1. John Mason's questioning philosophy certainly aligns with those of inquiry-based learning. Instead of funnelling the questioning into eventually guessing what's on the teacher's mind, encouraging them to instead disclose what comes to mind inverts the funnel/allows the discussion to enter new realms. Mason had his students develop resilience and resourcefulness through experiencing various challenges so that they "will know what to do when they don't know what to do". This is precisely what inquiry-based learning preaches: the meta-cognitive process.
2. I am an advocate of creative thinking when I teach, so I will definitely try to incorporate many forms of questioning in my classes. I will be sure to change it up so that students do not become dependent on you asking the same thing repeatedly whenever a stumbling block is reached. As mathematics is all about making and justifying conjectures, asking how many ways can you find the answer versus simply finding the answer promotes creativity and gives students who approach problems differently a chance to provide their input (instead of the answer, an answer). To discourage rote memorization, ask how do you know this and have them justify their answer. Inquire about not just the question at hand, but special cases (will that always be the case/when might that be the case), shortcuts, generalizations, elegance, increased/decreased number of constraints, etc... Being wary of student answers that are actually questions when they ostensibly appear as answers (teaching by listening) is something that I will need to practice. Lastly, moderate the frequency and duration of interventions depending on the skill level/dynamics of the classroom.
Saturday 5 December 2015
Group Micro-Teaching - Reflection
Our group's mini-lesson was separated into three parts, where each segment was taught by one of our members. The topic and course introduction mentioned which mentioned the tenth grade level (to get 'students' in the mindset) and mentioned the topic of the day (first lesson in a new unit). I asked some review questions, had the learners provide the answers for the class (teacher-guided). Afterwards, I used their extensive ninth grade knowledge of linear function to introduce the concert of rate of change (zero, positive, negative) based on the slope. Students seemed to understand the relationship well, but when I threw the curve ball (quadratic function - non constant rate of change) one student said they've never seen that before. I made sure to ask the class if they had any questions before passing the mic to Arshbir, who ran the second phase: a Kahoot quiz. She explained that these question were drawn from the textbook and to use your instinct combined with the new knowledge acquired to answer. Some students were confused with axes that weren't labelled or by questions that have more than one solution. Since there were a number of unforeseen questions, she made an informed decision to cut the quiz short (an excellent adaptation, in my opinion). After the virtual activity was completed, Sissi explained the second activity: groups working creatively to come up with answers given a situation contained in an envelope. I think this activity was well received, it's unfortunate that we didn't have time to see more than one of their answers (or take up the Kahoot quiz). Overall, I think our mini-lesson went well.
As with most of these mini-lessons, the challenge for the audience is to un-learn what they're mastered over the years. We had some questions that the vast majority of tenth grade students would not ask (especially for a new topic). An analogy would be trying to teach a math major how to add to integers: they may ask about how the binary operation of '+' is defined, its properties, and which field are we working with. Most first graders would not ask these questions (maybe Gauss) which affect the timing of the lesson.
As with most of these mini-lessons, the challenge for the audience is to un-learn what they're mastered over the years. We had some questions that the vast majority of tenth grade students would not ask (especially for a new topic). An analogy would be trying to teach a math major how to add to integers: they may ask about how the binary operation of '+' is defined, its properties, and which field are we working with. Most first graders would not ask these questions (maybe Gauss) which affect the timing of the lesson.
Friday 4 December 2015
Dave Hewitt: in-class video reflection
Once you got past the British accents and terminology changes, (take 2 vs. minus 2) watching Hewitt's atypical teaching style was very informative for aspiring mathematics teachers.
The first lesson had the teacher use a meter stick to tap arbitrarily spaced areas on the walls of the classroom to teach the real number line in a far more effective manner (via repetition). Even though the loud, repetitive sound appeared to be disruptive to learning, I found it to be conducive because it created an association between the number and its place in the room (relative to other ones). As opposed to teaching the number line (and some basic binary operations) the traditional way, (a two-sided horizontal arrow) the meter stick method of using the classroom as a tangible I believe enables more students to 'picture' the concept. Moreover, when the learners were asked to solve chains of binary operations (+/-) they were immediately able to not only solve the problems using their own 'meter' stick, but were able to construct multiple paths beginning at x and ending at y. A few criticisms of this technique were: some students will be confused with the usage of ellipsis (...) - why is the distance from 1000 to 15 the same as from 15 to 1? The choral approach can also mask certain students that may need help: if 90% of the class answers correctly, then those who are incorrect/unsure are drowned out and may be hesitant to voice their uncertainty.
The second lesson was a creative approach to solving equations of one variable (I'm thinking of a number). Starting with a simple example, having the students describe their thought patterns and repeat those results aloud draws the others into those patterns and helps everyone reinforce the connections between each operation and its inverse. Without explicitly introducing the notion of "x" or parentheses, the students were (very quickly) able to extend their newfound knowledge to a much more difficult problem. When converting the oral version of this more complex question to the board, (algebraic form) Dave used different and consistent sound cues when converting a word to its algebraic equivalent (parentheses, horizontal fraction, "x"). As in the previous lesson, creating an association in the student's mind can assist them when they attempt to recall a concept later on.
The first lesson had the teacher use a meter stick to tap arbitrarily spaced areas on the walls of the classroom to teach the real number line in a far more effective manner (via repetition). Even though the loud, repetitive sound appeared to be disruptive to learning, I found it to be conducive because it created an association between the number and its place in the room (relative to other ones). As opposed to teaching the number line (and some basic binary operations) the traditional way, (a two-sided horizontal arrow) the meter stick method of using the classroom as a tangible I believe enables more students to 'picture' the concept. Moreover, when the learners were asked to solve chains of binary operations (+/-) they were immediately able to not only solve the problems using their own 'meter' stick, but were able to construct multiple paths beginning at x and ending at y. A few criticisms of this technique were: some students will be confused with the usage of ellipsis (...) - why is the distance from 1000 to 15 the same as from 15 to 1? The choral approach can also mask certain students that may need help: if 90% of the class answers correctly, then those who are incorrect/unsure are drowned out and may be hesitant to voice their uncertainty.
The second lesson was a creative approach to solving equations of one variable (I'm thinking of a number). Starting with a simple example, having the students describe their thought patterns and repeat those results aloud draws the others into those patterns and helps everyone reinforce the connections between each operation and its inverse. Without explicitly introducing the notion of "x" or parentheses, the students were (very quickly) able to extend their newfound knowledge to a much more difficult problem. When converting the oral version of this more complex question to the board, (algebraic form) Dave used different and consistent sound cues when converting a word to its algebraic equivalent (parentheses, horizontal fraction, "x"). As in the previous lesson, creating an association in the student's mind can assist them when they attempt to recall a concept later on.
Sunday 29 November 2015
Group Micro-Teaching: Lesson Plan
Topic: Graphing Relations
Grade: 10
Partners: Arshbir, Sissi
Grade: 10
Partners: Arshbir, Sissi
Objective/Goals: Students will be able to describe a possible situation for a given graph and sketch a possible graph for a given situation.
Time: 15 minutes
Strategies to help learn: Class is set in groups to allow students to work together. Also, it is inquiry based learning; students will develop critical thinking and collaborative skills
Hook: Kahoot quiz serves as a hook
Materials required: Computer, colorful envelopes (group activity), large paper for the group activity, markers
Assessment: The group activity will act as formative assessment of their knowledge, and the kahoot will serve as a check of prior knowledge.
Assumed prior knowledge: It is assumed that students will know the axes of a graph, as well as the coordinates of a graph, and be able to plot a graph given the points
Development of idea/skill:
- Start off with Kahoot Quiz
- state that students are able to get into groups (MAX: 2) so this allows students to collaborate
- Group Activity: groups receive envelope and either have to create a situation for a given graph, or create a graph for a given situation
- if time permits, each group gives their answer
- Conclusion/come together and talk about underlying message of lesson
Conclusion: Go over key points of the lesson (i.e. rate of change)
Further extensions/Applications: Extra envelopes for students who finish earlier, talk about speed in physics (driving and the speed you drive)
- exponential growth can be related to decay and half-life, and also finances (analyzing stock market history…?)
Sunday 22 November 2015
Hewitt article: Arbitrary and Necessary
Hewitt defines arbitrary as something someone could only come to know it to be true by being informed of it by some external means - whether by a teacher, a book, the internet, etc... If something is arbitrary, then it is arbitrary for all learners, and needs to be memorized to be known. Necessary in this context refers to aspects of the math curriculum where students do not need to be informed - things which learners can work out for themselves, depending on their respective realm of awareness.
For any given lesson, if I believe that it will save the students time in the long run, I will teach a topic as arbitrary as opposed to necessary (even though less 'math' appears to be involved at the outset). As interesting as it would be to have students devise their own nomenclature when learning something new, imagine the challenges they would encounter attempting to communicate with the rest of mathematical culture subsequently. My justification for this teaching style comes from an experience I had as an undergraduate. My Number Theory professor had an assigned text for the course (A Classical Introduction to Modern Number Theory) and a recommended text (An Introduction to the Theory of Numbers). During the course, I learned from the required text only due to time constraints. Later on, I was keen on learning more so I looked up G.H. Hardy's book. The issue was that Hardy's text was written over a half-century prior to the one I was familiar with, even though similar topics were contained within. I found myself doing much less math while reading the Hardy book no due to lack of material contained within, but simply due to notational differences. I had to create a rudimentary translator so that I could follow along while reading the theorems. Once naming and symbolic conventions have been established, it's prudent to adhere to them so that you are free to focus more on building mathematical knowledge using those conventions. If time permits and I am confident that, if my students' realms of awareness are sufficient, I'd devise an activity (perhaps involving appropriate givens) to have them derive a concept (necessary); I would provide guidance afterwards to ensure uniformity. Either way, students maximize their mathematics in the classroom, either by using conventional terminology to solve problems or by deriving new concepts using their current cognitive abilities.
For any given lesson, if I believe that it will save the students time in the long run, I will teach a topic as arbitrary as opposed to necessary (even though less 'math' appears to be involved at the outset). As interesting as it would be to have students devise their own nomenclature when learning something new, imagine the challenges they would encounter attempting to communicate with the rest of mathematical culture subsequently. My justification for this teaching style comes from an experience I had as an undergraduate. My Number Theory professor had an assigned text for the course (A Classical Introduction to Modern Number Theory) and a recommended text (An Introduction to the Theory of Numbers). During the course, I learned from the required text only due to time constraints. Later on, I was keen on learning more so I looked up G.H. Hardy's book. The issue was that Hardy's text was written over a half-century prior to the one I was familiar with, even though similar topics were contained within. I found myself doing much less math while reading the Hardy book no due to lack of material contained within, but simply due to notational differences. I had to create a rudimentary translator so that I could follow along while reading the theorems. Once naming and symbolic conventions have been established, it's prudent to adhere to them so that you are free to focus more on building mathematical knowledge using those conventions. If time permits and I am confident that, if my students' realms of awareness are sufficient, I'd devise an activity (perhaps involving appropriate givens) to have them derive a concept (necessary); I would provide guidance afterwards to ensure uniformity. Either way, students maximize their mathematics in the classroom, either by using conventional terminology to solve problems or by deriving new concepts using their current cognitive abilities.
Saturday 21 November 2015
Math Fair @ the MOA
This week our class attended a math fair at the Museum of Anthropology (MOA) on the UBC campus. The students who designed the exhibits were sixth graders working in pairs from West Point Grey Academy, a K-12 private school in Vancouver. Their class had visited the MOA a few weeks earlier to gain inspiration (many entries were based loosely on real exhibits).
Since I was only able to stay for 45 minutes or so, (I had a class at Scarfe immediately after this one) I was only able to visit a few student exhibits. The first one was based on Aboriginal canoe carving: you and a partner take turns carving either one or two canoes per turn, given a set number to start with - the person who ends up carving the special (last) canoe is declared the winner. The students read us an interesting back story on prepared cue cards and even provided us with variants of the game (e.g. the special canoe has termites so the winner must avoid carving it) and hints/tips to winning. The second exhibit was based on the mythological hydra where you played a hunter who had to destroy a creature with three heads and three bodies with your sword (the heads can live independently of the bodies and vice-versa). The trouble was that if you destroyed a body then two heads would grow in its place, if you destroyed a head then a body would grow in its place, etc... Again, the students were very well-prepared and had a few of their worked solutions available (hidden) and some hints (also hidden) for those who were stuck. They even had prizes for guests who were able to solve the puzzle without resorting to hints! The final entry had visitors develop a strategy for determining which (indistinguishable) paper maché cabbage weighed more than the other eight - there were nine in total. The challenge here was that you weren't allowed to touch the cabbages and you were only allowed to use the balance beam scale twice.
Overall, I was very impressed with the amount of research the elementary school students performed, how well they each knew their topic, and how well they all conducted themselves in front of people twice their age.
Since I was only able to stay for 45 minutes or so, (I had a class at Scarfe immediately after this one) I was only able to visit a few student exhibits. The first one was based on Aboriginal canoe carving: you and a partner take turns carving either one or two canoes per turn, given a set number to start with - the person who ends up carving the special (last) canoe is declared the winner. The students read us an interesting back story on prepared cue cards and even provided us with variants of the game (e.g. the special canoe has termites so the winner must avoid carving it) and hints/tips to winning. The second exhibit was based on the mythological hydra where you played a hunter who had to destroy a creature with three heads and three bodies with your sword (the heads can live independently of the bodies and vice-versa). The trouble was that if you destroyed a body then two heads would grow in its place, if you destroyed a head then a body would grow in its place, etc... Again, the students were very well-prepared and had a few of their worked solutions available (hidden) and some hints (also hidden) for those who were stuck. They even had prizes for guests who were able to solve the puzzle without resorting to hints! The final entry had visitors develop a strategy for determining which (indistinguishable) paper maché cabbage weighed more than the other eight - there were nine in total. The challenge here was that you weren't allowed to touch the cabbages and you were only allowed to use the balance beam scale twice.
Overall, I was very impressed with the amount of research the elementary school students performed, how well they each knew their topic, and how well they all conducted themselves in front of people twice their age.
Math Art Project - Borromean Cube
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.
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.
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.
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).
Tuesday 29 September 2015
Reflection: my personal experiences as a math student
My most inspiring mathematics teacher wasn't a member of the county school board nor were they a family member. Instead, he was a Professor Emeritus who volunteered his time during evenings at the local University to introduce more complex problems & problem solving methods to young students who were keen and/or were not feeling challenged enough by the standard curriculum. Although the goal was to prepare for the annual Waterloo math contests (of varying difficulty based on your grade level) what I (and I believe most students) got out the experience was a better understanding of the grand scope of math and its sheer power. I attended these free extra-curricular sessions from ages 12-16; I definitely would not have studied math as passionately/intensely in the years that followed without his invaluable guidance. Looking back, I can say that he also influenced how I went about explaining math to other students in various environments (tutoring, learning centre groups, tutorials, etc...).
One the flip side of the coin, any teachers that were responsible for perpetuating the stigma that mathematics is difficult to understand and is only for the select few through banal, uninspired lessons and/or discouragement all fall into the least favourite category for me.
One the flip side of the coin, any teachers that were responsible for perpetuating the stigma that mathematics is difficult to understand and is only for the select few through banal, uninspired lessons and/or discouragement all fall into the least favourite category for me.
Monday 28 September 2015
TPI Reflection
Prior to last Wednesday, I had not heard of the Teaching Perspectives Inventory (TPI) test. My responses to each question yielded scores of 36, 32, 39, 37, and 16 in the teaching perspective categories of Transmission, Apprenticeship, Developmental, Nurturing, and Social Reform, respectively. The (downward-biased sample) standard deviation was computed as follows:
x bar = (36 + 32 + 39 + 37 + 16) / 5 = 32,
s^2 = [(36-32)^2 + (32-32)^2 + (39-32)^2 + (37-32)^2 + (16-32)^2]/5 = 69.20,
s = 8.32
The TPI summary page deem any perspective score exceeding [x bar + s =] 40.32 are dominant, while those less than [x bar - s =] 23.68 are recessive. I did not have any dominant perspectives, but I did have one (strongly) recessive one, namely Social Reform. I was not surprised to see these results; I believe that most mathematics taught at the secondary level is intended to improve the individual first, before they move onto more advanced work that may affect our society as a whole sometime down the line (pending impact). Two internal inconsistencies of note are the Belief subsection of the Developmental perspective and the Action subsection of the Nurturing perspective. Both sub-scores were lower than the other subsections within each category. Also, the highest subsection in 3 of the 5 categories was Intention. These observations could be explained by the short amount of time I spent taking the test or the fact that I have yet to establish consistent sub-perspectives as a teacher candidate.
Do the TPI test results perfectly describe me as a teacher? Certainly not, but it displays a fairly accurate portrayal of my pedagogical tendencies. It would be interesting to retake this test sometime after I finish my practicum and then again 3-5 years down the road to see how these stats evolve.
x bar = (36 + 32 + 39 + 37 + 16) / 5 = 32,
s^2 = [(36-32)^2 + (32-32)^2 + (39-32)^2 + (37-32)^2 + (16-32)^2]/5 = 69.20,
s = 8.32
The TPI summary page deem any perspective score exceeding [x bar + s =] 40.32 are dominant, while those less than [x bar - s =] 23.68 are recessive. I did not have any dominant perspectives, but I did have one (strongly) recessive one, namely Social Reform. I was not surprised to see these results; I believe that most mathematics taught at the secondary level is intended to improve the individual first, before they move onto more advanced work that may affect our society as a whole sometime down the line (pending impact). Two internal inconsistencies of note are the Belief subsection of the Developmental perspective and the Action subsection of the Nurturing perspective. Both sub-scores were lower than the other subsections within each category. Also, the highest subsection in 3 of the 5 categories was Intention. These observations could be explained by the short amount of time I spent taking the test or the fact that I have yet to establish consistent sub-perspectives as a teacher candidate.
Do the TPI test results perfectly describe me as a teacher? Certainly not, but it displays a fairly accurate portrayal of my pedagogical tendencies. It would be interesting to retake this test sometime after I finish my practicum and then again 3-5 years down the road to see how these stats evolve.
Monday 21 September 2015
Reflection: Instrumental vs. Relational debate (Richard Skemp)
Asking everyone
to take a stance on a topic and then defending that stance was something that I hadn’t experienced in a classroom setting for many
years. Initially, I was surprised how
many people were on the ‘instrumental’ side of the coin (approximately one third of my colleagues). It was entirely possible,
however, that those that (in class) were on the instrumental side of the debate
didn’t actually have that belief (and vice-versa).
During our
discussion, which I thoroughly enjoyed, many different viewpoints and counterpoints
were presented from the groups’ brainstorming lists. Many items were subsequently
argued accurately from both sides, but (for brevity) I’ll focus on just one. The ‘achieves that results’ argument (pro-instrumental)
involved teaching the Pythagorean Theorem via simple exercises involving right
triangles yielding immediate results and thus fluency. The 'branch out to other problems' pro-relational rebuttal hypothesized:
what if a student turned the paper over only to encounter non-right triangles
and then proceeded to apply the PT incorrectly (where the Cosine Law is
necessary)? Motivating PT by introducing
the more general setting and explaining how PT is a degenerate case of
CL1 gives the learner a better understanding of the meaning behind
the raw calculations.
This brings us
back to the original question that Skemp posed: should instrumental and
relational mathematics be considered two different subjects? The jury is still out…
1 For a triangle with sides a,b,c>0 and
corresponding angles A,B,C we have:
c2
= a2 + b2 -2abcos(C) [CL]
c2
= a2 + b2 [PT]
Hence, [CL] = [PT] <=> C = (pi)/2.
Wednesday 16 September 2015
# of squares on a chessboard
Many (seemingly daunting) problems in mathematics can be broken down into smaller, more manageable ones where the results can then be collected to solve the original question. Here, we partitioned the original question into 8 smaller ones and then compiled the results in an efficient manner by employing the sum of squares formula. Some possible extensions for this problem are:
- number of squares in a larger board (9x9, 10x10, 100x100, NxN)
- number of squares in a rectangular board (MxN)
- number of non-square (PxQ) rectangles instead of squares, P<Q.
- number of cubes in 3 dimensional board (8x8x8, NxNxN)
Richard R. Skemp - Relational Understanding and Instrumental Understanding
3 things that made me stop during my read-through:
1.
“There is this important difference, that one
side at least cannot refuse to play.” (pg. 4)
- a wonderful analogy describing the compulsory teacher-student bond that is forged over the years
- a wonderful analogy describing the compulsory teacher-student bond that is forged over the years
2.
“But his relational understanding, by knowing
not only what method worked but why, would have enabled him to relate the
method to the problem, and possibly to adapt the method to new problems.” (pg.
9)
- possibly the best statement supporting relational understanding from the entire article
3. “There is more to learn – the connections as well as the separate rules – but the result, once learnt, is more lasting.” (pg. 9)
3. “There is more to learn – the connections as well as the separate rules – but the result, once learnt, is more lasting.” (pg. 9)
My stance is that relational understanding is superior to instrumental
understanding. Why? Much of our world
consists of quick fixes and people demanding (near) immediate gratification,
with minimum effort. Instrumental
mathematical understanding caters to this by deadening the topic - only
skimming the surface and omitting the overall schema of knowledge in favour of
individual (often detached) components.
This practice can result is errors on an exam by students who otherwise are
quite competent (the area units when l is given in cm and b is given in
yards). Moreover, a base in instrumental
understanding could yield errors resulting in much larger consequences later in
the workforce (e.g. confusing 1% with 1bp when dealing with large sums of
money). I’ve seen many circumstances
where students churn out an answer (involving many correctly executed steps to
get there) and assume they are correct overall only to find out later that they
made a small error somewhere along the way.
Everyone is prone to mistakes, (we are human) but if the learner had a
more complete ‘mental map’ they would realize that although the steps seem
correct, their final answer is nonsensical (e.g. a negative length). It’s not worth the risk to only teach instrumental
understanding/‘plug and play’, because so many math problems can only be solved
by straying off the beaten path from A to B; without the proper map you could
end up lost. Relational understanding,
on the other hand, enables you to traverse most landscapes since it connects
(seemingly) disjoint components together to create a web of knowledge.
Monday 14 September 2015
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