Our group of three (Pacus, Simran) attempted to create a Borromean Cube using three different colours of paperclips. The model cube consisted of 81 paperclips (81 / 3 = 27 of each of colour) where the green, red, and blue clips were oriented to be parallel to the x, y, and z-axes, respectfully (i.e. two clips are orthogonal if and only if they were a different colour).
It was fascinating looking up the history of the Borromean ring (the base unit of the cube) and how different cultures (independently) discovered the ring without having any knowledge of Knot Theory (branch of Topology) as it wasn't formalized until the late 18th century.
We had a significant amount of difficulty constructing the full cube due to our inability to bend/unbend the small paperclips and fit the ends into confined spaces (picture trying to fit a rope through a small loop without having any slack or being able to easily bend/unbend the rope). Instead we settled on a 'frame cube' which was comprised of (4/9)*81 = 36 paperclips (12 red, 12 green, and 12 blue). Our cube looked similar to the model cube except ours had the interior missing; it took on the appearance of a 3-dimensional cube drawn on a 2-dimensional piece of paper.
Our class presentation was an enjoyable experience - we outlined some Borromean Cube history, showed/discussed our creation, created our own link using our arms, (as there were three of us!) and even had the class attempt to create the base unit/link using the clips. Most of our colleagues were able to figure it out with little or no assistance from us. Overall, our group had fun with this math art project (personally, I haven't combined math with art in many years). If we were to build another cube (or similar) we'll have to remember to choose more pliable materials.
Saturday, 21 November 2015
Wednesday, 18 November 2015
SNAP Math Fair?
I would certainly run a SNAP-style Math Fair at my practicum school, New Westminster Secondary School, because of the diverse academic ability of the student body. NWSS offers an array of streams catering to all types of learners: Honours, Non-Honours, Apprenticeship & Workplace, and International Baccalaureate Standard Level & Higher Level. The non-competitive nature of the SNAP fairs I consider to be highly appropriate for adolescent learners for two main reasons: it removes the incentive to seek out parental/other adult assistance, and (more importantly) it focuses the event on the math, not the trophies. The problem-solving nature of SNAP encourages those students at NWSS who may not possess a heavy theoretical knowledge base to solve a variant of a problem that is relevant to their own lives and is appropriate for their age/skill set. Having the students involved in the presentation helps their confidence not just with explaining mathematics but with speaking in front of others in general (recall, the students initially only pose the problem, not the solution). I would partition the NWSS fair into academic streams so that visitors could see how similar problems (generalization/special case) are approached by different learners (e.g A&W and IB). Since the SNAP fairs encourage group work, it's very possible that the visitors (who come from diverse backgrounds themselves) can better learn from certain members of a group (e.g. someone who is proficient at geometric explanations) or from a specific stream (A&W vs. IB) than others. A key benefit to drawing the problems from archives of professionally-constructed books is that (quite often) there are many different ways of extending, simplifying, and explaining the solution.
Wednesday, 28 October 2015
Micro-teaching Reflection
Choosing a topic for our 10-minute lesson was the easy part. I have been playing Settlers of Catan for about five years now, and am very familiar with the rules/strategies involved. I immediately thought that it would be both visually stimulating (colourful) and would involve some mathematics (probability). Writing a lesson plan that would adhere to our template, however, was a whole new challenge. Eventually, after the outline was complete, I rehearsed the lesson at home (including a tv commercial style introduction) and was satisfied that it would work in front of our group. It's safe to say things didn't go as planned. I found out that (unlike my ghost students in the trial runs) real students ask unforeseen questions that add time to your lesson. Since we compressed a standard hour-long lesson plan by a factor of six, each minute was very precious. I ended up rushing through the theoretical material, as I had only allocated two minutes for the final group activity/end goal of placing the first settlement on the board. Choosing a less ambitious topic may have been easier to achieve more '3s' on the peer assessment cards, but I do enjoy a good challenge.
My colleagues enjoyed the game setup, had fun with the java applet, the applied probability component, and were able to understand the instructions due to the clarity of my voice. Some constructive criticisms (that I agree with) were: to have more game pieces/manipulatives (hexes, settlements, etc...) on hand to move around and the aforementioned time management issue. I also forgot to mention some future directions during the conclusion (a second lesson based on this one).
It just goes to show you how far even a micro-lesson can stray off the beaten path. It was really eye-opening for me in that it made me think about what could happen with an hour-long lesson that was constructed in haste, without considering contingencies.
My colleagues enjoyed the game setup, had fun with the java applet, the applied probability component, and were able to understand the instructions due to the clarity of my voice. Some constructive criticisms (that I agree with) were: to have more game pieces/manipulatives (hexes, settlements, etc...) on hand to move around and the aforementioned time management issue. I also forgot to mention some future directions during the conclusion (a second lesson based on this one).
It just goes to show you how far even a micro-lesson can stray off the beaten path. It was really eye-opening for me in that it made me think about what could happen with an hour-long lesson that was constructed in haste, without considering contingencies.
Monday, 26 October 2015
Battleground Schools: Mathematics Education
Over the past century, mathematics education had been firmly tied to two horses: conservative and progressive. Much like the medieval torture, the party that is hurt the most when the horses are whipped is the party that is tied to the ends: education.
The Progressivist Reform of the early 20th century challenged the established ideals of knowing math versus doing math - claiming that students must practice doing math via inquiry-based questions if they want to add to their knowledge pool. As a free-thinker who likes to ponder math in my own way, I associate with the progressive mantra of understanding; I try to emphasize that in my own teaching (vs plug-and-play).
New Math, headed by the School Mathematics Study Group (SMSG), a subsidiary of the American Mathematical Society (AMS) was successful in reverting math teaching back to a highly conservative model. The impetus of which was USSR's launching of Sputnik in 1957. Influenced by the Bourbaki group's disregard for geometry (or any diagrams) in favour of abstraction - New Math supported minimal progressive features as it assumed that every student was a potential rocket scientist (certainly not a viable/realistic viewpoint - learners should be treated as individuals).
The Math Wars over the NCTM (National Council of Teachers of Mathematics) standards followed for the next generation. I (unknowingly) was a part of this, as my elementary/high school was throughout the 1990s and early 2000s. Looking back, I was enrolled in classes that followed the back to basics approach, incorporated standardized testing, encouraged flexible problem solving skills, represented mathematical relationships in multiple forms, and/or embraced new technology (personal computers, internet resources). As we move further into the 21st century, I'm worried that with the advent of modern technology students will end up becoming so dependent on it that their (mathematical) imagination will be stunted. Many beautiful results in mathematics were discovered by thought alone (and later formalized with ink and paper). Imagine if Cardano had simply typed a general cubic polynomial equation ax^3 + bx^2 + cx + d = 0 into a supercomputer and saw the end result instead of depressing the cubic by removing the x^2 term, performing a clever substitution, etc. I'm not saying that we should compute the sample variance for 1000 data points by hand, but we must exercise caution when technology is introduced - it should be used as an aid to walking (e.g. cane), not as a replacement for your legs.
After a 1996 study revealed that the USA ranked 28th out of 48 countries in math, it was determined that deeper conceptual analysis was required to rise in the rankings. It's appalling how there are so many (unqualified) math teachers operating today who only emphasize rote memorization of material and only focus on questions and question styles that will be on the next test instead of broadening and strengthening the student's understanding. As aspiring mathematics teachers, we must be ready to embrace change as waves of pedagogical reform cascade through our classrooms.
It may be possible to adopt a custom hybrid approach with select features from both philosophies should be implemented in the classroom as the teacher deems appropriate, as there are multiple ways to learn new mathematical concepts. A potential problem with the hybrid approach is you risk bifurcating the class into those who desire mathematical rigour and do not receive it fully and those who would benefit from a more exploratory approach and who have difficulty handling abstract theories. Is it even possible to teach in this manner, given time constraints and diversification of students in a given class? A second, more radical approach would be offer two streams of math: one for those more comfortable with conservative ideals, and another for progressive-minded students. Some issues with this strategy are: once you're on a track, how easy it is to make a switch if you think they other stream is more suitable, how easy is it to switch back, when is it too late to switch (back), or will learners' minds be less complete if they're taught one side of the math coin only?
Will one side win out eventually, or will it be like our political parties - at the mercy of the swaying breeze?
The Progressivist Reform of the early 20th century challenged the established ideals of knowing math versus doing math - claiming that students must practice doing math via inquiry-based questions if they want to add to their knowledge pool. As a free-thinker who likes to ponder math in my own way, I associate with the progressive mantra of understanding; I try to emphasize that in my own teaching (vs plug-and-play).
New Math, headed by the School Mathematics Study Group (SMSG), a subsidiary of the American Mathematical Society (AMS) was successful in reverting math teaching back to a highly conservative model. The impetus of which was USSR's launching of Sputnik in 1957. Influenced by the Bourbaki group's disregard for geometry (or any diagrams) in favour of abstraction - New Math supported minimal progressive features as it assumed that every student was a potential rocket scientist (certainly not a viable/realistic viewpoint - learners should be treated as individuals).
The Math Wars over the NCTM (National Council of Teachers of Mathematics) standards followed for the next generation. I (unknowingly) was a part of this, as my elementary/high school was throughout the 1990s and early 2000s. Looking back, I was enrolled in classes that followed the back to basics approach, incorporated standardized testing, encouraged flexible problem solving skills, represented mathematical relationships in multiple forms, and/or embraced new technology (personal computers, internet resources). As we move further into the 21st century, I'm worried that with the advent of modern technology students will end up becoming so dependent on it that their (mathematical) imagination will be stunted. Many beautiful results in mathematics were discovered by thought alone (and later formalized with ink and paper). Imagine if Cardano had simply typed a general cubic polynomial equation ax^3 + bx^2 + cx + d = 0 into a supercomputer and saw the end result instead of depressing the cubic by removing the x^2 term, performing a clever substitution, etc. I'm not saying that we should compute the sample variance for 1000 data points by hand, but we must exercise caution when technology is introduced - it should be used as an aid to walking (e.g. cane), not as a replacement for your legs.
After a 1996 study revealed that the USA ranked 28th out of 48 countries in math, it was determined that deeper conceptual analysis was required to rise in the rankings. It's appalling how there are so many (unqualified) math teachers operating today who only emphasize rote memorization of material and only focus on questions and question styles that will be on the next test instead of broadening and strengthening the student's understanding. As aspiring mathematics teachers, we must be ready to embrace change as waves of pedagogical reform cascade through our classrooms.
It may be possible to adopt a custom hybrid approach with select features from both philosophies should be implemented in the classroom as the teacher deems appropriate, as there are multiple ways to learn new mathematical concepts. A potential problem with the hybrid approach is you risk bifurcating the class into those who desire mathematical rigour and do not receive it fully and those who would benefit from a more exploratory approach and who have difficulty handling abstract theories. Is it even possible to teach in this manner, given time constraints and diversification of students in a given class? A second, more radical approach would be offer two streams of math: one for those more comfortable with conservative ideals, and another for progressive-minded students. Some issues with this strategy are: once you're on a track, how easy it is to make a switch if you think they other stream is more suitable, how easy is it to switch back, when is it too late to switch (back), or will learners' minds be less complete if they're taught one side of the math coin only?
Will one side win out eventually, or will it be like our political parties - at the mercy of the swaying breeze?
Wednesday, 21 October 2015
Micro-teaching: Settlers of Catan basics
Objectives: To introduce students to the multiplayer (3-4) board game entitled: The Settlers of Catan. Learners will be able to initiate a game by completing the first move.
Materials Required: a standard (3-4 player) Catan set, jsettlers (electronic version of the game), laptop
Procedure:
a) Opening Hook {30 sec}: Present the game with a commercial-style introduction [e.g. are you tired of looking at the same black and white chessboard and losing game after game online to grandmasters? Then try this (relatively) new strategy board game from Germany...] Today, we're going to go over some basics; by the end of the lesson you will all be able to perform the first move in the game with confidence.
b) Prior Knowledge Check {1 min}: ask students if:
- they have heard of the game before (if yes, ask if they have played the game before)
- they are familiar with the probability of each outcome of rolling 2 dice
c) Game Details {5 min}:
History - designed by Klaus Teuber (German), first published in 1995
Outline the board setup (distribution of the 19 resource hexes, 9 port hexes, and 9 blank water hexes, distribution and placement of the 18 number tiles, rolling a "7"/the robber) - be sure to reference the booklet's sample board & distribute the 'building cost' cards)
Outline of your pieces (5 settlements, 4 cities, 15 roads) and ways they can be used
Objective of the game is to acquire 10 victory points (combination of settlements - 1 pt. each to a max of 5, cities - 2 pts. each to a max of 4, longest road - 2 pts, largest army - 2 pts, victory pts. - 1 pt. each to a max of 5, notice that can't focus on only one category)
Explain how a game begins (2 initial settlements/roads) and what happens upon each roll
d) Strategies for placing the first settlement:
- total probability of the resource number tiles
- scarcity of resources (3 ore vs. 4 wheat or low number tiles on all ore tiles)
- variety of resources/numbers (try to predict what you're going to be left with)
- symbiosis of resources (check your 'building cost' card)
e) Strategies for placing the first road:
- where you want to settle next (expansion)
- blocking opponents
- try to predict where your opponents are going to go (don't want to waste a road)
Inquiry Questions {1 min}: What is the most valuable resource? Why is there no "7" number tile?
Participatory Activity/Formative Assessment {2 min, 30 sec}: Have the group of 4 collaborate to find the best spot on a given board and place the first settlement & corresponding road there (justify your answer). Explain why you agree/disagree with their decision.
Conclusion: Now that you understand the basics, you are well on your way to being able to play a full game! Next time, we'll go over the turn in more detail and introduce another important aspect of the game - trading.
Materials Required: a standard (3-4 player) Catan set, jsettlers (electronic version of the game), laptop
Procedure:
a) Opening Hook {30 sec}: Present the game with a commercial-style introduction [e.g. are you tired of looking at the same black and white chessboard and losing game after game online to grandmasters? Then try this (relatively) new strategy board game from Germany...] Today, we're going to go over some basics; by the end of the lesson you will all be able to perform the first move in the game with confidence.
b) Prior Knowledge Check {1 min}: ask students if:
- they have heard of the game before (if yes, ask if they have played the game before)
- they are familiar with the probability of each outcome of rolling 2 dice
c) Game Details {5 min}:
History - designed by Klaus Teuber (German), first published in 1995
Outline the board setup (distribution of the 19 resource hexes, 9 port hexes, and 9 blank water hexes, distribution and placement of the 18 number tiles, rolling a "7"/the robber) - be sure to reference the booklet's sample board & distribute the 'building cost' cards)
Outline of your pieces (5 settlements, 4 cities, 15 roads) and ways they can be used
Objective of the game is to acquire 10 victory points (combination of settlements - 1 pt. each to a max of 5, cities - 2 pts. each to a max of 4, longest road - 2 pts, largest army - 2 pts, victory pts. - 1 pt. each to a max of 5, notice that can't focus on only one category)
Explain how a game begins (2 initial settlements/roads) and what happens upon each roll
d) Strategies for placing the first settlement:
- total probability of the resource number tiles
- scarcity of resources (3 ore vs. 4 wheat or low number tiles on all ore tiles)
- variety of resources/numbers (try to predict what you're going to be left with)
- symbiosis of resources (check your 'building cost' card)
e) Strategies for placing the first road:
- where you want to settle next (expansion)
- blocking opponents
- try to predict where your opponents are going to go (don't want to waste a road)
Inquiry Questions {1 min}: What is the most valuable resource? Why is there no "7" number tile?
Participatory Activity/Formative Assessment {2 min, 30 sec}: Have the group of 4 collaborate to find the best spot on a given board and place the first settlement & corresponding road there (justify your answer). Explain why you agree/disagree with their decision.
Conclusion: Now that you understand the basics, you are well on your way to being able to play a full game! Next time, we'll go over the turn in more detail and introduce another important aspect of the game - trading.
Monday, 19 October 2015
The Giant Soup Can of Hornby Island
This is an interesting problem as it teaches students a technique that they may not have seen before, estimation.
I. Each Campbell's soup can is 51cm in height (h) and 41cm in width (w).
II. We are given that the water tank can is in exactly the same proportions as a soup can.
III. Assume that both cans are cylinders (i.e. ignore any lips, dents, or bulges)
IV. The volume of a cylinder is given by: V = (pi)(r^2)(h).
V. We are given the height of the bike in the photo.
from I, h = (51/41)(w) = (51/41)(2r) = (102/41)(r), since the width/diameter of a cylinder is equal to twice the radius, r.
from V, let the height of the bike be b meters. The question now is, how much of the radius of the water tank can does b represent? Assumptions:
- the bike wheels in the eyes of the camera are circular (i.e. the bike is not leaning)
- the bike's right handlebar is leaning against the tank (i.e. the bike is close to the tank)
- the base of the bike wheels are below the base of the tank (if the ground were transparent)
Then it appears that b slightly less than r, so we are justified in setting b = (0.9)(r) as our estimate => r = (10/9)(b)
from II, III, and IV, the volume of the water tank can,
V_w = (pi)(r^2)(h).
= (pi)(r^2)[(102/41)(r)]
= (102/41)(pi)(r^3)
= (102/41)(pi)[(10/9)(b)]^3
= (102/41)(pi)[(1000/729)(b^3)]
= (34000/9963)(pi)(b^3)
if b = 1 meter (estimate), then V_w = 10.7211 cubic meters.
I. Each Campbell's soup can is 51cm in height (h) and 41cm in width (w).
II. We are given that the water tank can is in exactly the same proportions as a soup can.
III. Assume that both cans are cylinders (i.e. ignore any lips, dents, or bulges)
IV. The volume of a cylinder is given by: V = (pi)(r^2)(h).
V. We are given the height of the bike in the photo.
from I, h = (51/41)(w) = (51/41)(2r) = (102/41)(r), since the width/diameter of a cylinder is equal to twice the radius, r.
from V, let the height of the bike be b meters. The question now is, how much of the radius of the water tank can does b represent? Assumptions:
- the bike wheels in the eyes of the camera are circular (i.e. the bike is not leaning)
- the bike's right handlebar is leaning against the tank (i.e. the bike is close to the tank)
- the base of the bike wheels are below the base of the tank (if the ground were transparent)
Then it appears that b slightly less than r, so we are justified in setting b = (0.9)(r) as our estimate => r = (10/9)(b)
from II, III, and IV, the volume of the water tank can,
V_w = (pi)(r^2)(h).
= (pi)(r^2)[(102/41)(r)]
= (102/41)(pi)(r^3)
= (102/41)(pi)[(10/9)(b)]^3
= (102/41)(pi)[(1000/729)(b^3)]
= (34000/9963)(pi)(b^3)
if b = 1 meter (estimate), then V_w = 10.7211 cubic meters.
Sunday, 18 October 2015
2 Letters from Students - 10 years later...
Positive:
Mr. Dickson,
You were my favourite math teacher in high school because you taught the subject with charisma and made the lessons fun. Math was always my favourite subject and you helped me understand just how powerful it can be as a problem solving tool when used correctly. Furthermore, you volunteered your time after class to outline some potential mathematics-oriented career paths for those considering post-secondary degrees. I ended up studying Actuarial Science as an undergrad (where many courses expanded on topics from your Data Management class) and now I'm working in the property/casualty reserve department at Manulife in Toronto. Thank you for inspiring me and setting me on a path to a rewarding career!
Negative:
Mr. Dickson,
As an artist, I always struggled with mathematical concepts and logical thinking. You were a tough marker that emphasized not only the correct steps/answer, but proper form (something that I also had difficulty understanding). Even though your lessons were organized, you catered them mainly to the keener students and I found many of the challenge problems inaccessible without many hours of practice and/or help from outside tutors. I know that you're only one man (vs. a class of thirty) but not everyone shares your passion for math; I would have been more interested in learning if you incorporated some interdisciplinary (e.g art) ideas into your lessons. It was a real struggle to pass your class; I hope that you consider some of my suggestions to help reach more non-math students as you refine your teaching techniques.
Mr. Dickson,
You were my favourite math teacher in high school because you taught the subject with charisma and made the lessons fun. Math was always my favourite subject and you helped me understand just how powerful it can be as a problem solving tool when used correctly. Furthermore, you volunteered your time after class to outline some potential mathematics-oriented career paths for those considering post-secondary degrees. I ended up studying Actuarial Science as an undergrad (where many courses expanded on topics from your Data Management class) and now I'm working in the property/casualty reserve department at Manulife in Toronto. Thank you for inspiring me and setting me on a path to a rewarding career!
Negative:
Mr. Dickson,
As an artist, I always struggled with mathematical concepts and logical thinking. You were a tough marker that emphasized not only the correct steps/answer, but proper form (something that I also had difficulty understanding). Even though your lessons were organized, you catered them mainly to the keener students and I found many of the challenge problems inaccessible without many hours of practice and/or help from outside tutors. I know that you're only one man (vs. a class of thirty) but not everyone shares your passion for math; I would have been more interested in learning if you incorporated some interdisciplinary (e.g art) ideas into your lessons. It was a real struggle to pass your class; I hope that you consider some of my suggestions to help reach more non-math students as you refine your teaching techniques.
Subscribe to:
Posts (Atom)