intersubjectivity involved with matters of the mind. As a teacher, I make observations of students working and I make inferences about their thinking based on my own ways of knowing the Curriculum. I must remind myself that these are my inferences and that I have no way of knowing the thinking an other because I am not them. At best, my pursuit as a teacher must be to work with them in a constant state of negotiation of meaning; not to direct their thinking until I judge it to be a mirror image of my own.

There are plenty of free downloads of the essay online. I encourage you to read it. Afterwards, come leave your thoughts in the comments.
The 12th graders and I started a new unit/project/problem/whatever last week. It started with this image:

What is this?

We took some guesses before someone correctly identified the image as the readout from a stationary bike machine. If this was the readout, I nudged, then:

What did the course look like?

I let them play with it for a while. They identified some important questions they had and information they wanted:
  • What do the dots represent?
  • How can we quantify this?
  • Does speed matter?
  • Did the bike have gears?
  • What do we mean when we say, "what does the course look like?
Ultimately, they agreed upon some answers and, as a result, gave themselves some space to work within. This part takes time and I really think we need to give it the time it deserves. If we don't let students pose and answer those questions, the rest of the unit suffers. I let them know that we would examine some simpler cases first and return to the unit question somewhere down the road.

So, the next day they came back and and I had them map the course for these two:
This took another class period for them to sort out. By the end, though, they started to feel pretty good about the fact that the dots were a measure of resistance which they connected to the slope of the course at that point. Towards the end of class, someone asked, "What would it look like if you were going downhill?' They talked for a bit and devised a system to handle that, so the next day I gave them this:
This allowed them to have conversations about what it meant when the resistance was positive, what it means when resistance is zero, and what it means when resistance is negative.

It's fun to watch the students construct an understanding of antiderivatives. The three-day span reminded me of a few things:
  • the importance of listening to how students are thinking about and making sense of problems
  • our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
  • allowing them the time to sort things out together

I feel like there are a couple ways to go with this trajectory next:
  1. Stay with resistance, but use a continuous curve
  2. Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
  3. Stay with discrete and resistance, but use a more complicated readout

I'm leaning towards #1. What do you think would be best for the students?
As I have posted about before, I really want to do a problem-based unit this year in which students attempt to answer the question, "How far away is the horizon line?" I'm getting ready to start the unit in about a week, so I have been thinking about it a lot lately. Mostly, I have been thinking a lot about the idea of a line that is tangent to a circle and how students might conceptualize that. The "visual" that I get in my head when I think about this horizon line question is this:
This morning I was sitting around the house with my girlfriend, and I decided to see what visual she might come up with and what ideas she might have about tangent lines. First, I asked her to draw the visual that comes to mind when she thinks about the horizon line problem. This is what she drew:
Pretty fantastic, right?!?! Certainly more artistic than my picture! It was interesting to me how differently two people might be thinking about the same scenario.

Then I asked her to draw a circle. Then I asked her to draw a line that touched that circle in only one point. She drew line #1 below (I added the numbering to make some distinctions here). Our conversation went something like this:

ME: Tell me why you decided to draw it that way.
HER: Well, cause it would only touch the circle in one point.
ME: What would happen if you continued your line?
HER: It would cross the circle on the other side.
ME: So would that work?
HER: I guess not....Well, I was thinking about this (draws line #2).
ME: Why did you decide not to draw that one?
HER: It just seems like it would touch the circle in more than one point.
ME: What if we zoomed in? (I drew the picture on the right)
HER: Hmmm...not sure. I still feel more certain that line #1 would only touch in one spot.

To me, this was really interesting. I wonder about how students think. I wonder about their mental models. And, mostly, I wonder how much we actually listen to them and respond to how THEY think. It can be tempting to tell students about a tangent line in the context of this problem, but that would be a missed opportunity for rich discussion. Perhaps more importantly, it would be imposing a way of thinking on them that is incompatible with how they are currently thinking. You hear a lot of people say that they don't like math. I wonder how much of that is due to the fact that they have learned that math doesn't care about their ideas, that math is always right, and that they need to learn to think more like math.

Ernst von Glasersfeld:

"I have a vivid memory of how our teacher started off in geometry. Chalk in hand, he made a small circular splotch on the blackboard and said, 'This is a point.' He hesitated for a moment, looked at the splotch once more, and added, 'Well, it isn't really a point, because a point has no extension.'

Then he went on to lines and other basic notions of geometry. We were left uneasy. We thought of grains of sand or specks of dust in the sunlight, but realized that, small though they were, they still had some extension. So, what was a point?

The question was buried in our struggle to keep up with the lessons, but it was not forgotten. It smouldered unresolved under whatever constructs came to cover it, and did not go away. In the course of the next few years it was joined by some other bubbles of uneasiness. When we came to infinite progressions, limits, and calculus, we were tacitly expected to think that there was a logically smooth transition from very small to nothing. We were told that Zeno's story of Achilles and the tortoise was a playful paradox, an oddity that did not really matter.

I did not like it, but I had decided to love mathematics anyway. Some of my schoolmates, however, concluded that mathematics was a silly game. Given the way some of it was presented, their reaction was not unjustified.

In retrospect, decades later, I realized that there had been quite a few occasions where the teacher could have resolved all those perplexing questions by one explanation. Shortly after the point episode in that geometry class, the teacher introduced the term 'equilateral triangle.' It was in the days when wooden rulers and triangles were used to draw on the blackboard. The teacher picked up one of these contraptions and showed it to the class. 'This is an equilateral triangle because its three sides have the same length.' As he was holding it up, he noticed that one of the corners was broken off. 'It's a little damaged,' he said, 'it would be an equilateral triangle, if you imagine the missing corner.' He missed a most appropriate occasion to explain that all the elements of geometry, from the point and the line to conic sections and regular bodies, have to be imagined. He could have explained that the points, lines, and perfect triangles of geometry are fictions that cannot be found in the sensorimotor world, because they are concepts rather than things. He could have told us that, no matter exactly a physical triangle is machined, it is clear that, if ones raises the standard of precision, its sides will be found to be not quite straight and their length not quite what it was supposed to be. He could have gone on to explain that mathematics - and indeed science in general - is not intended to describe reality but to provide a system for us to organize experience. I do not think that many students would be unable to understand this - and once it was understood, the domains of mathematics and science would seem a little more congenial."

Something to think about as we start a new school year.
After being on vacation, I wanted to ease myself back into blogging with this short post inspired by a journal article I recently read. I hear people use the teacher/coach analogy a lot in discussions about teaching. Most often, it is used as support for practice in math classrooms used towards developing skills. The analogy has always kinda rubbed me the wrong way but I was never sure exactly why. Ernst von Glasersfeld explains it best:

"Since the days of Socrates, teachers have known that it is one thing to bring students to acquire certain ways of acting – be it kicking a football, performing a multiplication algorithm, or the reciting of verbal expressions – but quite another to engender understanding. The one enterprise could be called “training”, the other “teaching”, but educators, who are often better at the first than at the second, do not always want to maintain the distinction. Consequently, the methods for attaining the two goals tend to be confused."

I have attached the whole article for anyone who wants to read more.

File Size: 559 kb
File Type: pdf
Download File

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.
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'm going to try my best to summarize a lot of what I have been thinking about lately in terms of curriculum design and pedagogy that supports students in "doing mathematics." Although there is more to it than what I write here, the following three pillars seem to be a good place to start:

1. Each lesson should revolve around a single task, prompt, or question

Students can't "do mathematics" if we don't offer them an opportunity. I think it is most helpful to think of this as a shift from the traditional lesson format of lecture, guided practice, individual practice to a more fitting lesson format of launch, explore, summarize. The traditional lesson format is designed precisely to have students mimic the teacher's way of knowing and doing mathematics. The second lesson format shifts the responsibility of doing, inventing, and creating to the student. By offering students this responsibility, we also give them the opportunity to develop mathematical habits of mind

2. Each task, prompt, or question should be student-driven

Many people interpret student-driven curriculum as the notion that students decide what to study. While I think that might be a possibility (by having students generate and pursue extensions of the task, prompt, or question), I prefer to think of this as curriculum that is designed to respond to a student's current way of knowing by always pushing on their equilibrium point. In this way, it becomes difficult for a curriculum to stand entirely on its own, independent of the student community. While curriculum can/should serve as a reference, it is ultimately up to the teacher to craft a lesson that, in a response to an assessment of a student's current mental model, puts the student in a position of cognitive conflict. This iterative process results in a connected mathematical trajectory that is student-driven.

3. The "resolution" of each task, prompt, or question must come from the students

Mostly, I mean here that no body other than the students should provide resolution (ex. textbook, teacher, video, etc.) When students see that the validity of their thinking will be measured by something/somebody other than themselves, they will always end up judging their thinking against a "greater" authority. In these situations, the "doing" becomes solely about figuring out how to arrive at someone else's understanding. When a task is open, the "doing" becomes more about arriving at a resolution that makes sense to the individual student/group. There may be differences in opinion and reasoning among students, which is good. In order for a pair, group, or class to reach a resolution, the authority must come from the logic and mathematics of the students. I would even suggest that any task that prompts students to uncover a pre-existing solution will inhibit their ability to engage in "doing mathematics."

Even though much of this sounds dramatic, I truly don't think it is too far removed from what is possible in even the most standards-driven schools. Thomas Romberg writes of a problem-based approach similar to this in which he sees students "studying much of the same mathematics currently taught, but with quite a different emphasis." I think this is a fitting description of what I propose here. I would push for much of this type of design to occur within a problem-based environment, but I don't think it is necessary. One could just as easily work with students to study, abstractly, a unit on exponential growth, trigonometric ratios, or any other traditional standards-based unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.