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.
Wednesday, 28 October 2015
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.
Sunday, 4 October 2015
Stocker - Math that Matters
Mathematics, in its essence, in 'neutral'. Any instructor can change the context of most any problem without actually changing the mathematical content within.
The author's intentions were to recruit/motivate those students who would not be ordinarily stimulated by math in its pure form. For that purpose, Math That Matters is fantastic. A large proportion of people who claim social studies to be their favourite subject and/or are interested in social justice and/or have difficulty understanding the why behind traditional mathematical lessons would benefit from the style of problems in this text. The mathematically inclined, on the other hand, will be motivated to learn more math regardless of the social justice spin put on the problems. It's not that the 'mathies' don't care about social justice, it's that they are indifferent with respect to context in acquiring their mathematical knowledge. One potential downside to Math That Matters is that if a student who normally wasn't passionate about math embraced the SJ spin graduates to a class that removes all SJ context they may revert to their previous less-motivated selves since the topics (in their head) become interrelated. Another negative is if a student is more interested in astronomy, atomic theory, endangered species populations, genetics, or any other non-SJ application, they might tune out if the focus it too heavy on one area of application.
You can certainly use these elementary school style lessons as a blueprint to write similar ones for secondary school students. There are even more possibilities with older students because of their larger (math) knowledge base, increased maturity level (more adult subject matter becomes acceptable), and greater understanding of social issues. Most advanced (highly theoretical) math branches would be difficult to connect to SJ issues (or any other topic). At the secondary level I would say that trigonometry would be challenging and graphing of functions or statistics would be easier to connect with SJ issues.
The author's intentions were to recruit/motivate those students who would not be ordinarily stimulated by math in its pure form. For that purpose, Math That Matters is fantastic. A large proportion of people who claim social studies to be their favourite subject and/or are interested in social justice and/or have difficulty understanding the why behind traditional mathematical lessons would benefit from the style of problems in this text. The mathematically inclined, on the other hand, will be motivated to learn more math regardless of the social justice spin put on the problems. It's not that the 'mathies' don't care about social justice, it's that they are indifferent with respect to context in acquiring their mathematical knowledge. One potential downside to Math That Matters is that if a student who normally wasn't passionate about math embraced the SJ spin graduates to a class that removes all SJ context they may revert to their previous less-motivated selves since the topics (in their head) become interrelated. Another negative is if a student is more interested in astronomy, atomic theory, endangered species populations, genetics, or any other non-SJ application, they might tune out if the focus it too heavy on one area of application.
You can certainly use these elementary school style lessons as a blueprint to write similar ones for secondary school students. There are even more possibilities with older students because of their larger (math) knowledge base, increased maturity level (more adult subject matter becomes acceptable), and greater understanding of social issues. Most advanced (highly theoretical) math branches would be difficult to connect to SJ issues (or any other topic). At the secondary level I would say that trigonometry would be challenging and graphing of functions or statistics would be easier to connect with SJ issues.
Saturday, 3 October 2015
Dishes Puzzle
Algebraic Solution:
Let x be the number of guests => the total number of rice dishes is x/2, since x guests divided by 2 guests per rice dish equals x/2 rice dishes. Using similar reasoning with the other two food types, there must be (x/2) + (x/3) + (x/4) dishes in all. But, we are given that there are 65 total dishes. Therefore,
(x/2) + (x/3) + (x/4) = 65
=> (6x + 4x + 3x)/12 = 65
=> 13x = 780
=> x = 60
Thus, there must be 60 guests in total, consuming 60/2 = 30, 60/3 = 20, and 60/4 = 15 dishes of rice, broth, and meat, respectively.
Non-Algebraic Solution:
Notice that the number of guests must be divisible by 2, 3, and 4 => the number of guests must be divisible by lcm(2,3,4) = 12*. Each group of 12 guests consume (12/2) + (12/3) + (12/4) = 6 + 4 + 3 = 13 dishes. [Picture a table with 13 dishes on it and 12 chairs around it] Since we are given that there are 65 dishes in all, there must be (65/13) = 5 groups(tables) of 12 guests, for a total of 60 guests.
*computation of the least common multiple (lcm) of 2, 3, and 4: both 2 and 3 are prime, 4 = 2x2 => lcm(2,3,4) = 3x2x2 = 12.
Cultural Context:
In certain cultures, food sharing is socially acceptable. Some students may be confused by the fact that people are sharing food at all (why doesn't each guest get their own dish?) - those familiar with pay-by-the-plate events. Others may ask why all of guests don't line up to eat from three large dishes (buffet-style). If you explain that the question is merely hypothetical (doesn't have to take place in China), then learners should have an easier time grasping the style of eating outlined (culture-free).
Let x be the number of guests => the total number of rice dishes is x/2, since x guests divided by 2 guests per rice dish equals x/2 rice dishes. Using similar reasoning with the other two food types, there must be (x/2) + (x/3) + (x/4) dishes in all. But, we are given that there are 65 total dishes. Therefore,
(x/2) + (x/3) + (x/4) = 65
=> (6x + 4x + 3x)/12 = 65
=> 13x = 780
=> x = 60
Thus, there must be 60 guests in total, consuming 60/2 = 30, 60/3 = 20, and 60/4 = 15 dishes of rice, broth, and meat, respectively.
Non-Algebraic Solution:
Notice that the number of guests must be divisible by 2, 3, and 4 => the number of guests must be divisible by lcm(2,3,4) = 12*. Each group of 12 guests consume (12/2) + (12/3) + (12/4) = 6 + 4 + 3 = 13 dishes. [Picture a table with 13 dishes on it and 12 chairs around it] Since we are given that there are 65 dishes in all, there must be (65/13) = 5 groups(tables) of 12 guests, for a total of 60 guests.
*computation of the least common multiple (lcm) of 2, 3, and 4: both 2 and 3 are prime, 4 = 2x2 => lcm(2,3,4) = 3x2x2 = 12.
Cultural Context:
In certain cultures, food sharing is socially acceptable. Some students may be confused by the fact that people are sharing food at all (why doesn't each guest get their own dish?) - those familiar with pay-by-the-plate events. Others may ask why all of guests don't line up to eat from three large dishes (buffet-style). If you explain that the question is merely hypothetical (doesn't have to take place in China), then learners should have an easier time grasping the style of eating outlined (culture-free).
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