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.
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