Wednesday, July 23, 2014

"__BL" : Education's Obsession With Labels

Last week there was an interesting twitter discussion on the nature of projects versus the nature of problems.
It occurred with specific reference between the differences of PBL (project-based learning) and PrBL (problem-based learning). If you follow this blog or scan the provided tags you will find PBL does occupy some space here. There is also a large amount of posts detailing "tasks". This is a rather artificial term I use to refer to a piece of mathematical work to be done or undertaken.

To me, the potency of all of these ideas is lost on many teachers. Not just my work, but the work of math educators worldwide. (and yes... even some textbook writers. Shock). Teachers love to label different approaches and then subsequently develop (or collect) resources that carry that specific label. Some labels just repackage old ideas while others aim to describe a pedagogy as much as their content. I think such is the aim of both PBL and PrBL. The only problem is that the label shifts the focus away from what is important: the pedagogy.

It is probably easiest to lay out the harmful effects of labels in a bulleted fashion.

  • Labels create commercialization.
Once there is an established "type" of resource, the entire educational machine revs up to churn out books, memberships, sample materials, presentations, and professional development. It isn't long until the original inputs for the label is diluted to suit mass production.
  • Labels attract connotations.
It doesn't take long for the rich understanding (and well meaning) behind a label to become simplified and vilified by those (as ignorant as they may be) opposed to the idea. Just today a chart was tweeted that labeled PBL mathematics as "fuzzy" and opposed to memorization. Ignorant over-generalizations take on lives of their own, and labels create easy targets. 
  • Labels put the focus on one stakeholder.
Whether it is PBL, PrBL, student-centered, or teacher-centered (to name a few), labels highlight but one piece of the educative puzzle. You can't honestly say that a certain type of room is teacher-centered? Or student-centered? An isolation of one of these players renders the entire process null. It never begins. All the players (including the content and culture) are co-implicated in an educational setting. Lecturing is not teacher-centered, if anything the teacher is just a passive mouthpiece void of any initiative. Their role is then to pass on pre-conceived knowledge. That doesn't sound teacher-centered, it sounds more like teacher-proof. 

People sit in the middle and say that their classrooms are "learning-centered". Well... duh. What well-meaning teacher doesn't want (and even think) this to be the case. A "deceit-centered" or "ignorance-centered" classroom is either non-existent or pathological. Why even label that?

Finally, and most importantly:
  • Labels often highlight the resources at the expense of pedagogy.

In the specific case of PBL and PrBL, we are debating what attributes make a educational artefact a problem rather than a project (or vice versa). What that does is pull the focus away from the pedagogy behind the label (the spirit in which they are to be encountered by the students), and place it on the specific instance of content. This means that teachers attempt to collect these artefacts, and once their repertoire is robust enough, they can then execute the "type" of classroom under that label. (i.e. If I can only find enough good problems, I could run PrBL).

This generates a mindset of attainment in teachers. We see it during curriculum renewal; we see it during internships. Teachers scrambling to attain the "stuff" needed to keep up. Labels pull us away from an attunement to the pedagogy behind the resources. Some do a better job at embedding the two mindsets. Recently, the phenomenon of 3Acts has generated a whole new label. The inherently great thing about these problems is the pedagogy was built within the content. It created a potent mix that could take relatively humdrum things like stacking cups and printing paper and create engaging classrooms. It wasn't the content (like the label might insinuate), it was the pedagogy behind it. 

I continue attempt to throw off labels; I try and describe to others how I teach and run my classroom (or at least attempt to). Maybe "Occasion-based learning" (OBL) or "Discourse-based learning" (DBL). Something that resists definition and implies that learning happens in the places where content, pedagogy, and curiosity meet. 


Wednesday, June 25, 2014

Problem Posing with Pills

My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are always honoured. 

Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don't like using tricks, but if they are "discovered" or "re-invented" (to borrow a term from Piaget and genetic epistemology), then we use them. 

Several posts on this blog have been born out of conjectures offered in class. Many task ideas come from shifts in tasks that we were working on. By far the most profitable, conjecture-rich structure has been that of large whiteboards. There is something about the organization mixed with the rich problems and communication obligations that opens student minds. Regardless of class structure for that day, if a conjecture arises, I get everyone's attention:

"I need your attention please. [Insert name here] has a conjecture to make."

After the conjecture is made, we have a conversation around its feasibility and even vote on its validity. Some groups (or individuals) devote themselves to refutation, while others remain in whole-hearted affirmation. It creates an interesting (yet non-competitive) dynamic. Some students even like having their conjectures disproven and get right back to modifying them to make them stand. 

When I feel like I have loosened the curricular pressure, I take my students through explicit problem posing (conjecturing) exercises. We use the process documented explicitly in Brown and Walter's book. I complete this exercise with a task that is solvable in 5-10 minutes and has many discrete attributes that can be changed. That leaves plenty of time to pose new problems, exchange, attempt solutions, and discuss. 

Side note: The Art of Problem Posing is a must read for any teacher of mathematics at any level. It's algorithmic encapsulation of the fluid process of posing problems jives well with beginners and is extended easily for experts. Like seriously, stop reading this post and get that book. L-I-F-E C-H-A-N-G-E-R.

On this particular day, I gave my grade 9s the "Poison Pills" problem from Stella's Stunners. (The initial solution offers avenues into factors and multiples)
The problem has nice places for students to create conjectures. For example, students quickly realize that two poison pills can ever be adjacent. They can use these certainties to build on their solution. 

After we solved the problem and discussed, I asked them to change attributes and exchange new problems. These are the new problems they created:

  • We split pills in half and put in two containers. Take pill from #1, place in top of #2 and then eat from bottom of #2.
  • Two players. We used three containers. Take one pill from bottom of all three. Ingest one and place the other two back in whichever column you like. Goal is to be last one alive.
  • Switch the bottom and top pill. Then take two from the bottom and put them on top. Eat the next pill.
  • Instead of taking two pills, you take three. Eat the third and place the first two back. 
  • Take three pills and eat the second. Return the other pills if you are alive to do so. Don't take third pill if you die when you eat second. 
  • Split the poison pills into six half-pills of poison. You need to eat two of them to die. 
  • Have 16 pills (12 good, 4 poison) and 12 prisoners. Look if pattern still exists.
  • Odd prisoners eat first pill and pass the second one back to the top. Even prisoners pass the first pill and eat the second. 
  • Add one antedate pill that can make one prisoner invincible to poison.
We managed to solve a few of them, but had to leave some for them to work on in their problem journals. 

There are three large benefits to encouraging problem posing in class:

Mathematical Intuition
Students were able to recognize and reason why new problems were too easy to even spend time on them. This is a great thing to see as a teacher. Four months ago, these same kids would have happily accepted work that resulted in no cognitive struggle, now they ditch their new problems because they see quick solutions. 

Mathematical Complexity
Students quickly discover that some of the hardest problems come from a simple switch. Their instinct was to change almost everything about the problem until the original was a distant cousin and fleeting memory. Some groups found that shifting one small attribute can create a difficult problem. The beauty and intricacy of mathematics shows itself. 

Mathematical Ownership
Students would much rather work on problems they authored. It makes me think of a quote from a recent presentation: "You know what students are interested in? Their own thinking". Students took these problems home and created new ones from them. I even had one student bring back an idea for a card game based on his problem. This ownership can be authored into problems in subtle or extreme ways. 

I remind my students that a good mathematician will try and keep problems alive. We are so used to math being about killing the problem--problem solving. While this is a noble pursuit, I am more interested in resuscitating problems and extending their mathematical lives--problem posing. What can we do to revive or extend this curiosity. A good class, in my books, leaves with more problems than it started. 


Sunday, June 15, 2014

Garbage Can Task

The following task happened by accident:

I was about to introduce a problem to my Math 9 Enriched class that we were going to complete with group whiteboards. Before I could introduce, life got in the way. Students wanted to know about their most recent examination. As I launched into a speech on their performance, a student got up to sharpen their pencil. She walked right in front of me. I made a comment, and she replied that the garbage can should be in the back corner where it would be more convenient. 

I told her that having it by my desk was most convenient for me. Then another student said:

"Why don't we put it in the middle of the room? Wouldn't that be the most convenient?"

In this class, we call this "breaking the math". Students are always welcome to stop our class and make a conjecture. When this leads us into further problems, we joke that the conjecturer broke the math. 

I then flipped the question (to many groans from students) and asked where we would place the can so that if every student had to travel directly to it, we would travel the least amount of distance collectively. 

After setting some parameters about the room, we whipped up an idealized model on the board (pictured below). We decided that the can should be on a grid intersection and the distance between each student is one meter. Also, the students travel as the crow flies. I placed dots where the students were sitting around the room. 

A few really cool ideas began to emerge. It should be mentioned that I foresaw the close parallels to the Road Building task. I anticipated that the Pythagorean theorem would need to be used. I didn't let them know this until one group unearthed the massive amount of calculation that was necessary. 

Once this was common knowledge, groups turned their attention to symmetry. They tried to place the can in a spot that created as many congruent triangles as possible. This enabled them to cut down on their calculations. I over heard the verbiage of 2-3 triangle and 4-5 triangle. They began to name the triangles based on the length of the legs. 

One group noticed that any seat in the same row or column with the can didn't require a calculation. They then decided to set their sights on finding the placement that was collinear with the maximum number of students. 

We had a conversation about the meaning of "center". The geometric center of this rectangle may not be acting as the center of the people placed within it. I saw parallels to measures of central tendency, but decided that it was not in the class' best interest to switch to statistics at this point. 

After answers filtered in, students started posing their own problems. Many started to pose problems around designing seating arrangements to meet certain criteria:

Design a classroom that only needs one calculation.
Design a classroom where every student needs their own calculation.
Design a classroom where the center of the room is the best place for the can.
Design a classroom where the corner of the room is the best place. 

I have a lot of curricular freedom with these students, but this problem would be a good one to practice the Pythagorean theorem. I introduce the idea with the simpler Road Building task, and then solidify knowledge with this one. 

One student asked what would happen if the can didn't have to be on the floor. You should have heard the groans as we pursued this latest instance of "broken math". 


Sunday, June 8, 2014

Basketball Golf Task

The other day, a future teacher asked what one piece of advice I would give to a soon-to-be mathematics teacher. I immediately had several. I settled on one that I felt encapsulated my belief both in and out of class:

Honour curiosity

In class, this finds me wandering through student suggestions and constantly posing new problems that create relevant challenges. Curiosity (both student and teacher) keeps a vibrant ecology going, and I would argue that the intellectual tension so often provided through curiosity is necessary for a positive ecology to thrive.

Outside of class, this has me interacting with my curios online and with others. The purpose of this blog was to document and elaborate on my educational (specifically mathematical) creativity. This is such an instance where a simple problem popped into my head and I forced myself to see it through. Who knows, it may become an important piece of a student's learning someday.

For no apparent reason I became curious whether it was easier (mathematically speaking) for a basketball to go through a hoop or a golf ball to fall into the cup. It was an innocent enough question--a starting point.

I shared it with a couple colleagues and we began to discuss strategy. We immediately placed it within our neat boxes of curricular units. I said that it would be a great example of scale. I would find the diameters of the large items (basketball), the diameters of the small item (golf), and then find the scale factors between the balls and holes respectively.

She said it would be a great idea for area and percent. She wanted to find all four areas and then find the percentage of the hole that each respective ball would cover. We both thought this was a great start and took to Google.

My strategy
Basketball - 9.07" diameter
Hoop - 18" diameter

Golf Ball - 1.680" diameter
Hole - 4.25" diameter

SF = Basketball / Hoop = 9.07 / 18 = .50 (two significant digits)
SF = Golf Ball / Hole = 1.680 / 4.25 = .40 (two significant digits)

This told me that the basketball diameter was approximately a one-half scale model of the basketball hoop while the golf ball was approximately a four-tenths scale model. Thus, it is easier to sink a golf ball.

Her strategy
Area Basketball = 64.61 (units omitted)
Area Hoop = 254.47 (units omitted)

Area Golf Ball = 2.22 (units omitted)
Area Hole - 14.19 (units omitted)

Ball / Hoop = 64.61 / 254.47 = .25 = 25%
Ball / Hole = 2.22 / 14.19 = .16 = 16%

This told her that the golf ball took up less of the hole than the basketball did of the hoop.

Regardless of strategy, this question poses some interesting extensions if you are willing to search for them. Enabling this curiosity is the critical piece to effective mathematics teaching. I'm curious about a men's basketball. The stats above are for a female ball, the men's ball is an additional inch in diameter. How much harder is it to sink a guy's ball? 

What if we combined the strategies and took the scale factors of the areas or percentages of the diameters? Would our answers be any different?

Two basketballs will squeeze into a hoop simultaneously How small would the golf hole need to be to create this exact phenomenon? How wide would the hoop have to be to create the same ratio that exists in golf? The PGA is wondering about expanding the golf hole, is this a good idea? why or why not? How wide would a basketball hoop need to be to  match the new 15 inch golf hole? 

I could see this task fitting nicely into a unit on area in the middle years. (I like how the relationship between 1/2 diameter and 1/4 area can be explored). That is beside the point of this post. The goal is to encourage teachers to view themselves as creative beings. Follow your queries and develop them. Don't be embarrassed to share; this blog is filled with posts I am sheepish about.

My favourite teacher once told me that he was having trouble with curricular reform because he wasn't creative. This was coming from one of the most creative men I had ever learned from. I think this is more common than we think. Share, collaborate, critique, and honour your curiosities. They just might make the difference.


Saturday, April 19, 2014

Conceptualizing Drills

I have students in an enriched class that demand for me to give them more practice. I tell them that we practice mathematics with daily class activities. They don't want practice, they want repeated practice. They are accustomed to receiving repeatable drills to cement understandings. 

I have learned to compromise with this demand. I do believe there is a place for basic skills training in mathematics, and would raise an eyebrow at anyone who claims these unnecessary. I do, however, also believe that the heart of mathematics is problem posing, problem framing, and problem solving. 

Here is how I've infused an ounce of conceptualization into regular drills. (I use this for both practice in a large group discussion, small group rotation format, take home work, as well as unit exams.)

The work begins like many math classrooms with a set of problems to do. In this post, the topic at hand is solving equations (at the Grade 9 level).

I'll give ten or so to show the possible variety in structures, and then begin to ask questions that allow students to think deeper about the rules they just employed. Most of these questions focus on flexible use and mathematical communication. 

Here's a question from my most recent unit exam on solving equations:
The question then reads:

Fill in the blank with the number that makes this equation as simple as possible. Explain your choice.
Once you've explained your choice, go ahead and solve the equation. Show all work. 

The results were fantastic. It was excellent for me, as a teacher worried in skill development as well as deep, conceptual growth, to see that these students were grappling on a deep level with the content when probed to do so. I was assuming that many students might choose "3" to match the denominators on the left-hand side. This scared me, because I felt that I was baiting my students into mistakes. Turns out, not a single student responded with "3". The most popular responses were "2" (foreseeing the first inverse operation), "6" (choosing a LCM of all denominators present), "5" (fully simplifying the fifths), and "1" (assuming that eliminating fractions is always easiest).

Great insight. 

The exam questions are a nice break from traditional assessment while still affording the convenience and balance of a pencil and paper test. My favourite format for these conceptual drills is a small group jigsaw where each group answers, explains, and rationalizes their actions to the larger group. It sparks great discussion. 

Here's a question used in the unit on solving equations:
The question then reads:

Change a single digit from the equation above to make the problem as simple as possible. Explain why you made the choice, and then proceed to solve the equation. Show all work.

Popular choices include changing the "2" to a "1" and shifting the "5" to a "6". These moves both have ample justification and spark great conversations. Eventually the topic of fractions came up, and students said that they would like to avoid them altogether. That led me to the natural extension:

Is there a number that we can replace "2" with to avoid fractions altogether? How many of these numbers exist? How can we find them?

The discussion skyrocketed from there. 

It causes me pause to think about why discussions like these don't happen more often. Is it a time issue? Do teachers see them as wastes of time? Do teachers struggle with the dimensions of problem posing necessary to see beautiful math staring them right in the face? Is it downright confusion of the purpose of mathematics?

**TANGENT: I think teachers don't practice looking for mathematics. We waste our time trying to appear mathematical by partaking in various stereotypical mathematical whimsies such as an undue infatuation with Pi day and the obligatory kudos to binary clocks. There is more to mathematics than surface niceties. 

It is one thing to preach balance but to continually teach at the poles. One day we work on a task and "construct" mathematical knowledge, and the next we "lecture" and "practice". Learning doesn't operate on this notion of average--flip-flopping will only confuse students. We need to develop a curriculum and supporting pedagogy that lives between the two worlds at the same time. Procedural and conceptual are not nearly as mutually exclusive as they are mutually dependent. 


PS. For another foray into this conceptualizing of drills see David Coffey's worksheet adaptation

Friday, March 14, 2014

Algorithms and Flexibility

I was given a section of enriched grade nine students this semester. I decided very early on that the proper way to enrich a group of gifted students is not through speed and fractions. They came to me almost done the entire course in half the allotted time. This essentially alleviated all issues of time pressure.

The beautiful thing about this is we are able to "while" on curiosities that come up during the class (Jardine, 2008). I am not afraid to stop and smell the mathematical roses--so to speak. In a recent tweet I explained it as the ability to stop and examine pockets of wonder. This has been a blessing because our curriculum has become far less of a path to be run and more of the process of running it.

We do introductory tasks each day to get our minds thinking mathematically. We started with various estimations and have since moved into a series of numerical flexibility tasks. Here, I write a problem on the board and they are to give me the exact answer. Only two rules:
  • No calculators
  • No standard algorithms
I am careful to always write these problems horizontally, because vertical alignment is too tempting. A typical problem might look like:
5 x 38
15% of 420
The students have done remarkably well working flexibly. I have them thinking outside the standard methods they have been so successful with.
One recent flexibility task highlighted subtraction. In the midst of students explaining their strategies, two very fruitful things occurred.
One student became frustrated with the banishment of algorithms.
"How do we know if we even used an algorithm?"
I re-directed the question to the class and the response was very cool. We settled that an algorithm had three characteristics: a set structure, easy repeatability, and stability (what I called robusticity).
With these three characteristics in mind, we collected solution strategies. There were only two. First, a student explained how they separated each place value and subtracted corresponding numbers. As he got answers (3000 - 2000), he kept a running tally. In the end, the tally after the units digit was the answer.

The second student separated each digit into a subtraction problem and then stacked them on top of each other and performed the final subtraction after all four were isolated and completed. It was at this moment that I pounced and provided an alternate algorithm.

What if we just combined these numbers back into their original place value and if they were negative, we would represent it with a bar over top. I showed them the familiar addition algorithm and the response was immediate.
Many wanted clarification, and others chimed in to explain. Others wanted to know if they can use this for the rest of their assignments. I told them if we were going to use it, we better develop algorithms for counting and the basic operations. I set them into groups to decipher them.

Algorithmic language came out right away. Things like, "add a zero on the left" and "reset the far right column to 9 nought" (nought was our verbal way of reading the bar notation).

For two days we wrestled with the possible algorithms; when they are polished by the students, I will post them and link from this post.

From the outside, this task seems like a colossal waste of time. The classroom was thrown into chaos as the students worked to re-align their worlds after this major perturbation. From the inside, it showed my students the value of algorithms if they are understood. Something that these students have abandoned in favour of guaranteed right answers. Aside from the natural engagement that speaking in these numbers provides for this class, it does illustrate the fact that "knowing why" it works is a prerequisite for "knowing that" it works.


P.S. After the fact, a student used "nought" notation to give an answer in class. Another classmate noticed that the number (in this case 36) could be represented by a different nought notation number. I then had the class work to find out how many ways each number could be represented in our new notation. The results (purposely omitted here) were quite surprising.

Jardine, D. W. (2008). On the while of things. Journal of the American Association for the Advancement of Curriculum Studies, 4. 

Sunday, January 26, 2014

Road Building Task

The Pythagorean Theorem is often taught in isolation. It has connections to solving equations, but often appears in curriculum long before other equations involving radicals. It also has unique ties to both radicals as well as geometry.

Despite these connections, the theorem has developed the reputation of a surface skill. It involves the  repetition of the rule alongside numerous iterations. Something so fundamental to geometry is reduced to a droning chorus of:

" 'a' squared plus 'b' squared equals 'c' squared "

This task aims to find a middle ground between where critical problem solving and collaboration can meet the technical precision of the outcome. 

The task is originally from James Tanton's Solve This! book. In a recent Twitter conversation, a couple people were looking for good Pythagorean Theorem tasks. It was through this conversation that I began to look for extensions / alterations. 

The task:

You are building roads between four towns. The only requirement is that there is a route between every pair of towns. It doesn't have to be a direct route. What is the shortest distance of roads that needs to be built?
I originally set the towns on the vertices of a 10m by 10m square. This seems like a good place to start building familiarity with the problem. I also included a handout that provided some suggestions to test as well as spark imaginations. These starters give the task a low floor and the extensions provide a high ceiling. 

I give this task to randomly selected groups with large whiteboards. This fits into the structure of my class, but it could be given in any matter that you are comfortable. I encourage parings or groupings so students can build communal lines of thought and action. 

Student work (with original task):

As evidenced, the original task provided opportunity for iterations, but didn't provide much opportunity for branching out. Students needed to be coaxed past the original suggestions. To increase the robustness of the task, allow me to offer some suggestions: 

Begin by having students work on a grid. Whether this is with an overhead transparency or a large whiteboard, the grid will ensure they work with integers if they place towns on grid intersections. Try extending the task in the following directions:

  • Move the towns off of the vertices of the square. Recommend that the students still stick to integer coordinates--at least to begin. This variation will allow students to see the right-angle triangles that exist on the grid regardless of position. They may also get a sense for lengths of diagonals, and whether or not they are congruent. 
  • Add more towns. A simple alternative that might open up the discussion of what it means to be co-linear. Various regular polygons can be explored for patterns. 
  • Mandate direct routes. Instead of having indirect routes, make direct routes between town mandatory. This may open up conversations around number of required roads, the patterns created by regular polygons, and even--although it's a stretch--the number of distinct sections created by the roads. (an extension into inductive reasoning and counter-examples).
  • Create and place a central depot. This could possibly be my favourite extension. Have students place a point that will serve as a central depot where each town must connect to directly. Where is the best place to put it? This could possibly lead into some type of regression analysis.
The task requires students make intelligent decisions with the theorem in mind. They will begin to run iterations of the theorem as they attempt to place towns. This not only builds fluency, but blankets it in critical thinking. It is in this sense that the task meets the middle ground between fluency and critical problem solving. 


** If there are further variations or extensions from readers, please do not hesitate to comment. 

On a side note, analyzing the distances between collinear towns provides a nice visual for simplifying radicals. Use this visual to show how 2*root(2) = root(2) + root(2) = root(8)