Assignment 3

Geo Sketchpad

From Natalie's workshop today, I had realized what a powerful tool the geosketchpad could be to bring out mathematics. One amazing feature was that the program did not make us use that much numbers, and had us constructing shapes and questioning basic properties of triangles and such. It's definitely a tool that I would love to incorporate into the classroom.

A few worries from the seminar that I had however were that I don't feel very competent with my geometry and am not sure if I would be able to incorporate the program into the classroom well. Secondly, as a pre-teacher, one of my main concerns would be time and I wouldn't want to stray away from the PLO's too much.

However, I'm very glad and excited that I had the opportunity to experience and play around with this technology. Hopefully, I'll figure out a way to effectively incorporate this into the classroom setting- even if it's just a method to introduce geometry or a way to wrap up the unit and solidify a student's understanding.

Thinking Mathematically Problem Solving!

Had trouble uploading the problem, so I just decided to type it:

Working on Problem: 31

WORK ---------------------MENTAL THINKING

Try it: a) 1 5 7 10 11 15 16 20 24 25 29 31 b) 5 10 15 20 25 30 31 c) 5 6 10 15 29 24 25 29 31

Try working it backwards. To be the person to reach 31, must land on 25 to ensure a win because if not- will end up landing on a number between 26-30. This leads to the other player adding to 31.

Try out hypothesis: d) 1 3 7 8 13 15 19 23 25 29 31 (Eureka, it works!!) (PLUS, this works every time AS LONG AS you start off with the number 1.. and make sure you don't lose count.)

Try working from it backwards with consistent train of thought... If person who lands on 25 is the winner, then he/she would want to land on 19 to ensure a win. And keep doing the same pattern, and realize we're just subtracting six from the number each time. Ending up with the pattern of wanting to hit the numbers 7, 13, 19, 25, 31 in order to secure a win. In other words, we want to touch x = 1 (mod 6)

Question number two: What if 31 was some other number? e) Reach 33: 3 8 9 10 14 20 21 22 27 28 33 f) Reach 24: 4 5 10 15 16 17 22	We can try to work backwards again or can we see if we can incorporate modular arithmetic? For example, if we want to reach 33, should we try x=3 (mod 6) where x is 33? Or if we want to reach 24, should we have x = 22= 4 (mod 6)
Question: What if it goes from 1-6? g) Reach 31: 3 9 10 12 17 23 24 25 31	Well, since it was in mod 6 when it was from 1-5, is it now in mod 7? So let's set up our general equation and check: x = 31 = 3 (mod 7)

Question: What if the numbers were 1,3,5 or 2,3,7?? Ex) 1 2 7 10 13 18 19 22 25 26 31

Try out with 1, 3, 5... Can we still use generality? YES! Why? Because, as long as we start at 1, can always reach x = 1 (mod 6) because of how the numbers work--> at your turn, will be able to chose number that makes it become 7. To make this more clear, if we start with 1, then need to add 6 to become 7. If player 2 says any of the numbers (1, 3, 5), we can find the complement of it that adds up to 6- making it 7. 1&5, 3&3.

** Unfortunately, I got really confused if we extended it to 3 players and came to the conclusion that we can have 2 people gang up on the other player to make sure the 1 person does not win. But I'm not sure if we can have a guaranteed winner.

*** Tried to make a tree (like a probability tree) with testing out 2, 3, 7 to see all the possibilities. But it got really messy and I was unable to figure out the winning strategy if we only use number 2, 3, 7.

Stories of My Short Term Practicum

1.
I had the opportunity to observe a variety of classes- as I am sure many of my peers also had done. However, it was very interesting to note down all the different strategies seen in the various classes, different management strategies and teaching strategies that one can incorporate perhaps into their own teaching style.

In one English course, the teacher had incorporated various teaching strategies that were engaging as well as relevant to the topic. They had worked in groups of four to discuss the topic at hand, with one person being the "recorder". After, each group had a presenter where they represented their group, went up in front of the classroom to speak into a "mic". It gave the students an opportunity to realize how there are various way of communicating. As well, it made me realize the IMPORTANCE in giving students the chance to have an outlet to explore the subject in different manners and to present information in other modes than merely note taking.

In another class, there were three different levels of Japanese in the room- grades 11a, 11b, and grades 12. Though they were learning different material, they had the ONE teacher in the same room. It made me realize the importance of staying on TOP of things, being really organized, and able to be multitask as a teacher.

I had also attended two Social Studies courses, and had realized how teaching styles and management can be different- yet, both teachers had a good control of the classroom. One teacher, a relatively petite lady, had really rowdy kids- however, when they were in HER class, not a peep was heard from them other than when she asked for participation. They were all on task and it was evident that the class as a whole was learning. In the other class, it was the polar opposite- it was a total RUCKUS! However, it was evident that the students knew where the teacher's boundaries were and would not step over the line. There was a lot of banter between the teacher and students, yet, material was still being learned. So from this experience, it makes me reflect and wonder what kind of style would I want to incorporate into my classroom? Which method would work for ME and my students?

Though there was much that I had observed and it is impossible to note it all down at the moment, another fantastic approach that I saw was co teaching. Or having teachers from other subjects (english) come into a science class and help students learn about note taking and how to grasp information from a textbook. It was a great approach to help students realize that learning should not just be ONE subject on its own- it should be interdisciplinary. If possible, I would love to try and combine mathematics with other subjects (such as science or language)!

2.

On my short term practicum, I had taught two classes. One grade 8 science (on osmosis and diffusion) and a grade 11 math class (on graphing quadratic functions). For the latter, I had quite the learning experience...

Though I was into the second week of the practicum, I had not met this particular class yet. So teaching the grade 11 class was a bit nerve wracking as I did not have any idea what to expect. My sponsor teacher started off the class by introducing me and telling them that I would be teaching them for the day - making me realize that the students had no forewarning that I was their student teacher until that day. After the short introduction, my S.A. went to the back of the room to let me start teaching.

With a no bell system at the school, I promptly closed the door and began to take attendance. I had learned from my first class that it was important to not let late comers disturb the flow of my class and that I should wait until I was ready and found a natural pause before I opened the door and let them in. (I had learned this the hard way and found myself being constantly disrupted the first time I taught).

After I had let the latecomers come in, I began teaching students on quadratic functions. I plotted out the basic parabola with them first and then had them explore transitions and transformations. By the time I had gotten to horizontal shifts, I realized that the pace I was going at was much slower than I anticipated. How could I manage to teach everything to the students by the end of the class?

Then, in the middle of my teaching, an earthquake drill went off and we were disrupted for at least 15 minutes.

After the drill, the energy of the class had gone up quite significantly and it was difficult to have the students get back on task again. However, being stubborn and thinking that it was REALLY crucial for the students to learn what I had planned, I sped up my teaching a bit more and finished up the transformation quickly and assigned homework for them to do, and then dismissed the class.

A few things that I had learned from the classroom- or finally realized what my UBC teachers have said is really true... Time is my worst enemy. In the future, I must plan for LOTS of extra time in case students struggle with a certain concept, or just for disturbances such as drills. Secondly, I had not followed up with the late students- and we need to make sure the students know that lates are NOT allowed. And as well- even though I'm rushed for time in teaching, if I ever ask a question, I need give some wait time for the students to absorb my question. Lesson plans are constantly being changed and as teachers, we need to be flexible and find a natural stopping point if we cannot teach all the material in one lesson. And finally, with a no bell system- I must be really AWARE of when to dismiss my students!

Group Micro Teaching Reflection and Summary

Overall, the micro teaching gave a great opportunity for us to try and test out our skills in explaining math techniques. Unfortunately, I was sick and did not get a chance to participate in my peer's teaching, however- it was a great opportunity to try teaching in a safe and constructive environment.

One thing I wished for would be that instead of a 15 minute microlesson, the lessons were 30 minutes long instead. That way, we would have a better sense of what a "true" lesson at school would feel like.

For my group's microteaching lesson in particular, there were quite a few points that we needed to improve on and a few points that went really great as well:

Plus-
- the class was very actively engaged and on topic in general
- it was a nice way to introduce the topic
- it allowed a chance for the students to shine on the stage
- the Mario and Luigi worksheet was humorous yet very applicable

Minus-
- some of the worksheets were not very consistent and at times too challenging and confusing for a grade 8 student
- the students may not understand what the focus of the activity was because we did not introduce the activity well
- our introduction did not go the way I wanted it to and I felt that it may have been confusing for the students
- the students began to dominate the discussions after awhile, and as "teachers", we should learn to facilitate the conversation and guide them in a way that was suitable for all students to understand
- our timing was really off and it would have been much better if we made the worksheet shorter since we lacked time to conclude the activity nicely

Interesting-
It was a great experience to try "coteaching" with other colleagues and I hope that in the future, I will have the opportunity to try this in real life. It is definitely challenging to communicate with one another as we all have our own unique styles and way of classroom management. However, with practice and confidence, I am sure that students will be able to benefit in learning from different teachers.

Group Microteaching Lesson Plan

Topic: Graphing

Bridge: Tell the students that we are going to work on graphing today. Assign them into groups and hand out worksheets for them to work on!

Teaching Objective: To allow students the chance to work in groups to think and hypothesize over graphing. To promote a bit of worksheet guided independent learning. For students to work on their communication among peers to reach a conclusion for the answer. And to facilitate discussion among the small groups.

Learning Objective: Students will be able to create lines of best fit for particular graphs and realize that graphs are not always linear. As well, they will learn that there is a practical reason for graphs!

Pre-test: Ask if anyone knows how to graph data points. Ask if they have seen a graph and know what should go on the x and y axis.

Participatory Activity: Hand out the different worksheets to different groups to have them explore graphing techniques.

Post-test: Have students come up to the board, draw their graph and describe what conclusions they derived from it. The students will explain the answers they obtained from the experience.

Summary: Conclude with we can graph many different shapes and they all explain data in a very visual friendly way. As well, it is important skill for us to be able to graph plots because we can predict the general trend and see relationships. Graphs can be very applicable to daily life. In the upcoming unit, we will focus in more detail on the linear graph.

Strengths and Weaknesses of "Dividing by Zero"

Strengths:
- The activity was fresh, interesting, and different
- It allows a person to explore their creative side and makes one realize that math does not mean LIMITED by numbers!
- It gives a chance for one to delve and try to understand why dividing by zero is an area that no one want to step into from a different perspective
- It gives a chance for a student to explore in a fun way

Weakness:
- If we choose a word that has a different meaning in a mathematical context, it may make it more confusing for a student.
- It may not be really relevant in mathematics and there is a chance that nothing is learned or gained from it. It's just a "hoop to jump through", leading to a student that is fixed minded
- It could make a student that is not interested in poetry discouraged when they expect to be learning "math"

Interesting:
- I first realized that Math and poetry can bode well in a Harold and Kumar film...

I’m sure that I will always be
A lonely number like root three

The three is all that’s good and right,
Why must my three keep out of sight
Beneath the vicious square root sign,
I wish instead I were a nine

For nine could thwart this evil trick,
with just some quick arithmetic

I know I’ll never see the sun, as 1.7321
Such is my reality, a sad irrationality

When hark! What is this I see,
Another square root of a three

As quietly co-waltzing by,
Together now we multiply
To form a number we prefer,
Rejoicing as an integer

We break free from our mortal bonds
With the wave of magic wands

Our square root signs become unglued
Your love for me has been renewed

- Asides from poetry, I want to try and mix different subjects with mathematics! (Like BIOLOGY. =) )

Divide, Zero, Dividing by Zero

DIVIDE

When one says the word DIVIDE, a series of random words flash through my head. Divide, as in using simple arithmetic with calculators. Divide means math. Divide means fractions. Divide makes one thinks of splitting pie into EQUAL PARTS. Divide also means to share things EQUALLY and fairly. At the same time, the word DIVIDE makes me think of an argument. Arguments between families, arguments due to language barriers, a space and gap between nations causes divides. Is division the opposite of multiplication? Grouping things into groups and finding the number of groups- "divide yourself into groups". Divide and conquer- what does that mean?

ZERO

Zero. Absolute ZERO. Nothingness. A vacuum. Zero = 0 = a nice round number~ A nice round number that has nothing IN it, it's empty inside that circle. Maybe that's why the notation 0 is zero? Zero. Coke zero. What is coke zero? Is it truly possible to eat something with ZERO calories? Is there a catch to this? Zero. Nothing... Does it mean that the number zero has no meaning to it? Do kids understand the concept of ZERO? Of NOTHING? Nothing we've experienced has really had NOTHING in it... except for in a vacuum. The word ZERO. Also a character's name in a Japanese anime. Why was he named as ZERO? What significance did it have? On the number line, zero isn't a positive. Nor is it a negative. It's stuck right in the middle. What is dividing by zero mean? Would there ever be a scenario that you would want to add or subtract by zero? Zero is such a SAD number. Zero is a WHOLE number, but not a natural number. Is it not found in nature? Zero. What a nice round number for a number that is EMPTY~

DIVIDING BY ZERO

You want to split something by nothing?
No! Don't you even dare dream of it.
It's ludicrous, it's blasphemy, it's absolute catastrophe!
Chaos and confusion, that's what it will do.
Dividing by zero- are you absurdly insane?

ONE pie, split among two,
Split it into halves and each get a part.
ONE pie, split it among one,
Then the person gets a whole pie to themselves.
ONE pie, split among ZERO...
How can you split a pie with nobody there?

Thinking about it makes my head pound and hurt.
So don't EVER waltz into that gray area.
Don't dare dream about dividing by zero.
It's just simply utterly impossible.

*********************************************************************
Here is the first free write with the word DIVIDE:

Divide, division. Wars- splitting up. What causes a divide? And argument and disagreements. There can also be physical divides- divides like being divided across an ocean. Divide = PIE. How many ways can you split a pie up into different pieces? Divide- the OPPOSITE of multiplication? Like add is "opposite" to minus. Divide... a fraction. Divide, a line with two dots, one on top, one on the bottom. Having a divide is a sad sad thing. A dividor would keep a person very organized! Eight divided by two, how many groups of two can fit into 8? Divide also makes me think of cliques and groups with language barriers.

Here's the second free write with the word ZERO:

Zero. Coke zero. Does that mean there is zero calories? Is it possible to have zero- absolutely NOTHING? Is there any possible way to have nothing? You'll still end up with air... except in a vacuum. Zero. A number in the "middle" of the number line, not a negative, not a positive. Just stuck there. Zero- is it written like 0 because there is a WHOLE in the middle with nothing in it? Zero, such a sad sad number~ All lonely, no one to hear. Nothingness. Zero, why are there main characters in anime where the hero is known as Zero? Is there a special significance around it? Zero is a WHOLE number, but it's not a NATURAL number. Why is it not a natural number- do you not find nothing in nature? Zero is funny number since x^0= 1. And anything times zero is zero. Zero- such a funny word to say, yet, it is a word that makes me feel sad, alone and empty~

Citizenship Education in the Context of School Mathematics

From Elaine Simmt's paper, we take a look at what mathematics education has in correlation to civics educations.

In truth, I originally thought that school mathematics had little relationship with citizenship. Furthermore, through my high school life (up until Grade 10 at least), I had grown up with the understanding that:

1. Mathematics is a "set of facts, skills and processes".
2. Mathematics are facts and fact.
3. Mathematics is either right or wrong.

These are three of the instructional stances and strategies in mathematics that may conflict with citizenship education according to Simmt. I believe that depending on the teacher, the students' insight and use of mathematics could be greatly affected. By teaching in a way where the instructors look for the "right" answer and look for a specific computational skill and algorithm, students ay lose sight of the fact that mathematics is seen all around society. But training our students into looking for information and "stripping from word problems" the facts, word problems are not serving their purpose. They are merely simple math questions in disguise and students lose sight that mathematics and citizenship are integrated with one another- especially in today's world. By asking closed questions, the impression that mathematics is either right or wrong is given to students. Do we as teacher candidates want our students to think close mindedly? By asking questions that are either right or wrong, our students grow to learn not to question the results of mathematics and not think on a higher level of the Bloom's taxonomy.

In order to prevent this from happening, Simmt offers suggestions that promote active and critical participation in society. These include:

1. Variable-entry prompts and investigations: problem posing.
Problem posing is a difficult concept to grasp- however, it is filled with benefits. By giving very open ended questions where the students can construe their own understanding, their knowledge on the individual level becomes more solid. As wll, the students will need to "pose, negotiate, and judge the appropriateness and adequacy of their own and classmates' questions and solutions". By doing so, students not only work on their mathematical skills, but also develop skills in taking an active and critical participation in society that is expected of citizens.

2. The Demand for Explanation
By helping students reazlie that math is not merely about the RIGHT answer but about the "truth" and having them realize that there are multiple truths as long as it is backed up by explanation- their skills in articulating thoughts and assertions are built. By having my students explain their reasoning, I would furthermore be able to teach my students via construetivism. I believe that in this way, we would be able to educate for citizenship with independent and individual thinking.

3. Mathematical Conversations
Through discussing with their peers and teachers, students can perhaps develop the ability to solve concepts and problems through communication. By having interaction with one another, this would help them realize that the skills developed in mathematics is applicable in society as well.

From Simmt's paper, I have realized the importance of not only helping students learn mathematics in terms of "general numeracy" but also as a subject that the power to change society. Hence, though we may not realize it immediately, mathematics plays a great role in citizenship education as it helps us understand society AND develops active and critical individuals.

Pages 33-66 The Art of Problem Posing Reading

I thought that the reading from The Art of Problem Posing was quite interesting and a bit difficult.

Firstly, language was a bit difficult to read through and the details was a bit hard to grapple on. But in general, I really enjoyed the authors use of analyzing how we could approach theorems and "truths".

I can see a few strengths and limitations in the what-if-not approach and using it in a classroom setting. Firstly, by using this technique, it would help students to think on their own and to try to formulate their own thoughts. By using this concept in class and modeling it, the students will be able to perhaps develop this technique and approach the questions in this method. By doing so, they could perhaps make their own discovers, understand concepts much more deeply and gain further insight. Another strength to this method would be that students would not be taking facts at point blank, and coinciding with Brent Davis' lecture, it would promote growth-minded thinking. By questioning what is true, one can figure out and perhaps even make new discovery! As teachers, I believe that using the what-if-not technique and incorporating it into the lesson is very important if it is possible.

One limitation from using this technique is that at the high school level, most students are already familiar with taking theorems as facts. They may not appreciate questioning everything and become more confused in the class setting. As well, due to class and time constraints, using this approach may take much longer than expected and one simple "fact" may take a whole lesson to help students test out their "hypothesis". Furthermore, there is no real end to using what-if-not and if there is poor classroom and time management, this could pretty much go on forever. As well, I believe that the mindset of the students is really important to have them engaged with this approach. Otherwise, they would not take much out of it and only want the "fact" and not want to think too much over the concepts.

But all in all, I believe as teacher candidates, we should try to incorporate this form of problem posing into the classroom. The method is a great way for students to start thinking and questioning and probing into mathematics. By using this technique, some students may find that math isn't as boring as expected and that there is much to discover and much to probe. Furthermore, by using this form of probing questions there are a variety ways of approaching the theorem (for example, with Pythagorean theorem, the students were able to look at it geometrically as well as numerically).

Though the what-if-not approach is an interesting one and it sounds appealing, I would like to test it out in the classroom setting to see if I can apply it in reality.

10 Questions on The Art of Problem Posing

After reading pages 1-32 of "The Art of Problem Posing", I had a few questions that I'd like to post:

1. When should we incorporate these problems in class? Would it be good for a hook/part of the strategy/or is it best to have it in the conclusion?

2. How much time should we spend discussing these questions in class? Will these questions even be beneficial for the students? (Has there been studies done?)

3. What age group should we be posing these students to? At certain ages, would they not just rather have answers than be left with the frustration of not having their questions answered?

4. Should it be up to the teachers to form the problems or should students be problem posing? In either case, how do we get into that "frame of mind"?

5. How would problem posing relate to grades in school? Will they get marked for having "good" questions or making contributions in class? (Otherwise, they might not deem problem posing as something relevant)

6. How do we make sure that the problems as not too broad or too narrow and that we are able to keep on track with the lesson plan?

7. Has this method actually been used in class?

8. Should we have problem posing projects?

9. Is problem posing relevant and important in the eyes of the B.C. Education Ministry?

10. Though there are no such thing as "stupid" questions, are there such things as bad problem posing questions?

From the View of Two Hypothetical Students

Dear Editor,

I'm here to send a complaint on the education system in B.C. In particular, my mathematics teacher.

I have one math teacher, Miss Lo, that I see as dreadfully inefficient in her teaching styles and wish that she would change the way she teaches. Firstly, her teaching style varies too much. Instead of having straight forward notes and examples with homework, she covers a variety of teaching methods. From using group works, projects, and puzzles- the format is a bit too over the place and I wish she would just stick to the classic method of using the good old computer to type up notes. Like really, is it NECESSARY to teach us with so many different styles? It gets a bit too confusing.

Secondly, the quality of homework she gives us is much too difficult! Though she only gives about a page of homework a day, I think it's too hard to understand. Is it really essential for a student who does not plan to become a mathematician to do such abstract and complicated thinking?

Finally, I dislikes the discipline that she invokes in class. The classroom regulations are so rigid and there is no gum chewing allowed in class- we're only allowed to drink water. Like really, who can learn without eating in class?

These are just a few complaints I have with this one teacher and I wish she would switch her style of teaching.

Thank you very much for your time.

Student who dislikes Math Class.

________________________________________________________

Dear Editor,

I am also a student of Miss Lo's class and I would like to write a reply to "Student who hates Math Class" letter.

Firstly, I enjoy the way she incorporates different methods of teaching. It keeps the class entertained and occupied. Furthermore, the reason she does so is because it will allow students of different learning methods to understand the concepts and helps us look at mathematics in different perspectives. Hence, I think that Miss Lo is doing a good job of teaching with different styles and using different forms asides from merely note taking and lecture.

Secondly, though I DO enjoy eating. I respect Miss Lo as a teacher and she needs to set up some rules to make sure the class is tidy. There are times to eat and other times, we should listen and work in class. PLUS, since we're all supposedly busy learning in class and doing group work, there shouldn't be time to eat anyways. We should be eating at lunch.

Finally, I personally enjoy the way Miss Lo gives out her homework. By giving us only two to three challenge homework questions, we are able to try to push ourselves and have a further deeper knowledge of mathematics. Though it is at times very difficult, she provides morning hours for help if you need it, and as well, she does not mind that we work in groups as long as we write out our own homework in our own words after.

So in response, I believe that Miss Lo is the best teacher ever!

Student SMILEY! :D

_________________________________________________________________

My Hopes and Worries:

From what I've written, I hope that I'll be able to teach with various methods and that I'll be able to have the students involved in mathematics whole heartedly. A few worries that I have would be that because I want to teach in so various methods, they may find it confusing. In order to overcome this obstacle, I hope to be very well prepared and give students clear instructions and notes by the end of the lesson. As well, I hope that I don't end up being too strict a teaching. So goal wise, I hope that I'll be able to be entertaining yet educational in class and make it a learning experience for both the students and me. I would like to make math a fun, applicable and yet challenging experience.

Marked Classroom Instruction

Mr. Dave Hewwitt's approach to classroom instruction is different from what I have seen in my past years- both as a tutor/cram school teacher and as a student. It is an interesting concept, though I am not sure if I'll be able to incorporate it in my own teachings. His teaching approach is quite interactive and engaging- leading to the impossibilty of being unattentive in class.

I appreciated the fact that the classroom environment he created was one that was independent and his teaching methods were decentralized. This approach would take time for the teachers inexperienced and experienced alike to cultivate- as the roles of a typical student and teacher are redefined. As students, they are no longer merely sponges, but rather math scientists. I enjoyed watching this clip as it allowed me insight to how one may approach a class in order to have students learn independently.

One key thing that I noticed and will need to work on if I wanted to use this method is having a LOT of patience. Without the patience, Mr. Hewwitt would not have been able to go from the simple basic building blocks to subtly incorporating higher learning concepts into the lessons. It was very encouraging to note however, that once the basic foundations were set in place (over repetition and drawing out the basic concept), more difficult concepts seemed to come more easily to the students.

One method of technique that I find particularly attractive is his use of incorporating the whole classroom to answer his questions. Instead of picking on ONE student, all the students have the chance to call out the answers in unison- nullifying the chance of a student feeling picked on or being ignored. This led to the classroom atmosphere to be a cooperative learning environment where students worked together as a team to strive towards the same goal instead of a competitive one where they did individual learning. I believe that by doing so, Mr. Hewwitt was successful in forwarding the progress of the students as a whole. Furthermore, this allowed the chance to have the students to self check their concepts and to make sure that they understood what was going on in class.

I believe this is a very ENGAGING and progressive method to incorporate mathematics into the classroom setting.

Group Summary of Assignment 1

We chose to pose similar questions for the 3 math students and 1 math teacher that we interviewed. Although we asked many more than the ‘5 Burning Questions’ required, the interviews were able to be summarized and contrasted nicely.

The Math teacher we interviewed has been teaching for over 10 years and now mainly focuses on the senior math courses, generally Grades 11 and 12. He is well known among students and faculty and is apparently quite popular within the school environment, many of his former students still keep in touch through email and visit with him surprisingly regularly.

Our student interviewees happened to run the grade gradient and we were able to have one A student (graded at approximately 90 – 95% across all her high school math courses), one B student (regularly scores approximately 75 – 80% on tests) and a C student (who wasn’t terribly concerned with her math scores). It seems presumptuous and slightly disheartening to be classifying our students by their scores, but we thought that it might help to add a sense of significance to their attitudes towards math. Also important is that the A student and the B student are both in Grade 12 Math and the C student is currently in Grade 10 Math. All 3 students are in the Principles of Mathematics stream and have no interests to seriously pursue Math after high school.

The interview with our C-student was shorter compared to the others and was therefore fairly stilted. She informed us that she was only studying math because it was required and would probably have stopped if it had been optional except that her parents are making her. She also indicated that she didn’t like math because ‘[she wasn’t] good at it’ and that she had a peer tutor to get her through the course. However, she there are times where she does enjoy math classes and those are the ‘rare’ times when she feels like she understands a concept. The classes she does enjoy are Music and Science because she ‘gets it’.

Our B-student is a former D-student whose marks were forcibly pulled up after she decided to do math by distance with a private tutor. She indicated that she had been having trouble understanding concepts brought up in class and was too afraid of looking stupid to acknowledge her confusions in class; having a tutor allows her to ask questions immediately and one-on-one which greatly increases her confidence and ultimately her skills. She said that she was constantly unsure of herself and felt lost in the big classroom where many of her peers understood the concepts and it just made her ‘feel dumb’. She anticipates taking math in a higher level institution only if the program she chose required a math prerequisite. She seems to support the idea of some group work within mathematics to add to the lectures to allow her some time to listen to her peers’ ideas, commenting that ‘math just seems so lonely’.

The Grade 12 student who had been pulling an A-grade had a slightly different take on her math class than I had expected. Her attitude towards math was not a question of like or dislike but of competition. This was a class that she felt she could compete in and makes an effort to do so. She likes having other people consider her to be good at it and so she works hard for her grade. She is taking Math because she feels that it is an essential skill for people to have and that it provides her with the option of taking sciences in post-secondary. What surprised me was her attitude towards rectifying confusion during math lectures. She said that she was too intimidated to ask questions during class and would often just relegate her attention to copying down the notes so she could review on her own or ask her teacher after class. According to her, math class would be more accessible if teachers would take small breaks during class to diverge attention elsewhere for a short while. This would allow her time to digest the information and refresh her mind so she could return to the lectures with a renewed concentration. Her ideal math teacher would be kind, understanding and fun.

The interview we had with the teacher was equally informative, if not more so. He likes to teach from a relational standpoint and stresses that a student who aces tests is not the same as the student who really understands the material. It was important for us that he did not think that it was a challenge to get through the curriculum in the proposed time line. In fact, he thought that there was a lot of time and the challenges of teaching were at a far more personal level. He warned us to ensure we are friendly towards our students but not be their friends because teachers are still in a position of authority and must have their students respect them as such. He also advised us to take our practicum seriously and treat it like a ‘real job’, not just as a ‘practice run’.

Our group really enjoyed this project and found that the varied and sometimes surprising answers to our questions helped us to be more aware of how everyone else views math. It is easy, as students who had relatively successful highschool math class experiences, to forget the challenges that others may have. We expect that this will help us in our practicum and future teaching posts to be aware of the difficulties some of our students face and to encourage them and adjust our teaching accordingly.

Individual Reflection on Assignment 1

From this assignment, I had the opportunity to hear opinions from the perspectives of multiple teachers and students. This was a great opportunity and I had learned many things from this experience.

A few notes that I found really interesting from the students perspectives included:

1. Teachers need to bring his/her own personality to the classroom.
Though I already know that all teachers have their own teaching styles and they are can be equally effective, I forgot how crucial it is to a student's learning. I was surprised at how much emphasis the students had put on this point, and at times even stressing personality and enthusiasm was more important than the material itself! I'm glad many students think this way, as it strengthens my resolve to be a nurturing teacher that sincerely cares about them and is passionate about their future learning.

2. The speed of the classroom instruction should be slower.
I have a slight fear that I might teach too fast for all the students to keep up. In particular, I'm worried that I will speed up in my teaching because of the time constraints to finish teaching the curriculum material. But after talking to these students, I realize that I need to worry less about cramming all the material into their knowledge, but instead, focusing on pacing myself to make sure the students understand what is going on in class. And if necessary, I should making myself available before or after school hours.

Things that I found interesting from the perspective of teachers:

There were many tips that I had picked up from teachers during this assignment. One thing that I found useful was teaching by modeling. In particular, to show how we answer our questions by working through it in class, thinking out loud (in a clear and precise manner), and doing it that way instead of having the students copy things line by line. By doing it this way, there is an emphasis on how we can find solutions in mathematics and not just emphasizing the results. A second point that finally got ingrained into my head was the importance of being FRIENDLY but not being friends with the students. It's OK to be nice, but as teacher candidates, we need to be able to take control of the class and be responsible for them. As well, though I hear it often, I had never realized the value of lesson planning until talking to the teachers. Although a lesson plan doesn't set the lesson into stone and there are always adjustments to be made, it is really important to have guidelines. I had also want to make sure that as a teacher candidate, I treat the practicum not just as a practice run, but as a real job. Because it IS a job and we are responsible for the growth and development of the students. I want to be able to act in the students' best interest.

Although many of these points are things that I had learned or heard before, I had never made a personal connection with it. Though the facts were in my head, sometimes, it was not something that I valued as much because I did not make the concept my own. However, because of the interviews, I had the opportunity to talk and see the opinions of students and teachers firsthand, and as a result, I realized how important it is as teacher candidates to make sure that students and teachers have the same goals. These concepts will be a quite the challenge that I look forward to working on within a month!

On an added note, we had a few more presentations the following day, and I just wanted to remark on a few thoughts that came to mind. With Mike's presentation, they had interviewed teachers and students who had undergone the "Learn on your own Pace" math model. This was a very interesting concept and I believe that one should look into and consider this a great option. It provides flexibility to students who already know what they want to do with their lives and know that math is something they merely need to finish. It also allows students who strive academically to work ahead. It was an interesting proposition that shows other methods of teaching asides from lecturing in class. A second note that I want to reflect on is Jill's group and how they discussed the use of technology. I had found it interesting how many teachers view using technology as a hinder to the lesson. It is something to consider- will the students be able to learn mathematical concepts by using computers in mathematics or will they become distracted and lose the main point and purpose? There is a fine line that we need to be aware of and though technology can be a great aid and tool in working alongside mathematics, it is tricky to teach with technology and not have the students lose the focus and objective of learning a certain concept.

Battle Ground Schools

Battleground Schools written by Gerofsky summarizes three "battles" concerning mathematics in the 20th century that occurred in North America. In particular, there was the Progressive movement, the New Math Reform movement, and the "Math Wars". In general, all three of the debates concern different stances in mathematical education and how we should approach teaching students the subject.

The first period, the Progressivist Movement, emphasized and criticized school mathematics as "meaningless memorized procedures". The crux of the argument was that though the students studied and knew the algorithms to arrive at a solution, there was no flexibility because the students would not know any alternative methods to achieve the answer and did not understand the concept (the WHY) of why one would use the procedure to solve the answer. With this line of thinking, Dewey (one of the leading figures of the Progressivist Movement) revolutionalized the idea of the Montessori classroom. The focus of this classroom was not to memorize and implement, but rather to experiment, inquire, and interpret by having students actively engage in activities. Though many teachers stuck with the tradition of lectures and homework exercises, some classrooms had put Dewey's inspirations into practice.

Over time, the New Math movement began. The trigger for this was the space race between Americans and the USSR. The main fear was that Americans were losing in the space race due to the fact that "school mathematics was not keeping pace with ... research level mathematics". With the goal of creating a generation of "elite rocket scientists", mathematics in education began to change once again. The whole curriculum from grades K-12 was changed and the goal was to "create a unified, logical, highly abstract algebraic structure based on set theory". In order to help the students become familiar with mathematical notations and concepts used in the sciences, many topics from university were taught in from grades K-12. However, there were major problems with the New Math movement. Many teachers had little knowledge of the new material and would have had trouble teaching children. For homework, parents would not be able to aid their children with their homework even in elementary grades. It was difficult to justify why students were taught this material as not every child planned or wanted to become a rocket scientist. In the end, New Math was scrapped as a misguided experiment.

Finally, we come to discuss the Math Wars over the NCTM Standards. This is a "battle" that has started in the 1990's and is still occurring. The focus of the Math Wars once again is, "How do we best educate our young in mathematics?". Should we focus on using a progressive approach or using a traditional method of teaching? With the NCTM Standards in 1980's, there was a focus to have "flexible problem solving skills", using mathematical relationships in different forms, and to use technology as an aid to solve math problems. It was a progressive step to help students appreciate the "beauty of mathematics". However, by 1990's, a "backlash" occured and there was protests against the NCTM Standards. There were people who prefered the traditional and conservative style of teaching.

Hence, over the years, though there has been drastic changes and movements in education and mathematical pedagogy, the question still remains, is it better to have instrumental or relational understanding of mathematics? Is it better to lean on one side or to have a balance of both? I believe that there are both positives and negatives behind conservative and progressive stances in mathematics educaion and a balance should be struck between the two. However, without trying and without experimentation, one can not achieve this goal and "perfect" this way of instruction. When the general public is concerned, change is not always welcomed with open arms, and I believe that is why "battles" over mathematics education is always brought up. One thing that I had not considered much before but am much more aware of now is that the STANCE of which form of teaching is "better" depends much on what goes on with the world around us and the goals of the nation. Do we want to create a generation of rocket scientists? Do we want our children to be "practical" and robots? Would we like them to be philosophical and inquisitive? The culture around us will dictate what the public requires and desires. And what the public desires will inevitably end up effect the way math education is taught.

Making Changes in Instruction

Two Memorable Math Teachers

Throughout my life, I've had various math teachers- whether it was professional teachers in elementary, high school, and college or the various tutors or friends that had helped me along the way. Today, I'd like to talk about two teachers that had made an impression on me.

My earliest childhood math teacher and I believe the most memorable one is my mother. As a child, I remember her teaching me the basics such as addition, subtraction and multiplication. On a semiweekly basis, we'd go over the material through mandarin textbooks that we brought over from Taiwan cram schools. When I was learning the material, it was frustrating at times for two reasons. One, I had trouble reading the Chinese. And secondly, when I asked her WHY the formulas or algorithms were the way it was, she more often than not could not explain, and told me to memorize it. What was worse was when she told me the reason was "just because that's the way it is". I remember one instance where I was learning about multiplying negatives. In particular, just memorize the fact that two negatives make a positive and THAT was just the way things worked. It was very frustrating not to know the concept and half the time I learned math, I made up my own reasoning for why things worked.

Another memorable teacher I had was in high school. He had a tendency to give an example or question at the beginning of class for us to ponder over, and then went through the classic lecture approach to teaching. But what was different and more engaging to the lesson was the atmosphere he had set up when teaching us using this method. He had set a relaxing and at ease atmosphere that allowed us to ask questions if we needed to and gave us a chance to have class discussion. As well, while he approached the examples he wanted to give to us in class, it was not simply a process of copy and pasting information from his notes to the overhead/whiteboard. Instead, he showed us his way of thinking and worked out the problem with us during the class. By doing so, it helped us stimulate our minds and try to work out the problem in our own way and see if we could obtain the same results as he did. I believe that by giving us the tools we needed to work with first, and then having us work on a problem together in class and seeing if we used the same method of solving a problem- he showed us that there was always multiple ways of deriving an answer.

From the two different experiences, I want to try to reflect on the way I plan to teach. Though I believe in integrating the method of instructional and relational learning, I need to find my balance. I believe that perhaps giving the students the tools to solve the problem is very important, but what is equally important is to give them the concept behind using the formulas and such. As teacher candidates, we should not be having our students blindly using formulas without understanding deeply why the formulas work. As well, giving the opportunity to vary up the lessons such as through visual aids and group work is something that I hope to continuously work on.

Reflection of Origami Microlesson

Summarizing my peer's evaluation of my lesson:

From my microlesson, the summary and post-tests were too rushed due to the lack of time. For the participatory event, instead of each person taking a turn and then testing it, perhaps each student should have a sheet of paper and participate in folding the crane. As well, it was a bit hard to just let students learn by looking at others doing the steps. Though my explanation of the procedure is clear, I must consider the timing and if a crane may be too complicated to fold in 10 minutes or not. One thing that I didn't notice before was my behaviour. It is good that I have nice eye contact and am friendly, but I use the word "so" too much and need to get rid of that "pause" word.

Write up of my own evaluation of lesson:

I believe that the structure of my lesson and the way I flowed through the BOOPPPS went well. I realize I enjoy telling others about the history of what I'm teaching to others a bit first and it was good that my peers were interested in it. I'm happy with the participation my colleagues had when doing the post test and it was a good way for me to assess how much they learned from the lesson and where I may need to be more clear in instructing my students.

However, if I were to reteach the lesson, I would definitely need to consider how to manage my time better and figure out a way to teach this lesson within the 10 minute time span. As well, even if my time was coming up, next time, if I was short on time- I would not rush my summary and instead, I would tell them the summary and perhaps only have them fold up to half a crane in the post test. Another thing I would have ready would be a diagram with the instructions put up for students that may prefer to learn from paper rather than an instructor or for people that already know how to make certain basic steps of origami. It is beneficial for students to be able to have alternative ways to learning. Finally, to save time, I should keep my introduction less short so I could get into origami folding more quickly!

From my peers, I believe we all agreed that time management is something I need to improve on. Though it is important to go at a pace where students can keep up with, I will then need to be more selective on what I want to teach. As well, though I never realized it before, I have replaced all my "uhm" with "so", and I should get rid of use that word in my oral vocabulary more often.

In summary, though there are many things I need to work on, a few that I will try to improve on first would be my timing and selection of what material to go on. To incorporate more methods of learning during my lesson. And finally, correcting my vocabulary.

Instrumental Understanding vs. Relational Understanding

BOOPPPS! Project

Min-Chee's ORIGAMI ACTIVITY!

Bridge: The bridge in this activity will be showing my students three different objects, one at a time. First, I will take out an origami rose and ask them what they think it is. Then, a crane, and finally a jar of paper stars. Ask them what in common these objects have with each other. Then talk a bit about the meaning of origami.

Teaching Objectives: From my lesson, I hope that I will be able to incorporate my lesson in such a way that my students will be able to learn in a very hands on fashion and through modeling.

Learning Objectives: The students will be able to learn how to fold an origami object based on their levels and as well, develop an understanding and appreciation for the joy of this folding art.

Pretest: Ask if they have had any folding experience before, and if they did- what have they folded?

Participation: I will demonstrate how to fold the object, and in a circle, each student will take a turn folding.

Post Test: Hand out origami paper to every student, and assess them by seeing if they are able to fold the objects on their own. If they need a helping hand, have the student next to them help them out a bit since teaching one another is a great method of learning.

Summary: By the end of this lesson, the students will hopefully be able to fold an origami creature. If not, they will be able to learn at least a few basic steps of folding.