Trig 12

http://docs.google.com/Doc?docid=0Ae4-u3X8YvW7ZGY4NzU1Y2NfM2duNDM1dHZj&hl=en

Assignment #3: The other parts

Part 1 and 2:
http://erwinlee1980.blogspot.com/2009/11/assignment-3-math-project-on.htm
Part 1 and 2 (my pictures are here):
http://paulsiumaed314a.blogspot.com/2009/11/assignment-3-math-project-tessellations.html

Math and Sound Lesson Plan

Math and Sound Teacher Notes

Grade 12

Trigonometric Identities

Paul, Stanley, Erwin, and Gigi

Purpose: To show the trigonometric identity:
sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2)
in an auditory manner.

Students will hypothesis, predict and test which part of the wave is associated with the pitch and which part is associated with the beat or the volume.

Description of Activities: Students will be using Audacity (a free program) to generate sinusoidal sound waves. Playing two waves together is like playing the sum of the two waves. (This shows the additive property of waves.) Students will be investigating the beat phenomenon and the trigonometric identity: sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2) . They will be following their worksheet to guide them through the activity.

Sources: http://en.wikipedia.org/wiki/Beat_(acoustics)

Estimated time: 1-1.5 Classes or 1 Class and the rest is due for homework

Students are required to produce a page of answers and supporting graphs. Students will conclude cos(2π (f1-f2)t/2) is the part responsible for the amplitude (and thereby the beating) of the wave. The pitch of the wave is the part sin(2π (f1-f2)t/2) .

Marking Criteria: Students will be marked on the correctness of their answers. Students will also be given a mark on their participation in class out of a 4 point scale worth 25% of their total mark.

1- Did not participate

2- Participated 20% of the time

3- Barely meets expectation for participation

4- Meets expectation for participation

5- Exceeds expectation for participation

mathworksheetsound

Math and Sound Investigation

1) Open up Audacity.  Go to Generate : Tone to create a tone also known as a sound wave.  You can delete a sound wave by highlighting it and pressing delete.   It is not recommended to have more than two sounds waves in your workspace at once.  Try generating sound waves with different amplitudes, waveform and frequency. 

2) At a fixed point in space, a basic sound wave (pure tone) can be represented by a sinusoidal function.  One of the functional representation is Asin(kt). 

Fill in the blanks with the following words: amplitude, frequency, volume, pitch

The _________________ of the function represents the loudness or _________________ of the sound.

The _________________ of the wave function represents the _________________ or how high or low the note is. 

Copy and paste the above two statements with their answers onto separate file.  You will print your answers at the end of class to hand in.

2) Investigate:

Generate a sine wave tone. Choose a frequency (call this n)
Generate another sine wave with frequency n+1.
Play the result.
What does it sound like? Record your answer.

You may hear a pitch change and a beating frequency.
Delete the wave with frequency n+1
Repeat the above and investigate with frequency n+2, n+3, n+10, n+20, n+100, etc
How does the pitch change as the second frequency increases?
How does the rate / frequency of the beat change?
Record your answers.

3) On graph paper, hypothesis what the wave would look like.

4) When you are listening to the waves together, it is like adding two sin waves.
sin2πf1+sin(2πf2) =
What is the trigonometric identity associated with this sum?

5) Hypothesis which part of the wave is associated with the amplitude (the beating) of the wave and which part is associated with the pitch.

6) Clear your workspace.  Make two sound waves with different frequencies.  Record this frequency.  Using your hypothesis in question 5 predict the frequency of the beat and the frequency of the pitch.  Use appropriate frequencies in which you can hear the beats.

7) Test your prediction by listening to the waves.  Record how many beats you hear in 10 seconds.  What are the experimental values of the beats?  Is your hypothesis in 5 valid?

8) With the trigonometric identity found in 4, label which part is responsible for the beats and which part is responsible for the pitch of the waveform.

9) Plot out the sum in a graphing program or graphing calculator.  You may need to plot twice: choose an appropriate scale for the beat and another for the pitch.   Include your plots with your answers to hand in at the end of class.

10)  Find two frequencies f1, f2, in which a period of the beat is 10 times larger than the period of the oscillations (part responsible for the pitch).  Show your work.  Graph the sum of sin2πf1+sin(2πf2) .

 

Extension Questions:

Go to http://www.mindspring.com/~j.blackstone/dist101.htm.  Scroll down to the applet.  Play around with the applet and record how the sound changes.  Does the perceived pitch change?  In other words, if you tried to hum the note does it change as you change the wave?

Math Project

Part 3

Worksheet
Teacher Guide

Memorable Moments from Practicum

I sort of can't find where the sheet I wrote was so I'll try to just type for eight or so minutes.

Memory 1:
I was asked if I was a new student in the grade 9 class science class. Granted my sponsor teacher had a policy of not outright saying this new person was but just showing by interactions and presentations, I think this student wanted to spite me. She was a bit catty and chatty. I had dressed up very professional the last day and this second day I was wearing nice pants and a leather jacket. Anyways, I decided not to play her game if she was playing a game and just told her I was a student teacher. My sponsor teacher said that there are sometimes students coming in and out and peer tutors helping out in the class. I knew this was going to happen.

Memory 2:
I had taught my math sponsor's two classes. The first was a grade 11 class and the 2nd was a grade 9 class. I wanted to do TPS with the students. I made a mistake with the grade 11 class and they went crazy. They didn't do their seatwork and didn't want to present to the class after. I talked with my FA and SA after the lesson. I learned now I need to set expectations and stress these expectations. I needed to stress that they needed to work and at the end I would choose randomly a person to go and present to the class. I deliberately wrote out the words I was going to say so that I was emphasize on the SHARING part of TPS. My SA stressed that if the students are working harder than the teacher, the teacher is working too hard. Thus, with all this in mind, my grade 9 class went much smoother even though they were grade 9s. I also learned from the ToC that day, if you confiscate a cell phone, keep it hidden in a drawer or in a pocket or the phone will be stolen or lost.

Group Microteaching

(from a long time ago!)
Group Microteaching Lesson Plan

Bridge: Our hook? Everyone loves the out of the norm problem solving. We'll have the students pair up and work on one of three problems for 5-7 minutes. They'll be using string to make lines of best fit.

Learning Objectives: Students will turn data from word problems based on real life and make a table. Students will have hands on experience on making a best fit line from the data they plot. Students will then extrapolate from the fit to answer other questions.

Teaching Objectives: Students will work in pairs. Students will be responsible for presenting their solutions.

Pre-Test: Ask students if they remember graphing from previous years.

Participatory Activity: There are three different worksheets for this activity. Each worksheet will have the students graph out a different type of graph.
Linear Function: Data is already given to the student. Students will plot out the relationship between payment for a plumber's rates at different hours.
Reciprocal Function: A class wants to buy the teacher $100 gift. The amount each student has to pitch in depends on the number of students that want to pitch in.
Quadratic Function: You want to carpet the floor. What is the area of the square of the carpet if we vary the length of the square?

Students will have to answer questions about their scenario and graph out the data. Students will plot out the data and do a best fit. Students would then have to answer questions based on their fit.

Post-Test: One student representing a different worksheet will present their scenario and their graphs. They explain and discuss their graph and give answers to their questions.

Summary: Students are learning from their worksheets and their peers. They will learn hands on one of the ways to do a best fit curve. At the presentation, students will be able to see the different types of graphs. Students will also learn the different types of real life problems, the mathematics they know already can handle.


Group Microteaching Reflection

I was feeling quite nervous and excited that day as we had to present in front of the whole class and that it was the last day of class before our short term practicum. One of our biggest enemies is timing. How will we gauge whether the majority of the class has finished their worksheet and that we should move on? If they are finished or not finished, they still look like they are busy chatting away. I think if their was a blank backside to their work sheet, we could get them to turn it over and sit quietly or for them to draw on it. We may have given them 2 extra minutes but in a 20 minute activity, that is a lot of time.

We didn't tell our class that this was a grade 8-9 activity. When the students presented, we had our trustworthy students answer in the class. In a real class situation, I learned from my practicum that before I hand out the activity, I would have to make sure all students are responsible for presenting and that they would be randomly called up to present their findings. Our first student that presented gave answers above and beyond the grade level and using big mathematical terms to explain the problem. I was mortified and a bit tired so I was quite stunned. I did not know what to do. The student became the teacher and was messing up the time allotement for the activity. If my mind was more cleared, then I would have stepped in and ask the student just to explain the scenario and how it relates to the graph that he has drawn. We wanted more on how the curve looked: Was it straight? Did it increase or decrease? All in all, that was probably our biggest problem of the presentation. In our responses, other students noted this problem. They did however also noted the relavency of the activity to real life.

Liouville Problem-Two Column Approach

I forgot to do the case for 1
So 1 has 1 factor, that is 1
so the sum of the number of factors squared is 1.
Sum of the cubes of the factors of 1 is 1.
So that is the 1 = 1. The sum of cubes of the factors then becomes larger than the sum of the number of factors of the factors as you go higher one.


Thoughts: Well if I was a high school student, I would be happy with such findings. We were like are we supposed to prove it or?
I figured Stan will have some nice elegant proof.

Reflections on Dividing by Zero

The concept of dividing was already mentioned by Brent Davis in our lectures already. His lecture reminded me division by a fraction is like scaling. We had to come up with ideas on dividing on the spot so I think some one said reproduction.

I particularly like the free write. It allows for free thoughts to go through people's minds. The poem part is a bit hard. Maybe one can extend it to art or some other non-mathematical media.

For marking, I think it can be done on a 3 or 4 point scale for effort / meeting expectations. That way the teacher doesn't have to mark something so suggestive.

Poetry: Divide by Zero

Note: I can't find if there's a minimum word limit to this.

When dividing by zero,
you put a bar or line over zero.
This bar over zero
is like a teeter totter.
Rock the sides up and down,
the slope tries to approach the answer.
The actual answer to dividing by zero is the slope of the bar
when it's fallen off and leaning on the zero vertically.
Which side you ask?
It could be either.
So is it negative or positive?
You tell me.
|0 and 0|

A Taste of Pi (Part I)

Comments: I really liked this Saturday morning event so far. I observed how to teach math and observed the students reactions and interactions. There were a good number of females. I took four pages of notes. The topic of yesterday's talk were:
Speaker: Dr. Jonathan Jedwab, Department Of Mathematics
Title: "Games, counting, and curvy lines"
Speaker: Dr. Andrew Blaber, Department of Biomedical Physiology and Kinesiology
Title: "Weightlessness and fainting astronauts"
Date: October 17, 2009Time: 9:00 -12:30
Place: IRMACS Presentation Studio

I wrote my observation notes relating to teaching on the left column. These notes will be written in italics.

We started off with having a student introduce the speaker. I thought this was interesting and reminded me of the student conferences. It gives students a sense of ownership. I like how the introducers adlibed. The introduction talked about the speaker's history, what influenced them to go into this profession and their hobbies. It's a bit more than what I would hear at a professional conference but this makes the speakers more relatable and human.

Dr. Jedwab introduced the game of Set. This game had figures on cards. There were four attributes (number, colour, shading and shape). We saw the cards. The slides were very nice. He introduced the rules.

The Math of Set
Dr. Jedwab introduced the vector representation for the cards. (Each card can be represented by (i,j,k,l) where each of the letters are either 0,1 or 2.) He also introduced modulo 3 (in words... no slide) by saying it's just the remainder after dividing by 3 and giving an example. (That would be a teaching choice so the talk progresses.) At this point there was a big yawn.

3 cards form a set iff their vector representation x,y,z:
x + y + z = (0,0,0,0) (modulo 3).
If the attribute were the same, the sum would be a multiple of 3; if the attribute was all different then they would be 0,1,2 which add up to three.
He gave examples of vector representation of three cards in a set. Perhaps he should get the students to do this part.

He talked about the history of how it was created (coding information for researching epilespsy in German Shepard Dogs).

He presented the scenario: if there were only three cards left on the table and all other cards have been put into sets, what do you do? Student: You yell set. The sum of all the other cards satisfy (0,0,0,0). We need to know the sum of all 81 vectors modulo 3. I think he can have students come up to show different ways to show this sum.
Example: Partion 27 sets by fixing the last 3 digits
1 1 2 2
0 1 2 2
2 1 2 2

My answer: look at how you would generate sets orderly. Examine the x column only. You will see that you can group it in 0, 1, 2. Or 0, 1, 2, will always alternate and thus the x column will sum up to 0 modulo 3.

0 0 0 0
1 0 0 0
2 0 0 0
0 1 0 0
1 1 0 0
2 1 0 0
0 2 0 0
1 2 0 0
2 2 0 0


How many sets are there?
By example If you fix 2 cards, you uniquely determine the third one. (OK, he just gave us to that right off the bat. I would have hoped for students to come up with other answers for this.)
How many ways are there to pick 2 cards from 81?
81*80/2.
He showed the over counting 3 times.
So it's actually
81*80/(2*3). (What a nice introduction to combinatorics.)

He showed us an example of 20 cards and have us find sets in there and there turned out to be none. What about 21 cards and no sets? What's the maximum? It turns out the maximum amount of cards you can have without finding any sets is 20. ( I felt a bit of the What if not theory in this section.) 21 cards must have one set.
He showed how you must prove this fact. He used a smaller case example to show.
Constrict the cards to two attributes. There can be four cards maximum with no sets.
We must show
a) There exists four cards in which no there are no set
b) For five cards, there must be a set.

He showed four cards that didn't form a set: (I'm going to just put down their vector represetation)
(0,0) (0,1) (0,2)
(1,0) (1,1) (1,2)
(2,0) (2,1) (2,2)
(insert pic)
He used lines to represent a set. (Questions that arose was, does this affine geometry influence this game of set or does the rules influence these lines? It turns out we can just look at the geometry and its self consistency to get the same picture. You can also just have the rules define the lines. They kind of asked the chicken or the egg question here. However, it can be either or and both is just as legit.)
3 lines form a set iff they fall on a line. Very visual
Here we start off with object and we assign numbers and then we assign it to geometory.
Object -> Numbers -> Geometry Just like history mentions Dr. Jedwab.

Now we want to show any 5 points implies that they must lie on a point.

Pick a point (bottom right corner). 4 lines go through this point. One of these has 1 point. He's going too fast. (I couldn't follow so I think I'm going to ask him to type up his arguement). I think he could have placed a coin on that point.

I think let's look at the bottom right corner. Pick that point. Now choose a point on each of the lines going through it making sure you don't make a line with those other points. Do this for 3 other points. Now you notice if you place one more, three of them must form a line.

This is the Pidgeon Hole Principle. Real life! n holes, n+1 objects=> One hole must have 2 objects. (Anecdote about having a name to this principle. However, this is the one of the few things we really know and not just made up.) Very powerful.

Mathematicians solve 1 problem, wrap it up and then try to generalize.

3D version (3 attributes).
a) 9 cards
b) any 10 cards must contain a set
(I'm going to just transcribe my notes...)
Lines and planes. Every plane that goes through a point looks like lines on the 2D version. Every plane has 2 points (?) at least fix two points... all planes in 2 points => 2 planes. 7 more points not on this plane / line => one of other plane. 2 more points = > 5 points on one plane => 1!
Anyways I bought his arguement. It's kinda about placement and sort of follows the 2 variable case. He went a lot faster with this one. He said he would be fishy about this one.
1h has gone by.
Affine gemeotry

4, 10, 20 are the maximal caps

5 variable (2002) - 45 is max. It's an obvious question to ask. They used Fourier Transform.
6 variable (2008) - 112 is max. Talked about how research is new and how one goes into a hole for 3 years and come out with an answer. Talked about pure math (like number theory) where it wasn't tainted but just research for the beauty but now it's the backbone for e-commerce.

Good Question asked by audience. If motivation is Set then it defines the line. If motivation is geometry then the line is defined by the thing that makes sense.

He's being fishy: What we know has run out. we must create new math! => Needs to make sense. This means that it has to be internally consistent.


Postive:

• talked about history of math


• audience participation


• jokes


• dancing


• principles of how to show max (noted mistakes grad students make in not showing the two)


• added a lot of relevant real life annecdots


• made math fun


• showed what math research does / how math is developed in the past and currently


• very visual and colourful

Minus:

• some parts could be left for students to figure out instead of giving out the answer (of course he had to do this because of time constraints


• the punchline (of the 2 variable case) was given too fast and there were too many parts. I think it would have been better if this was written up and posted on a slide

Interesting:

I would like to introduce this game and use it to present the variety of concepts that were presented. I think I would give the students time to go home and think about the problems and the different ways of solving it.

So that was my three pages of notes.

Astronauts and Weightlessness and Fainting

Basically this was a talk about physics, "biology" and space using math as a tool. Dr. Blaber's research was testing astronauts in weighlessness and when they got back to earth on blood pressure, heart rate and blood flow in the brain. He showed us the tools and the graphs being generated by measuring his graduate student. (Go guinea pigs.)

He introduced weightlessness with a lot of physics. I thought there was too much text on the slide. He talked about hairs in ear sensing where gravity is. I thought htis was interesting.

Then he talked about pressure in the capilaries. I liked his diagram with the backup flow. He related Blood Pressure (P), flow (Q), and resistance (R) as

delta P = Q R

which is like

V = IR

At this point the chairs was so comfortable I noticed a few students falling asleep.

This system depends on heart rate, stroke volume, and resistance (how big the arteries are... there's things that control the size of them).

The blood flow's velocity doesn't go to 0 in the brain. If it does then you faint or have a stroke.

(Neat) Resonance: Can get standing rate of with 6 breaths per minute. This is for a think blood flow int the brain. This is why people have breathing exercises.

There was talk about hyperventilation. It stops the blood flow or slows it down?

If CO2 goes up, the capilaries dialte and eliminates Co2 faster than it is being produced. (Breath out faster)

Pressure at tow is = 2 * Pressure at heart.

Then he showed how the blood pressure at his grad student's finger changes as he changes the position of his hand (above head, at heart level and as low as he can sitting). This was very neat.

We saw how the blood flow changes after his student stood up from sitting in the chair for over half an hour. The blood flow dropped dramatically and the heart rate went up (fastest response). The heart beats was broadcasted in number and audio form.

Good Undemo Idea: He had his student clench his leg muscles. This is what they tell pilots to do when they reach high g. (This relates to blood flow and constricting it in the legs so blood doesn't rush to the legs I think.)

Dr. Blaber talked about the astronauts he worked with and showed pictures. Students were very interested in this. He also then talked about witnessing the shuttle landing and how the shuttle slows down. There was also talk of the Colbert treadmill on the shuttle and Buzz Lightyear returning after being in space for 15 months. Everyone was attentive as soon as he started talking about space.

http://dsc.discovery.com/news/2009/09/11/shuttle-landing.html

http://www.nasa.gov/multimedia/imagegallery/image_feature_1472.html

Positives: very visual, showed how all disciplines relate to each other, space is interesting

Negatives: too much information on the first few physics slides. Cognitive overload

Interesting: I had some good undemo ideas from testing the heartrate by sitting and standing and clenching leg muscles.

How can my class demonstrate democratic citizenship?

Simmt suggests in Citizenship Education in the Context of School Mathematics that mathematics plays an integral role in social, political and economic structures. Knowing mathematics would be a powerful skill for one to integrate into society. Simmit wants our youth to "understand and critique the formatting power of mathematics in society". She suggests instead of teaching math as fact, skills and processes, teach math as a creative process for exploration and give students responsibility for explainations.

In my classroom, depending on what type of class I will be doing, it is easier to do the latter suggestion than the first because of time constraint. For the first suggestion, Simmit suggests posing open ended problems. My fear that the students would not be comfortable enough with the mathematics yet to take on such a challenge. However, I can see myself dedicating 3 or so teaching days to open ended group problem solving to help train their minds for how mathematics can be done. I am more opened to giving my class responsibility for explainations. I may do something instrumentally and ask why I may have done that. Another way to engage the class in the dicussion of explaination is to show two ways how students do the question (in which the results may or may not differ) anad ask the class which solution is correct (or are both correct) and to show which parts are correct mathematically and which are not.

Taste of Pi

From the goodies of the math listserv:
http://www.math.sfu.ca/atasteofpi/
Taste of Pi!

Even thought "registration" is over, I e-mailed them saying I was a student teacher and would like to attend so I got in.

What-If-Not-Approach

The What If Not Approach is a powerful way of thinking about mathematics concepts. Instead of just focusing on one certain "thing" about a topic, it looks at other concepts around the topic. This is where I have a problem. I am not that creative at looking at things a different way. To me, the What If Not Approach is a question one poses to start the exploration on a topic. I am not used to asking the question, “What if not?” I find that I get stumped when I ask myself that question and draw a blank. It's like a stumbling a block. To quote Erwin, “students might not have the mathematical ability to grapple with "cycling" two negated statements and combining them into an alternate statement because it doesn't seem natural.” This was basically my problem. For example, in the geometric board, I was not able to see anything about the amount of pins there would be if the board was a different shape. It didn't even occur to me that is what would happen. I was thinking maybe the same square but the diagonal of the square would be the diameter of the circle.
In our micro-lesson, we sort of have a “what if not approach”. We are not simply, having the students graph some points and fit it to a straight line. We are exploring the possibility of what if the relation and graph were not linear. Students will either explore a y=1/x, y=x^2 or y=mx+b relationship and report back to the class their findings. That way, the students will see that graphing is a technique suitable for all type of relationships between variables.

10 Questions / Comments for the Authors

1) It's a hard exercise to come up with all those ideas after seeing one line of math. I certainly wasn't able to think up of all those questions and ideas on x^2 + y^2 = z^2

2)What are the different things students think about while playing with the geometric board?

3) How has the study / resources of this topic changed from the first edition of this book to the third?

4) How clear must the question be when we pose it?

5) Should we enlighten the students what questions to ask when we give them a question?

6) How much exploring should we allow for a student?

7) If the student has problems with the basic operations (addition/subtraction/multiplication/division), how much should the student explore on the particular math topic?

8) How long should we leave a child with a geometric board? How much exploration (guided by questions) should a student do?

9) I like how when approaching the topic one:

1. makes the observations
2. asks the questions
3. comes up with the conjecture

10) I like how the authors pointed out internal versus external exploration. Often we focus on internal exploration but don't ask the external questions that uses the skills from a certain topic. A lot of external questions were presented when exploring the Pythagorean triple.
.

Timed Writing: Into the future

After teaching 2000 students....
Maybe you are the worse teacher, the best teacher, the average teacher?

From the point of view of an admiring student:
What a great day in class today. I love Math. Ms. W is right! Math is fun. I love how she shows us all these interesting ways to do and to think about math. I never understood multiplying fractions until now... She showed us a few to think about fractions today! She's always so funny but clear. Also she makes sure we understand b efore we go to the nexxt important step. I'm never lost in that class and I'm very focused. If only all the teachers have this engaging personaility.

From the point of view of a student that hates me
What is the teacher doing? Why do I have a woman teacher for math? All the other teachers in math are men. Math is in the morning and I'm too tired to do math at 8. All I ever hear is blah blah blah, I think I sleep in that class a lot. She likes to pick on students too much. Why does she ask so many questions? Doesn't she know I don't care about this subject? I'm pretty sure she does. She tries too hard. How does she have so much energy in the morning. I wish she'd stop picking on me. That's so embarasing, I don't want to look stupid in class. I don't want people to notice me. I hate math at 8 pm in the morning. Just give me my homework and I'll eventuallyy figure it out. Eventually.... Actually, I hate homework. Good thing she doesn't always check. Why did I get a C last time on my test? I did everything right. I think she just marked it wrong. I hope I can get a petition for the students to fire her... haha. Math is not fun... Math is evil!

I am most afraid of making mathematics boring and having students trying to kick me out of school.

I aspire to be someone that makes math fun.

Dave Hewitt's Video Response

Dave Hewitt has a very interesting way of teaching students. I thought in this way, the students can try to guess what is going on and then have an immediate reassurance of being correct. However, to be honest, I had no idea what he was doing at first. I thought he was showing distances at first with the counting. I also thought that maybe he was marking a spot as 4 or 5. After I saw that he was just trying to get the kids to count with him on a large number line, I understood what he was doing. Then he had the students start at different places like 700 or 1,000,0000 and have them count forwards and backwards from that spot. This was probably review for the students. The finale was when he started at 0 and then had the student go forwards and backwards from that number. I think this showed that minus 1 was just 1 less than 0.

I thought teaching grade 9 was interesting. He had "taught: them how to solve for a variable. My only problem would be what if a student answered but the question was wrong. Also not everyone was participating in the speaking portion of it. However, he had the students repeat it many times and had them pick out certain words to highlight the patterns of it. Then he had the students highlight another point. Thus he's guiding the students to speak the material he was teaching. This was very interesting. The other positive I found in this section was that he would be able to have the students speaking to learn it and reinforce the concepts. Thus, the material is not only heard, but also spoken and read.

Battleground Schools

Susan Gerofsky enlightens the reader in her article in Battleground Schools about the curriculum changes throughout the years.


Key negative ideas society holds about mathematics that affected these changes:
-is inhuman-only appropriate and necessary for small elite(flip side means it's not appropriate nor necessary for the general public)
-those who like mathematics are weird, different and do not fit into society
-it is acceptible and not shameful to be incapable of doing and understanding mathematics.

Progressive Reform (1910-1940)
Goal: Meaningful mathematics curriculum for preparation of democractic, industrial society.
Chicago Movement wanted to unify algebra and geometery, and bring pure and applied math to the curriculum. Dewey wanted students to also do and experiement mathematics. Students would be more involved and active in the classroom.


The New Math (1960)
Goal: Train mathematical students to beat the Russians.
Mathematics became more abstract. Set theory, abstract algebra, linear algebra, calculus were taught in the public stystem. However, teachers and parents were not qualified to teach this material.


NCTM Standards (1990-now)
In the 1970's and 1980's, there was much standardized testing. National Council of Teachers of Mathematics (NCTM) developed their own set of standards. The changes emphasized problem-solving, ability to represent mathematical ideas in different forms, technology such as graphing calculators, and ability to communicate mathematical concepts. They also wanted to develop relational and instrumental skills and an appreciation of the power and beauty of mathematics.

Math Interviews: Reflections

I found the responses from all the teachers are varied. The teacher I interviewed was my grade 4, 6, and 8 teacher. I remember him most for caring about the students and his discipline. Since he taught in the middle school, he would teach math and science to the students. Thus, he uses games as an incentive for students to keep on task and for classroom management. I now see that Bus Driver is a game which engages the whole class, even the struggling students. It helps the struggling students practice their multiplication tables and gives them an achievable challenge or goal. Bus Driver also makes math fun.

For the students we interviewed, I found it interesting the math and logic puzzles and challenges were the highlights of the students instead of any particular curriculum subject. They seemed have problems with concepts that were similar. For example, after plotting out y = x^2 for the student, we asked the student to plot y = 1/(x^2). He plotted out 1/x. He seemed to mix up the concept of reciprocal and inverse also. I would be clear on definitions and make sure I do not use the wrong terminology. I would not use the term, take the inverse of 6 when I mean take the reciprocal of 6.

Math Interviews: Group Summary

We interviewed a grade 8 mathematics teacher from a middle school in the Coquitlam school district via e-mail. He had expressed the biggest challenge was trying to teach the basic operations such as addition, subtraction, multiplication, and division to the struggling students such that the students would not give up. Consequently, he would also have to keep the top students from being bored. For the top students, he allows them to be peer tutors and provides math challenges or puzzles for them. The math class schedules goes as: class game of Bus Driver, presentation of the math challenge, lecture (25 minutes) and assigned questions (30 minutes). Bus driver is a game with multiplication flash cards. These cards are never shuffled. The bus driver is the winner from the last game. They start off facing off another student in the class. Whoever is the fastest at answering the next flipped up card is the winner for that round. If it's a tie, the teacher keeps flipping the cards (sometimes many at a time) to see who is the fastest. The winner is the bus driver and goes to the next student. The game ends when everyone has a turn and is proclaimed the bus driver and has their name written on the board as the bus driver. Even the struggling students love participating and sometimes may guess the answer ahead of time to win.

By interviewing two students in different grades, we were able to observe some interesting similarities and differences in their responses. First half of the interview questions dealt with what the students like and not like about mathematics. When we asked them what their favourite parts in mathematics were, the grade 9 student responded that he likes to work with integers because they are straight-forward and it is easy to remember the rules. The grade 11 student responded that he likes algebra for a similar reason, but he also likes riddles and logic puzzles. It is interesting how the grade 9 student likes the straightforward and easy concept, while the grade 11 student likes the challenging puzzles and riddles that allow him to think beyond the simple rules and concept.

We then asked them to share some of the challenges they encounter in mathematics classes. The grade 9 student responded that one thing he finds really difficult is translating the word problems into the equations. Also, he is confused when the same symbols are used to represent different things. Similarly, the grade 11 student had a problem with understanding the idea behind the concepts and rules. For example, he has difficulties with understanding the differences between the inverse function and the reciprocal function. Some of these areas of difficulty in math reminded us of the articles that we have read by Skemp (1976) and Robinson (2006) stressing the idea of relational understanding instead of instrumental understanding. We thought that the students are having difficulties in those areas, because they lack the relational understanding of the concepts.

They also expressed an interest in class when the teacher used different media for explanations (that is, anything but the chalkboard). The use of geometric shape blocks on the projector depicting larger shapes and ideas was found as interesting. Presenting students with “challenge-of-the-week” (COW) puzzles and problems got the students interested in math beyond the daily lessons. With the grade 11 student, humour made the classroom more relaxed and the math lessons more interesting–but we can’t all be comedians! In the students’ opinions, logic puzzles would be an interesting addition to math classes. Real-life applications of math would also make the classes more practical. When asked about group work, the reactions were mixed. The grade 9 student did not prefer group work too much because of the added distractions from getting the work done. The grade 11 student has had math projects, such as designing a water slide using cubic functions, and enjoyed the idea of working with others, finding value in comparing answers and thought-processes with others.

Memorable Teachers

Self / Parents: For most of elementary school and up to grade 9, I learned mathematics on my own (besides using compasses in geometry). Since my teachers would not challenge me enough, I would learn multiplication, division from my parents and exercise books they bought me. I would understand the mathematics instrumentally and relationally. The only part of the problem would be I wouldn't be doing math as fast as my friends who were in Kumon. School was for refining my math speed / skills. If I didn't know how to do something, I would ask my parents. The only difficult concept I could not learn on my own was long division in grade 2 or 3 and the base concept in grade 5.

Grade 10/11 teacher: I really enjoyed my grade 10 and 11 IB math teacher. He was very fun and funny. He would promote that math is fun and had a big smile on his face. Sometimes he would make some jokes and tell stories of his past. Sometimes he would make math jokes and math cheers. He's memorable for bringing the parabola dance into the classroom. Most of the students are comfortable with him.

http://www.youtube.com/watch?v=URziKVxG0kA

Response to article

It should be every teacher's desire for the students to understand mathematics at a higher level. Students should not only be able to compute answer but also use manipulate math concepts they learned to do other questions. For this reason, I thought it was interesting that the students did not have a good grasp of the higher mathematics with the lecture style than the hands on approach. Our class really prefers to teach instrumentally. Perhaps this is not so teacher centered teaching but rather student centered learning. This suggests I should let students have more time to learn the material rather than just lecturing a lot. Once when I was doing my volunteering, I had to teach a whole class on grade 10 math. I did the think pair share with the class as an activity during the lecture. The students had to multiply polynomial fractions and simplify. I had the students talk to their neighbours, share and change their answers. When they were done sharing, I asked for different answers and polled how many got each of those answers. Then I told them showed them how I would do the problem and where it could have gone wrong to show the other answers. I do question this method in whether it would help them have a higher level of understanding.

Cool Math Art Project Idea

http://math.slu.edu/escher/index.php/Tessellation_Art_Project

I really like some of the examples they gave.

Sending Secret Messages with Light (Microteaching)

1) Things I thought went well:

• asking for verification of understanding of the subject
• would refresh what needed to be known

• interactive
• drew analogies with already known concepts

2) Ways to improve:
I have taught this lesson a few times with my friends. I may have "lost" a student but I wouldn't know because the students kept nodding their head. Asking specific questions on the facts that needed to be known would have allowed me to see whether they knew the subject or not.

3) Reflection on peers feedback:
One peer enjoyed the participatory activity of roleplaying Alice and Bob and thought the involvement was great. However, another peer suggested more participation. The visuals helped to explain the lesson. Another peer thought the material and presentation was very suitable for the topic. All the peers would have liked more theory or notes on the topic. They agreed that the objectives were clear. Quantum cryptography is a difficult topic to understand. I thought this lesson and activity was appropriate for the time and the topic. I prepare some resources and notes for the students after if this was the sole lesson. However, it would be a great introductory lesson or a good activity to reinforce quantum mechanics.

BLOOOPPS

1) BRIDGE: “hook” or introductory activity
Ask students, what can you do with light?
You can send secret messages with light!
2) Teaching OBJECTIVES:
Have everyone involve in the active learning / the game.
3) Learning OBJECTIVES: “SWBAT” 
Students will be able to relate eigenvectors to the polarization vectors of light and the implications.
Students will be able to relate measurement as a matrix acting on a vector.
SWABAT explain that measurement on light will change the polarization unless it is already an eigenvector.
4) PRETEST: Ask orally about understanding of light and quantum mechanics.
5) PARTICIPATORY
Lay down ground work rules on light.
Explain how Alice and Bob (Quantum Cryptographic System) work.
Get students to pretend / role play Alice and Bob.
Ask what information is being sent classically vs what information is kept as the message.
6) POST-TEST
Ask students one by one what they learned.
Key questions:
What about measurement allows for light to be a medium for sending secret messages?
What is the classical (verbal) information that is communicated through between Alice and Bob?
What is this information used for?
What part of light does the message information come from?
7) SUMMARY or conclusion
Alice and Bob are actual machines of a quantum cryptographic system. Such systems exist and were used in the Swiss 2007 election to encrypt the ballots.

Reflections on Skemp

Skemp presents the idea of faux amis to introduce the problem with different meanings of understanding and extends it to that of mathematics. I would relate the “instrumental understanding” to knowing simply which rule to use and how to compute the answer. “Relational understanding” is knowing the concept which leads to explanation with more than just the equation. One of the examples given was a “teacher reminding a class that the area of a rectangle is given by the area being A = L * B” (area is length times the breadth) (1). I had difficulty with how an elementary school teacher would teach the relational understanding of the concept of area. The only way I could think of was a picture of the rectangle with a grid drawn over it. If one was to be very thorough about it one would have to remind the class that length was a measure of how much space a line took up and that area was like the same concept but in two dimensions. One would also have to help the students understand why one has to use multiplication. One of the gems of Skemp's article is the quote: “Unless the two sides (who have a different interpretation on what was going on) stop and talk about what game they think they are playing at, long enough to gain some mutual understanding, the game will break up in disorder and the two teams will never want to meet again”(2). I would agree it would be quite important for the student and teacher discuss the end goal of the lesson if they are different. The frustration that would mount up for the student would cause them to have problems each lesson and may cause them to steer away from the subject in the future. It would help the student to understand why they were learning such things and for the teacher to understand the needs of the student. As a teacher candidate for physics, I would hope to teach the student relational understanding of the subject. This would help them have the skills to analyse and tackle new problems. If they only understood the subject instrumentally, they would only be able to take the same type of questions they learned and look for some equations that would give them the answer; they would not be able to teach a peer how the equations applied to the physical situation. As a music enthusiast, music taught “as a pencil-and-paper subject” compared with students “taught to associate certain sounds with these marks on paper” (3) was a very strong example depicting the importance relational understanding. The first group would miss the idea of music completely and miss the goal of learning music. They would not know music at all. Skemp states, “it is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this” (4). As a new teacher, I am more enthusiastic on students learning which would leave me not remembering students want to feel some sort of accomplishment. In this education system, marks are perhaps used for students to judge each other and a source of pride and self-confidence. However, I would like to at least empower the students to be at a point where they can say “I can do this”. Skemp states that the difficulty with teaching relational mathematics is “the backwash effect of examinations” (5). In the current education system, measurements of success stem mostly from examinations. This results in students just wanting to cut corners and learn mathematics the quick and fool-proof way (instrumental learning). Physics education groups have been developing questions testing the relational understanding of physics instead of computations. I think students would switch from just learning instrumental mathematics to relational mathematics if relational mathematics was tested. Overall, I learned mathematics and physics relationally and would be frustrated at just an instrumental understanding of these subjects. However, I would have to be mindful and plan carefully how to teach the concepts and problem solving without the students just filtering all the information and learning just the computational part of the lesson.

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