#6 Math vs. Stats

Math vs. Stats

This last week in class we talked a lot of statistics and the standards that middle school students have that involve stats. During this time we got into the age old debate of is stats math.

Personally I do not think that stats is math. That may just be because of my terrifying time spent in STA 312. However, I think that if we take a step back and look at the definitions of these words we can see that they really are very different.

Mathematics- the abstract science of number, quantity, and space. Mathematics may be studied in its own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied mathematics)

Statistics-  the practice or science of collecting and analyzing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample.

When we look purely at the definitions we see some similarities as well as some differences. Some similarities are the words science and quantity. The differences are just about everything else. And so when you think about it the only thing math and stats seem to have in common is that they both deal with numbers. So would you say that basketball is the same as football because they both deal with a ball?

While this is something that we often talk about in the math center I figured we couldn't be the only ones and so I found a cool blog post written by a stats grad student about how the world views those with math degrees compared to stats degrees. (Blog) She talks about how she was initially was studying math and then changed to study stats. I think that she makes a lot of good points how how the world differentiates these people. I also think that it is odd how people often say that math and stats are the same thing and yet people said vastly different things when she told them she was studying stats compared to studying math.

However, as much as I know that while those who understand one or the other may say they are vastly different, the rest of the people who do not like or understand math and stats will continue to say they are the same. This being said I know that in most public schools math classes often have a few stats standards that they need to cover as well. And so maybe they aren't so different after all, at least according to the policy makers.

#5 --- the 5e's

In my earth science teaching class we talk about how lessons should involve the 5e's:
engage, explore, explain, elaborate, evaluate. This is something that I think could be helpful when it comes to planning math lessons as well. In fact I think that this is something many math teachers already do...

ENGAGE: this what we call the warm up in math, the only difference in science is that it generally leads to discussion where as in math it is generally a review. This is something I think could be improved. Warm ups are often thought of as a way to get the students brains working again and thinking about math. I think that sparking a discussion could be helpful to not only get students in the mind set to do math but to also talk about it.

EXPLORE: This is something that I think needs to become more apparent in math. Having students discover the information is something that is often done in science but rarely done in math. I think that there should be "labs" in math as there is in science. This would allow the students to move and work on real world problems. This way students might be able to understand that this information can be useful in the real world and make the class more fun.

EXPLAIN: Another topic that is generally over looked in Math. We often tell students about equations and formulas without explaining why they are the way they are. I think students could benefit from knowing why they do things and not just how to do them.

ELABORATE: This step is the one that is often pushed on to the students as homework. This means students generally do a few problems in class and then have some to do at home. The problem I always had with this was that the problems we do in class are simply and generally stick to the pattern of whatever we are learning in class. Then when the students get home that is when things get tricky. This is something that I think teachers need to change a little bit, students often struggle with the work they are supposed to be doing outside of class.

Evaluate: This is obviously something that is frequently done in math and all subjects.

And so I think that it is important to think about using the information you learned in other classes and bringing it to the table when it comes to talking about math. 

Post #4: Programing

A few classes a go we walked through a 20 step program (http://studio.code.org/hoc/1)  that walked us through using computer programing to finish tasks in the context of video games. This was the best experience I have ever had with programing. In my experience programing has always been like a foreign language to me and therefore trying to make anything has always been a struggle. 

Then we went to a website called scratch where we worked on creating a "program." Before we started to create a game that actually involved math we were asked to simply play around and make something. Brittany and I were able to create a cat and mouse game. You start the mouse moving and then control where the cat goes in hope of "catching" the mouse. While this is not a very complex game the program behind it was much harder than you would think. 
Here is a link to what we created --> http://scratch.mit.edu/projects/49234414/

Overall I think that this is a great resource to become more familiar with. I am working on trying to create a game that is more math centered but I am still struggling to become familiar with scratch.

This entire process has reminded me a lot of my experience with LaTex. LaTex is a program that is frequently used in the math world and I absolutely loath it. I started working with it in Math 210 (our first proofs class) and immediately I struggled with it. I would constantly work through the math required for my proofs and then be unable to make them look presentable due to programming issues. Eventually the class ended and I did my best to forget about it. I am now in Math 310 (another proof class) and it is all coming back to haunt me. My lack of LaTex skills are really hurting my ability to write good proofs. 

While I understand that middle schools are not going to be asked to use a program to write a proof I think that it would be very helpful to them to introduce them to some form of computer programing. Our world has become so based on technology that I doubt they will make it through without having to program something or other. Math class is a great place to give them a peak into the world of programing because there is a surprising amount of math involved in programing.

And so if given the chance I would love to have a lesson in which the kids are introduced to programing (however, that means I need to work on my programing in order to be able to help them).

Post #3: Quiz by Exhaustion

This week I went and observed in Trever Kuzee's classroom. After the usual warm up and problems with partners, we did what was called a quiz by exhaustion. Each student was given a quiz with 10 questions. They were all about the simplification of expressions. At anytime the students could come up to me, Mr. Kuzee, or Kevin (the other student there for observation) and we would correct their quiz. We would circle the questions that were right. The goal of the whole day was for everyone to get a 100%.

This was a great was to give the students immediate feedback. Some students went through the whole quiz and then came up to have it corrected. Other students would do one question and then have us check it.

After awhile we would circle individual terms that were right. This was a great way for students to get to see exactly what they knew and didn't know.

I loved this because this was the last day before break so the students were a little antsy. This was a great way to get the kids up and moving around.

Here was the quiz they took:

Write each expression in simplest form.
1. h+6h
No one really had problems with this one.

2. 5c-3d-12c+d
This one gave a lot of kids problems. They would often get -17c+4d
They had trouble with keeping track of the signs. I think that in response to this Mr. Kuzee should really review adding and subtracting integers.
* a few of them were trying to combine c's and d's. To this Mr. Kuzee brought out the analogy that c's are like socks and d's are like apples. And so we cannot combine socks and apples so we can't combine c's and d's. I liked how he did this. (He also referred back to when the kids did algebra tiles but I think putting it into real life things made it easier for them to understand).

 3. -y+9z-16y-25z
This question gave the students similar problems as #2

MEASUREMENT Write an expression in simplest form for the perimeter of each figure.

This was a great way to combine their previous talk about perimeter with today's lesson about simplifying expressions. The main issue that students had with this was combining -3 and 6 to get 9.

Simplify the expression below: Use models if needed.
5. 2(2x-1)+(-3x+7)
6. (-5x+4)+(-9x-2)
7. (11x+2)+(-8x-2)
8. (-9x-10)+(-5x-4)
All of these problems brought up the same issue, of working with integers.
9. Find the sum of (x+3) and (-x-4)
With this question many students did not understand that finding the sum meant adding
10. GEOMETRY Write and simplify an expression to present the perimeter of the triangle shown.

Mr. Kuzee was really surprised by the results that this quiz brought up. Most of the students were not struggling with the idea of combining "like terms" but with working with positive and negative integers. He was frustrated because this is something that he said they did at the beginning of the year. 

So while this "quiz" did not give him the results that he expected, it did show him exactly what he needed to focus on and review with the students.

As a whole I really like this method of assessment, it is low risk because the students are allowed to come up and get their work checked at any point. And it gives them immediate feedback. The students were able to see right away if what they thought was right was in fact right. While I know that students often have the answers to some of the questions in the backs of their books, this method creates better feedback.

The feedback is better in this situation because when the students are given just the answer a few things can happen:

(1) they could just copy the answers from the back and not try to actually do the work at all

(2) they could get the right answer by luck instead of actually having down the work right.

(3) they could get the wrong answer but not know how to change things so they can get the right answer. 

The quiz by exhaustion does a good job of eliminating most of these options.

Also having us check their answer immediately also helps it so the students can have an immediate response. Too often students are given homework or a quiz and by the time they find out if they go the right answer they have forgot what they did in the first place.

Post #2: Fractions

This week we discussed the idea of when first teaching fractions it is possible to not actually discuss fractions. While I understand that fractions have a certain stigma associated with them I also think that they are a necessary evil. However, I also think that it is important to introduce them in a way that the students understand. A great example of this would be to use pizzas. 

This is an example of how you could show the students the different types of way to divide a pizza. I would take circles and cut them up into even pieces: specifically 1 piece, 2 pieces, 3 pieces, 4 pieces, 5 pieces, 6 pieces, 8 pieces, and 10 pieces. I would label what each piece represented in terms of a whole pizza. Then the students could compare the different size pieces, for example they might notice that 2 fourths is the same as 1 half.

The students could then talk about using the pieces to look at physically what it looks like to add and subtract fractions.

There are also games that involve fractions as well as pizza:

http://www.softschools.com/math/fractions/games/
This game has the students placing a designated amount of pizza on different plates. At first it seems to be a useless game because it has students do this like put ⅓ of a pizza on one place and ⅔ of a pizza on the other. However, I liked it when I was given a pizza cut in 6 pieces and I had to put ½ on one plate and ⅓ on another plate. Still, this game is not the best for students in terms of fun, I think that I would use the idea in my classroom instead of the actual game. You could have students make their own "pizzas" and call out a fraction and have the students try and make it with their pizza slices.

http://mrnussbaum.com/tonyfraction/
I liked this game more than the last, it requires students to pick the right size pizza and then top it with the correct amount of the specified toppings. I think that this would be a fun game for the students to play because you are only given a certain amount of time and you have to see how many pizzas you can make correctly. I think that it would be interesting if there were toppings that over lapped on some slices. I would be interested in both how to create a game like that as well as how the students would react to that type of question.

http://www.primarygames.com/fractions/start.htm
This game gives the students a pizza cut into slices in which some of the pieces are missing and asks them how much is left. I think that it is a little simple for students, I would like to see how they would do if the pizzas were not cut. They would have to simply estimate how much they thought was left. This way they get to chose the fraction increments and explain why they chose what they did. Again this is something that a teacher could incorporate into their classroom without the actual game. You could use it as a quick warm up or do a few and discuss how the students answers compare as a class. 

This is an example of what I mean, you would ask the students how much they thought was left of the original full circle.

Then to challenge the students perhaps you could do something like this:

If the circle started out as a full circle, how much of it is left?

I think that a better game that could be developed to work in this situation would be perhaps be something that involves a restaurant and trying to help costumers. The students could deb in charge of either making sure they get the correct amount of pizza or making sure they there is the right amount of toppings on the pizza (that reminds me of the challenging pizza question you had on one of our activities). 

Actual pizzas would really interest the students and therefore might also be able to be used, although that would be expensive.

Blog Post #1: Integers

Over the past couple class periods we have been talking about integers. First we discussed using a timeline to show how positive and negative numbers are related. I really enjoyed this activity because it allowed students to apply negative and positive numbers to real life situations.

This is an example of what a finished timeline might look like. You could have students compare their timelines to the other students as well as to yours. Then students could get with a partner and ask each other questions about their time line. An example of this would be how much older is my sister than my brother. I would have students first answer this using the timeline and then write out the expression using integer. The answer to this question is 10. Which can be calculated by 7- -3. This allows students to think about integers in a real life situation. It is also a way to get students comfortable working with a number line.

Another activity we did involving integers was working with an interactive number line. This means the students create a number line on the floor of the classroom. This allows students to physically use the number line and become comfortable with it. Once they are comfortable with it you can have students walk out questions you ask them or have them make up problems of their own.

Here is a video that talks about using a rap to teach the number line. (https://www.teachingchannel.org/videos/math-teaching-techniques) While I am not a rapper myself this is a great idea. I thought that this teacher really got the students to understand the material and he made it fun as well.

All of these activities give students a way to put integers and using the number line into real world context. Too often students think of the number line as simply a tool, and especially as a tool that they do not need. However, all of these activities have a way to strengthen a students knowledge of the number line.

The timeline activity allows students to think about their lives and time as a whole in terms of numbers. Positive and negative numbers really have strong meaning in this case because positive things are those that have happened since they were born and negative things are those that happened before. This is a good way to start students off thinking about integers in terms of positives and negatives because it makes it about them.

The interactive number line gets the students physically involved. I love the idea of asking students how they would present adding and subtracting positive and negative numbers on the number line because it forces them to think about why they do things and not just how.

The video uses rap to try and get to the students. Math is a subject that many students struggle to understand and when you give them a strategy to help them remember things it can really help. And so students might be more interested in learning a rap then simply learning math rules.

When it comes to talking about positive and negative numbers there are countless different methods and activities that you can use. In the end you have to figure out what works best for your students. The students could be struggling to understand what exactly it means for a number to be negative, and in that case the timeline activity may be helpful. They could learn though physically doing things and so an interactive number line may be the way to go. Or they could have no desire to learn this material and may need something to make it more enjoyable for them, and so then maybe use the rap.

This blogpost helped me to realize that when it comes to teaching you can plan all you want but in the end you need to figure out what works best for your current students.