Common Core

 An interesting article about the Common Core from my West Michigan news WZZM 13 by Sarah Sell and Bob Brenzing.

 Click here for the article

I'm not sure how much of the opposition I agree with.  When the article paraphrases Melanie Kurdys, it says, "They say they no longer have input in their child's education and can't even help them with homework."  The Common Core really does not change a parents input in their child's education, and parents may not be able to help with homework because they don't have enough mathematical understanding to help their children.  Asking your kids to stop learning just because you can't help them, seems selfish to me.  If you're saying, "My student struggles and I can't help them," then my suggestion would be to talk to your student's teacher.  Many schools have after school programs to help students with homework.  We also live in a world full of technology and the internet can be a valuable resource for both you and your student.  Wolfram Alpha has helped me personally, with many homework assignments.

Please read the article and tell me what you think.  If you have other politically neutral articles about the Common Core, I would love to take a look at them.  I would like some good resources to give parents a clear understanding of their child's education.
 

Proof of Euclid's Number Proposistion 26

Proof of Euclid's Number Proposition 26

 

I loved writing this proof because I am able to clearly understand a theory of Euclid's.  Many famous mathematics come up with such complex theorems that as an undergraduate mathematics student I can barely understand the theorems, much less the proofs of those theorems.  I appreciate that this proposition is simple and the proof is simple, but the implications are vast. 

In the future I would look into proving more complicated number theory propositions by Euclid.  This type of mathematics appeals to me and number theory would be an interesting topic for me to explore more.

A Review of Mathematician’s Lament by Paul Lockhart

    Paul Lockhart makes many interesting points in his book Mathematician’s Lament.  He argues that mathematics education in schools today is far from where it should be.  The author thinks that mathematics is an art, cheapened and stripped down to formulas and definitions.  Although I think reform is needed in mathematics classrooms today, Lockhart’s idea of reform is not practical.
    Lockhart’s main theme of the book is that mathematics is an art form and should be treated as such.  Just as art is taught with a blank canvas, students should be given a blank canvas to explore mathematics as they please.  Although I agree that mathematics is so much more than procedure and “plug and chug” formulas, can anyone really expect a middle schooler to work on mathematics for an hour every day given no guidance or instruction?  There has to be some structure in a mathematics classroom especially for younger students. 
    The author makes it clear that we stress notation too much in our classrooms.  What I don’t understand is how he expects students to create and explore mathematics with no base to start from.  You have to crawl before you can walk and you have to be able to add numbers before you can create mathematics.
    In “Mathematician’s Lament” the author states that there should be no mathematics curriculum, no standardized test, and no regulation of mathematics teachers cross country because teachers are being too bogged down by scores to actually teach mathematics.  There are several alarming implications to this.  First, if there were no regulations then teacher performance would probably decline because there is no way to keep them accountable.  Second, there is a sad truth that many teachers do not have enough mathematical understanding to guide students through any and every mathematics the students will explore.  Teachers teach procedures mostly because they were taught procedures.  There are ways to encourage teachers to branch out, but taking away all standards and expectations is not one of them. Finally, students who can achieve high level mathematics will never be taught the mathematical background needed to succeed.
    Lockhart writes that mathematics is a fun art form that has only a few real life applications.  This point breaks my heart, because I believe mathematics is fun, but I also believe that mathematics is important.  He makes it seem like the only purpose of mathematics fun, making it essentially unimportant to teach in schools.  In his argument to reform, he goes so far as to make mathematics a trivial subject.  This seems far from accurate.  Where would our world be without mathematics? Computers, medicine, science, technology, business and more wouldn’t even exist without mathematics.  So yes, Lockhart, mathematics is fun, but it is also vital to the world we live in.

Unit Plan: Honors Senior Project

7th Grade Mathematics Unit Plan Outline: 

Number Lines and Negative Integers

 

Common Core State Standards: 7.NS.1

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
1. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
2. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
3. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
4. Apply properties of operations as strategies to add and subtract rational numbers.

I Can Statements:

1. I can name real world examples of two numbers that combine to make zero.
2. I can tell you the distance between two rational numbers p and q on a number line.  
3. I can associate directions on the number line with subtraction and addition.
4. I can use integers and rational numbers in all operations.
5. I can tell you why adding a negative number is the same as subtracting a positive.
6. I can define the vocabulary words integer and rational number.
7. I can tell you how far away a is from b and where a+b is on a number line.

Activities 

Score Keepers

   a. Standard alignment: NS.1.a, NS.1.c, NS.1.d

   b. Use a jeopardy game with negative and positive score values to introduce adding and subtracting negative numbers  This activity is meant for the whole class to do together.  The teacher will guide the score keepers in keeping score.  Point out times where the score is zero because negative points balanced with positive.  

   c. 1-2 Class periods

   d. Materials: Projector

Lesson Plan

Chip Board

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Use red and black checkers chips to represent positive and negative numbers.  Have students practice subtracting and adding negative numbers using their checkers.

   c. 2-3 Class periods

   d. Materials: Black and red checkers, student page

Lesson Plan

Student Page

Debt and Negative Numbers

   a. Standard alignment: NS.1.a, NS.1.c

   b. Use debt as an example of real world negative numbers.  Use national debt as an example.  Ask question like, if x amount is the US debt, how much would it take to break even?  Use a number line to describe debt.  Also have students explore adding and subtracting debt.   

   c. 2-3 Class periods

   d. Materials: Debt Power point, at home assignment

Lesson Plan

Slide Presentation

Student Page

Number Line Introduction: Thermometers

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Use an activity with traditional thermometers to introduce number lines.  This will give students a real world examples of number lines.

   c. 1-2 Class periods

   d. Materials: Student page

Lesson Plan

Student Page

Performance task: Create your own Timeline

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Students will work on creating their own timeline.  They will use personal life events and write about the mathematics involved

   c. 2-3 Class periods

   d. Materials: Student pages, rubric, student peer review, markers/colored pencils, glue/tape, poster board/graph paper/11x17 paper.  

Lesson Plan Day 1

Student Page Day 1
Lesson Plan Day 2
Student Page 2

Who Dun It?

   a. Standard alignment: NS.1.a, NS.1.b, NS.1.c, NS.1.d

   b. Set up 4 stations, one for each role.  Students will work on solving the mystery, the undoing of -7.  This activity will help review the unit while letting the students have a fun mystery activity.

   c. 4 class periods

   d. Materials: Caution tape, graph paper, poster paper, coloring utensils, rulers, student pages

Lesson Plan 

Rainy Day Game

(Thanks to Dr. John Golden)

MARCH MADNESS - to Teach Mathematics

My knowledge of basketball goes about as far as my one year on the 6th grade basketball team.  But, what I lack in basketball knowledge I hope I can make up with my knowledge of mathematics.  When I filled out my bracket this year (only to please my overly competitive family) I wanted to use mathematics to help me win the coveted prize, a gift card to my dad's favorite restaurant (you can guess who picked out the prize).  I have two purposes to this post.  First, I want to tell you how I used mathematics to fill out my bracket.   Second, I will give some ideas on how to use march madness in your classroom.

Math and my bracket:
The first thing to know is that there are MANY options for the march madness and the chances of 
guessing a totally perfect bracket is pretty much ZERO.  

For the first round there are 2^32 ways of guessing, that would be
4,292,967,296.  This is because there are 32 games and 2 possible outcomes for each game.  Yup, 4 billion ways for the first round to turn out.  The second is less, but still 2^16 is 65,536.  Continuing on, there are 2^8 for the third round, 2^4 for the fourth round, 2^2 for the fifth round, and finally 2 ways for the final round to turn out.

We multiply these numbers together to get the total number of ways the tournament could turn out and we get....a number too big to fit on my calculator.

9.22 x 10^18

So based on WikiAnswers I found the length from the earth to the moon at it's farthest point from the earth.  From this, I discovered that 9.22 x 10^18 paperclips would make it to the moon and back 363,755,010 times. 

That's why my bracket already isn't perfect, as you can tell below.

Asking your students what the probability of having the perfect bracket is could be a great way to get the real world into your classroom.  

You can also ask them to find out how I came up 363,755,010 times to the moon and back.  This is a great way to look at and explore conversion lengths.

To fill out my bracket I decided to use statistics from the past to try to predict the future.  I found statistics on this website:
http://statistics.about.com/od/Applications/a/March-Madness-Statistics.htm

There is more than one way to apply these statistics.  You could just go by percentages and have the team win who has the higher percentage.  I decided where there was a 70% winning rate for one seed against another to pick about 3/4 of these teams to win and 1/4 of them to loose.  This wouldn't be the statistically perfect way to pick, but I wanted to do something a little different. 

Next, I followed these statistics from the same website as above,

• "Only once (4% of the time) has all four #1 seeds have made it to the Final Four.
• Three times (12% of the time) no #1 seeds have made it to the Final Four.
• 14 times (52% of the time) a #1 seed has won the entire tournament.
• The lowest seed to win the tournament is a #8 seed.
• The lowest seed to make it to the Final Four is a #11 seed."

I checked my bracket to see if it followed these statistics too.  Essentially this means that I made all the number 9 seeds win the first round.
http://www.washingtonpost.com/wp-srv/special/sports/ncaa-march-madness-bracket-guide/
 

Finally, I used these to help pick my final four and championship game.  I used the winning percentages to finalize who would win in the final four.
http://fivethirtyeight.com/interactives/march-madness-predictions/

There are no perfect statistics to use and very few perfect brackets, but you can be a much more informed competitor in your office if you know how to use mathematics.

Good luck and have fun!
 

 

Sensible Mathematics: Second Edition by Steven Leinward

Recently I read this book for my mathematics capstone course.  My summary and review of the book is below.  

Sensible mathematics is a book focused on empowering leaders to push for better mathematics school programs.  The book is written towards school leaders, but gives an interesting perspective for a future teacher like me.  Leinward explains that the common core state standards are the first step in creating a better math program.  It takes leadership and teacher support to implement the common core effectively.  He gives many reasons why change is important.  One example is the way society is changing.  This change demands a different mathematics classroom.  Students must be prepared for the workforce where calculators and excel spreadsheets are readily available.

    According to the author, one person can make a lot of change in a school’s mathematics program.  Schools must provide support for their teachers, but teachers must also provide support to each other.  Just one of these teachers has the power to influence a mathematics program.  As a teacher, helping my future colleges and challenging them to try new thing in the classroom is very important.  There are obstacles according to the author, like the fear of failing.  It is important though to encourage new methods and ideas in the classroom.  As a teacher, if I lead other teachers to try new strategies and share new strategies with my colleges, I am being a helpful leader, according to Leinward.    

    I also learned a lot about the shift in mathematics education from this book.  I would recommend it to leaders and administrators more than I would teachers, but it does provide great arguments for change.  The most interesting part of the book, for me, was looking at examples of lessons that promote sensible mathematics.  He shared one teacher’s story of a classroom exploring the speed at which toy cars go. The students discovered that under the conditions they were testing, the car speed was much slower than the advertised speed.  They used proportions and experiments to come up with data and sent it to the company that made the cars.  The company suggested they look at different situations to test in.  I can only imagine the excitement the students had when they heard back from the company, and to do more math testing.  This is the type of mathematics that I want to teach, mathematics that is real world, fun, and as the title says, sensible.   

Timeline of Mathematics

Below is a link to my prezi I created on the history of mathematics.  It provides a timeline and brief explanation of the topics we have covered so far in my MTH 495 class.

What fascinated me most about creating this timeline is there is a huge gap in time were mathematics wasn't making any strides.  I'm not a historian, but there has to be some reason for the 600 year gap.   Also what is interesting to me is how fast mathematics has moved since 600 AD.  It makes me excited to see the new things coming in the mathematics world!


History of Mathematics