Puzzler

To all my fellow math lovers, today I was posed with a question by my friend Ms. K. Ms. K is a math major like myself who finds puzzling math problems very intriguing, she came across this problem when her mom called and asked her how it worked. This problem came from a grade 7 text.

So, three guys walk into a hotel (sounds like a joke right? keep reading). The receptionist at the desk charges the men $30 to stay at the hotel for the night; each man pays $10. Later the receptionist realizes that he made a mistake and should have only charged the men $25. The receptionist then gives the bellhop the extra $5 to bring back up to the men. On the way up to the room the bellhop realizes that he doesn't know how to split $5 into three equal parts for the men, he decides to give each man back $1 (for a total of $3) and keep the remaining $2 for himself as a tip. If the men originally paid $10 and got back $1 each then that means they really only paid $9 each. $9 times 3 men is only $27. The bellhop keeps $2 for himself, making a total of $29.

What happens here? Why is there a difference in the numbers? If you add up $25 + $3 + $2 = $30 but( 3 X $9) + $2 = $29. What explanations can you come up with? The only explanation I could come up with was that if you write all the dollar amounts on a number line then there are 31 numbers (including 0) and there are 30 spaces (for the cents) in between them. Therefore I think this problem has something to do with integers, but that's open for debate.

In keeping with our problem solving activities of recent classes I thought that this problem was appropriate for us to think about. If anyone else has any suggestions as to the discrepancy in the numbers feel free to comment on this post, I will gladly take any ideas you have. I hope this puzzling question is as intriguing for you as it was for me.

Reflecting on the text

During our classes we have touched on several sections from the text book. There is one in particular section that I found interesting that I didn't really think about before. This section is called Helping Children Master the Basic Facts.

When I learned how to add I don't really remember the exact methods that I was taught but according to the text, students now have a variety of strategies to use for addition. One-More-Than and Two-More-Than facts give all the numbers of 1+n and 2+n (where n is any number between 0 and 9) twice on an addition chart. Facts with Zero covers all the numbers 1 to 9 twice and 0 once. Doubles are all the numbers added to themselves to cover the numbers 0,2,4,6,8,10,12,14,16, and 18. Near Doubles (also known and doubles-plus-one or doubles-less-one) covers the numbers 1,3,5,7,9,11,13,15,17, and 19 twice. The Make-Ten Facts are the numbers of 8 or 9 plus any given number. Generally you can make a set of 10 and then add on the rest. For example 9 +4. 9 is 1 less then 10 so if you take one from 4 and place it on the 9 then your new equation becomes 10+3 which is easier to add. The remaining six facts do not fall under any of these strategies. Under the commutative property these six facts are 5+3=8, 6+3=9, 6+4=10, 7+3=10, 7+4=11, and 7+5=12. Lucky for us that there are four more strategies that help students cover these facts. Doubles-Plus-Two, Make-Ten Extend, Counting On and Ten-Frame Facts.

The same is true for Subtraction. When I learned to subtract I remember having physical objects such as counters, blocks and even my fingers that you could use for help. Now students can Think-Addition when trying to do subtraction, what plus n will give you x (n+_=x). To me this seems logical and I wish that someone had told me that when I was younger. Another method is Subtraction Facts with Sums to 10, make sets of ten and then add the remaining number.

I think these methods are incredibly useful for younger students. Students who may be having difficulty mastering addition and subtraction using one method may try several others before finding the right one to match their learning style.

Math in The Simpsons

I'm not really sure what to write about for this week so I decided to write about something that I find interesting and that may be interesting to other students in our class and even perhaps interesting to junior high and high school students.

The Simpsons is a popular TV sitcom that is not necessarily directed for young children but is aimed at older children and adults, some may find the content humorous and entertaining while others may think the content isn't fit to be on TV and the common catch phrase from Bart Simpson, "Eat my shorts", is nothing more then teaching young children to be brazen and ignorant. Some others may also think of the cartoon within the show Itchy and Scratchy is nothing more then a form of violence that teaches children that is is ok for a mouse to shoot at a cats head with a cannon, place a cat in a blender and then serve him as a drink, pour spiders on his head that eat his flesh, or hang him by his intestines into a volcano of hot lava.

I do understand where the parents of young children would be concerned about these issues however The Simpsons is the longest running sitcom of all time. It has been running in prime time television for the last 20 years, they must be doing something right. In fact The Simpsons has a large reference to various academic subjects including mathematics. It contains over one hundred references to math that range from arithmetic to geometry to calculus. Many of these references are aimed to poke fun at innumeracy.

Al Jean is the current Executive Producer and Head Writer of The Simpsons and has been involved with The Simpsons since the show began in 1989. He graduated from Harvard University in 1981 with a Bachelor of Mathematics. Ken Keeler was a writer for The Simpsons from 1994-1998. He graduated from Harvard University in 1983 with a Bachelor of Applied Mathematics and later received his Ph.D in Mathematics in 1990 (also from Harvard). J. Stewart Burns graduated from Harvard University in 1992 with his Bachelor of Mathematics, and his Masters Degree from UC Berkeley in 1993, he began working on The Simpsons in 2002 after he got his start on another cartoon show, Futurama.

Other members of The Simpsons staff have a physics degree, Ph.D in inorganic chemistry, and Ph.D in computer science. These multiple areas of academics are represented directly in the show by Professor Frink, the local Springfield Scientist.

examples from the show:

Bart the Genius (1/14/1990)
Bart reads a math problem out loud, and then day dreams about it.
Bart: 7:30 am an express train travelling 60 miles per hour leaves Santa Fe bound for Phoenix,
Train conductor: Ticket please!
Bart: I don't have a ticket!
Train conductor: Come with me, boy.
[ Train conductor drags Bart off, numbers circle around Bart's head]
Train conductor: We've got a stowaway, sir.
Bart: I'll Pay! How much?
[the train engineer is Martin, shoveling numbers into the engine]
Martin: Twice the fare from Tucson from Flagstaff minus two-thirds of the fare from Albuquerque to El Paso! Ha ha ha ha!

Dead Putting Society (11/15/1990)
Lisa: And I'm studying for the math fair, If I win, I'll bring home a brand new protractor.
Homer: Too bad we don't live on a farm.

Dead Putting Society (11/15/1990)
Lisa, armed with a measuring tape, helps Bart play miniature golf.
Lisa: The basis of this game seems to be simple geometry. All you have to do is hit the ball...here.
[The ball is hit, gets bounced around, and goes into the hole.]
Bart: I can't believe it. You've actually found a practical use for geometry!

Springfield (12/16/1993)
[After putting on Henry Kissinger's glasses, found in a men's room toilet]
Homer: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.
Man in stall: That's a right triangle, you idiot!
Homer: D'Oh!

This carries on throughout all 20 years that the show has been produced, and it covers many topics from the basics of mathematics, 2+2=4, in one show to talking about the Pythagorean Theorem and 3 dimensional space. The Simpsons is intended for a more mature audience and should not been shown to younger children without adult supervision. It contains many inside jokes and academic content that is directed at an older audience

(This information was written with the help of Simpsons Math http://www.mathsci.appstate.edu/~sjg/simpsonsmath/)

Problem Solving

In the past few classes we have been looking at problem solving, mainly for the purpose of our own problem solving session in class, but also to teach us how to use it in the classroom. I think problem solving in the primary/elementary years of school is very important.

When you ask students to solve problems in class it can be done in several different ways. The first way is that you can give specific details for a problem and students will have figure out the answer based on the information they are given. Another way is to give students open ended questions that have multiple answers, students are able to look at all different perspectives and possibilities. Some students will have their mind set on one answer and one answer only and not be able to broaden their perspective to look at the problem in a different way.

Getting students to work in pairs or groups is a good way to get students to look at the problem in another way. When students bounce ideas off each other they are likely to come up with their own solutions and solutions that neither student would have came up with on their own.

Grouping students is also a good way to get them to understand a new concept, I remember in school my teacher used to explain a new topic and give us a few example problems and then group us into pairs or groups of three so we could discuss the topic and explain it to each other, this method gave us a new view if we didn't understand it or explaining it to our group helped us remember it ourselves. Then we could work on problems together to make sure everyone got the concept.

Observation Day Math

Each Friday of this semester the students of Memorial University's Primary/Elementary Consecutive Program participate in observation days at various local schools. So far I have completed five of these observation days and have seen math being taught during four of them. I am shocked when classmates come back to class and tell us that they are yet to see any regular math happening in classrooms.

As a math major I am very pleased to see as much math happening in schools as possible, I get quite excited when I able to get involved in students' learning experiences. During the first observation day I was able to observe a class of grade 2 students who were learning about patterns. The teacher first explained to the students what the elements of a pattern were and then what the core of the pattern was. The teacher then modeled several patterns using connector blocks and asked the students to tell what the elements were and what the core was. Once the students had seen many examples of patterns the teacher then placed them into groups and gave each group different manipulatives to make their own patterns. Students were given connector blocks, colored wooden blocks, linking chains, etc... Once they had mastered the process of making patterns the teacher then made a request of what type of pattern she wanted each group to make. For example a pattern that contained 3 elements and had a core of size 4 repeated 5 times could be in general ABBCABBCABBCABBCABBC.

The second observation day I was in a grade 1 classroom where the teacher let me get involved in her math lesson. I stood in front of the class and held up paper plates with colored stickers on them, the plates had 1-10 stickers on them. Some plates had all one color and some plates and two different colored stickers on them. Each student had their own cards with the numbers 1-10 on them and had to count the stickers and then hold up the number that represented how many stickers were on the plate. This was nice because it familiarized students with the numbers 1-10 and also started them on addition. For example a plate with 6 blue dots and 2 red dots had a total of 8 dots.

The third class I observed was a special education class, this class I did not see any math since Fridays are "games day" for the students. These students are only in this class for one or two periods a day and usually need more help with language arts activities. However I did get to see some very interesting games that were educational for the students, there was a game that focused on phonics, a game that focused on sentence formation, picture bingo, and of course one of my favorites, Scattergories.

The fourth week of observation I was in a class of grade 6 students. This was a great experience for me. First thing in the morning the teacher gave me the guide to the math book and asked if I would correct a few questions that the students were supposed to have done previously and then teach the next section. I was thrilled about this :) He gave me sufficient time to get my thoughts together and understand what I was supposed to be teaching them, then while the students were in music class I was able to test out the SMART board for the very first time and see how it worked. Once the students returned I started correcting the questions and then taught them how to compare very large numbers (in the millions). As I was conducting the lesson the teacher wrote down comments about my lesson, and during lunch he gave me a sheet full of wonderful feedback. I thought this was a great idea, it let me know what I was doing right, what I should keep doing, and what I need to improve on.

Last Friday was the fifth week of observation and I was in a grade 3 classroom, this was a bit of a difficult day for me. The regular teacher was out sick and I was observing a substitute, this was nice because I got to see what it was like to be a substitute walking into a classroom and having to pick up where someone else left off. However the students do tend to give substitutes a bit of a harder time, they push the limits to see what they can do without getting in trouble. The math lesson was supposed to be on estimating and rounding off numbers to the nearest 10, but I don't think it was a very successful lesson. This seemed to be a new topic for students and they were very confused as too why certain numbers round down and not up. This was not explained to them very well because the substitute hadn't realized it was totally new to them. I think If I was in his shoes I would have noticed the students were having difficulty and went back and explained how rounding works in a different way so more students would understand better.

Overall I have seen quite a bit of math during my observation days and quite pleased with the results, When I become a teacher I would hope that I would be able to not only conduct a math lesson but bring a little bit of math into the other subjects as well, like in language arts you can read books that have some math in them, or even in the younger grades you can count how many days you have been in school since the start of the year.

What is mathematics?

I was one of those students in school who always ask "why?". Sometimes I didn't get an answer as to why things were they way they were, sometimes I did get the answer and even sometimes I got a delayed answer that may have just had to wait a class or two. My experiences growing up were good and I was really interested in math, so when I got older and went to university I decided to find out "why" for myself. After four years of a math degree I still can't give you an exact definition of what math is because math is complex and many people look at math in different ways.

Throughout elementary school students were expected to learn how to add and subtract, and multiply and divide using a given algorithm to find an answer and then they were tested with pencil and paper on how well they had mastered this procedure. The same is true for junior high and high school. Math wasn't made to be fun, it was taught as one subject that was separated from all other subjects. Students mostly thought that math was memorizing formulas and then solving problems with these formulas to get one particular answer. Chances are if you were a student who solved a problem in a different way then was expected, you may have gotten it marked wrong. But the truth is that math is not a subject that should be separated from everything else and it's not just having a formula and answering a question to produce an answer, in fact there are many branches of math such as geometry, patterns, counting, sequencing, algebra, graphing, etc.. Some of these involve formulas and some don't, and if they do involve formulas then there may be more then one way to solve the problem. Many students would try to memorize what they were doing and not fully understand it, therefore they would get very confused and end up not liking math.

In his talk "What Kind of Thing is a Number?", Reuben Hersh states "Mathematics is neither physical or mental. It's part of culture, it's part of history, it's like law, like religion, like money, like all those very real things which are real only as part of collective human consciousness". I think this is a good way to describe math, it is not internal or external, it is both, it's a concept that people follow, and it's involved in your everyday life whether you realize it or not. He later states "A good math teacher starts with examples. He first asks the question and then gives the answer, instead of giving the answer without mentioning what the question was." As a teacher I hope to give my students the ability to understand what it is they are doing and let them think for themselves, I would like to per mote cooperative learning and let the students discuss among themselves why things are they way they are, and make meaning of the answers they are producing. I will at first give them a question and give them time to think about it before talking about the question as a whole and then showing them the solution.

During class we read the book "Math Curse" by Jon Scieszka and Lane Smith. I highly enjoyed this book, I usually get excited when I see books directed at younger children about math. The main reason for this is because I don't remember reading anything like this when I was young and I think it is a great way to get young students involved and interested in learning math. Math is a universal concept that is involved in everyday life and this is represented in the book when Mrs Fibonacci tells the class "You know, you can think of almost everything as a math problem..." This book has one student in a whirl-wind of thoughts about math and she thinks about EVERYTHING as a math problem.

These are my thoughts and views on what math could be but there is still alot to learn and knowledge to be gained in the coming years of my career.

Classroom Visitors

During this past week Patricia Maxwell from the Department of Education had visited our classroom. She brought with her some very useful advice and creative ideas.

Now that I have seen the manipulatives that she brought with her I do recall more of my own memories from primary and elementary school. However, the first object she showed us I was not familiar with. This was a Geoboard. A Geoboard was a large flat square that had pegs on it that formed each smaller unit on the board. The Geoboard can be used for many different concepts and difficulties in a primary or elementary classroom. It can be used for shapes, counting, graphing, and calculating values such as area and primeter.

Next Patricia showed us some fun games you can play with students to make learning math more fun. The first game was a game that was laid out almost like a bingo card. There were several rows and columns of numbers that you needed to connect to get four in a row. At the bottom of the card were the numbers 0-9. You pick two numbers that multiply to give you a number on the card that you want and then the next player has to use one of the previous numbers and can change the other one to get the number they want. This continues until one player connects four numbers that are consecutively placed on the board. This could be used for adding, subtracting, or multiplying. In my own experience I had not encountered this before, the only fun game we used to play was regular bingo. This was for practice of number recognition.

The next game consisted of cards that had numbers written on them and the tens and ones value of another number. You went around the classroom asking who had a certain number that was written on your card in tens and ones and they must respond with their number and then ask the question again. This is done in a round so all students would get a turn.

The next item she showed us was connector blocks or linking cubes. These bring back many memories for me, we used them frequently in school. They are small, square, colored blocks that snap together. They can be used for making patterns, counting, color identification, graphing, 3D shapes, rotations and transformations.

She then showed us a useful webpage that we can use to help students get a better understanding of difficult concepts or just extra practice. This website is the National Library of Virtual Manipulatives (http://nlvm/usu.edu/). It can also be used as a tool for demonstration if materials are not available in your school.

I found this to be very helpful, it game me a start in thinking creatively about math and how it can be made more fun for students to enjoy instead of the traditional form of work sheets.

As well last week we got the pleasure of getting a crash course on how a SMART Board works. I have never seen a smart board until this semester and I was amazed to see it in action. With the way technology is changing and having such a diversity of students in each class a SMART Board is definitely the way to go. The learning possibilities are endless.

Mathematics Autobiography

To start off this entry I would like to say that my history with math has been quite extensive and it has been a major part of my life for the last 4 years. I have had good experiences and a few bad experiences but once you get over the bad ones it becomes quite useful.

As for the most part I do not remember much of what math looked like in my primary and elementary classrooms. I remember mostly in kindergarten the focus was mainly put on learning geometrical shapes from pictures and 3D objects. Entering the primary years of school I remember having workbooks that had tear out pages in them. These workbooks were used in class as practice and brought home as homework that had to be completed and passed in the next day. The pages of the workbooks contained small pictures of objects that were used for counting, adding and subtracting. Also I remember having an Abacus in both the primary and elementary classrooms. Small colored blocks that snapped together were used in classrooms to help students create patterns. In the elementary years I recall yellow blocks that came as 1 single block, 10 blocks joined together or a big block that contained 100 smaller blocks. These blocks were used to help students with ones, tens, and hundreds digits. I also remember having to bring home a multiplication table and memorize it until I knew it.

My best memory surrounding math with in grade 5 when we only had to know the times tables for single digit numbers and I had figured out a neat trick for multiplying any 2 digit number by 11 before most of my friends had any idea. For example 11x53=583 this is found quickly by taking 53 and adding the digits 5+3=8, then place 8 as the middle number. This worked well as long as the digits did not add up to be more then 9. For digits more then 9 I had another method. For example 11x67=737, this is easily shown with simple addition that looked like this: Take 67 and add a zero (also meaning multiply by 10) to get 670 then you have one 67 left to make 11 so you add the remaining 67 to 670 to get 670+67=737.

This memory greatly affected my love for math because I realized my abilities to manipulate numbers quickly and easily at a young age. Because I was good at math I liked it, and because I liked it I worked hard at it, and of course because I worked hard at it I got better at it.

My abilities in math were probably influenced by my teachers but I do not know how my teachers in primary and elementary school felt about math, if they enjoyed math then I’m sure they included the curriculum whenever possible, and if they didn’t like math then they didn’t seem to mind in my mind because I enjoyed it so much.

Assessment was done in a traditional way for the most part. Tests were given on paper with questions where you had to calculate the answers. They usually included an answer that was right or wrong. However through practice in the classroom the teacher would observe students to make sure everyone had the right idea on what to do and to further help those students who may have been struggling.

As I progressed through junior high and high school my views about math did not change. I had a teacher in grade 9 who explained math very well in a relevant manner and caught my attention to realize that math was more then just numbers. It was also in grade 9 when I really realized that I was good at math. Our school took part in the Pascal Math Competition from Waterloo University in Ontario and I took frisat place in our school.

In high school I was also blessed with a great math teacher that I had for all 3 years. He gave notes and questions but also brought in visual aids when available and let us work in pairs sometimes to figure out questions together. He would bring in models of unusual shapes like cones and tetrahedrons that could be taken apart so we could see how to get surface area; he would bring in the overhead projector and show us how to mark different aspects of a graph by using different colors and taught us how to use a graphing calculator but also kept our math skills fresh by not letting us use a calculator for simple operations. To make class fun we would have mini competitions, for example, when learning to use a graphing calculator to generate a list of random numbers we would see who could do it the fastest or who could get the biggest number, the winner would get a prize. This prize could be something as simple as a mini bag of Halloween or Christmas candy, depended on the occasion.

When I arrived at university I was unsure of what degree I was about to tackle but again my love of math and a great professor got me through math 1000 with high success. I decided to try math 1001 and then realized that it was the subject that I enjoyed the most and put most of my effort into. I didn’t complain about having to do assignments (in fact I usually had them done ahead of time) and I enjoyed explaining how to do problems to my friends. I continued with math in university and completed a Bachelor of Science with a major in Pure Math.

Below are a list of the courses that I have completed at Sir Wilfred Grenfell College and Memorial University. Most of these courses have been enjoyable and very successful, others have been quite difficult but well worth it in the end, and others I feel were so frustrating and confusing that I didn’t learn anything (although I’m sure I really did).

Math 1000 – Calculus I

Math 1001 – Calculus II

Math 2000 – Calculus III

Math 2050 – Linear Algebra I

Math 2051 – Linear Algebra II

Applied Math 2130 – Technical Writing in Math

Pure Math 2320 – Discrete Math

Stats 2510 – Stats for Physical Science

Math 3000 – Real Analysis I

Math 3001 – Real Analysis II

Pure Math 3320 – Abstract Algebra

Pure Math 3330 – Euclidean Geometry

Pure Math 3370 – Intro Number Theory

Pure Math 3340 – Intro Combinatorics

Applied Math 3260 – Ordinary Differential Equations I

Applied Math 3202 – Vector Calculus

Pure Math 3240 – Applied Graph Theory

Pure Math 4340 – Combinatorial Analysis

Pure Math 4341 – Combinatorial Designs

Today I feel that I have a great understanding of some math concepts and still enjoy it just as much or more than I did before. I try to make math a part of everyday life and I feel that I have good logic skills due to my background.