"Why do I have to learn this?"

The all too familiar question that I so often experienced from the students over my last two student teaching experiences... "Ms. P, I'm not trying to be rude, but why do we have to learn this?" Looking back on this experience, I realize that my students asked this question because, as Zull (2002) states, "we will always be motivated to learn things that fit into what we want and to resist those that don't..." (p. 65). If students don't see the material as information that aligns with what they want in their lives, then they most likely have little motivation to learn the information. This all makes sense, but I have never thought of it in terms of cognition before. This ties together so many aspects of, in my opinion, an effective lesson. The need for real-world application and the need to incorporate student interest isn't simply because it appeases the administration, these things are necessary for student motivation. And once the students are motivated to learn, there is no telling what potential the lesson has! In addition, providing opportunities for students to use the content, in my case- math, to find errors in each other's work is necessary. This simple task is a real world application of math... you can use math as a tool to make sure you aren't being taken advantage of at the store, or by the plumber, or by the car salesman. Would you rather pay the $23 you expected, or the $46 you were accidentally charged by the salesman? Without math, calculating this error would be impossible. We participate in these tasks all the time, but we might not think of them as a direct application of math until we stop and reflect.

In my classroom, I hope to spend a lot of time discussing the fact that math is a tool that we can use to understand the world around us. Math isn't just a set of procedures you should memorize to move on to the next course, math is a set of strategies that can help us make brilliant findings and invent amazing technology. Without math, you wouldn't have the cell phone you use so often, or the laptop you play on instead of doing your homework, or the medicine that you take when you get sick. Without math, we would know very little about the world around us. And perhaps educators hesitate to have these discussions, "because what if the students ask a question that I can't answer?" I have definitely thought of this scenario and fear that my students will lose respect for me. They will wonder how I can expect them to learn the material if I can't even answer their questions. I think that it's okay for teachers to not know the answer sometimes. Especially when trying to help the students relate the material to the real world. I also think that an appropriate technique for handling the situation would be to take a little time to help the students find the answer. Use the opportunity as a learning experience for both myself and my students, since "the learning cycle is about life itself" (Zull, p. 48).

Interesting article on why we learn math (other cool resources linked at the bottom of the article)...
Why Do We Learn Math?

Seeing Patterns

This week I covered some basic statistics with my Algebra I students. I love the brief discussion on week 6: day 4 of Dr. Zadina's workbook. She addresses the fact that the brain is designed to see patterns and to see differences in patterns. This is math in a nutshell- the study of hidden patterns all around us. (Cool article here). So at what point does math get so hard for some students? If the brain literally wants to see patterns and give us a red flag when there is a deviation from the pattern, why is math content sometimes so difficult grasp? All of this made me start thinking about ways that I can present the content to my students in a way that allows them to see patterns and develop the relevant processes from the patterns that they are seeing. Keeping in mind that I simply don't have time to allow my students to "reinvent" concepts on their own, I decided to try a few things that would allow them to come up with at least come up with a process on their own. I know that so often we just hand our students a process or a list of steps and show them how to work the problems, and let's face it- it's the easiest way for us to make sure that we taught the material. But this week I was determined to try.

We started the week by quickly reviewing mean, median, mode, and range. The majority of my students remembered these concepts with no problem. So then we moved on to discuss outliers- what are they and why are outliers relevant to a set of data? More importantly, how do they affect the data? I would show the students a set of data and tell them the outlier. It was up to them to tell me why the particular value was an outlier. I thought this was a great way for their brain to recognize a pattern- the outlier is much less or much greater than the rest of the data set. For example, if all of the numbers in the data set are around 30 and one value is 1,000 this is a deviation from the rest of the data. The students quickly picked up on this and they understood that this value was the outlier. They also realized that this value sometimes greatly affected the mean of the data. If all of the values are near 30, then adding that 1,000 to the data would make the "average" appear higher than it really is.

On the day that we covered standard deviation, I made an effort to give my students the chance to determine the process we needed to use to calculate the standard deviation. I gave the students a handout with a couple of completely worked out examples. I gave them some time to work in a group to try and figure out what was happening with the data set in order to determine the standard deviation. It would have been so easy for me to give them a list of steps and show them exactly what to do. I just felt like it would be much more meaningful for them to try to come up with that process on their own. So after 15 minutes or so of discussion, many of the students were coming up with various processes. I gave them a new example that wasn't worked out, it simply had a data set and the answer to the standard deviation. I asked them to try their steps out on a new example to see if they came up with the correct standard deviation. After a while, we came together as a class to fill out a six-step graphic organizer.

As we filled in the steps, I had the students explain things like "we figured out that you take the square root of 16 because 4 squared gives us the value in the previous step... so then we made sure it worked for both examples." Doing it this way probably took more time than if I had just handed them a set of steps at the beginning of class, but they were SO excited that they figured this out on their own. There was discussion, friendly competition, some disagreement, and it required the students to justify their answers and analyze patterns.

Focusing on Improvement

In Chapter 8 of our text book, Dr. Zadina discusses the need for talking about results with students rather than only focusing on "number correct." This discussion reminded me of math Sprints that have become an increasingly popular way to work on pattern recognition as well as fluency in basic math skills. I tried a couple of sprints with my classes this week, and they really enjoyed them! 

The students have one minute to complete the first page of problems. It begins with "read, set, go" much like a race. After one minute, the teacher calls out the answers to the problems; the students follow along on their paper and say "yes" if they wrote the correct answer. If they wrote the incorrect answer, they simply draw a line through their answer but they don't respond orally. Next, the teacher asks how many students got 1 correct, 2, 3, etc. until only one student's hand remains. The class claps or snaps or gives some other acknowledgement to the winner of that road. The students then take a little time to complete the remaining problems on the page to continue working on the particular skill. The students then get another page of problems and one minute on the clock. After one minute, the teacher again calls out the answers as students respond with a "yes" if they wrote the correct answer. Students add up the number of problems they answered correctly, but instead of writing how many they got correct at the top of their paper they should calculate and write their improvement. The sprints are designed so that students will more quickly recognize the pattern on the second page of problems, which leads to an increased number of correct problems. Instead of recognizing the student who got the most problems correct, the class recognizes the student who showed the most improvement. Chances are, this is a student who wouldn't normally be recognized for a strong performance in math class. Not only did my students get a chance to work on their math skills, they had the opportunity to engage in a friendly competition with their classmates as well as the opportunity to recognize students in the class who sometimes just slip by without being heard. 

Here is a link to a more detailed explanation of Sprints: 

Math Sprints

Activating Multiple Pathways

In week 4 of Dr. Zadina's workbook, she discusses ways in which teachers should try to activate multiple pathways for students so that they can create a more intricate network of the information in the brain. She suggests selecting a lesson and composing a list of various assignments that the students are free to choose from. Keeping in mind her discussion of executive functions in chapter 7 of our textbook, I thought it would be interesting to also incorporate some of those ideas into the lesson.

The topic I chose to work with was box plots. I wanted my students to understand how to read, create, analyze, and compare box plots. The lesson consisted of a brief introduction and a teacher led example of creating a box plot. During this time, we had a whole class discussion about how to read a box plot so that the students would be familiar with this before completing the other activities. After discussing how to read a box plot, we completed an example of creating a box plot. I didn't feel as though this was something that the students would be able to "discover" on their own, which is why I decided to lead an example. I then gave the students a list of assignments that involved reading, creating, analyzing, and comparing box plots. This page outlined their assignments for two class periods. At the end of the second class period, the students understood that they would be responsible for the concepts and would be completing a quiz. Some of the assignments involved worksheets. Some involved task cards in which the students receive 10 or so cards; on the bottom of each card is a problem, on the top of each is an answer. The students choose a card from the stack and work the problem presented, they then look for the answer on the top of another card. Using these cards, they are able to self-assess because they know that if their answer doesn't appear on any of the cards, they made a mistake and need to re-work the problem. We recently made accounts on Khan Academy, so some assignments involved watching a video and completing the assignment using their laptop. Some of the assignments involved creating a foldable in order to learn more about box plots. I asked the students to complete 6 out of the 12 or so assignments on the paper. The page also outlined the focus of each assignment; for example, task cards - reading box plots, Khan Academy activity 1- comparing box plots.

I explained that they were responsible for choosing which assignments to complete and making sure that they kept an eye on time. After the quiz at the end of the second period, we had a discussion about which activities they enjoyed the most and which activities helped them understand the material the best. My students made comments like "well since I really understood how to read box plots right after you did an example, I thought it would be better to spend my time on making the box plots and comparing them instead of doing the assignments that just wanted me to read them." It was awesome! So the activity even involved metacognition; I felt like my students were thinking about their own learning and getting to know the ways that they learned best. Is it by working basic problems and leading to more difficult ones, or is it working difficult problems and figuring out the mistakes they are making? I felt like it was a great assignment and I would definitely try it again!

Fish to Infinity

A brief discussion by Dr. Zadina in chapter 6 of our book reminded me of a Sesame Street clip involving numbers and counting. The textbook piece states "cognitive neuroscientist Kathy Mann Koepke stresses the importance of early intervention and suggests that in early childhood, we should teach more than counting aloud and include attaching numbers to items and talking aloud about numbers." The Sesame Street video gives a great description of what numbers are and why we use them in every day life.

I think that even at the high school level, my students lose sight of what numbers are and why we learn to use them. When students get to algebra and begin using variables along with numbers, it is still important that they remember the very basics of numbers and operations. In order to help my students build math skills in a meaningful way, I have tried to make it a point to give them opportunities to form connections between the math being learned and the real world. The video shows why numbers are such an important tool, which seems like a simple concept. But as math gets more and more complex, I think that it becomes increasingly important to connect the math back to real life situations and show what a useful tool math can be.