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.
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