Summary of Dave Hewitt's Video

The video on Dave Hewitt’s teaching method was quite fascinating. His non-stop interaction with the students really made it seem like he was just a guide and the students were really figuring out the concepts themselves. It was remarkable how simple he makes it seem, keeping all the students engaged and getting all of them participating. A few observations I saw was that he always made the ideas concrete, such as tapping the board using the ruler. He would stay on the same concept and made sure that the students got the concept and he did it in such a way that it seemed like the students weren’t thinking that it was very repetitive. He made sure that everyone was participating and in fact from what I noticed, when he asked questions, he would wait a while and let the students who raised their hand up a little later to answer the question. Throughout the video, we got to see a lot of the students participate and he even gives everyone a chance to answer the questions as one unified voice. This is something that I found very intriguing as everyone feels part of the learning process and it does show if there’s different answers in the class. Because of all these the students seemed so attentive and very much active in class. It was also great to witness him not directly saying whether a question is right or wrong when a student asks him. He tries to deepen their understanding by not just saying the answer, but rather by getting them to think carefully of how they came to that conclusion. The video definitely gave me some great ideas in how to keep participation and a continued discussion going in future classes.

Summary on Battlegrounds Schools

In the book “Battleground Schools”, Susan Gerofsky discusses the progression of mathematical education within the last century. She discusses 3 reforms that occurred in mathematical learning in particular: the Progressivist Reform, New Math and Math Wars. Early in the 1900’s, the public started to realize that the traditional way of teaching “meaningless memorized procedures” was not ideal in student learning. The public started to realize that the students’ knowledge of mathematics was very narrow and only was meant to answer how rather than the why in problem solving. With the emergence of the need for technology, schools started to implement some of John Dewey’s work of allowing the students to discover mathematical education for themselves. The schools started to adopt an independent way of learning to broaden and personalize students’ ways of learning.
But by the 1960’s, the mathematical education turned into a competition as the Americans were worried about losing the space race. As a result, the curriculum was enhanced and the New Math reform began. The emphasis of this reform was to promote students from K-12 to have a larger mathematical capacity so that more of them may continue in university and hopefully create more of rocket scientist type of students. This came to the expense of the students who weren’t more inclined in math and also the teachers who did not have the math knowledge required to teach these subjects.
The Math Wars over the NCTM Standards era brought an end to the need to create rocket scientists but instead focused on “back-to-basics” way of reaching and learning math. This era of learning emphasized a national standard in learning followed y a standardized test. This would allow teachers to have more freedom in the way they taught their class but still allow all students to learn the same topics. This way of learning encouraged developing different ways of solving problems as to better understand a topic, being able to think about it in different forms. This reform has gotten opposition and support from many groups and to this day both groups are debating on this topic.

Personal Reflection for the Interviews

Since our interviews, I've been thinking about how I would like to run my class. Stemming from our discussion with one of the math teachers, the idea of being a guide to student-centered and group-centered learning has come to the forefront of my thoughts. Ideally I'd love to teach in this manner as the students learn concepts and discover new ideas through group discussions and deep thought. This would promote higher learning and a deeper understanding of each topic rather than just being told how to solve problems. But I've been struggling with this idea because of a few things. First of all is the fact that I've been so used to just being lectured and told what to do by the teacher, the other way is unfamiliar to me and not having had great experience with it makes me think twice about it. Also having heard the 2 teachers talk about how the pressure of provincials have made them move towards the lecture style way of teaching, makes me think that it is really difficult to do it. But having heard that one of the teachers is starting to try it and is seemingly having success with it gives me hope and hopefully my worries regarding this style of teaching will be eased by his success.

Heather Robinson's Article summary and reflection

Summary of Heather Robinson’s article:

In Heather Robinson’s “Using Research to Analyze, Inform, and Asses Changes in Instruction”, she talks about the role of her graduate studies in improving her teaching method. She talks about how she discovered that she was just lecturing and giving the students a one dimensional way of thinking about a question, so that when the finals came out, a lot of her “better” students were failing or just barely passing the finals. She discusses that she wasn’t engaging her students to learn but more teaching them how to take tests. She begins to implement less lecturing and also promoting higher learning within the subject. She began to ask more thought provoking questions like what would happen if the question was changed this way or why it acted in a certain manner? She also implements a “jigsaw” way of learning where the students are grouped into expert groups and basically learn and discover certain topics by themselves. This promotes an outpouring of ideas by everyone and makes sure that everyone in the group gets the topics covered. Her new style of teaching seemingly pays off and her students do better in the final exams.

Reflection on Robinson’s article:

This particular reading is actually something I’ve discussed with a current high school math teacher. I volunteered with this math teacher and we discussed exactly what Heather Robinson talked about. He also took a graduate course in education and he implemented a few things that she discusses. Things like the jigsaw groups were something I particularly caught sight of. Upon hearing about it, I thought it was a great idea. Having students lead each other and educate each other puts more responsibility on them and they would find ways more interesting to themselves when they teach each other. At the same time it’ll be easier to manage and it does promote the quieter students to be more interactive and it promotes teamwork. The only thing I found with it and I’ve discussed this with that teacher is the time restraint. It would be hard to implement it because ideally you would give students a lot of time to discover new things for themselves. But now that the curriculum has seemingly been minimized and now that the provincials aren’t as imposing, this way of learning could actually be tried. It is something I’ve definitely been thinking about and hope to one day use it myself.

Interview Reflection

From our interviews with the teachers and students, we got a good sense of what to be aware of and how to better ourselves in becoming effective instructors. Though some of the ideas from the teachers and students are different there are a few things that stand out. Firstly was the provincial and its seemingly “narrowing way of teaching”. Both teachers agreed on this and though both wanted to expand the students’ understanding of mathematics, the provincials restricted them into just teaching rules and equations. One of the teachers even just referred this method of teaching as “training” the students into passing tests and exams. This evidence of training even shows up with the students’ answer on how important they think high school math is later on in life. They were trained to thinking that it’ll be very useful and even if it is in certain areas, they couldn’t really explain why or where it would be important.

Another difference that stood out was the analytical versus computational component of mathematics. The students all agreed that they rather just learn the equations and they don’t need to know the concept whereas the teachers were fully for analysis and expanding those equations. It seems like the students have been so immersed in the computational part that they don’t see the other alternatives and are seemingly dismissive of the idea of analytical math. So we thought that’s something that we as teacher candidates are probably going to struggle with, trying to broaden the students’ minds even though they might just want the equations.

The most significant thing that we found through these interviews is the similarity in both the teachers’ and the students’ response on how to engage students. They all agreed that to be able to engage the students, teachers must be energetic and be relational with the students. The students must be able to be comfortable with the teacher and like the teacher in order for “something to stick” with the students. At the same time the teacher must show that they are enthusiastic with the subject in order for the students to feed off that excitement and be attentive to the teacher. This above all else is what engages the students. From our interviews with the students, they remember more of the teachers rather than most creative lesson they’ve been taught. They learn better from the teacher’s attitude towards the subject rather than “gimmicks” to making learning math fun. That’s why one of the teachers always tried to eliminate the idea of math being tough and always stayed positive.

Interview with Math teachers and students

We interviewed 2 teachers and 3 students and asked them 5 questions. I've summarized the points we got from their answers. To give a brief background, Teacher A is an older teacher who's been teaching Math for at least 20 years and is more of the instrumental type of teacher. Teacher B on the other hand is a much younger teacher. He's been teaching for about 8 years and is starting to move towards relational way of teaching. So here's the interview:

1) How important do you think high school Math is in getting a good job? Which topic is more important than others?

-Teacher A says for the most part, not that important. Outside of engineering and commerce, basic arithmetic or grade 7 is good enough. He is hoping that high school math be more geared towards using math in everyday settings and life skills, like paying bills, balancing cheques, taxes and so on.

-Teacher B says it’s important to about grade 10 math. Being able to do fraction work, being able to read graphs is important in the work place. He mentions accounting specifically as the major motivator to continue taking high school math since it does require more math than other jobs. He also mentioned that it’s the process of learning and doing things step by step in high school math that’s significant rather than the math itself. Just the way of figuring out problem solving and translating it to the workplace is what counts.

The students on the other hand think that in general you’ll need high school math in getting a good job. They mentioned that we use math in our everyday lives but mostly mentioned pre-high school math in their examples. But still they believe that high school math will be extremely beneficial in getting good jobs

2) Are there any tips you can provide so that we can engage students into wanting to learn Math?

-Teacher A just said to try to make the questions into real life situations. Other than that he suggested to just be relatable to the students and have a good working relationship with them. In that way, if they like you and respect you, they’ll be more attentive and will be more willing to listen and something is more likely to stick with them.

-Teacher B suggested to always keep it positive. He wants to get rid of the “tough math” talk within the classroom and just give the students encouragement. He also suggested to move away from lecture type class and more into a class where the students teach themselves and build off each other and the teacher as just a “guider”.

What do you expect from your teacher so that you can keep engaged and wanting to learn Math?

-From the students we talked to, the things we got from them is to just be energetic and upbeat so that the students do stay engaged and be more attentive. At the same time the students said that no matter how enthusiastic the teacher is if the students don’t get the material, they would not be engaged. So breaking down the ideas into an understandable material is crucial for the students.

3) Should students have Math homework everyday? How much homework?

-Teacher A goes by a rule of 10 mins per grade increment and then dividing it according to how many subjects the students are taking. He gives students time at school though to work on those problems as well. In the following class he takes up some questions and sometimes some of the questions lead in to the next lesson for class.

-Teacher B too gives time to do work in class. He assigns 5 questions and they work on it in groups in class. That way they get to work in their groups and they’re actually keener on doing the work and also the teacher can check the work right away.

For the most part the students agreed that homework should be assigned in order to make sure that they’re caught up with what’s being taught. But one students said it should be up to the students so that the more responsible students won’t have to do as much work since they’re already caught up.

4) Do you emphasize more on computational or more analytical mathematics?

-Both teachers said that because of the provincials they have to teach computational heavily. They both ideally would like to be more analytical but since it’s a private school, the parents just want the grade results. So they actually feel like they’re just “training” the students rather than teaching them.

-Likewise the students all preferred the computational part of math. Since they are concerned about tests they just want to learn the equations and don’t care much of what theories are behind the equations

5) What is the most creative lesson plan you’ve had?

-Teacher A emphasized the importance of how the teacher’s energy has a lot to do with how the students learn and react. He doesn’t believe in “gimmicks” or special activities in order to make the students learn.

-Teacher B mentioned a few things such as making brides and slides using quadratic and cubic functions. But the one he liked the most was giving students a protractor and a ruler and making them measure the height of a flagpole. This was prior to teaching trigonometry, so the students actually got creative and some of them actually came close to using trig without knowing it.

-The students however reinforced teacher A’s thoughts. They actually don’t have anything that standout to them, they just said that they just learned it because they had a good teacher that they liked to learn from.

My 2 Most Memorable Math Teachers and Forming an Idea of Teaching

With a short period of time I wasn't able to write as much as I wanted to but it did jog memories of 2 memorable teachers that will hopefully influence my teaching techniques in a way.

One of the most memorable math teachers I had was actually the teacher who ignited the idea of teaching into me. He was my senior school math teacher. He was very knowledgeable, rarely saw him look through a book, and was very open to questions. He commanded great respect from the class and he in turn reciprocated that respect. At the same time he would always have a lighter side, alway open to joking around with the students and would try to include humour in his teachings.

Another teacher I had was in university. He wasn't as effective as a teacher to say the least. He was monotone, didn't seem to like to talk to the class. He wasn't too clear on his lessons and wasn't receptive to student's questions.


What I've taken from these 2 teachers is communication with the students is key. I'm pretty sure my university prof was more knowledgeable than my high school teacher, but he was not as approachable as my high school prof. Having that approachability is key so that if the students don't get the lessons or have any concerns, they will tell you rather than be lost and not have you know about it. Also a tiny thing like actually facing the class to project your voice is a big difference in these two teachers and something I look to emphasize on. Obviously I want to make the students feel comfortable and to express themselves, so having humour in the class is great. What I really took from my high school teacher is to have a good balance of humour and command. There's a time and place for jokes, appropriate jokes mind you, and there's times where you just have to buckle down and be serious. So these things are what I'm striving to take into teaching a class.

Microteaching - How to throw a football

Throwing a football Lesson Plan

1) BRIDGE:

One of my big passions is football. I love watching it, but I love playing it more. Whenever I say I love to play football, a lot of the response I get is “I don’t know how to throw a football”. Who here would say that? Now physically, you can throw a football, but my guess is when you say that, you mean throw it with a spiral and make it look good. That’s what I plan to teach you today.

2) Teaching OBJECTIVES:

-Being able to ask questions that allows me to figure out the students’ understanding

3) Learning OBJECTIVES:

-Learn how to grip the football

-Arm motion (semicircle)

-body movement (leaning forward to get power)

-finger movement (to get a spiral)

-release point (dependent on how high you want your ball to go)

4) PRETEST:

-“who has thrown a football before”

-“who has thrown any ball before one handed, say dodgeball?”

-connect it with throwing point, arm motion and body movement

5) PARTICIPATORY Activity Ideas:

-start with ball grip

-then just arm motion

-work up to whole body motion

-actually get them to throw in pairs (if there’s enough room and appropriate setting)

6) POST-TEST:

-See how students threw

-Ask if there are any difficulties

-check if there’s something the majority is not doing

7) SUMMARY or conclusion:

Since we’re in a confined space, we’re not able to throw as freely as we want. But having the basics of throwing, you can now start working on throwing at moving targets, working on accuracy as well as throwing further.

Response to Skemp's Article on Relational and Instrumental Understanding

The article on relational understanding and instrumental understanding brings up a great point in the subject of mathematics. In so many areas of math, so much emphasis is put on rules in figuring out problems. It seems a lot of the people I interact with have an instrumental understanding of math, as formulas and equations, and find the subject difficult because of the number of rules associated with it. So it’s interesting when Kemp states that “instrumental understanding I would until recently not have regarded as understanding at all” (pg. 2). Knowing when the rules work is only part of the understanding and the rest of it is how and why it works, so I would agree with Kemp’s previous regard of instrumental understanding. Although this type of understanding is not as effective it does seem that more students would prefer it this way. “As soon as this is [rules for getting answers] are reached, [the students] latch on to it and ignore the rest” (pg. 4). So trying to teach a different way or trying to expand previous knowledge to your students might not be effective. So it does encourage to try to teach a relational approach immediately if that’s what you intend to do later in the course. It seems the less they can know but still pass the better, “if pupils can get the right answer…, they will not take kindly to suggestions that they should try for something beyond this” (pg. 5). There’s also the fact that relational understanding would require more work and more time and students will probably complain and lose interest in the subject after a certain point. It is also more appealing to students that often with the instrumental method, “the rewards are more immediate and more apparent” (8), giving the students a short-term pleasure, though may cause long-term turmoil. Though the students might want the instrumental way of understanding, as teachers we must realize that long term, it is better for them to know how things work. If there’s something that goes wrong during problem solving, they are able to “produce an infinite number of plans for getting from any starting point to any finishing point.” (14) Kemp’s analogy of traveling from point A to point B also applies to teachers. There are times when you are explaining a subject and different students will not understand it, so knowing a different route helps you get to point B in order for your student to get there as well.