Assignment 3

Part 1 – (8) Develop a project of own choice based on the in-class library. Topics include Goldbach’s Conjecture (from novel), infinities, fractals, paradoxes, etc.

Grade: 11

Purpose:

This activity is both a history project as well as an activity to have the students think mathematically. These mathematical concepts can be very difficult to grasp and by doing this project, students will have to go beyond instrumental comprehension to fully understand the given topics and produce the desired results. Students will get the opportunity to work in small groups and present their findings to the rest of the class in a short presentation.

Description of Activity:

Project

In groups of 2 or 3 you will choose a topic from the provided list. Feel free to research and choose your own, however, please confirm your decision with me before proceeding.

• Goldbach’s Conjecture
• Infinities
• Fractals
• Paradoxes (Choose only one to work with)

A paradox is a statement that goes against our intuition but may be true, or a statement that is self-contradictory. Examples of some mathematical paradoxes are:

• Zeno’s paradoxes of motion (the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox or the stadium paradox
• Russell paradox (Barbershop paradox)
• Greeling paradox
• Petersburg paradox
• Galileo’s paradox
• Etc.

Once you have made your decision, the following tasks must be completed:

1. Research your topic through the internet or by using books from the school library. I will provide a list of books that are available in the library.

2. Produce a visual representation of your topic. This can be in the form of a poster, tables, comic, performance, etc. Include a brief history of the topic including the person who developed the concept.

3. Present your work to the class in a short presentation of no more than 5 minutes.
Marking Criteria:

Your grade will be dependent on your understanding of the mathematical paradox which will be conveyed through your final product and presentation. Effort and contribution to the team will affect your final mark. There will be a peer evaluation at the end.

Due Date: 2 weeks from day assigned.
This project is expected to be completed primarily outside of class.

Sources:

http://www.stormloader.com/ajy/paradoxes.html
http://www.suitcaseofdreams.net/Content_Paradox.htm
Mazur, Joseph. Zeno’s Paradox: Unravelling the Ancient Mystery Behind the Science of Space and Time.

Part 2 – Evaluation of the project

This project is an enrichment project designed to make students think mathematically about concepts not ordinarily encountered throughout life. Benefits and weaknesses of this project are listed below.

Benefits:

Students will learn of various mathematicians who have contributed to mathematical knowledge over the centuries.

Students will have to critically think about concepts of math outside of what they usually experience.

Students will have to work in groups to accomplish tasks.

Students will have to present their findings in a clear manner such that peers will understand.

Weaknesses:

Not all of the topics are connected to the grade 11 IRP’s. Some of the mathematical concepts may be more suited for grade 12.

Students may have a difficult time comprehending some of the paradoxes.

Students may not comprehend the purpose or necessity of this project and may view is as another mundane task they must work through.


Trying out the project as a sketch

Our group has decided to tackle this project by focusing on Zeno’s paradoxes of motion.

History of Zeno of Elea

The Greek philosopher Zeno of Elea, born around 490 B.C., proposed four paradoxes that are still discussed 2500 years later. He wanted to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved.
Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. His four famous paradoxes include the dichotomy paradox, Achilles and the tortoise paradox, arrow paradox and the stadium paradox.
http://mathforum.org/isaac/problems/zeno1.html


Part 3 – New Math Project Design

Grade: 11

Purpose:
The goal of this math project is for students to examine cyclical patterns in an artistic, fun and unique way. The students are presented with a classic problem that has been posed in an abstract way, to see if they can discover the numerical relationship. The students will be challenged to recreate and experiment with the ‘celtic knot’ pattern and hopefully they will begin to see that math exists is science and in art too. This is an enrichment project that promote deeper thinking in the students.

Description of activities:
1. Follow the guide (see below) for producing a ‘celtic knot’ and produce a 4 by 3 knot. Decorate it as necessary to show how the pattern loops.
2. If the knot has m vertices along the left side and n vertices along the top side, determine the formula/algorithm for determining the number of loops present in an m by n knot.
3. What other patterns of loops can be made by combining loops of various sizes. What are the limitations of this pattern, if any?

Guide:
The (1) blocks are placed at the four corners of the pattern (see below), each rotated 90 degrees from each other. The (2) blocks are placed next to them, cascading along the edge of the knot pattern to form a perimeter. The (3) blocks are placed in the middle to fill the space and complete the knot design. The picture below illustrates how this works. Under ‘Graphic’, there is a drawing of a completed knot for reference.


Sources:
There is no source for the creation of this knot pattern. This is original artwork that shall now be examined for mathematical inquisition.

Length of time that project will take:
Students are given 1 week from the date assigned to complete this project. If time permits, a small section of class time may be devoted to beginning the project. Otherwise, it is to be work upon outside of class in groups of 2-3.

What students are required to produce:
1. The students must draw a 4 by 3 ‘celtic knot’ and decorate it to make it easy to see.
2. The students must derive a formula for determining the number of loops in an m by n knot.
3. The students must create a new knot pattern by opening & closing loops to create their own knot pattern.

Graphic:


m=4, n=4, # of loops: 4 (highlighted as purple, red, green & blue)

Marking Criteria:
1. /4 - Completed drawing of 4 by 3 ‘celtic knot’.
2. /4 - Formula for the # of loops.
3. /2 - Originality of knot created.

Thinking Mathematically - P192 Sequence

I could not find a direct method in finding out the final digit. However, I found a shortcut. It is most easily explained by taking a closer look at page 2. So how does this shortcut work.

Take a look the example at the bottom of page 2. I am taking the 1st and 5th bits and putting it through an exclusive OR gate.

Exclusive OR gate:
inputs output
x y z
0 0 0
0 1 1
1 0 1
1 1 0

The output is 1 only if one of the two inputs are 1. So, I am taking the 1st and 5th bits and putting it through an exclusive OR gate. Then I take the 2nd and 6th, 3rd and 7th... Using this method I can skip 3 lines of binary strings.

I tried extending it and found that it works for every 9 bits, 13 bits and 17 bits. 13 is a special case though. If you are going to do every 13 bits you need an Exclusive NOR gate which looks like:

Exclusive NOR gate:
inputs output
x y z
0 0 1
0 1 0
1 0 0
1 1 1

I believe this also works with 25, 33, 45, 57, 73, 89... with 25, 45, 73 needing a Exclusive NOR gate.

You can see a pattern by subtracting each working number from the number prior. Starting at 9 you add 4 + 4 + 8 + 8 + 16 +16 + 32 + 32... Notice all of the numbers are 2 to the power of something.

----------------------------

I have something to add, a better explanation following my above method.

Given a bit string of any length, it can quickly be reduced by skipping bits! The problem is, which bits can you skip?

This is how it is done:

1. Take any bit in the string, n
2. Take a second bit, n + or - 2^m (Stay within your max string length)
3. Compare these 2 bits using an Exclusive OR gate, as written above
4. The output is your single resultant bit 2^m rows down

Example: A string of 20 bits

11101 00110 10100 00101

Take the first bit and another bit 2^m bits away. If m is 4, 2^4 = 16. If m is 5, 2^5 = 32. 32 is too large so take m = 4.

Looking at bits 1 and 17 (1+2^4) you compare them using an Exclusive Or gate. --> 1 + 0 = 1
Bit1 Bit2 Output
1(1) 17(0) 1
2(1) 18(1) 0
3(1) 19(0) 1
4(0) 20(1) 1

The new string is

1011

This is now easy to reduce to:

110
01
1

And we are done!

from Practicum

#1

MacNeill has what is known as an Advisory class right in the middle of every day before lunch. The interesting thing about Advisory is that in grade 8, a group of students are assigned to a single teacher. For the next 5 years that same group of students stay together with the same teacher for Advisory class. The goal is to create a comfortable environment and relationship between students and teacher, where students can feel safe and get help when needed involving school and possibly life situations.

Advisory runs for almost 30 minutes every day and what happens within depends on the teacher. Some teachers have the students do silent reading of the book of their choice. Most teachers however let the students to self study/homework.

#2

On the last friday, I dressed up as Geordi LaForge for Halloween. All the other student teachers said they'd dress up too but most bailed out. I was extremely dissapointed. However, my outfit was awesome and I was pleasantly surprised how many people knew the Star Trek Reference. One of my sponsor teachers even dressed up as Captain Kirk! We would have taken the prize if there was a staff costume contest.

Aside: I went to a conference earlier in the week about the adolecent brain and apparently adolecents do not read facial expressions the same way as adults...

On that day I got to supervise a math test with a TOC. One of the students, I got along well with but he completely bombed the test. During the test he asked me a question which I couldn't answer for him because of the test. Then he starting saying how poorly he was doing and how he really did study. In my mind I wanted to help but couldn't. On my face I had what I considered my pained want to help but can't expression.

What the student said to me was, "Mr. Thiessen, don't laugh."

Prime example of misreading of expressions.

On a different note, the class before I ran a review session. I prepared a practice test which covered all the material needed. If you could do this test you could do the test no problem. I gave them the numerical answers at the end of the hand out and even posted an answer key at the front of class with detailed solutions. What surprised me was even with an exam lingering 2 days away, students do not suddenly become more motivated to do math. Lesson learned.

Also, it doesn't matter if you are in grade 10, 11, or 12, there will be students who don't know how to graph y=mx+b

Division by Zero

Approaching zero,
Entering the dark.
To the blackest black,
We dive and divide.
Improbable,
But not impossible.
To exist at all,
Throughout space.

.
.
.

During the free write, the first word was 'divide'.

One into another. Over another. Fraction. Dum doodle dee. There is a line with a dot above and bellow. Or you could see a backslash. A divider! Two sides to something, ............ Cut a cucumber or an apple into 2, 3, 4 pieces. Oh it's a miracle. Big number over smaller number could be bigger or smaller. Divide those people into groups.

The second word was 'zero'.

The name of a character in Code Geass... What is 0? Is it nothing? Is it very very small, approaching nothing? What do you get when you divide by 0? Calculators say undefined. But what is a number divided by something so incredibly small? Something so big... Infinity. That's right, Infinity. Yeah, capitalize that! Where does 0 exist? Where in the Universe, Galaxy, Solar System, Star, Planet, Continent, Household, Pocket?

Would I use this in a 9, 10, 11 classroom? Depends on the class and how I feel they learn. Writing a poem about zero is interesting, but is it useful? When doing the free write it wasn't as if I had a revelation about zero. I didn't come to any new conclusions. I just wrote what I already knew with the knowledge of limits as x approaches 0. This is what would probably happen if my high school students were given the same assignment. They would write what they knew already without learning anything new or coming to an understanding about the topic.

Math lesson reflections

Our lesson on logarithms did not go as planned. Our original plan was as follows,

1) Ralph do a short history as a hook
2) Write up some of the rules on logarithms
3) Derive multiplication and division laws for logarithms
4) Have the students do two quick questions to review the logarithm rule blog(x) = log(x^b)
5) Have an engaging question for the class to do

For 15 minutes this was way too ambitious of a time-line. The history went a bit longer than expected taking up ~50% of our time, not giving much time to work on the lecture portion and student centered activity.

The big issue from what I gleamed from the comments was that there was not enough activities for the students. So how would I change it for the future?

I would cut the history to 2 minutes max and quickly write up the basic rules of logarithms, such as for log(0), log(1), when base is the same as what is being logged. For this type of presentation, we can assume that we are just doing a review of the rules and the students have already learned it before.

I would cut out the derivations for the multiplication and division laws. In fact, I don't think I'd include those laws at all for this presentation. For 15minutes we need to keep it extremely short. What I would do is write a function on the board, starting with y = x and ask the class, 'without using a calculator how would I graph y = log(x). The purpose of this activity would be for the students to recognize that when x <= 0, there would exist no y. When x = 1, y =0. When x = 10, y = 1 etc. I would then get them to do the same graph but with a different base instead of 10 and see how the different graphs compare.

I could then give them an equation such as y = x^2 or x^2 - 3 and we could repeat the above process. They can check their results using a graphing calculator after wards. The purpose of this activity would be for them to recognize that a log of a negative number or 0 is always undefined and a log of 1 is always 0. Also, to graph it without a calculator, the students would have to recognize how to choose easy number to work with when dealing with logarithms. For example, if you were working with a base 2 logarithm, you would want to solve for when the inside is 2^n.

Example, for y = log(x^2 - 2) [base 2]
If y = 1, then x^2 - 2 = 2. Solving for x you get 2. Next you do for y = 2, then x^2 - 2 = 2^2 = 4. Solving x you get root(6). It's even easier to graph the logarithm if you were t first graph y1 = x^2 - 2 first, and then look for key locations on y1 such as when y1 = 0,1,2,4,8,16, 1/2, 1/4, etc.

That's all I have to say.

Booppps Lesson Plan - Exponentials and Logarithms

Bridge (The Hook) - The history of logarithms, who invented it and why was it invented.

Teaching Objectives - To cover/review some of the laws of exponentials and logarithms before going over a more challenging problem together as a class. If there is additional time we have an extra problem that we can challenge the class with.

Learning Objectives - Have the students apply various log rules to solve a question, displaying their understanding.

Pretest - Go over the rules of exponentials and logarithms and then have the students answer a couple simple one or two step questions.

Participatory Activity - Work through together as a class a challenging logarithmic problem. Graph, without a calculator, y = log(x+2) - log(x-1) where the log is base 3.

Post Test - The participatory activity can be considered as the post test since there is only 15 minutes for the lecture. If there happens to be time we have an additional question to give.

Summary - Tell the class there will be an exam next week!

Flatlander

Just finished the book 'Flatland'. I'll talk about it more later, for now I'd like to just put up this link which someone sent to me a long while back: http://www.tenthdimension.com/medialinks.php.

I remember reading a Sci-Novel a while back which involved a 3-D world and a 2-D parallel world. This was a long time ago and could possibly just be part of my imagination... I'm going to see if I can find.

Mathematics and Citizenship Education

The article "Citizenship Education in the Context of School Mathematics," by Elaine Simmt, states that mathematics has a role in citizenship education and is crucial in the development of informed, active and critical citizens in our society. "Mathematics will help us understand ourselves... [and] understand the world in which we participate." Mathematics is embedded into our world from business, economics, time, space, communication, transport and war and our youth must be educated to understand and critique the 'formatting power' of mathematics.

So, how to we teach math to accomplish this. Right now, math is usually taught as:
1) a set of facts, skills and processes - creating consumers who know how to leave a tip.
2) as facts and fact - pushing around numbers and writing them in different ways suiting the teacher.
3) as right or wrong - reinforcing the notion that mathematics must not be questioned and creating no confidence in individuals.

Elaine gives possible methods such as:
1) focusing on problems and questions arising from student interaction with math using a strategy called 'posing variable-entry prompts'.
2) having the students explain their reasoning attempting to prove their assertions. The necessity of student contribution.

The example in this article about making rectangles using 36 square tiles reminds me of the recent video we saw in class about solving two challenging exponential/logarithmic problems using 2 or more methods.

These types of questions would likely take up a lot of time and therefore I would probably limit each unit to one, possibly two activities of this nature. It would depend on the unit and how much worth I see the students actually getting out of the activities.

These activities encourage group work and negotiation, forcing the students to pose questions and hypothesis on how to approach open ended questions. I like the idea of groups of students writing down their solutions on poster paper and then having to present their results. I completely agree that it is important for students to present their results explaining their reasoning's and approaches to the problems. Also, being in groups will provide them more confidence when discussing the results.

What-if-not

What-if-not? Yes, what is that?

It is a strategy to assist you in problem posing. It is a systematic approach that has five levels.

• Level 0 Choosing a Starting Point: ie. Pythagorean Theorem
• Level 1 Listing Attributes
• Level 2 What-If Not-ing: Take an attribute and ask yourself, 'what if that attribute was not true and was instead something else?' List as many alternatives as you can think of. ie. What if the Pythagorean Theorem was instead written as a^2 + b^2 <>
• Level 3 Question Asking or Problem Posing: What questions come to mind when you write down this alternative form? When is it true? Is it significant?
• Level 4 Analyzing the Problem

You can also combine your What-If-Not's to form even more possibilities.

How can I use this technique in preparation for my groups micro-teaching lesson next Wednesday on Logarithms and Exponentials? At the moment, I'm not really sure how it will be of use. However, I am sure we'll be able to use this problem posing strategy to think of some questions that could be useful for the presentation.

As a level 0, we could possibly begin with one of the rules for logarithms, such as the Power Law, or Multiplication/Devision Laws. From there, who knows where it will go?

The strengths of the What-if-not method is that it is a pretty direct way to formulate a lot of questions from a single starting point. It is a tree that keeps expanding and is interconnected with itself to create even more possibilities. Another positive aspect is that fact that it is a systematic approach to problem posing. It walks you through the steps making sure you really analyze the starting point, the topic you chose, by listing all the attributes you can see. For every question you get a what-if-not question and you are forced to analyze to prove/disprove (find out when true/false) each statement before continuing. You will really able to flesh out the topic you are looking at using this method.

The weaknesses of this method is that it can take a long time, and I mean a LONG time. And after putting a lot of hard work into the process, you may find yourself with nothing to show for it. For example, you may create a lot of what-if-not statements similar to Attribute 8 on page 50. Those arn't going anywhere and will prove to be a waste of time (If I'm wrong about that last sentence, please let me know). Nothing useful may come out of the process meaning I could have watched an episode of House instead of using the strategy.

I'll try using this method during the preparation for the micro teaching lesson. I'll see how it goes.

10 Questions or Comments

1) From the text, 'The Art of Problem Posing', I like the line on page 3, '"The answers Gertrude, what are the answers?" - whereupon Gertrude allegedly responded, "The questions, what are the questions?"'

It reminds of a scene in the book 'The Hitchhikers Guide to the Galaxy'. Deep Thought, a super computer created by a pan-dimensional, hyper-intelligent race of beings, was faced with the task of figuring out the answer to Life, The Universe and Everything. 7 and a half million years later he eventually comes up with an answer, 42. Deep Thought then replies, "I think the problem, to be quite honest with you is that you've never actually known what the question was."

There are all these answers and facts out there but there isn't only one correct path to these answers. So what are the questions?

2) x^2 + y^2 = z^2. An interesting way to begin because it is so open ended. You could choose anything, there are an infinite number of possibilities. The book asks you "What are some answers?" and the only way to come up with any is to pose some questions your self. I simply chose some simple numbers so I could move on with the reading. After reading some of the answers written though it got me thinking about the vast number of responses, and the fact that the responses didn't have to do with math at all.

3) When I see x^2 + y^2 = z^2 I think of a circle first then Pythagoras and right angle triangles. It's a tool to get an answer. The triangles that come to mind are the standard 'special' triangles with ratios 1:1:root2 and 1:2:root3.

4) On page 14 they show a triangle with squares on each edge. I have seen this before, but I have never known the reasons, it seems pointless. How does it help? After writing this question I think I may have figured it out. Is it simply a reminder that to relate each edge length you must square each length first?

5) Do you ever give us any of the answers to the questions you pose? Some of the points brought up are interesting but I guess it isn't the purpose of the book to give answers. That's what the internet is for eh?

6) The hand shaking question on page 19. When I hear this problem I instantly think 'the number of people, choose 2'. nC2, where n is the number of people shaking hands. So, with 9 people, we get 36 handshakes.

Yes, I am one of those guys who came up with the question, "How many handshakes were there altogether."

7) It has been so long since the first time we actually learned the properties of isosceles triangles and Pythagoras, etc. It makes me wonder, how did I first learn these concepts? What went through my mind when it was introduced to me? Did I ever think of interesting questions such as "can I make a bicycle hub out of congruent isosceles triangles?" When I read this chapter today, I certainly didn't come up with any questions like that. I agree with what is written on page 22 whole heartedly, those who have been exposed to geometry "find great difficulty coming up with much more than observations relating the equality of the lengths to the equality of angles."

8) If I was given a Geoboard of 25 nails I would get an extremely stretchy rubber band and try to make the most complicated looking picture possible using all of the pins without the elastic ever crossing over itself.

9) Coming up with unique questions for these pythagoras, geoboard, isosceles and handshake problems is tough. I'm sure after teaching a few courses in math and seeing what students come up with I'll be more capable of thinking of questions, both in and out of context.

10) The handy list of questions on page 30. Add ons:
What is the origin of the formula? Is it continuous? Discrete? Is there an average? Are there visual representations? For whom is this useful to? Practical or theoretical? Can we prove it? Do we need to prove it?

The Distant Future...

Letter from a student who liked you.

Hello Mr. Thiessen,

My name is Slarty Bartfast and I was in your class 10 years ago in grade 10. You changed my life. I wasn't always a fan of math. In fact I hated it! After your class I was inspired. You showed me how math could actually be and taught. Your student centered lessons really got me involved and had me understanding the material in no time. It made me continue onwards into grade 11 and 12 and beyond.

You most likely recognize my name because I've been on the news constantly this past week. The man who successfully made the calculations for the first intergalactic warp drive system. It was actually my PHD paper.

Thanks Mr. Thiessen. Because of you, mankind can now travel to the stars.

- Slarty


Letter from a student who hated you.

Mr. Thiessen,

My name is Troy McClure, you may remember me from high school grade 10. Math, I hated it and due to that I didn't give you a chance when I walked into your class. Math was tough, equations and equations of nonsense. Plug this number into x and sub that number into y. It made no sense and that's how I wanted it to stay.

During your lessons you'd often talk about understanding this, and understanding that. I didn't care about any of that. I knew that once math was no longer a prerequisite I would no longer enroll into it.

I didn't want to discuss ideas in groups or do any projects. The bare minimum to pass, that's all I wanted. This was the last mandatory year for math and once I finished high school I was set to pursue my future career in TV broadcasting.

Which is what I did. No thanks to you,

Troy McClure


My hopes and worries...

I hope to be a teacher who passes on an understanding of math to my students. I didn't fear math, I enjoyed it, and I hope I can pass that feeling on to my students by being a effective and engaging teacher. However, I know there will be students who will never like math and I hope I can inspire them enough to get them through the courses.

Dave Hewitt - Classroom Orchestra

Today we Dave Hewitt demonstrate a alternative method of teaching. A strategy that involve the whole class, one that is dramatic and like an orchestra. In the classroom we saw him tap the walls with his rulers and have the class add and subtract depending on the direction Dave was tapping in. They would all reply by speaking in unison. Clockwise was add, counterclockwise was subtract. His goal was to have the students see that pattern of what is going on.

Dave eventually introduced variables as a starting number. He drew a row of boxes on the chalk board and designated one as x and chose another box as the destination. He then demonstrated that he could do various combinations of left and right taps to get to the desired destination.

Example: x -> x + 2
This could be solved as three taps right and one left or two taps left and four right, etc

On the board he wrote:
x + 2 = x + 3 - 1 = x - 2 + 4 = x + 612 - 610 ...

He did this all without describing what '=' was too. He wanted them to learn by association.

Some good things about it is that it starts simple and goes to complex and it is an introduction to patters. Also, it is interactive and engaged the whole classroom. Also, the classroom was well managed.

However, there are many negatives to this teaching lesson. There would be a good number of children who will be very bored due to the extremely slow pace. There was no note taking involved which could be seen as both good and bad. Bad for some since a few people may get nervous not having anything written down to refer to. I would also like to see this orchestra method used for a more involved topic other than adding and subtracting which is something the students should have already learned.

Certain topics should be taught using traditional style (quadratic equation), some using constructivist style (geometry) and others by using this Orchestra method (counting). I guess the trouble is figuring out which is best suited for each lesson topic.

On a different note, I was talking to Mike after class and we were talking about alternative ways of teaching math. Us both being Physics students too, Mike brought up math labs (ie. Physics). I thought, why not? Especially kinematics, the math involved in physics is very simple and uses math from grades 8 and 9, and a little from grade 10 in the means of trig. I definitely want to explore this more, teaching math but using physics related questions for applications and word problems.

Math Interview Reflections

After listening to the many groups talk about their interviews I have concluded that the most useful and interesting questions were the ones regarding teacher strategies and classroom management. All those questions asking students their feelings about math and such resulted in what I expected. Among students, not much has changed. Well, I am only 5 years removed from high school and that’s not a lot of time to result in large changes. Every school still has a wide range of students with various competencies in math which usually reflects how they grade their teachers. Also, most students don’t know what the purpose of math is other than a requirement for getting by high school or needed for doing taxes. If that was true though, math could be seen as useless since there are computer programs to do our taxes for us now a days. The applications and purposes of math must be better explained to the students during the lessons.

Before I talk about what I found interesting from some of the other groups I’ll go in detail about the responses I got from our own groups interview. The person we interview is a new teacher who just went through the UBC education program last year. He studied math and physics and got a full time job right after graduation at a fairly new school in Surrey.

The first question we asked was about his transition from the practicum to the full time job. He explained how the full time job is a lot more work. Lesson prepping is the most time consuming component of teaching and all of that has to be completed outside of the classroom. I didn’t expect the degree and quantity of work that he made it out to be. The number of different courses he has to teach his first year is amazing, and for all of them, new lesson plans. I didn’t ask, but hopefully he gets lots of help from his fellow educators.

The second question asked about the amount of time the class is devoted to lecture and how much towards seat work. He responded with 20-40 minutes of seat work depending on the lesson. He reinforced the notion that it is very important to vary the class with activities such as math bingo. Many students can’t last the full 75 minutes that a class goes for and need to get up and move around. This is similar to one of the other groups interviews. One teacher said they like to start with a bus driver game, move on to a challenge question, lecture for 25 minutes and end with some time for work. Variety seems to be a common trend among math teachers these days and I think it is a good thing. Another thing was peer tutoring. I like the idea of peer tutoring because lots of times students have different perspectives and methods on how to approach a problem which can be extremely useful. Also, it may be easier for some kids to ask for help from a peer than a teacher. In my old school one of my math teachers was trying to set up a peer tutor system where the tutors would get paid 10 dollars per hour from the school. I believe it got going after the year I graduated.

The third question asked about any visual methods he used in class. To him, algebra tiles are stupid and I’m going to have to agree with him. It depends on the subject. Subjects such as geometry and surface area where kids can physically measure objects are visual methods really useful. Consistencies of notation was the fourth question. His advice, follow the notation in the textbook. Students get lost easily with simple changes such as f(x) = g(x).

The final question asked about special needs students and the alternative teaching strategies used when working with them. As a new teacher he says it is extremely difficult. Be patient and offer time outside of class. Also, use peer tutors!

Keeping students interested is a must as a teacher. I liked one of the responses from one of the other groups. To keep the top students interested, give them challenging problems and have them act as tutors. As for the struggling students, as long as they don’t give up, you’re doing a good job. Keep giving them encouragements as much as possible.

From these interview sessions I have learned a lot of useful information which I should be able to use in my own teaching practices. To end this blog I will mention the secret final question of the interview: What is your favourite pie?

If you answered 3.14… you are on the right track.

Battlegrounds School Response

This article talks about the two main stances in Mathematics Education. The two dichotomies are the Conservative approach and the Progressive approach. To keep it short, the Conservative method is what most people know of when they think math class, lectures, homework, memorization, tests and repeat. The Progressive approach is to have the students come to understand math, to see the patterns emerging from lived experiences and to stimulate problem solving skills.

In table M.1, one of the columns is what should drive curriculum design in public schooling. For Conservative it reads: for the masses, the need to do simple, practical calculations. For Progressive it reads: to expand learners’ modes of thought and develop flexible problem-solving abilities.

This article is pushing for the Progressive approach and lists complicating factors which are preventing the switch from Conservative to Progressive. It goes into different times of our history when a change of mathematics teaching was attempted.

I completely agree that people are math phobic in North America. Many people, particularly parents remember math being extremely difficult and meant only for a small elite or the nerds, and mad scientists who are unable to cope with the world of human interactions. Also, these people feel no shame at being incompetent at math. Children become scared of math before they even start and if they find it troubling they are told it’s ok. This fearful approach is the completely wrong approach towards math and has to be fixed.

Who said war wasn’t good for anything... Apparently, because of war, math curriculums were changed to better prepare students in math due to the perceived need of an increasingly scientific and technological populace. Between 1910 and 1940 a Progressivist Reform occurred due to World War I. John “Dewey challenged the Cartesian split between knowing and doing, or abstract and applied knowledge.” Dewey’s progressive education involved that students must form and test hypotheses and perceive patterns and relationships. His ideas only won acclaim in very progressive schools.

In the 1960s due to the USSR beating the US in the space race, “The New Math” was created. This was another progressive teaching method but it didn’t go over very well. Many math teachers themselves did not know this new math and were incapable of teaching. The new math “supported understanding over fluency,” and “inquiry and sense-making over absorbing and applying facts.

In some ways conservative methods of teaching math works. It works for some but not for all and because of that, people have been trying to change the curriculum for the better. But what gets in the way of this change? The math illiterate with their fears. Also, traditionalists who claim that children are being short-changed by teachers experimenting with their education when they use progressive methods.

The traditional methods of teaching works for some students but still too many are being left behind. Progressive methods must be integrated with the Conservative methods to accommodate a larger portion of students.

Response to Heather J. Robinson Reading

"First Comes Learning," is the motto at Robinson's school and it is a good motto.

Before I begin talking about what I like and dislike about her changes in teaching methods I will first talk about those grades listed on Table 4.1. Am I reading it correctly?! From my understanding, in the final exam 56 out of 64 students failed the exam when she did primarily lecture style lessons. That's 88%. When she changed over to the new instructional method, the semester B failure percentage was 38% and semester A failure rate was 24%. My math teachers all taught similarly to Robinson's original teaching style and I'm positive the failure rate was no where near 88%. Maybe it was due to the demography of the area she taught in?

Robinson really changed from one teaching style to the extreme other end of the spectrum. What I get from this article is that just like how different students require different methods of being taught to learn, some teachers are much better at teaching one way than another. Lecture style teaching has been used for a very long time in high schools and for many teachers I bet they saw success year in and year out. Does this mean they are better at teaching? Who knows!?

The problem with lecture style teaching is that there will always be some, the number of students varying each year, who cannot effectively learn through that method. Robinson's goal is to "leave no child behind." However, I think she's a bit excessive. I'm not sure exactly how many hours a week she teaches for but since it is semester I'm assuming about 5 or 6. Only 1 of those hours she states as using for lectures. Wow, that is an extremely small amount of time. I cannot even conceive of teaching like that since I have never experienced anything like it before. If I go to my practicum it would be awesome if my supervisor taught like that because I would love to see it in action and see how it is possible.

Some things I really liked about her school and teaching methods is the time reserved after school for homework and getting help and the peer tutoring that developed. That 45 minutes reserved for getting help from the teachers would definitely be really useful for many students. But at the same time, for the students that don't need help and have to wait around before club activities begin have it a bit rough. Kind of like punishing them for doing well. Peer tutoring is good. When I teach I would like to set up some form of peer tutoring outside of class under the supervision of teachers.

Time to bring this all to a close. Robinson has some great ideas but I think she's going a bit overboard. She has seen great improvement from changing teaching styles in just one year but is it because she was not a very effective lecturer, the demography of the city or the fact that this new teaching style really suits her and her sample of students? For a math class, less than 20% of the course being lecture format seems a bit small. Especially for grades 11 and 12 I have a hard time seeing all the material being covered by only lecturing for about 10 or 12 minutes a day.

I'll have to wait until I teach a class for the first time and find out for myself. What will work and what will fail miserably. Time will tell.

Memorable Math Teachers

Teachers from my past...

Mrs. Law from my high school, how could I ever forget her. I believe she taught me in grade 9, 11, 12 and calculus. 4 out of 6 math courses, there is no way I would not remember her unlike my grade 8 and 10 math teachers. Anyways, as a math teacher, I thought she was good although I didn't have anyone to compare her with. I like math, I was good at math, math was my thing in high school.

Many of my class mates on the other hand thought differently then me. Their complaint was that she just lectured day in and day out. And it's true, that is what she did. For me though, that was all I needed from her, she would lecture I would take it in and work it out in my head and if I knew the stuff I would go ahead and do homework in class. I can understand where my class mates are coming from though. Pure lecture was not enough for many of them and they needed something more or different.

Another teacher that comes to mind is Anmar Khadra from UBC. I think the course was for either 1st or 2nd order PDE's. That guy had so much energy and had such passion when he taught our class. He wanted everyone to get involved and there was no way anyone would be able to get away with sleeping in that class. He did a good job in teaching too, at least for me.

In both cases, a lecture teaching style was used and they both relied on instrumental learning for the subjects. Before I came to the education program I didn't know there was a different way to teach math. I still think instrumental learning and lecture style teaching is necessary in math, but I don't want to be like my past teachers and rely solely on those techniques. I just finished reading the Robinson article and she really went from one extreme of teaching to the other. I'll have to find the correct balance for myself.

Booppps Lesson Plan - My Reflections

The micro teaching assignment has come to a close and now I must reflect on what I learned. I will begin by giving my impressions on the lesson immediately after presenting.

What I thought went well:

The props were well prepared and I clearly stated the objectives. I discovered that most everyone had had some experience with baseball but none had pitched before. I had everyone participate by giving each person a baseball to hold. As I showed the different grips, I had them follow along, making sure each person had the correct grip before continuing. For each grip, I tried to give a really brief explanation on how they worked, however I did not linger very long on each since I had a lot of grips to get through.

The first grip I showed them was the 4-seam fastball. After showing a total of 3 grips I tested their knowledge by asking them each to again show me the 4-seam fastball. I explained that if you are going to remember anything from today, remember this grip because whenever you throw a ball of around the same size, this is the proper grip to use. One of my students then said "really? That doesn't seem natural to me. When I throw a ball, the natural way I grip it is like this." This actually led into the next grip I was going to show because the grip he displayed was a palm ball, a variation of the change up. I thought it was cool how one students comment/question allowed the flow of the lesson to run more smoothly.

How would I improve the lesson?:

I needed more time. To solve this problem, the best method would be to cut down the number of grips being shown. If I were to do this lesson again, I would not show the 2-seam fastball and the splitter, giving more time to show the curve ball. I felt a bit rushed and with more time (or less grips) this problem would be solved. I would also be able to add a post test at the end of the presentation and have a more flushed out summary.

Reflections based on my peers' feedback:

My peers feedback pretty much coincides with my own evaluation. Time management. They would have liked to have seen more time spent on the curve ball. I agree with them. Now that I think about it, the curve ball is one of the most interesting pitches in baseball. The way I set up my lesson was showing the different grips in the order that I learned them when I was young (except the splitter). My peers would have liked to have seen more closure to the topic and to include a post-test at the end.

One of my favorite comments I received was to provide more info on why each grip/throw is important and how they are used in baseball strategy. I liked that one, explaining a little bit about the strategy of baseball would have been a good edition to the lesson.

Booppps Lesson Plan - Peer Evaluation

Hello and welcome back to Thiessen's Math Blog. My last blog detailed my lesson plan on how a pitcher grips a baseball. My group consisted of 3 other people. I taught and was evaluated by Erwin, Ralph and Vincent. Below is a summary of what they said:

"The learning objectives were clear. Greg told us that we would be learning some of the various grips that a pitcher would use in a game. He bridged the topic by showing his interest in baseball and asking about our prior knowledge with the topic. We each said we played a little baseball but Greg then asked we if any of us had been pitchers. None of us had. Greg's lesson included a participatory activity by providing us each with a baseball to practice the various grips. In the middle of the lesson he asked us a question testing our memory on the fastball grip. He concluded the presentation by explaining how grips can change the rotation and speed of the ball. Overall it was a smoothly run hands on lesson. He showed enthusiasm for the subject and kept our attention.

"Some things Greg could work on is including more time at the end for a post-test to check in on our learning. He should also provide more info on why different grips are important and how they are used in baseball strategy. He needs a bit more closure to the topic and a little less complicated working with the different types of fastballs. The order of presentation of the grips could have been modified a bit too. More time should have been given to the curve ball."

Booppps Lesson Plan - Gripping a Baseball

For this micro teaching assignment I will be instructing the students on how a pitcher grips a baseball. I will be giving each of my students a baseball and they can follow me as I show and explain the various grips. I will tell them why the pitch is used and how it works. I will be drawing lines on each of the baseballs to make it more evident on how to hold the ball.

Pitching Grips for a Baseball - Teaching the Booppps Way

For the players, baseball is the loneliest team sport in North America. Why is that? In sports like hockey, football, and the other football each of the players constantly come in contact with each other. In baseball, when out in the field, rarely do the players get within a foot of each other. Every play begins with the man on the mound, the pitcher.

But even though the pitcher is all alone on that mound, he is dependent and can rely on the catcher calling the pitches and the 7 players supporting him from behind. Also, a pitcher is not without his weapons. A variety of pitches to fool the batters.

Bridge - Baseball is awesome! Particularly pitching.

Teaching Objectives - To get everyone participating. Time permitting I will try and show the students the following pitches: 4-seam fastball, 2-seam fastball, splitter, the 3 change-ups, curve-ball, and slider. The most important ones (in my opinion) are the 4-seam, change-ups and curve so I will get those out of the way first.

Learning Objectives - To have the students remember how to grip a 4-seam fastball and at least 1 other grip I showed them.

Pretest - Ask them about their baseball experience and if any had played baseball in the past.

Participating Activity - I will demonstrate each grip one at a time and have each person follow my lead with their own baseballs.

Post Test - I will ask each of them to demonstrate to me a 4-seam fastball and one other one I randomly select.

Summary - If you ever have to throw a baseball or similar sized ball in the future, hopefully you remember the 4-seam fastball grip. It is the most effective for any type of throwing of a ball around that size.

Relational and Instrumental Understanding - Response to Skemp Article

We were given an article by Richard R. Skemp to read outside of class. The name of the article was: Relational Understanding and Instrumental Understanding. The paper was first published in 'Mathematics Teaching', 77, 20-26, (1976). Wow that is old. When he wrote this, Richard R. Skemp was part of the Department of Education at the University of Warwick.

A quick breakdown, Instrumental Understanding is understanding without knowing the why behind whatever you are learning. You are given a tool or equation to solve a problem and then you use that tool to solve it. Relational Understanding is understanding by relating the topic to things you already know. Using reason and deduction to understand, giving the person a tool box of knowledge to solve a problem.

So, the assignment was to read the 15 page paper, select 5 quotes and respond. Here it is, enjoy thoroughly!

As a student, when it came to math, I would categorize myself into the relational understanding group. But it was not as if I would not do well in a topic if it was taught instrumentally. Whenever that occurred, my brain would just store that information somewhere in my head where it could churn until eventually connections would be made and the ‘why’ behind the topic revealed itself on its own. In high school, those connections rarely took much time. However, I realize that not everyone is math orientated and capable of making rational connections on their own and therefore require a teacher to show them.

“…for many pupils and their teachers the possession of such a rule, and the ability to use it, was what they meant by ‘understanding.’ – page 2, lines 11-13

As Skemp implies, memorizing does not equal understanding. In math, a teacher could give out formulas and rules and if a student managed to memorize the two he would probably do well on an exam. Not great, but well. To many, this result would be satisfactory, but this should not be. Pure memorization rarely becomes long term knowledge with the formulas and rules trickling away from the students minds after the course in question has ended. Little do they expect the hole they are digging for themselves. As math progresses and gets more involved each year, more formulas and rules come into play which overlaps what was learned previously. The result: overwhelmed students who dread math.

“I used to think that math teachers were all teaching the same subject, some doing it better than others.” – page 6, lines 17-19

“…there are two effectively different subjects being taught under the same name, ‘mathematics’.” – page 6, lines 20-21

Skemp’s article impressed me since it really made clear the difference between rational and instrumental understanding and reinforced the importance of rational understanding. He made me take another look at myself and realize that lots of my knowledge remains at “the intuitive level” [page 13, line 5] and the importance of taking that extra time to readdress those topics.

“…if people get satisfaction from relational understanding, they may not only try to understand relationally new material which is put before them, but also actively seek out new material and explore new areas…” – page 10, lines 22-26

Instrumental understanding is very one dimensional. Facts just are and there exists no depth to their meaning. Relational understanding fills that depth and lets the student know that there is indeed a purpose or a reason to what is being taught. Understanding often equals interesting and when something is interesting desires to learn more often arise.

“From the marks he makes on paper, it is very hard to make valid inference about the mental processes by which a pupil has been led to make them; hence the difficulty of sound examining in mathematics.” – page 12, lines 4-8

Skemp makes a good point; it is very challenging to discover all of the learning styles of every student in the classroom. Even if you did discover it, could you accommodate every learning style? From reading this article I would conclude that relational teaching should be used since it accommodates both types of students and hopefully the instrumental students would one day discover there is more to math and desire to know the reasons behind the formulas.

“…learning relational mathematics consists of building up a conceptual structure from which its possessor can produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.” – page 15, line 36-page 16, line 1

Relational teaching best benefits the students. In math, if a student understands the reasons and processes, he can more easily apply his knowledge when problems he has not seen before arise. Also, there may be more than one method to solving a problem and relational understanding would allow the student to possibly find those paths and reach the desired solution.