Unit Plan. Math 8: Fractions

http://docs.google.com/Doc?docid=0AVXQhrx0fazLZGR3bmJxd2RfNTRncGZ2ZndjOQ&hl=en

Since it doesn't format as-is on blogger, here's a google doc of my unit plan. enjoy!

Cycling Digits: An incomplete solution.

Short Practicum. Story!

I was asked to teach a lesson for October 28, 2009. That lesson was the quadratic formula. I was asked to teach this lesson twice in a row. Once in the period before lunch, and another the period after lunch so that I would be able to figure out what went wrong in the first period and fix it for the second period. This is the story of the first period: Block C.

In the days leading to d-day, I was busy writing out the lesson plan from which I would impart my knowledge to the students of the class. I would start off the class reviewing material from the previous day, hand out a quiz, and proceed with the lesson. This lesson consisted of presenting the quadratic formula, working through some examples and proving why the quadratic formula looks the way that it is. I had originally planned to prove the equation before moving on to do examples because that was the way I was taught the formula. Under advice from my sponsor teacher, and experience team teaching the students, I decided to take the advice and adapt my lesson plan.

On the day of the lesson, I was nervous. My palms were cold, sweaty and I had butterflies in my stomach. The bell rings to mark the end of the previous period. "Fifteen minutes", I said to myself as I packed my bag to go to the classroom across the hall from the student prep room. I opened the door. I crossed the hallway. I dropped my bag off to the side of the room. I unpacked my notes and readied myself, high on adrenaline. My sponsor teacher walks up to me and asks, "how are you feeling?". I replied, "I'm nervous". He offers me tea to calm me down, and I take him up on his offer. It helped. I review my notes, and the bell rings. D-day.

"Good morning!" The lesson began. Review questions were written up. Review questions were answered. The quiz was handed out. The quiz was answered. It was the right difficulty and length. The lesson was going well.

"Who thinks completing the square is tedious and painful to use?" Hands rose. "Here's a faster way of solving quadratic equations." Time to teach the quadratic formula. The quadratic formula was presented. Everyone was happy. Engaged to boot. Examples went up on the board. Lots of audience participation. Hands were freely rising to give the answers. The lesson was going well.

"Who wants to see why the quadratic formula looks the way it looks?" A few inquisitive looks. Not bad. I can proceed. I proceeded. I erred. Twice. I gave up the proof. The lesson wasn't going well anymore.

"Ten minutes" I thought to myself. "I've lost the kids. Time for a story." So I start off the story with "Did you know that. . ." The lesson finished.

So what did I learn? I learned that it doesn't take much to lose a class and one needs a back up plan to be able to bring the students back from a disengaged state.

Dividing by Zero, a Reflection

What do I think about the free write? The free write, I thought was a little loaded. Discussing division and zero separate from each other sort of avoids the idea of dividing by zero. I did manage to use some of what I wrote on the free write in my poem but except for a few lines, I managed to keep division by zero out of the picture. In fact, I think that I didn't manage to glean anything I didn't know before about division by zero from the free write. Maybe it's because that I've heard the arguments for why one cannot divide by zero. I think this exercise would work *with* a classroom that doesn't genuinely know why dividing by zero is so taboo.

What do I think about the poem? I think it'd be awesome to use this exercise in a high school class, paired with the free write. I think each student would be able to pull out some sense of why division by zero is not discussed from their own experiences with division and zero. It most likely would be fun and cool all at the same time. As a pedagogical exercise, I think that I didn't take away all that much from it apart from the idea that it could work in a classroom. I feel that I could probably explain this mathematical no-no to a variety of secondary school kids of all levels.

Dividing by Zero, The Free Write

Divide:

What does it mean to divide? How do you divide? What *can* be divided? Let's divide by 7. 7's a number that's very hard to divide with in the sense that you cannot easily tell if a number is divisible by 7. Other numbers like 2, 3 and 5 have mnemonics that you can easily tell. They're quick and painless to tell. 7 is much harder. I've gone too far from dividing. Does dividing mean counting how many somethings fit into a bigger something? I think that makes sense in the realm of integers. For non integers though it gets a little more weird. It's all well and done so I don't really know. I dont know what to write. I dont know what to write. Oh. Let's divide. a over b. b fits in x many times into a perfectly. and it's only x. it's not any number. it's cool like that.

Zero:

Zero. nothing. nada. zilch. Nothingness. Emptiness. like the amount of what i want to write. zero is kind of cool. when you multiply it to something, it reduces that something to zero. adding zero to something does not change that something. it's great for problem solving sometimes. but when you divide. what happens when you divide. zero over something is always and will be zero but what happens when you divide by zero? thats sort of a grey area in math. it's weird. it's like a black hole. a singularity. a void, even. a singular point. it's crazy. but we'll talk more about that later. let's talk about something raised to zero. something raised to zero is always 1. except zero raising itself to zero. it's one of those weird divide by zero things. crazy.

Dividing by Zero, a Poem.

How can you divide by zero

I want to know

Why would you want to divide by zero

I want to know

Is it practical is it trivial

I want to know

Is it fun is it boring

I want to know

How many zeros can fit

Too many says my head

Too little says my head

Just enough says my head

So many answers

So many suggestions

All numbers fit in my head

Is it zero is it one or is it fifteen-million three hundred-thousand two hundred and ninety-four

I do not know

I do not know

I want to know

I want to know

All I can say is that it cannot be any

Because if it is any then it is all

All are zero all are one all or all are fifteen-million three hundred-thousand two hundred and ninety-four

I want to know

I want to know

But I cannot know

But I cannot know

Team Teaching: Factoring Quadratics

THE GOOD:

My peers felt that the use of algebra tiles as manipulatives worked very well in explaining what the factored and expanded form of a quadratic looks like. It kept them very engaged and it was sort of a fun puzzle solving activity because finding the right configuration to get a rectangle was fun. It also helped students realized that there are some configurations that don't really work. As well, they found the lesson to be clear, concise and got the point across well enough.

I felt very much the same about what my peers felt. When Enrique, Nathan, and I were discussing how we would introduce factoring quadratics as such and got to discussing the algebra tiles, I saw that the activity would really engage my peers. I think this would totally work in a classroom one day. In fact, during my practicum, if I'm asked to do an introduction to factoring quadratics like this, I might adapt this lesson plan.

THE NOT-SO-GOOD:

My peers thought that time management was an issue. They also saw that there were some issues with classroom management as some students were chatting and not paying attention after the group activity had ended and some students felt that the group activity should have deserved a bit more time.

I saw the issue with time management as well. We tried our best to get our point across as quickly as possible but I felt that this lesson could have gone a full hour given the activities and the discussions that it could have garnered. I wanted to let my peers figure out the pattern between the two ways of calculating area of the rectangles they formed as this would show them, not only FOIL, but how to factor a quadratic polynomial. But as such, time was an issue and I felt that the point of the exercise was lost due to time constraints.

With regards to classroom management, I think I could have handled myself a little better. There were a couple of times where I "snapped" and shot the infamous "look" but I recovered quite nicely once I got the students' attention again. That might be something I want to look out for in my future practice.

Factoring Quadratic Polynomials: A BOOPPPS Lesson Plan

Bridge: Introduce quadratic polynomials by using algebra tiles and relating them to areas of rectangles.

Learning Objectives: Students will gain a basic, entry level idea of how to factor basic quadratic polynomials of the form ax² + bx + c.

Teaching Objectives: Teachers will be able to use group work to teach students the patterns of factoring quadratic polynomials using manipulative such as algebra tiles.

Pretest: Ask students what kind of simple geometric shapes can be made with the algebra tiles. Ask, in what ways, can one find the area of the rectangle. (Should be two distinct ways. More if the class is creative.)

Participation: Put students into groups of 3 with a set of algebra tiles. Have them form various rectangles and note, using the different ways, what the area of the rectangle is. Make note that these are the expanded and factored forms of the quadratic polynomial.

Post Test: Ask students if they found a pattern to the two ways that the area was calculated.

Summary: From here on, discuss the various other forms of quadratic polynomials as the group work. Discuss factorable and non-factorable quadratics in that manner. Talk about the quadratic formula and what it can do to factor any quadratic.

Mathematics and Civics

We need math. Math is all around us. We need math to function as citizens of our nation. Numeracy teaches us about logic, problem solving, and common sense. To be able to function as “good” citizens in our society, we need all these skills that math teaches us. We, as citizens, use the basic principles of what we learn in mathematics class regardless of whether we see it that way or not. I find it, as well, that people don’t see the math they do as math. If we can get people to understand that the math they use in counting money, figuring out travel time, and other life skills are, fundamentally, mathematical, then I think that people will want to study math more.

This article basically talks about what I’ve believed in for as long as I’ve been studying math. As we are constantly immersed in numbers, shapes, statistics, problems, and other mathematics related ideas, the skills taught to us in math class help us in our day to day lives. We *can* live as citizens of our society *because* we learned math.

In that note, I agree with the article provided for us. Mathematics is necessary in our tool kit of skills for life as citizens in the world. It's just that we need to help people realize this fact.

reflecting on WIN

The What-If-Not approach of problem posing interests me. It takes a commonly viewed notion and turns it upside down. It takes something true, and asks how we can change these attributes. This would lead to a great many questions that need to be asked, of varying significance. I think that this strength is also its weakness.

Asking what-if-not, as said earlier, leads to many new questions asked. In asking these questions, these lead us to new and profound ways of mathematical thinking. It sparks our curiosity because we ask questions that, in some cases, have nothing to do with the original idea. It is, however, bad as well in a sense. The depth of these questions relies significantly on how the problem poser thinks. Things might be left out. Even deep questions that let us form a new mathematical perspective may be missed because the problem poser didn’t see an attribute and change it accordingly. Also, changing an attribute a certain way could lead to a similar problem like this.

Given that, it would be very difficult to put this sort of thinking into factoring quadratics. But that wouldn’t stop me. I could ask questions like what if it wasn’t a quadratic? What if the exponent wasn’t an integer? What if it wasn’t just one variable?

These thoughts would all lead to very open ended discussions revolving the subject matter. I like the WIN approach. It challenges both the problem poser and the problem solver. These what if not questions can lead to deep and profound mathematical ideas that could very well reach outside of the mathematics curriculum and I think that would be totally awesome.

10 Questions on The Art of Problem Posing

1. I found the first section quite dry. But I do understand that there is an intended audience for this book.
2. How do you encourage students to pose their own problems?
3. In respect to the title of chapter 3, what does it mean to accept?
4. How appropriate is asking a broad question to students who have never really thought about math in terms of problem solving/posing?
5. How do I make the ideas outlined in these first 3 chapters of the book more relevant in my pedagogy?
6. With respect to internal and external exploration, which do you think is more effective?
7. When exactly is it appropriate to start a thought experiment such as problem posing?
8. Should I encourage students to pose problems for each other?
9. I love how they let us pose our own problems about a² + b² = c² and showed that there wasn't any problem, but we answered our own questions about the equation.
10. Overall, I think the rest of this book will be a very interesting read.

Think Ahead 10 Years. What Would Students Think of You?

Student who likes you:

Hey Mom

I had the most wonderful year learning Math 12 with the best teacher that I have ever had. He made sure that the mathematics that we were doing were relevant to our daily lives. He gave us clever stories about the mathematicians that 'roamed the earth'. And he made it fun. We didn't just do work individually in class; we did a lot of group work. We even got to make 3 dimensional models of things like cubes and 20-sided dice. I had loads of fun.

Also, he's so helpful when we're doing work because he walks around the classroom checking up on us every now and then. I find that it helps when he asks questions that lead to the answer and does not just tell me what the answer is.

Oh and his teaching style isn't always lecturing and having us do work. It's a combination of many things including video, interactive presentations, and all really cool stuff.

I really enjoyed my time in his classroom. Thanks for enrolling me into this school!

All the best

Student A

Unhappy student:

Hey Mr. Vicente

I figured that I would write you an e-mail discussing my opinion of your teaching style. In my opinion, I think that you speak too fast and cover the material inadequately. Sometimes you mumble and your jokes are never funny. You're almost never available after school because you're too busy with other school activities and I find it hard to ask for help in class. You always put me in a group with people who are smarter than me and I feel that I don't learn because they are always just doing all the work instead of me. I just wish that you would spend a little more time helping me with my difficulties.

On that note, I feel that when you ask a question, you always seem to pick on me when i look so confused. Is it that you want to humiliate me in front of my students? I don't appreciate that so much. Just give me a chance to understand whatever you say in the class before throwing me into the fire.

Sincerely

Student B

As a reply to Student A, I hope to be able to do whatever he or she mentions in the above. I think it'll be cool to have a dynamic and engaging classroom.

As a reply to Student B, I hope that whenever I single out a student, I hope that they understand that I do it so that they can accomplish something. When I put a weak student with good students together for group work, I hope that the good students teach the weak student, and hope that the good students don't do what they do to student b.

Teaching the Marked Case: In Reflection

The video we had watched in class spoke to me. I saw that simple repetition and building upon what was established in class worked effectively to teach the students the basic principle of the number line. Small steps helped ingrain the basic principles in the students and I think that the way the teacher taught was quite effective.

Using repetition, he was able to tell the students that steps generalized to more complex things. He was able to teach gr 9 students simple algebra by example and by asking the students what they would do. In effect, he was able to let the students flourish and figure things out by themselves by guiding them. The students, in effect, already knew what to do. He just let them know that they already knew. This, I think, would build confidence on the children.

As well, whole class participation helped the teacher see if there was confusion among the students. This also let the students reflect on what they had just said and allowed for them to correct their mistakes. I thought this was very effective rather than the teacher telling the students that they were wrong.

The teacher also used the classroom effectively to teach the students. He used the space outside of the blackboard to extend the 'number line' so that his students may understand that the number line infinitely grows.

Overall, I think that his teaching style works quite well to be able to teach students concepts and such. My question is, how would you scale this sort of teaching to higher levels like Math 11 or 12?

Battlefield Schools - A Summary

The article outlines two very different schools of thought in our education system, progressive and conservative. The conservative view on mathematics education focuses more on instructional and instrumental understanding of the material. This view sees that answers and facts are the way to teach math, treating it as a means to an end. In stark contrast, the progressive view sees that the journey to the answer is much more important and experimentation and understanding is the way that math should be taught. The rest of the article outlines how these two have “battled” for supremacy in how mathematics is taught in the K-12 level.

The progressive movement of the early 20th century saw a realization that experimentation and exploration would lead to more democratic citizens. Educators like Dewey found that, instead of dictating knowledge to the student, it was more effective to let students grapple with issues and problems and let them figure some things out for themselves.

In the 1960’s, a more abstract form of math, called the New Math, permeated the mathematics curriculum of the day. The launching of Sputnik into space meant that America was losing the space race and it was imperative that students in the secondary level ought to be trained to become future rocket scientists. This was seen as very radical and unhealthy for the state of mathematics education as it was conservative idea parading as a progressive one. It also did not account for the fact that not all students wanted to get into Mathematics and Science after finishing secondary school. Lastly, teachers could not teach the material at all because they were educated in a totally different than what the curriculum outlines.

In the latter part of the 20th century, the 80’s and 90’s, a third movement caused a great shift in mathematics education, the Math Wars. Conservative ideas of mathematics education were being branded as right-wing and radical while relational understanding and progressive ideas were seen as more desirable. These ideas are what we would see today happening in our current curriculum.

Conversations with Teachers and Students: A Group Retrospective.

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.

Conversations with Teachers and Students: An Individual Retrospective.

In the past few days, I have had a chance to reflect on the recent events that my group and I had with the math and student teachers we chose to interview.

In interviewing these groups, I found that there is a certain difference with what the students expect from the teachers and what the teachers expect from the students. The teachers we interviewed try their best to engage the students by including more relational oriented material. The students, however, wanted more instrumental learning because of provincial exams. In that note, the teachers found that they did end up teaching for the provincial examinations. With the death of the provincial exam would be a greater change in curriculum.

I think now that my job, as a teacher candidate, is to be able to find a happy medium between what the students want and what the teachers would like to teach. Reflecting on these interviews has given me an idea of what is expected of me as a teacher by the students and what I should expect of myself as a teacher for the students.

The Two Most Memorable Teachers in My Life

The first teacher I'd like to talk about is Michael Bennett. I had him as a professor for two courses at UBC and the reason why I found him so memorable is because he knew his subject but he wasn't so encumbered by formal mathematical writing. He would talk about squares of numbers and write them out literally as boxes. I found that quite novel, clever, and amusing. It got the point across. Also, he always knew how to get a student interested in his subject matter by his energy in the classroom. All in all, he made two seemingly difficult subjects enjoyable because of his teaching style.

The second math teacher I would like to talk about is Don Furugori. He was my high school math teacher for grades eleven and twelve. He always seemed to teach the material well enough so that an expected mark would be achieved. What I do remember from taking his classes was that it snows a lot in Ottawa. That really summed up his teaching style: very tangential.

Using Research to Analyze, Inform, and Assess Changes in Instruction: A Mini Commentary.

I felt empowered by Robinson’s article. I feel that her article was basically a success story of how one can change the dynamic of the classroom for the better after years of a certain way of teaching. Also, since she’s fairly new to the profession, I can relate to her better in the sense that I most likely will end up the way she ended up during her first four years of teaching simply because it is practical and that I would probably need to gain more experience before I can start experimenting with lesson plans. Ideally, I would like to start out with the kind of lesson plan that she implemented after her research but I won’t really know how that goes until I actually start teaching.

With respect to the topic of research, which she mentioned quite often in her article, I once heard that “if you know what you are doing, then you are not doing research”. I believe that to be true and that her research endeavour was started off as an experiment. She really didn’t know what to expect from changing her teaching style and she does admit that she was quite surprised with her results.

Overall, I think her article was very revealing and touched on very well how a teacher should always be developing his or her practice. Creating an engaging environment like that will create less stress for the teacher in the long run because there is less to worry about and if all works out like she outlined in her article, students are learning. If students are learning, then her job is a success.

Micro-Teaching: A Review

On Friday, September 18th, my peers and I got into groups of four to five to do some micro teaching. Here is a summary of what my peers and I thought about my mini lecture on the basic rules of composing photographs.

After reading what my wrote about my micro teaching session on how to make a compelling photograph, here is what I found for each section:

Bridge

They found my bridge to be concise and to the point.

Learning and Teaching Objectives

They found my learning objectives to be clear and well understood.

Pretest

My peers found my pretest to be very seamlessly integrated with the learning objectives and the bridge. They found that my use of visual examples via a laptop computer to be very well done

Participatory Activity and Post-Test

All my peers found my activity and post test were, again, seamlessly joined together and found the constructive criticism that I provided useful.

Summary

Was labelled as ‘good’

Strengths

They found that my session was well structured and flowed very well from one part of the lesson plan to the next.

Also, my use of examples and feedback during the post-test were mentioned to be quite good

Weaknesses

It was felt that, during the participatory activity, a single subject should have been used. I had let the students decide on what subject that he or she would like to photograph.

One of the more tactile learners would have preferred to touch the cameras I had brought while I was lecturing

Lastly, and most importantly, it was mentioned that I used the phrase ‘you know’ quite a bit. Notably, it stopped towards the end of the lecture.

Given that, here are my own thoughts about my micro teaching session:

I liked the fact that my microteaching session flowed seamlessly between through the BOOPPPS lesson plan. I quite enjoyed the fact that my peers had a hard time distinguishing where one section ended and where another started. I liked that I gave the students a chance to apply what I had just talked about by letting them take my cameras and shoot photos. All in all, I think that those aspects really worked well for my lesson.

Unfortunately, like all lessons, there were quite a few things that went wrong. Most of these things involved my own speech. I found that halfway through a sentence, I would say “umm” or as a peer noticed, “you know”. I tried my best to avoid such phrases but I found that I was speaking faster than I could generate words to express the ideas that I was trying to convey. As well, I feel that I had brought up too many topics to think about in a 10 minutes mini-lecture so I could not properly go in depth with the details. That is, I could not fully explain relationally why a compelling photograph had the elements I outlined and I feel that this made my micro-teaching lesson much weaker than I would have liked.

BOOPPPS Lesson PLan

In preparation for the micro-teaching assignment tomorrow during MAED314A, I have prepared a simple lesson plan based on BOOPPPS to outline what I intend to teach during tomorrow's activity.

Composition. How to Shoot a Compelling Photograph.

Bridge

• Everyone loves to take photos. With the dawn of the digital camera and websites to share photos like Facebook and Flickr, how do you create photographs that stand out from the rest?

Teaching Objectives

• Find balance between instrumental and relational teaching of proper composition in photography.
• Engage the students’ interest in the subject matter.
• Learn from the students too. That is, learn different opinions on what makes a good photograph.

Learning Objectives

• Students should have a basic and working knowledge of proper composition.
• Students will learn about lines in composition using some sample photos.
• Students will learn how to lead the viewer’s eye with lines.
• Students will learn the ‘rule of thirds’
• Students will learn to be more comfortable with ‘getting close to the subject as possible’ to fill the frame of the camera with their subject.
• Students will learn to shoot from unusual angles.

Pretest

• Show two photographs to the students.
• The photos are to have the same subject but one is improperly composed, while the other is properly composed.
• Ask the questions: “Which photograph do you find more compelling? Why does this photo stand out more than the other?”
• Segue way into discussing the learning objectives with the students.

Participation

• Students will be given one or two minutes to take photos of a subject of their choosing using concepts outlined in learning objectives.

Post-Test

• Check to see if students followed ideas that were discussed in the learning objectives.
• See if they applied the ‘rule of thirds’
• Note how ‘full’ the subject is in the frame of the photograph and if the student shot the photo at a clever angle.
• As an added bonus, the photos that the students took will be sent to them by the teacher via email.

Summary

• Reiterate the basic rules of composition discussed in the lecture
• Describe more advanced composition techniques such as ‘depth of field’ and ‘shutter speed’.

Relational or Instrumental? A Commentary on Skemp's Article

For our first MAED314A assignment, we were asked to write a short commentary on an article by Richard Skemp on instrumental and relational understanding of mathematics. As well, we were asked to choose 5 quotes from the article and comment on them.

Commentary:

In reading Skemp’s article, Relational Understanding and Instrumental Understanding, I was able to fully realize concepts that I had encountered in my own time of learning mathematics. Skemp makes note of two ways of understanding mathematics and says that both ways of understanding have merits. I’m inclined to agree with Skemp in so far as both types of understanding are necessary to be able to effectively teach mathematics. I feel that a compromise between the two schools of thought are necessary to cater to the minds of each learner in the classroom. I would like to say, however, that I would much rather prefer the relational way of understanding mathematics because I have always felt that, in the long run, it is simpler and more useful. Given that, instrumental learning and understanding of mathematics has its merits and I believe that Skemp has covered those merits quite well in his article. I agree that it is faster, simpler, and provides immediate satisfaction than its relational counterpart. As such, I am inclined not to simply dismiss this way of learning. The bottom line is that I agree with most of what Skemp has said in his article and that I, myself, have learned quite a bit through reading and digesting the ideas inside it.

Quotes:

“That he is a junior teacher in a school where all the other mathematics teaching is instrumental.” (11)

In some merits, Skemp is right. A junior teacher will find it difficult to apply a different teaching style in a school where the vast majority of teachers support a certain style. But my question is, why would it be difficult? Why can’t the junior teacher be a catalyst for change? There isn’t really a concrete reason why the junior teacher should teach a certain way. Unless maybe his or her job depended on it.

“From the marks he makes on paper, it is very hard to make valid inference about the mental processes by which a pupil has been led to make them”(12)

From the meagre amount of marking that I’ve done, I am inclined to agree. While it is easy to spot a mistake that a student can make and correct it, it isn’t always clear to see what the student was thinking while writing down the solution to a problem.

“So far my glowing tribute to mathematics has left out a vital point: the rejection of mathematics by so many, a rejection that in not a few cases turns to abject fright.”(12)

I hear this all the time from other students. “Math I hard. Math is scary.” Partly the reason I wanted to get into teaching was to help prevent this sort of mentality.

“At present most teachers have to learn from their own mistakes” (13)

I don’t yet know whether or not I should agree or disagree with this statement. I am inclined to disagree given the amount of support that a teacher has at his or her disposal. Besides, a teacher can learn from other teachers’ mistakes too.

5. “…nothing else but rational understanding can ever be adequate for a teacher.” (11)

I wholeheartedly agree with this statement. As discussed in the article, instrumental understanding is generally superficial and a teacher with superficial understanding of the given material, regardless of subject matter, cannot effectively teach.