Teaching Weblog - David Krebes
7/9/2023
Why am I a teacher?
I like to help people. It always makes me feel good to hear that a student did well on an exam. I enjoy being a small part of a student's success story. It means I have done a good job of explaining things, which is fun for me.
Also, I am impassioned to share my love of mathematics.
7/9/2023
What is my proudest accomplishment as a teacher?
This would be my student Justin, with whom I have worked for 5 years now. We have been working together since September 2018, currently for 3 hours every week. He is now in Grade 12. Together we have solved 143 challenge problems from journal Crux Mathematicorum published by the Canadian Mathematical Society. These are challenge problems such as appear in math contests, such as the Putnam, Cayley and Fermat. They require a high degree of ingenuity and a collaborative effort – we work as a team, producing both "proofs" and "counterexamples" involving extensive pencilwork. There is a strong rapport between us and we have fun together! His assessment of our work can be found under "Justin" under the "Testimonials" tab.
7/9/2023
How do I determine what my student needs and what is my teaching style?
If the student brings me problems to work on then I am happy to oblige. I like working on topics and skills that the student identifies as problem areas.
As we work through a problem, I frequently pause to ask the student for the next step. This can be hard for the student but I am forgiving. When we do this I can almost see my student learning.
Students need effective teaching that covers as much material as possible. We try to work at a steady pace. The student needs to know how to solve not just individual exercises but entire clusters of exercises. This is my aim. The complaint “we never covered that” of exam questions seldom arises with my students.
My students feel comfortable working with me, though I do challenge them and give them responsibility.
7/9/2023
What is gap analysis and how can I make it work for my student?
Gap analysis is the detailed investigation of the “gap” between where the student is now and where the student wants/needs to be, as well as an attempt to figure out how to bridge that gap. It is similar to the concept of variance analysis (below) but is looking forward rather than backward.
Do we really need gap analysis? Can’t we just do the math? I have certainly had some success with the latter. My experience is that just by thinking about math the student is sure to improve. But I do think that the kind of math which students work on in grade school is tailored to a goal-setting type approach. It is measurable in a way that research-type math is not. Why not make use of that measurability? Why not acknowledge it?
Therefore we are interested in the gap between present math scores and desired math scores. But more interesting and perhaps more illuminating is the gap between how the student works now and how the student might work to get higher grades. For instance I have a student who is too slow. The gap is the differential between his present speed and a speed at which he can finish the exam and have a little time to check back over his work.
7/9/2023
What is variance analysis and how can I use it to my student’s advantage?
Variance analysis addresses the question “What went wrong?” It is easier to correct problems which have already been identified than to speculate on possible problems that might arise in the future. To make variance analysis work the student must be candid about what has not worked for them in the past.
Roughly speaking, there are two kinds of things that can go wrong: mathematical and para-mathematical. “Mathematical” includes things like calculation errors, or being stumped on a problem. “Para-mathematical” are factors that support mathematical achievement. Running out of time on an exam, mis-reading a question on an exam, or not having good study habits are all para-mathematical factors.
There are also two steps in variance analysis: figuring out what went wrong and figuring out how to fix it. Sometimes there is no distinction, for instance in noticing a calculation error.
Is variance analysis too late? Not if the student has a cumulative exam coming up. Nor is variance analysis of para-mathematical factors, which persist from one exam to the next.
Why am I a teacher?
I like to help people. It always makes me feel good to hear that a student did well on an exam. I enjoy being a small part of a student's success story. It means I have done a good job of explaining things, which is fun for me.
Also, I am impassioned to share my love of mathematics.
7/9/2023
What is my proudest accomplishment as a teacher?
This would be my student Justin, with whom I have worked for 5 years now. We have been working together since September 2018, currently for 3 hours every week. He is now in Grade 12. Together we have solved 143 challenge problems from journal Crux Mathematicorum published by the Canadian Mathematical Society. These are challenge problems such as appear in math contests, such as the Putnam, Cayley and Fermat. They require a high degree of ingenuity and a collaborative effort – we work as a team, producing both "proofs" and "counterexamples" involving extensive pencilwork. There is a strong rapport between us and we have fun together! His assessment of our work can be found under "Justin" under the "Testimonials" tab.
7/9/2023
How do I determine what my student needs and what is my teaching style?
If the student brings me problems to work on then I am happy to oblige. I like working on topics and skills that the student identifies as problem areas.
As we work through a problem, I frequently pause to ask the student for the next step. This can be hard for the student but I am forgiving. When we do this I can almost see my student learning.
Students need effective teaching that covers as much material as possible. We try to work at a steady pace. The student needs to know how to solve not just individual exercises but entire clusters of exercises. This is my aim. The complaint “we never covered that” of exam questions seldom arises with my students.
My students feel comfortable working with me, though I do challenge them and give them responsibility.
7/9/2023
What is gap analysis and how can I make it work for my student?
Gap analysis is the detailed investigation of the “gap” between where the student is now and where the student wants/needs to be, as well as an attempt to figure out how to bridge that gap. It is similar to the concept of variance analysis (below) but is looking forward rather than backward.
Do we really need gap analysis? Can’t we just do the math? I have certainly had some success with the latter. My experience is that just by thinking about math the student is sure to improve. But I do think that the kind of math which students work on in grade school is tailored to a goal-setting type approach. It is measurable in a way that research-type math is not. Why not make use of that measurability? Why not acknowledge it?
Therefore we are interested in the gap between present math scores and desired math scores. But more interesting and perhaps more illuminating is the gap between how the student works now and how the student might work to get higher grades. For instance I have a student who is too slow. The gap is the differential between his present speed and a speed at which he can finish the exam and have a little time to check back over his work.
7/9/2023
What is variance analysis and how can I use it to my student’s advantage?
Variance analysis addresses the question “What went wrong?” It is easier to correct problems which have already been identified than to speculate on possible problems that might arise in the future. To make variance analysis work the student must be candid about what has not worked for them in the past.
Roughly speaking, there are two kinds of things that can go wrong: mathematical and para-mathematical. “Mathematical” includes things like calculation errors, or being stumped on a problem. “Para-mathematical” are factors that support mathematical achievement. Running out of time on an exam, mis-reading a question on an exam, or not having good study habits are all para-mathematical factors.
There are also two steps in variance analysis: figuring out what went wrong and figuring out how to fix it. Sometimes there is no distinction, for instance in noticing a calculation error.
Is variance analysis too late? Not if the student has a cumulative exam coming up. Nor is variance analysis of para-mathematical factors, which persist from one exam to the next.