'Twas the Night Before Exams

‘Twas the night before Exams and all through my room,

I sat quietly awaiting my ultimate doom.

The pencils were placed in the backpack with care,

In hope that at the paper I just wouldn't stare.

The students were nervous all stiff in their chairs

While visions of failure hung in the air.

And the teachers with their Scantrons and I with my note cards,

Prepared for a day that was sure to be hard.

When books began to fall creating a clatter,

My teacher looked up asking what was the matter?

Frustration had built inside of my body,

I responded to her trying not to be snobby.

Then what with my wandering eyes should I see?

My clock blinking bright, twelve thirty-three.

A full night’s rest, surely not ahead,

I crawl to the couch, resting my head.

Hours passed by filled with cartoons and news,

But this was not curing my exam day blues.

Go math! Go english! Now physics! Spanish two!

Everything chemistry and anatomy too!

All Gen Ed classes, no matter how small,

Go away! Go away! Go away all!

All folders filled with useless review

All protractors, pens, and everything due.

As I began to drift off, finally calming down,

Thinking of the future made me frown.

I prayed I would not sleep through my alarm,

Surely a snow day would cause no harm.

My eyes-how dull! My hair balled in a mess!

As morning drew near I tried hard not to obsess.

I tumbled out of my cave, my stomach filled with fear.

I am not ready for this oh my, oh dear.

And grabbing my backpack with nerves in my belly,

I embarked on my journey, my legs just like jelly.

All of this panic filling my head,

Soon developing a new feeling of dread.

Last minute cramming would do nothing to alter,

Should have studied but knew I would falter.

With a switch of the light and a quick little sigh,

I trudged to school, kissing my good grades good-bye.

The teachers passed out the intimidating tests,

All I could think is, I should have gotten some rest.

Then I yelled to my peers, with a loud hearty cry,

Happy exams to all, kiss your GPA good-bye! 

What is a Radian?

Last week I spent some time reviewing what the CCSS say about trigonometric functions. As I was reading I came across a standard that made me pause.

CCSS.MATH.CONTENT.HSF.TFA.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

So I'll be honest, radians are still a bit mysterious to me. So I did some digging around in Sameer Shah's Virtual Filing Cabinet, and I came across Kate Nowak's blog post encouraging teachers to ask their students "what is one radian?" I then decided that I would write a post doing by best job to explain what is one radian.

Roger Cotes is credited for the concept of radian measure in 1714, however the term radian did not come about until 1873. The radian is considered the standard unit of angle measure. Radian measure depends on arc measure on the unit circle.

The measure, in radians, of an angle subtended by a circle arc is described as the length of the arc on the unit circle, divided by the radius. Wait...what is an "angle subtended by a circle arc"?

So this means that one radian is the angle subtended by an arc that is equal to the radius of the circle. So in other words an angle with measure 1 radian has an arc length of 1 on the unit circle.

An angle with radian measure pi (which we know is 180 degrees) has an arc length of 3.14159...Which means there are pi radians in 180 degrees.

So again, one radian is the angle subtended by an arc that is equal to the radius of of the unit circle. 

Another observation that comes from the use of radians is that the circumference of a circle is 2pi times the radius of the circle. The circumference can be thought of as the length of the arc of the entire circle, or as my high school math teacher taught me, the distance it takes to walk all the way around a circle.  So this would mean that the circumference of the unit circle is equal to the radian measure of a 360 degree angle ,  Well we know that the radian measure of a 360 degree angle is 2pi, so that means that the circumference of the unit circle is also 2pi. When we say that the circumference of a circle is 2pi*r, we are really just counting the number of "radii" that make up the exterior of the circle. 

In most mathematics angles are generally measured in radians because radians are more "natural", there are no degrees or meters attached to this measure so this allows radians to be unitless. It is very important that students understand the concept of radian measure, not just because the Common Core State Standards state its important but because using radians makes math easier! Trigonometric functions are defined by ratios in the same way radians are defined by ratios, so it only makes sense to use radians while working with trigonometric functions. However, it is important that students have a deep understanding of radians before they begin to work with them, it is important for students to know exactly what a radian is. When students have conceptual understanding of ideas like this it will only help their understanding of other mathematical concepts. 

Enrico Fermi and Walt Disney

So for those of you who don't know already, I love Disney. Those closest to me might even call me obsessed. I even have a Mickey Mouse tattoo and a tattoo with the coordinates to Cinderella's Castle in Magic Kingdom. Ok, so I will admit it....I am obsessed. Last weekend I took my 17th trip to Walt Disney World in Orlando to celebrate Halloween and enjoy Disney's Food and Wine Festival. While I was in Disney I of course discovered new rides and new shows that I had never even seen before, each time I go down I am amazed at how much of WDW I still have yet to experience. So this got me thinking, how many days would it take to do everything in WDW?

So in case you are not very familiar with Disney World, here are some facts for you:

• On Disney property there are 4 amusement parks (Magic Kingdom, Animal Kingdom, Hollywood Studios, and Epcot), two water parks, a shopping district (Downtown Disney), and 34 different resorts, all which total 27,258 acres (42.59 square miles). 
• 50,125,000 people visited WDW in 2013
• In the 4 main amusement parks there are 71 different rides and shows

So now that I have decided to tackle my Disney Fermi problem, my next step was figuring out what "doing everything" means to me. I decided that this would mean riding every ride and watching every show (a show being some sort of entertainment that I would watch that was not a moving ride) in all four of the main parks. I decided to exclude watching parades and fireworks because those vary depending on the day and also the time of the year. I also excluded anytime spending shopping. So next I figured out what information I would need to solve this problem, and this is what I decided was important to know:

• Average wait time for each attraction (I used a 10 minute wait for each show)
• Ride length for every ride and the run time for each show
• Average time spent walking from ride to ride
• Time spent getting to the park (I used the Disney bus system because it is provided for free for people who stay at a Disney resort)
• Time getting into the park and leaving the park
• Time spent eating
• Park hours for each of the four parks

Note: I decided that for all of my approximations I would round up my values to account for bathroom breaks, longer than average wait times,  crowds etc. 

So most all of this information I was able to find online. I even found a website that lists the exact times for each ride and show in WDW. Another cool website I found complied average wait times for every ride depending on the crowd. Information I was not able to find online I just estimated using my experience of my many trips to the parks. So once I collected all of this information I compiled it into one huge spreadsheet and then began to add everything up. 

Except it wasn't that easy. I decided that I would figure out how many days it would take to complete each park first and then add all of those days together. Some parks took longer than one day to complete so on those extra days I had to add extra time for eating and travel time. This was the formula I used to calculate the number of days each park would take.

# of Days = [Travel Time + Time Spent Entering Park + Meal Time + Total Average Wait Time + Total Ride Time + Time Spent Walking] / [Park Hours]

Then if the number of days was greater than one I added more travel time, time entering the park, and meal time accordingly. 

After all of my calculations I came up with these values for the number of days it would take to do everything in each park:

• Magic Kingdom: 2.462 days
• Hollywood Studios: 1.26 days
• Epcot: 1.65 days
• Animal Kingdom: 1.26 days

 This means that it would take roughly 6.63 days to ride every ride and watch every show that WDW offers. Based on my experience I figured that my answer would be close to a week once I figured in the park hours, travel time, etc. Then I began to wonder how much it would cost to take this trip to Disney to do everything. So then I began another hunt for more information. I decided I would estimate this seven day trip using a family of four (I figured this was a common family size). 

These were the expenses I found:

• 4 round trip plane tickets from Grand Rapids to Orlando: $1,836.80
• 8 night stay at Disney's Caribbean Beach Resort: $1,781.68
• 7 days worth of admission to WDW: $1,358.94
• Cost of food (I used the Disney Dinning Plan estimate): $464.16

So to take your family of four to ride every ride and see every show in WDW it would cost roughly $5,441.58. And this doesn't even include any shopping. So after spending time working on this problem I decided to make myself a Disney checklist, and it is my current goal to finish this checklist. I guess that just means I have an excuse to visit my favorite mouse another time! 

Math Anxiety in the Classroom

My palms are sweaty, I feel nervous, my heart is racing, and I have completely forgetten how to do basic math...it must be time for a counting circle. I tell my self that I learned how to add and subtract numbers in grade school so counting in a circle with my peers should not be a challange for me, yet it always is.

Here is a quick description of the concept behind counting circles. To begin you start with a number on the board, it can be any number (positive, negative, even fractions). Then there is a number that you must count by, again this can be any number. So for example one counting circle we might start at 327 and count by -62, or another counting circle we might start at 3/7 and count by 1/9. The part that makes this a counting circle is that everyone in the class is standing in a circle and we all take turns saying the next number in the count. So the first person to start will say the first number (eg. 327+-62=265) and then the next person will count from there (eg. 265+-62=202) and so on until we have gone around the circle a few times. 

It is all just basic math that I, as a math major, should feel comfortable doing but each time a new counting circle starts I feel the same pit in my stomach, the sweaty palms, the fear of making a mistake. Even though in our classroom there is no pressure to get the correct answer (if someone is incorrect the circle keeps going from whatever number they said right or wrong) I still feel so much stress over these circles. So I decided to do some research on math anxiety. 

Students who have such a lack of confidence in their math ability that it actually affects their academic performance are said to have math anxiety. Most students feel anxiety especially in math class because in math there are so many different concepts that are completely foreign to them. A study by the University of Granada in Spain did research on 855 first-year college students with 23 different degrees that all require mandatory math classes. The found that 60% of those students had some for of math anxiety. Of those students who experienced math anxiety 47% were men and 62% were women.                                                                                                                                                                                                                                                                                    

Math anxiety and being "bad at math" are two different things, math anxiety produces physical and psychological responses in students when they are presented with math problems. Students who have math anxiety can experience nausea, shortness of breath, sweating, heart palpitations, and increased blood pressure when they are presented with math problems. They can also experience psychological symptoms such as memory loss, paralysis of thought, loss of self-control, negative self-talk, math avoidance, and isoslation. Standford University scientists say that the brain function of young adults who have math anxiety is different that those who don't. They conducted functional magnetic reasonce imgaing on the brains of 2nd and 3rd graders as they worked on math problems. They found that students who experienced math anxiety had increased brain activity in areas associated with fear and decreased activity in the part of the brain associated with problem solving. Other studies have also shows that math anxiety disrupts the working memory of students (The working memory of the brain is very important for problem solving). 

After finding all of this research and reading though articles I realized how important the problem of math anxiety is to us as teachers. How can we expect students to be able to problem solve when science shows decreased activity in that part of the brain if students have math anxiety? Math anxiety is a real problem in classrooms and can effect the success of our students. Some ways that we can help as teachers is to make sure that we are confident in our own math ability. If we are having some form of math anxiety our students can feel that and may be more likely to pick up on those anxieties. We can also help by making sure we know which students may experience some form of math anxiety.   There are surveys and questionaires (eg. Fennema-Sherman Mathematics Attitudes Scales) that can asses levels of math anxiety in students. If we are aware that some students have higher anxiety than others we will be better able to help those students (such as finding them a tutor, giving extended time on a test, extra practice problem etc). 

Math anxiety is a real problem and we will most likley be faced with this problem a lot as teachers. It is very important to make sure that we have the tools to help our students. There are a lot of resources out there on math anxiety, being aware of the problem is the first step to a solution. 

References: 

"Math Anxiety." Gifted Child Today (2007). Print.

Thilmany, Jean. "Math Anxiety." Mechanical Engineering (2009). Print.

"Math Anxiety: The Neurodevleopmental Basis of Math Anxiety." Education Week (2012). Print.

Bohrod, Nina, Candance Blazek, and Sasha Verkoutesva. "Math Anxiety." Anoka-Ramsey Community College . Print. 

Quadratic Function Think Aloud

The problem in the following blog post can be found in the April 2014 issuse of Mathematics Teacher.

The first think I see when I start this problem is the the function f(x) has three unknowns, x, b, and c. So right away I know before I can find f(4) I first need to determine what the values of b and c are. I also see that I am given two other pieces of information that might help in figuring out the values of b and c in this situation. The first thing I am going to do is use the fact that f(1)=9 and I am going to substitute 1 in place of x in my function f(x). 

So from this information I have found out that when x=1 then f(x)=1+b+c. I also know that f(1)=9 because I was given that information at the beginning of the problem. 

 Now I see that I still have two unknowns b and c, so I still need to use the other information I have to determine the values of these unknowns. 

I was given at the beginning of the problem that f(3)-f(2)=8. Even though I do not know exactly what f(3) or f(2) equal I do know that their difference is equal to 8. So now in the same way that I found f(1) earlier I am going to find f(2) and f(3). 

So now I know what f(3) and f(2) are equal to in terms of b and c, and therefore I can find f(3)-f(2). 

So now I have found that f(3)-f(2)=b+5. I remember that I was also told that f(3)-f(2)=8. 

I have a value for b, this is progress! Now that I know that b=3, I can use this information and substitute b into my earlier equation, b+c=8. (Remember that I found b+c=8 when I found out that f(1)=9) 

So now I know that b=3 and c=5. Since I know these values I can "complete" my function f(x) by subtituting in these values for b and c. 

From here I can easily figure out the value for f(4) by substituing 4 in for x in my function f(x). 

So I have finally finished my problem and found out that f(4)=33 when f(1)=9, and f(3)-f(2)=8. 

If I were doing this problem in my classroom here are some follow up questions I might ask:

1. Do the values of b and c change if we were given that f(1)=3? What if f(1)=-9?

2. How does the value for f(4) change if we were given that f(2)-f(3)=8?

3. Graph the functions f(x)=x^2+3x+5 and f(x)=x^2-3x-5 on the same graph. What happens to our original graph when b and c are both negative? 

The Meaning Behind the Numbers

It took me a long time before I realized I wanted to become a teacher. I should start out by saying I come from a whole family of teachers, and not just any teachers they are all math teachers. My grandparents are retired college math professors, my mom was also a college math professor and now  is teaching preschool in Ohio, and my dad is a 5th grade math teacher. So the oblivious choice for me to make would be to become a math teacher right? Wrong. Until last year I wanted to do everything but teach math. I wanted to go to med school, I wanted to be a psychiatrist, I wanted to be a biomedical engineer and design artificial organs and most recently I wanted to be an occupational therapist. After five major changes in two years I called my grandma and told her I had no idea what I wanted to major in and more importantly what I wanted to do for the rest of my life. She gave me some of the best advice I have had to this day, she said "stop running from your calling". I knew that she meant my call to be a math teacher, she always said it was "in my blood". So long story short, listen to your grandma because she always knows what's best for you. 

Once I did decided I wanted to be a teacher and found my passion in education I started thinking about what type of teacher I wanted to be and what I wanted to value in my classroom. Since I want to be a math teacher I knew that math absolutely had to be the highest priority in my class. Every teacher wants to change the lives of their students, they wants to make a difference. I knew, and still know, that I want to make a difference in my students lives through the math I teach. I want to make math approachable and realistic to my students. Everyone has heard (or maybe even been the one to say) "when will I use this?" in a math class. My goal as a teacher is to show my students how the math they are learning is important and that there is a reason why we are learning this. I want my students to see through projects and activities there is more to math that just numbers on a lined sheet of paper, or solving for some unknown, that math is about problem solving and reasoning. 

In my classroom I want to use activities like "101 Questions" by Dan Meyer or "Would You Rather Math" to  engage my students. I don't want them to just give me an answer I want them to tell me how they got there and why they believe that is the right answer. I want my students to learn how to problem solve and collaborate. I want them to learn how to explain their thinking, and what to do if they are struggling with a problem. If my students do a problem incorrectly I want them to be able to go back to what they know and try the problem again a different way. Life is so much more complex than simply solving problems from a textbook and therefore my classroom will be more complex than solving problems from a textbook.