Math 9 Unit Plan

Project Idea - Ancient Counting Systems

Counting Systems Through the Ages!

Grade Level: 

8-10

Purpose:

To gain a more thorough understanding of number systems.

Activities: 

Research a counting system that we don’t use in class.

Create a poster showcasing their research.

Compare and contrast the researched number system with our modern system.

Write a word problem with this number system.

Present findings to class.

Sources:

1. A nice website on ethnomathematics: http://www.tacomacc.edu/home/jkellerm/Ethnomath/

2. A website on Egyptian numbers: http://www.eyelid.co.uk/numbers.htm

Time: 

Two class periods where many resources will be made available. Students would spend approximately two hours outside of class working on this project.

Marking: 

60% for content (shows understanding of number system and its relationship to modern counting systems, evidence of research, word problem), 20% poster aesthetic (well organized, information is displayed clearly and creatively), 20% for in class presentation (showmanship).

Scientific American Project - Martin Gardner Bio

Martin Gardner

Martin Gardner was born on October 21, 1914. A prolific writer, Gardner has published 70 books to date on topics including: recreational mathematics (our interest here), pseudoscience, magic, philosophy, literature, and even a couple works of fiction. His most recent work of fiction is a story about Oz involving Klein bottles. Gardner is most fondly remembered for his column Mathematical Games, which ran in Scientific American for 25 years.
Gardner has no formal mathematics training outside of high school and even had a hard time passing his high school calculus course. His background is in philosophy, which he studied at The University of Chicago. When asked what he likes about mathematics Gardner responded: “There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known.”
After serving as a yeoman in World War II he became a contributor to Esquire magazine. His first submission was a short story entitled “The Horse on the Escalator.” From Esquire he became assistant editor of the children’s magazine Humpty Dumpty (Humpty was the head editor). After submitting a piece on flexagons to Scientific American in 1956 he was asked to head a monthly column focusing on recreational mathematics.
Gardner claims that the number of puzzles that he has invented could be counted on one hand. He could often be found scouring Manhattan bookstores for books of recreational mathematics that he would use for inspiration. Gardner believes that his column was so successful because of his lack of experience. While writing the column he would also be solving the problem for the first time.
In 1981 Gardner retired from Scientific American to focus on his other passions, mainly debunking pseudoscience. Gardner inspired many a future mathematician with his column which lives on in various collections that have been published since his retirement.

Scientific American Project Evaluation

This project is a good way to show students that math problems exist in popular culture and can be done just for the fun of it.  It’s also a chance for students to pick a problem of their own choosing that will hopefully be a fun challenge for them and not just another problem where they aren’t interested in the answer (except that it’s correct). One of the great things about the problems in Scientific American is that they are phrased in a way where you are actually interested in finding the answer. During class it’s hard to make questions that are interesting for students, and showing them that they can actually find math problems that are “fun” will hopefully improve their opinion of math and willingness to engage in the learning of it. While I was finding the solution to the problem, I wondered how easily I would be able to find the solution if I were still in high school and didn’t have a degree in mathematics. If I assigned this project to students I would worry that the problem they pick might be so hard that they get the impression that math is just something that they will never be able to understand and use fully. On the other hand, picking problems for the students doesn’t allow them to find problems that are interesting to them and takes away a lot of the benefit of the project.

I would most likely use this project as an enrichment assignment to be completed between units, since there is no specific topic that is being used or learned here, it’s simply a general interest project. It would be fun to have everyone present their project, but I don’t know if I would have the time to spend doing in-class presentations. Instead, maybe I would have them create a poster and put the posters up in the classroom. Another thing I could do if I had one class to spare is make this the basis of an in-class math fair where everyone would create a booth and go from group to group trying to solve the other problems.

To put together a math fair I could modify this project so that the end result is a booth rather than a report and poster. If the students were not in grade 12, I could adapt the project so that I choose a number of problems that I know are solvable for their grade level and allow them to choose only from my selection of problems.  Another idea would be if I had a grade 12 class and a lower grade class (grade 8-9) I could assign this project to the grade 12’s, have them make a booth with tools to allow anyone to solve the problem and bring the finished projects to my grade 8-9 class for the students to work on. I could make part of my evaluation of the grade 12’s that they present the problem in a way that is understandable and solvable for a grade 8 class.

Jacobean Locks (Thinking Mathematically, pg. 176)

Practicum Experiences

For my first lesson I wanted to make a "fun" and memorable class for the students.  I designed a human model of how to distribute a monomial into a polynomial.  Before the class I got signs, whiteboards and wrote out a narrative of the math story I would tell while the students acted it out.  I was nervous that students wouldn't volunteer or that they would think it was lame, but when I announced my idea a few students actually gasped with excitement.  It was a bit disorganized (my FA referred to it as "organized chaos") but overall a really fun activity that engaged the students in a way notes aren't able to.  After my lesson my FA told me how happy he was to see that I did something different and interesting in a math class and really encouraged me to keep trying out ideas like that.

My second experience that I wrote in my notebook I don't want to put on my blog for various reasons.  Suffice it to say I learned a lot about the kind of teacher I don't want to be.

Math Poem

Writing a poem about math can be beneficial to students, especially if they are creative and enjoy writing poetry.  It can make them see math in a different light and is generally a new way of looking at abstract ideas.  However, for students who don't enjoy writing poetry (I would say >50% of students fall into this category) it can further alienate them from math - the opposite of the desired effect.  Also for my own classes I need to take into account my own interests and strengths, neither of which is poetry.  Unfortunately my attitude toward poetry is beyond simple apathy - I actively dislike poetry.  If I were to assign this in a classroom I don't know if I could mask my distaste for the assignment I am giving, and when it comes to questions or marking I would be lost and disinterested.  I like the idea of connecting math to other areas, but I would pick something I have more personal interest in so that I can at least model interest in the assignments I am giving out.

Fermi Problems - Reflection

Doing a lesson in 15 minutes is difficult to say the least, and when we began to notice we were running out of time the first thing that got cut was any explanation of what we were doing and how it relates (in general) to mathematics.  We should have planned more how we were going to present the material in a more in-depth way to ensure we did not run out of time.  I think we planned very well what we we going to teach, but not enough of how we were going to teach it and exactly how long each part would take.  This is especially important in a short lesson, but is also applicable to lessons in a real class where I often remember teachers running out of time when they were only halfway done their lesson.  The only other thing I would have done differently in our presentation was to give a clear outline of what we were going to do and why.  I don't think we made our goals for the lesson very clear to everyone and some students were left not really knowing where the lesson was going or how it applies to mathematics.  Perhaps even giving an additional problem for the groups to work on that involved Fermi estimations and integrated other more traditional mathematics that they are used to seeing.  That might have really highlighted the applications of estimating and the importance of it in all areas of mathematics, and given the students a reason why they should care about Fermi problems (other than they're cool).
A few of the good things we did that I would not change were: including a small group assignment, giving an interesting "hook" in the beginning with a small historical anecdote, and leaving time in the end to hear all of the groups' answers.  If there had been more time I would have liked to have had a class discussion about Fermi problems and the different answers each group came up with.  Also, having all four groups working on a slight variation of the same problem really worked for our presentation and I think made the lesson more interesting than if all four groups had just competed for the "correct" answer.

Fermi Problems - Lesson Plan

Learning Outcomes: Be confident in making estimations, understand error in estimation, problem solving.

Hook: Fermi brief introduction and story about estimating TNT in atomic bomb.  Talk about in everyday life when we use estimation: tipping, tax, time, distances (driving).

Pretest: How many 10-dollar bills would it take to cover the surface of Canada?  Answer:950 trillion!

Lesson: Understand the problem -> Think of a plan -> Carry out the plan
Go through the example and how we would solve it using those three steps.

Activity: In groups of 3-4, hand out one problem to be solved by the group in the same way we have just shown.  Each group will have basically the same problem, but one group has access to computers, one has access to calculators only, one has no computer/calculator, and one has limited information from the problem. 
Question: How many songs does a radio station play per week?  Groups 1-3 will be told it is a classical station.
After 5 mins, have groups share their answer and how they got it.  After they've all shared talk about error and which group they think was closest in their estimation.

Post-Test: Assign a few problems for homework.

Summary/Conclusion: Scientists often look for Fermi estimates of the answer to a problem before turning to more sophisticated methods to calculate a precise answer. This provides a useful check on the results: where the complexity of a precise calculation might obscure a large error, the simplicity of Fermi calculations makes them far less susceptible to such mistakes. (Performing the Fermi calculation first is preferable because the intermediate estimates might otherwise be biased by knowledge of the calculated answer.)

Mathematical Citizenship

Mathematical citizenship is about creating citizens who are not just adequately educated in the subject of math, but who also show a fundamental understanding of problem solving, analysis and the power of numbers in everyday life.  In teaching students I hope to encourage a sincere dialogue in the classroom where they are not just trying to get the right answer as quickly as possible.  Mostly I want students to question what they are learning, be it from their peers or from me.  "Why are we learning this?" seems to be the central focus of most of the students, but it would be great if students would instead take observations from outside the classroom and ask questions about those observations in terms of math.  One of the most pervasive examples of math in everyday life is in statistics given in the news.  Questioning the validity of opinion polls and statistics is an important skill that will be of immeasurable value later in life.  In the current year, the question of weather or not H1N1 is a real danger we should be worried about can be partially answered in terms of statistics if a student is confident/adept enough in mathematics.  Teaching students to think critically is something that will hopefully be encouraged in all areas of study, but in mathematics we really have an opportunity to encourage problem-solving skills along with the subject matter.  Having students bring in articles or anything they find in their outside life that relates to math, and sharing what they bring in with the rest of the class, might drive home the idea that math is not just contained within the classroom.  Occasionally taking a short break from pure mathematical discussion to talk about different topics that are related to math might also help to develop mathematical citizens, and possibly even develop a bit more interest in the subject matter.

What-If-Not Technique: Thoughts

In looking at the list of questions I compiled using the WIN technique, it becomes clear that one of the most pressing issues in estimation (in my opinion) is what differentiates an estimate from a wrong answer?  How can we know if our estimate is at least in the right ballpark?  Before doing this “what-if-not” I had not even thought of addressing these issues in our lesson, but they would probably be of value to discuss.  Defining what something is not (i.e. An estimation is not just a wrong answer) can illuminate what it truly is.  In this case an estimation has the chance of being correct, is designed to be as close to the real answer as possible, should be easier to compute than the actual answer, and is known to be incorrect.  In contrast, a wrong answer is thought to be correct and (presumably) takes just as long to compute as the right answer would.  Covering these points in the lesson will make sure students know to make it clear that their estimations are not actual answers, and show them that a balance between accuracy and ease of calculation should exist in estimation.

Writing out “what-if-not”’s seems useful, although time consuming, and certainly seems to facilitate thought that would otherwise be ignored.  It can lead to interesting thoughts and discussions and even possible questions for exams that will get students thinking on a more in-depth level about the topic at hand.  One of the best parts that I saw in doing this WIN strategy is that connections between other topics seem to appear out of nowhere and suddenly a topic students may have learned about and left alone for a few months can be reintroduced in a new context.  The weaknesses of this are mostly time related, both in the time I would need to spend creating these questions and the time it would take in class to go over the interesting connections I made in my line of questioning.  It would definitely deepen their understanding of a topic they already have a firm grasp of, but for students who are still struggling to understand the basic topic this addition of more questions may only add confusion to the already fuzzy concept.

What-If-Not Technique: In Use

Our microteaching lesson is going to be covering the topic of estimation and give an introduction to error in estimation.  Some assumptions about estimation are:

1) The estimation is not an exactly correct value

2) The estimations was easier to calculate than the correct answer

3) The estimation is relatively close to the actual value

4) Small individual errors can add up to larger errors when everything is added together

5) Estimation involves using incorrect numbers in place of correct values which are hard to use or unavailable

~1: The estimation is the exact answer.

-If one estimates the correct answer, is it still an estimation or was the method a calculation?

-Can you ever know your estimation is correct without checking it explicitly?

~2: The estimation was harder to calculate than the correct answer.

-Will this only be true in trivial questions?

-Would there be a point of estimation if it is harder and less accurate?

~3: The estimation is nowhere near the actual value.

-How close does an estimation have to be to be of use to us?

-How will we know if our estimation is way off?

-Are there any checks we can use that don’t involve actually calculating the value?

~4: Small individual errors don’t give any error when everything is added up.

-Are there cases where the answer has no error but the individual steps had error?  Is there a pattern to these cases?

-Should small steps be estimated in alternatively higher/lower fashion in hopes that errors “cancel” out?  Could this strategy backfire?

~5: Estimation involves using incorrect numbers in place of correct values, even if they are available and easy to use.

-Is estimation of value even if the real answer is easy to calculate?

The Art of Problem Posing - 10 Questions

How much time should be devoted to trying to actually answer the questions posed?

Is there a distinction between posing problems that test knowledge and posing problems to gain knowledge?  Should there be?

If a student asks an insightful question that can only be solved using mathematics well beyond his/her background, should the teacher answer the question without justifying it?

Does problem posing by students tend to lend itself to a less linear style of teaching?

How can classes be kept focused when asking such broad and unrelated questions?

Do students benefit from brainstorming questions with peers or is a more solitary approach generally best?

Do more academically talented students tend to ask questions that are more or less insightful than their peers?  Do they think of problem posing as a useless exercise when they already understand the material?

Does it get overwhelming for students to have more questions than answers?

Have any in-depth studies been done on the effectiveness of students using these problem-posing techniques to increase mathematical understanding?

How can we assess how well a student problem-poses? Should we assess it at all?

In the year 2000...

A Twitter Account of a Student who Hated Me

Geoff95: Thanks for wasting my time with math and trying to convince me it's worthwhile.  #mathsux
Geoff95: How come I need to know how to calculate the optimal plot of farming land in the year 2019?  What is this, Apocalypse Now?
Geoff95: 50 questions on how to calculate logarithms really helped me out when I was trying to figure out what I should do with my life.
Geoff95: Miss Dicker is so boring to listen to, and her class had nothing to do with my life.  #IHateSchool
Geoff95: Miss Dicker only likes the smart kids in her class and ignores everyone else.  Fml. #IHateSchool
Geoff95: @MissDicker You think you're funny in class but you're not.  You try too hard.
Geoff95: lolz ya. RT@Rogerrr Math is for nerds neways. Eff it.
Geoff95: Math sux. Whatever.  I'm over this. #mathsux

An Email from a Student who Loved Me

Hey Miss Dicker,

I just wanted to email you to let you know I finished my first year at university and I'm thinking of majoring in math!  I know I hated it when I was in your class but I'm starting to really like it.  I just wanted to write to thank you for being so into math when probably most of your classes were filled with people who didn't like it (like me!).  You really showed me that math could be fun and doesn't have to be scary.  If it wasn't for that I probably wouldn't have even taken a first-year calculus course.  Anyway, just wanted to thank you for your course and the time you put into teaching math to me!

Sincerely,
Prucilla


I hope that I will be that teacher that students like and appreciate, even if they don't love the subject I'm teaching.  I'm afraid I will be a teacher that tries too hard and that students can't relate to.  I'm also afraid I'll get too caught up in the math and forget that the students are people too and not just math-machines for me to teach.

Math Video on Constructionist Teaching

The classroom depicted in this video is innovative and certainly a lot different than any math class I have ever taken.  While there is a place for traditional mathematics teaching, having lessons where students are involved and the students can see math in a new way is an integral part of being an effective teacher.  One thing that particularly stuck out to me was the teacher's introduction to algebra within the context of a class that had nothing to do with algebra.  In fact, the students probably won't see algebra for another year or so.  Introducing students to concepts like variables early in their life makes the transition into using the variables a lot easier.  When I was in school when a new concept was brought up out of nowhere it would always be scary and hard to understand; we were caught off guard.  However, when I had previously seen something used - even if I didn't fully understand the concept yet - I felt that twinge of recognition and it gave me the confidence to think I could probably learn this concept pretty easily.  Additionally, seeing the concept in use before learning exactly how to use it gives motivation for learning it.  Knowing that what you are learning has come up earlier gives the impression that this concept is probably important and will be used again, and this is great motivation for students to actually want to learn something.

Interview Summary

What we Asked:

5 Questions for Students:

1. What do you like about other subjects that you don't find in math?
2. Do you think the math you learn will help you in your life and how so?
3. Do you think the grades you get in math reflect your knowledge or understanding?
4. Give reasons math should or shouldn't be learned if a student "knows" they won't use it in their future?
5. Which classes that you are taking currently do you think will be most useful in your life?
   
   
   
   

Battleground Schools

Summary:
Mathematics education is thought of in two different, and often conflicting, ways.  The first is a conservative viewpoint where math is thought to be unchanging, objective and inaccessible to most.  The second is a more progressive viewpoint where math is thought to be dynamic, exploratory, and of importance to everyone.  These two differing opinions generally correlate with the two political poles, and for years there has been unrest between the two regarding how, and why, we educate our children in mathematics.

Interview After-Thoughts

After interviewing three high school students and three high school math teachers I have learned a lot about the practicality of teaching mathematics.  In school we tend to get caught up in the theory of teaching, but this assignment really anchored everything back into reality.  The following is a reflection on a few of the ideas that jumped out at me most from the interviews and how those ideas relate to my idea of how I will teach.

Interview - Notes

These are my own (unedited, unrevised) notes I made on the tapes of the interview with three high school math teachers.  I thought it might give me some context for the reflection I wrote to go along with this assignment.  All three of the teachers' thoughts are jumbled together but I think knowing who said what isn't important for this interview.  The ideas themselves were important - not who said them.

Using Research to Analyze, Inform, and Assess Changes in Instruction

While reading Robinson's article Using Research to Analyze, Inform, and Assess Change in Instruction, I thought about my own impending teaching career and what this article means to me.  The following are two quotes that I have responded to from my own "new teacher" point of view.

“Small peer groups often provide students with an acceptance and a sense of cohesiveness that is invaluable.”

I sometimes overlook the value of peer groups and learning from each other as well as from the teacher.  It is tempting to think that students will automatically look to each other for help if they need it, but I agree with Robinson that it is something that needs to be strived for and actively nurtured in the classroom.  Students need that sense of community in a classroom and in our Principles of Teaching lectures it was mentioned that one of the signs of a “great” teacher is the ability to instill “a sense of shared project” among students.  That is exactly what Robinson seemed to be doing and the change worked.  Students did better both in class and on exams, both strong indicators of her success.