Occasionally, I give a task to my students before I have done it myself. Sometimes it is because the solution is fairly straightforward and I can see multiple ways to arriving at it without actually doing it. Other times it is because I want to have no impact on my students' thought pathways. The practice also makes class time more exciting as students reason through methods that I would not have though of--I am trying to move from a monotonous state of lesson planning to a more exciting one of lesson participation.
About two weeks ago, I was preparing a unit on surface area and volume for a new course in our province. I sent a tweet out asking for any "favourite problems" that people may have on the subject, and got multiple responses. Three or four of them have made their way (in various states of modification) in to my unit plans in the form of projects, problems, and tasks, but one in particular stuck with me. Although it would have been great to first "do" the problem alongside my students, curiosity got the best of me and I began to puzzle it out. The question, courtesy of Andrew Kelly (@bewdyrooster) is as follows:
First off, I love the way the problem was posed to me. He didn't include an elongated set of instructions which defined every aspect of the problem. I was left wondering various things:
We know the holes are square, what about the solid?
Would this work for solids with more faces than a cube?
Would the solids have to have a even number of faces?
Are the "punch-outs" centered?
Are they a certain ratio of the side area?
What is the original side length?
Good numeracy tasks are presented with sufficient vagueness to entice curiosity. This did the trick. I looked up "Menger Sponge" on Google, and began to define the characteristics of the problem.
The solid is indeed a cube
The punch-outs are to be 1/3 of the existing dimensions, for a total of 1/9 the area
The punch-outs are centered
I set out to answer the original question, but like all good problems, I discovered many more questions, problems, viewpoints, and applications as I went. I will discuss the applications to a high school curriculum later on. I began my thinking like this:
The cube has six sides, so the "punch-outs" would only have to be done 3 times (all the way through).
If I begin with volume, the removed pieces will consist of 3 rectangular prisms.
There will be an overlap in the center where all three meet; I will have to subtract that area twice.
Seemed pretty water tight, so I went to work. I was envisioning the first two sets of cutting to look like this:
My vision got complicated quickly. I had to develop a formula for the total number of prisms to be cut out. If that wasn't complex enough, I had to decide how many times the cutouts overlapped and subtract the volume not being removed. Do the corner pieces overlap more times than the side pieces? How do you classify a corner and a side as "n" increases? The entire process was quickly slipping out of the scope of my class until my wife came in to check on my progress.
I explained my thinking, and she had a glorious insight. Thinking of the "waste" as rectangular prisms was over complicating things; because the cut-outs had identical dimensions and were always cut out on the same plane, their overlap formed cubes. We (notice the pronoun change) could then view the entire solid as a sum of cubes made up out of the smallest "punch-out" dimension.
What was better, we could actually take the first, solid cube and build the next step out of 20 cubes. Because a 3x3 cube should be made up of 27 smaller ones, it was plain to see that 7 had been punched out. The task became much simpler, and suddenly accessible to a wide variety of high school mathematics. We began to view the cubes as sums of smaller cubes:
The transition moved our thought pattern from calculating volumes of rectangular prisms to counting cubes of a uniform volume.
A change in perspective led to powerful learning. My active participation in the "lesson" (bolstered by a different viewpoint) ended up unlocking potential for the problem.
The original question asked for a pattern in the volume and surface area of the cube. As the "n" increases the volume seems to approach zero while the surface area approaches infinity. As I continued building the models in Google SketchUp, many other curricular connections emerged:
What if we "built" a Menger cube, instead of poked holes in it?
If n=1 is composed of 20 n=0 cubes, and n=2 is composed of 20 n=1 cubes than you could set up a function to calculate the volume of the growing cube.
Infinite Sequences and Series
Again, with the building of the cube.
Could calculate the total number of cubes in a "n=x" cube by finding the "x-th" term in the sequence with a first term of 1, and a common ration of 20. (because it takes 20 cubes to build the next Menger model).
Students could calculate the volume of the cube by calculating the missing pieces.
It would give a visual representation of a cube combined with some intrigue.
How many cubes should the picture have? How many does it actually have? Can you find the pattern of construction?
Surface Area and Volume
Its original intention
Track the trends of the Surface Area in the lower iterations and Volume in the higher iterations
As you can see, the problems carries elementary and sophisticated extensions. I loved the problem in the beginning for its intrigue, but it wasn't until I participated in the lesson that I learned its true versatility. Facilitating an effective (and enjoyable) math class hinges on the teacher's ability to see potential in problems. In some cases, that vision is possible without ever doing the problem. In other cases, the most powerful planning comes through authentic participation.
This post marks a couple of milestones for Musing Mathematically. First, this is the 50th post overall. For some reason that seems significant. Second, this post marks the blog's first coined phrase--Atomic Skills.
I love the term atomic skills, but I can't remember when I started using it. I believe it was the result of my limited vocabulary attempting to explain the current disconnectedness of math education. An atomic skill is a foundational skill. An atomic skill is a skill that holds no real 'stand-alone' significance, but can build toward a very significant solution. Atomic skills are usually practiced in isolation of each other in a very repetitive fashion. In school mathematics, atomic skills often make the difference between a good and bad student. Students classify errors with atomic skills as "stupid mistakes".
The term "atomic" fits for two main reasons:
Atomic skills are the building blocks of mathematical problem solving much like atoms build our physical universe. They are the pieces from which elegant mathematics arise.
Atomic skills often cause the most explosive conflicts among mathematics educators. Traditionalists stress the importance of these skills, while reformers advocate for their incorporation into significant contexts.
(I use the terms "traditionalist" and "reformer" very loosly; I think all math educators fall somewhere on a continuum between the two)
The flagship for Atomic skills has always been, and will always be, the multiplication tables. For some reason, these basic facts have infatuated teachers, parents, policy makers, and flashcard manufacturers to no end. Successful students can perform the mulitplication tables automatically. By themselves, they represent a set of patterns; it is not until they are granted a context that they take on real significance.
To be clear, I agree that successful students should know their mulitplication tables. To further clarify, I believe that the incessant drilling of these facts is not the best way to attain them. In order for these atomic skills to be meaningful to students, they must be learned in a context.
Enter Project Based Learning.
I am not arguing for the elimination of atomic skills in education. I am calling for a re-invention of how they are presented. Students need to be presented skill-rich tasks that require them to work on atomic fluency to reach solutions. Too many textbook problems require a single atomic skill to solve--especially in the lower grades. Worse than that, when students begin to show deficits in mathematics we just heap on more repetitions of the same process that failed them originally. An expertly designed project or task provides a numeracy-rich setting embedded with large amounts of atomic skills. Students are required to use, and subsequently learn, basic skills to arrive at a larger and more significant solution.
My most recent stab at this is my "Classroom Remodeling" project. It is a real problem that I faced during my implementation of Project Based Learning in my class. The task is built with a degree of vaguity as well as some strict mathematical guidelines. The task materials are available for download here. It was originally designed in GeoGebra, but students found that paper cutouts worked better than the digital model. This also included work with scale.
Students need to use atomic skills to arrive at a final project that balances:
These three factors require students to repeatedly interact with various atomic skills:
3 digit addition
1 digit by 3 digit multiplication
Mulitples of 3 and 4
Conversion between inches and feet
Tax calculations (if included in the task)
The teacher's job is to elicit the deeper thinking as the groups progress. For example, a group may be struggling to find 2 more seats for students in their lecture. Maybe replacing a 30"x60" table with two 24"x60" tables would provide the same group capacity (4) while increasing the lecture capacity by 2. (from 2 to 4). How will that effect the cost? Will it put you over budget? Will there be space? Are circular tables an option? The thoughts take on an exploratory nature, but students are still required to master the embedded atomic skills. One alteration leads to a question in another area, but every stage is fueled by calculations.
Many of the problems, tasks, prompts, and projects in this blog are designed in this vein. The atomic skills are rehearshed within a much grander scope. This way students are working on automaticity of low-level skills while increasing larege scale teamwork, problem solving, and numeracy skills. It may be analagous to shoving an antibiotic pill into a bowl of yogurt.
Cleverly designed projects don't devalue the importance of atomic skills; they put them in their proper place--as subsidiary, yet explosive, skills.
My leisure time is often interrupted by educational thoughts. I am often sent scrambling to find a piece of paper after I have accidentaly encountered a mathematical situation that I feel would fit great in to the classroom. Last night an episode of Modern Family piqued an interest I have had for months.
In the show, a father is desperately looking for a skill that he can say his son excels at. He creates a list of candidates, but settles on baseball as the most likely avenue for this success. As he and his boy are heading out for their first game, he explains the "10,000 hour rule" to his wife. This concept, pioneered by Malcolm Galdwell in his best seller Outliers: The Story of Success, is based on the idea that mastery is a result of repeated exposure to the feat that you are trying to master. In his estimates, he concludes that at least 10,000 hours are necessary to achive the stage of "mastery".
Gladwell's book is filled with very interesting statistical analysis surrounding the concept of success. He calls into question the ideas of meritocracy--the self-made man. The typical math education blog may focus on these statistics and their possible implications to the high school curriculum, but that was not why I had to hit pause and make jot notes. I wanted to know if the 10,000 hour rule applied to education. Does teaching for 10,000 hours make you a "master teacher"?
The mathematics involved is insultingly simple, and far less interesting that the over-arching question at hand. Where I teach, there are 197 school days in the year. Although not every one involves direct contact with students, they all, in some way, are part of "being" a teacher. (Professional development, exams, interviews, etc). Each school day lasts around 6 hours. (If you subtract the countless hours on the field, rink, court, and stage). At this rate, a teacher should become a master after approximately 1,667 teaching days or just under 8 and 1/2 years on the job.
Teaching, however, seems to be governed by different "mastery" laws. Ironically, I think a master teacher is one that a) doesn't believe that mastery can ever be achieved, and b) continually re-evaluates their practice with the intention to assimilate new, and better, components. A teaching career is long enough to reach mastery and reset 4 times. (If we use the 10,000 hour rule). There is a certain appeal in this thinking. Teaching should be a series of mastery loops; a good professional continues to achieve a cyclical mastery throughout their career. This may take the form of new instructional or assessment strategies, a new position in a consultant or administrative role, or a stab at higher education.
Conveniently, public policy often drives major revisions of curriculum and instruction every decade or so; maybe this is karma's way of pushing us toward this cyclical mastery. One of my favourite educational quotes states, "perfection is an asymptote"; maybe what it's trying to say is that when it comes to the 10,000 hour rule, teaching is--by its very nature--an outlier.
I don't mind the staffroom. I share a prep period with my math department head, and we often engage in meaningful conversation about the ongoing struggle of curriculum renewal. It is an unbelievable support to have a leader who is so willing to learn about what the reform approach has to offer. Teachers do not spend near enough time learning--which is an ironic shame.
One morning, I walked into the staffroom and he greeted me with a question:
"What's new in math education today?"
I didn't pretend to be on the cutting edge of everything math education, but was working through a problem that I read before bed the night before. It came from James Tanton's book Solve This: Math Activities for Students and Clubs. The question itself was very subtle:
Tanton, James. Solve This: Math Activities for Students and Clubs. p. 5
Tanton simply asks, "The figures share a curious property. What is it?"
It only takes a short while to realize that the area is equal to the perimeter in these three figures. Most traditional mathematical training is focused on looking at the numbers and plugging them into some kind of formula. Area and perimeter are the first two that spring to mind. My interest in the problem was a little deeper, and I shared that with my questioner.
I explained a list of questions that I had documented on my BlackBerry the night before:
Does this property exist for shapes with more that 4 sides?
For how many triangles must this property be true?
Can I find any more?
Can I create a foolproof system to devise triangles with this property?
What must the side length of a pentagon with this property be?
Must the polygons be regular to have this property?
Do these figures share any other curious properties?
He immediately pulled out a pen and grabbed for a errant napkin on the nearby table and we started doing mathematics. His background in computer programming and my background in number theory allowed us to find higher order similar polygons with the property. The contagious nature of mathematics shone through. An English teacher, also with a background in mathematics, came over and inquired about the problem. The three of use poured over the project until the break ended. The experience taught me two things:
Teachers are curious people deep, deep down.
Math teachers do not spend enough time engaging in mathematical activities outside of their students' work.
Later on in the day I was contacted by another math teacher--a good friend--about the problem. In the middle of class I received an instant message from him saying that he heard the problem and was wondering if his solution was correct. We sent a series of messages back and fourth about his dilemma.
After school, I was discussing the problem with him again and two other teachers walked in and gave their input. Something motivated these teachers, from multiple disciplines, to do the problem. I think it was a potent mix of availability, pride, and anxiety. Most of these teachers had a bad experience with math in school; some were solving their first interest based math problem ever. It served as some kind of vindication for them.
The whole experience made me wonder; how much of professional development is spent modeling reform techniques? More often than not, teachers are herded into rooms to hear an expert preach about assessment or differentiated this-and-that. Why don't math teachers ever get opportunities to feel what real mathematics is like?
I think if we tried it, we may just discover who is cut out to be a math teacher. Learning is infectious; if students see teachers engaging in problems with each other, they will be more willing to engage in them personally. Teachers would be more open to change if they experienced the change themselves.
Tanton's book comes recommended not only as a resource for "students and clubs" but also for teachers. It can be purchased online.