I'm getting ready to plan a problembased unit on probability and I'm looking for a unit question that can drive our entire investigation. So, this morning I spent about an hour looking online and I was shocked by what I did NOT find. Probability has to be one of the most interesting an relevant topics in high school math and there was not much that caught my attention (and if it doesn't catch my attention, I'm pretty sure it won't catch the students' attention). Here are possible questions that I'm considering right now:
1. The Birthday Problem What is the probability that, in a room with "n" people, at least two will have the same birthday.
2. Probability Game I don't know of a great one, but I was thinking about using a "game" to drive our investigation (something like Pig...but preferably more complex)
3. Sports What is the probability that Team X will make it to the NBA Finals (or something along those lines)
4. Insurance Pricing Is insurance pricing fair?
The only thing that I really DON'T want to do is Casino/Gambling stuff. Other than that, any suggestion for a unit question is fair game...
p.s. I'm really tempted to create a tab where "we" can start to organize unit questions for reference. Good idea?
I have written about these "habits of a mathematician" before, but the list is changing, improving, and evolving so I wanted to share my most recent thoughts. Mostly, this list has grown out of a desire to show students that "doing mathematics" is not about trying to find an answer that exists at the back of a book (or elsewhere); doing mathematics is about creating, exploring, inventing, and authoring. I owe a lot to these people for their work before mine in helping shape this list: Look for Patterns
I am always on the hunt for patterns and regularity I bring a skeptical eye to pattern recognition I look for reasons why a pattern exists
Solve a Simpler Problem
I start with small cases I "chunk" things into small sub problems I take things apart and put them back together
Be Systematic
I experiment methodically and systematically I make small changes to look for change and permanence
Stay Organized
I use charts and other methods to organize information I organize my work in a way that can be referenced by others
Visualize
I draw pictures and diagrams to represent a situation I invent notation or representations to facilitate exploration
Experiment through Conjectures
I ask "what if..." I invent numbers to test relationships I use specialized cases to test my ideas I modify conjectures to continue/deepen exploration
Be Confident, Patient, and Persistent
I am willing to take risks I will explore even when I'm unsure about where to start I do not give up easily
Collaborate and Listen
I listen carefully to and respect the ideas of others I effectively use strengths of people around me I ask people for their thoughts when they seem hesitant I don't dominate a conversation
Seek Why and Prove
I seek to understand why things are the way they are I look for clues that help me understand why I create logical arguments that prove my ideas
Generalize
I look to understand ALL cases I invent notation that helps me generalize I algebraically represent the structure I see
I'm most interested in how a curriculum that is based on these might help students in "doing" their own mathematics. It's not enough to simply slap these up on wall and reference them often (although that might be a start). As I see it, there are some serious implications on curriculum, pedagogy, and assessment that go with these.
Curriculum
Most importantly, if we expect students to create their own mathematics we need to offer them the opportunity to do that. I think the best response to this is to center curriculum around rich, engaging, interesting problems (in short, a problembased curriculum). This can still be done within a traditional, standardsbased classroom but it definitely would suggest relying less on textbooks as an instructional tool. Here are two examples of what I see as possible problembased units. Assessment
At the end of each unit (or possibly periodically through the unit) students will select work(s) that demonstrate their use/growth in relation to these habits. They will fill out a reflection sheet (coming soon) and then submit that work to their portfolio. Three times a semester, students will evaluate their portfolio and will receive evaluations from a peer and myself (and possibly a parent/guardian?). Pedagogy
In the past, I have prompted students to implement these habits (both explicitly and implicitly). The more I think about this, I'm inclined to think this is doing them a disservice for two reasons: 1) I'm telling them what to think or encouraging them to understand in my way and 2) they aren't the ones creating mathematics. Essentially, all I should do is help introduce the habits, provide students with opportunities to do mathematics, and help give them feedback as they grow and develop.
As always, a work in progress...feedback welcome!!!
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 studentdriven Many people interpret studentdriven 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 studentdriven. 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 preexisting 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 standardsdriven schools. Thomas Romberg writes of a problembased 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 problembased 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 standardsbased unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.
In " Mathematical Thinking and Problem Solving," Judah Schwartz (somewhat) jokingly makes the suggestion that we should never ask students to engage in a problem/task that has only one right answer (if even a right answer at all). The idea really caught my attention. Schwartz knows that the suggestion may be unrealistic, but striving for that would have profound impacts on many aspects of math education. This idea resurfaced for me this week. I was finishing up a lesson sequence designed to get students thinking about exponential growth and percent change. We were working on this last problem (video) in that sequence: I was really dissatisfied with how it was went. I heard students saying "I don't know what to do" more often than usual, which was frustrating. I could be wrong here, but after reflecting on the lesson I couldn't help but think that the elusive "right answer" might have been the problem here. It is almost as if I was taunting students by covering up something that they were supposed to somehow (magically) discover. All of the students had the original dollar and they all had the final copy. It's not a stretch here to at least experiment with different percentages and see how close you can get…but nobody did that without me suggesting it. I'm still not sure exactly what went wrong here, but I would love thoughts and suggestions.
Helping students to trust their own thinking and feel comfortable experimenting is frickin' hard. That doesn't mean we should resort to easier, more mundane forms of math education that make us, as teachers, feel better about ourselves. UPDATE 3/22 I just came across this video (look for the one by Annie Fetter). It seems to be exactly in line with what I am thinking with this post.
Confession: I obsess over the details of curriculum design. For me, having the opportunity to design and create challenging, perplexing learning situations is one of the best parts of my job as a teacher. I also think that the details of these designs can have a lasting impact on our students. Last time I posted about this (read this first), the class left off knowing that population growth was not linear (as they had expected) but that, in fact, it was increasing by and increasing amount each period (as they put it). Here is how the sequence of curriculum design has gone since then and I'm wondering....did I get this wrong? 1. The next lesson was designed to get them thinking about multiplicative structures/exponential growth. We experimented with the problem below and then did a bunch of variations/extensions to eventually come up with this generalization (in their words): starting value (multiplier)^"iteration1" 2. All of the variations from the previous lesson were whole number "multipliers." So, I decided to follow up with this video (thanks Dan). We talked about how big the 100th dollar would be and how big the "nth" dollar would be. 3. Now that they had an idea about percent growth, I decided to push their thinking with this "sequel." I gave them the resulting dollar bill. As I have mentioned previously, I am really pushing myself with every lesson to put students into a place of cognitive dissonance or disequilibrium. With that in mind, I'm wondering:
1. Did I get this curriculum sequence wrong? 2. What would have made it better in terms of promoting disequilibrium?
I always start class with a "journal" prompt. Sometimes I like to use puzzles to start the day on a fun note. Yesterday, I gave the class this famous puzzle as our journal prompt (if you haven't ever seen this, try it before you read on):
Today we were talking about possible solutions when a student put the following solution up. "Four straight lines"…this is why kids are awesome.
"Piaget suggested that the creation of new cognitive structures springs from the child's need to reach equilibrium when confronted with internally constructed contradictions; that is, when perception and "reality" conflict."
"Nussbaum and Novick suggest a threepart instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)
We started our unit on exponential growth yesterday and I'm pushing myself really hard to take this approach to my curriculum design. Here's how it has played out so far:
1. I opened with this population counter and asked students to make their best estimate about what the counter would read at an exact date and time. They were just watching the counter so it was pretty difficult to get any data or make any calculated guesses.  
2. I asked them how they thought we should break down the problem and what information they would need. The responses were somewhat varied, but most hovered around something like the response at the right. It was obvious that they were assuming a linear/constant growth (their "perception").  
3. Based on their existing perception, I gave students (essentially) what they asked for and prompted them to make some predictions about the next few decades.  
4. Then, I revealed this slide. You can imagine their surprise when their constant growth didn't match up with "reality." I think this is what Nussbaum and Novick mean when they say "cognitive conflict" or when Piaget talks about "when perception and reality conflict."  
5. I asked them to revisit their predictions based on what they noticed in the new data. There was some variation in their response, but the common theme was something like the image on the right.  
This is where we left off for today. In my opinion, students have made the cognitive shift I was hoping they would. They came in thinking population was linear and left knowing that it isn't. They also have some intuitive ideas about how to make predictions based on how it actually is growing. It's sooooo tempting to step in and explain to them about percent increase, exponential growth, and (for some) derivatives. But, if I did that, I would completely rob them of the opportunity to construct that understanding for themselves. I do know one thing…they're about ready for this awesome video tomorrow!
My school's I.T. Guy would probably take issue with the title of this blog post but, for me, this is why kids are awesome...
Last night, "60 Minutes" ran a segment on Khan Academy and the influence it is having in improving mathematics education. Here is the letter I sent them today:
I just finished reading three selections from Alan Schoenfeld's "Mathematical Thinking and Problem Solving." The book is incredible and I would highly recommend it. It features eight different papers that are each counterbalanced by a written response or discussion. This may be a lengthy post but I wanted to record my thoughts here for later reference. In an effort to promote conversation about these, I would encourage you to find at least one quote that is interesting to you and post a thought or question in the comments.  
Research and Reforming Mathematics Education (Schwartz) NONTHREATENING REFORM 1. "If the important idea that one devises is to find its way into the way mathematics is taught, learned, and made in schools, then it is important that it not appear to threaten current practice." (p. 3) 2. "It should appear to augment, rather than replace." (p. 4)
REFORM CANNOT BE 'TEACHERPROOF' "However, let it be clearly stated and recognized, that any curricular development that takes place without the continuing central and focal involvement of teachers is almost certain to fall short of its potential."
RESISTANCE FROM LACK OF CLARITY "Less visible, but equally if not more important, is the role of the social network in legitimizing the adoption of new habits of mind. It is difficult for students well into a school career in which mathematics has been an endless series of incompletely understood calculation and manipulation ceremonies to shift gears and to exercise in class their curiosity and inventiveness. It is difficult for teachers and principles, who will be held accountable to superintendents and school boards to imagine mathematics classes in which mathematics is 'discovered' rather than 'covered.' It is difficult for school boards to imagine that the mathematics they learned and the way they learned it is possible not universal or eternal in its importance. All of these groups need support as they develop new 'habits of mind.'" (p. 6)
Doing and Teaching Mathematics (Schoenfeld) STUDENTS DEFINE MATH THROUGH THEIR EXPERIENCES 1. "In turn, our classrooms are the primary source of mathematical experiences (as they perceive them) for our students, the experiential base from which they abstract their sense of what mathematics is all about." (p. 53) 2. "The activities in our mathematics classrooms can and must reflect and foster the understandings that we want students to develop with and about mathematics." (p. 60) 3. "When mathematics is taught as received knowledge rather than as something that (a) should fit together meaningfully, and (b) should be shared, students neither try to use it for sensemaking nor develop a means of communicating with it." (p. 57)
'DOING' MATHEMATICS IN THE CLASSROOM 1. "The implicit but widespread presumption in the mathematical community is that an extensive background is required before one can do mathematics." (p. 65) 2. "I work to make my problemsolving courses 'microcosms of selected aspects of mathematical practice and culture,' in that the classroom practices reflect (some of) the values of the mathematical community at large." (p. 66) 3. "Mathematics is the science of patterns, and relevant mathematical activities  looking to perceive structure, seeing connections, capturing patterns symbolically, conjecturing and proving, and abstracting and generalizing  all are valued." (p. 68)
Classroom Instruction That Fosters Mathematical Thinking and Problem Solving (Romberg) THE CONSTRUCTIVIST LEARNER 1. "What is constructed by an individual depends to some extent on what is brought to the situation, where the current activity fits in a sequence leading toward a goal, and how it relates to mathematical knowledge." (p. 300) 2. "The 'intended' curriculum can only include our best guesses about what will both interest students and lead all toward development of mathematical power. At the same time, the 'actual' curriculum depends on teacher choice, and the 'achieved' curriculum depends on each student's interest and prior knowledge." (p. 300) 3. "Nussbaum and Novick suggest a threepart instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)
CURRICULUM DESIGN PRINCIPLES 1. "Classrooms should be places where interesting problems are explored using important mathematical ideas…This vision sees students studying much of the same mathematics currently taught, but with quite a different emphasis." 2. Romberg calls for a problembased approach centered around his five principles of curriculum design (conceptual domains should be specified, domains should be segmented into 24 week units, students are exposed to domains as they arise naturally in problem situations, activities are related to how students process information, and curriculum units should always be considered problematic).
