# Student Thinking #2

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We recently started in on sequences and the book opens the chapter with problems that require students to create a table and come up with a pattern.  The first one was about multiplying bunnies.  The problem stated that we start off with two bunnies and, each month, each pair of bunnies has two babies.  The problem wanted students to figure out how many bunnies there would be at the end of 12 months, as well as to determine the pattern and what family of equations described the growth.

I had one student who fairly quickly completed the problem.  One of the extension questions that the textbook suggested was for students to attempt to write an equation that represented the situation.  This student, after he graphed the growth, recognized that it wasn’t a linear growth pattern.  When asked if the situation could be represented by a straight line, he said no, it was something else.  However, when he was set to the task of determining an equation for the situation, he always came up with a linear equation.  When I asked him what the graph of his equation would look like, he knew it would be a line.  When I showed him the basic exponential equation (y=b^x), he recognized that was the form he needed to use, but even by the end of class all of the equations he was creating were linear equations.

I didn’t expect him to come up with the correct equation, but it was interesting to me that he was constantly writing linear equations.  If I had been able to sit with him, he probably could have been guided to finding the correct equation.  It just wasn’t possible for me to do that.  I know that it is a topic covered later on in the chapter, and in another chapter in the textbook, so I know that he’ll eventually have the opportunity to work with the concept more.

# Student Thinking #1 – Graphing

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Something I’ve noticed for a while in both my math class and my chemistry classes is this aversion to graphing.  In both cases, the graphs were simple linear relationships, but students will go out of their way to not graph.

I noticed it with my math students in a station activity that I had them work on.  Of the nine stations, three of the stations were ones that required them to graph.  When I reviewed the work papers later, almost all of the students had ignored the directions and simply solved algebraically.  When we began sequences and were working to build rules for either arithmetic or geometric sequences, students still had no desire to graph.  They would happily create a table, but were missing on what kind of relationship there was until they saw it graphically.

In my chemistry class, it became noticeable when there were two consecutive labs that had a graphing component.  One student flat out asked me why we had to graph (though I think the underlying comment was “why do we have to do all this work?”).  One of the questions on the second lab asked them to determine which value of the x-intercept was more accurate: the one from the calculations or the one from the graph.  At least 3/4 of the students said that the graph provided the more accurate answer, and many of those responses came from the students who were vehemently against generating the graph in the first place.

I think what I find the most puzzling is why students have such an aversion to graphing.  In math, of the many methods we discuss regarding solving a system of equations, graphing is certainly the easiest.  In chemistry, graphing your data is often much easier than any of the required calculations, even when you have to create a best-fit line to go with the data.  Perhaps students missed out on the discussion of why we graph data or given equations.  I think it is important that students see the benefit of graphing and that it does tell you something, either about an equation or a given set of data.

# 5 Practices for Orchestrating Productive Mathematics Discussions

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This book was an interesting read.  It really makes you think about both what you have been doing in your classroom as well as ways you can improve your teaching.  I think it is important for students to have authorship in their own problem solving methods.

I do think that implementing this method would require a lot of work up front.  If you teach the same topics every year, it is not the same level of work each school year.  I do think that the level of work that has to be put in to implement this method could be a detractor for many teachers.  It’s something that I definitely think should be implemented if you have a strong, collaborative PLC because you have both extra support and feedback.

# Classroom Culture, Challenging Mathematical Tasks, and Student Persistence

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This article discussed the importance of creating a classroom culture that supports students in persisting with mathematical tasks.  The goal is to create an environment where students feel comfortable tackling more cognitively challenging problems and persisting in finding a solution.

The key components of successfully accomplishing this are:

1. The way the tasks are introduced. If students are given preliminary experience, they were more comfortable in engaging in the task and persisting with it.

2.  Providing support to students. It is important to allow students time to struggle individually, followed by working collaboratively, as well as overcoming the tendency to give students “hints” and to use prompts that differentiate the task.  The authors make the point that you can provide support for students without detracting their opportunity to understand the math.

3.  Providing an extension task.  It allows students who have finished to explore the topic with more depth and allows those students who are still working adequate time to work with the task to a conclusion.

4.  The sharing of student solutions.  The authors note that is important to observe students as they’re working and to thoughtfully choose students to contribute.  This allows students to explain their strategies and see that they are making an important contribution to the classroom.

5.  The method of assessment.  Students who feel like they are being assessed competitively (compared to their peers) are less willing to participate. Using a criterion-based rubric encourages students to participate and persist, because they can see up front how they will be graded, as well as encouraged them to reflect on their learning.

Overall, the authors concluded that a positive classroom environment is not just rules and procedures, but ongoing and interactive support from the teacher.

# Chapters 7 & 9 – Images and Social Media

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Images are an extremely powerful way to convey a message.  I think it’s why the adage “A picture is worth a thousand words” is so enduring.

There are now a multitude of sites that now incorporate easy to find, high quality images.  The best known of these is Flickr.  Flickr is an online photo-sharing community that contains millions of photographs that cover a broad range of topics and interests.  Within Flickr, users can group, tag, share, and create presentations that can have incredible potential in the classroom.

The use of Flickr, and other sites like it, can extend across so many different subjects.  Having a strong visual can make conveying a new concept clearer and more meaningful for students.  Some students need that visual to really have a grasp of the material.  Using a photo-sharing website within my lessons could help a student understand the material.

With the option to tag and describe photos, it allows you can connect to a larger group of similar photos, allowing for a wider breadth of options.  Students can find and share photos that are meaningful to them, as well as find discussion points to share in class.  It also allows teachers to find classroom display ideas that can help them to create bulletin boards that help their students.  The ideas available to students and teachers are almost endless.

One of the most powerful things that can be done through Flickr is the idea of a virtual field trip. Due to geographical and economical constraints field trips are often unrealistic. While students are studying the Lewis and Clark expedition, they can be shown photos to enhance their understanding and help draw a better picture in the students mind.

Just as with all things connected, there are some risks.  The most obvious concern is appropriate content.  Because anyone with an internet connection and an email address can create a Flickr account, there are a great deal of inappropriate images on the site.  If you decide to use Flickr in the classroom, you need to be aware and set boundaries for your students.

Another powerful classroom tool is social media.  The textbook discusses the classroom potential of Ning and Facebook.  While I think that using social media in the classroom can be a unique way to build community and facilitate learning, sites like Facebook should be used with caution, if at all.  Because so many students are familiar with Facebook, it would seem to be a great option to use in the classroom, but it is too easy for students to get off track.  I think, for all the good that Facebook could provide, there are too many distractions for it to be a really viable tool.

Ning is a site that allows you to create your own social network, allowing you to easily upload videos, presentations, and articles.  This ability makes it a great resource for students.  Having a forum allows students to communicate with each other and with you as the teacher, allowing for greater collaboration.  It opens up a new way for students to have discussions, as well as providing a place for students to help each other with homework, as well as allow the teacher to chime in and make any necessary corrections.  Since the teacher can create the network, it limits the number of distractions, making it more beneficial tool in the classroom.

# Homework

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Homework has become a very conflicted idea in the teaching and learning process.  Some argue that homework is always beneficial, and we should always assign students homework to reinforce ideas.  Others argue that homework does more harm than good and we should just focus on in-class work.

My opinion sits somewhere in the middle of those.  In some instances, I think homework is a great thing and can really help students to practice and understand a new concept.  In other cases, I think homework isn’t as effective as it could be.

The first article I read, Fischer’s “Homework and the Gradual Release of Responsibility”, argues that homework can be beneficial if done in a strategic way.  The article states that homework can be used as part of a process they call the “gradual release of responsibility”.  They detail this process as, at the beginning of a lesson, the teacher assumes all the responsibility for performing a task.  As instruction progresses, the responsibility shifts from the teacher to the student and, ultimately, the student is responsible for performing the task.

The second article I read, Eren’s “Are we wasting our children’s time by giving them more homework?”, argues that homework is really only beneficial in math courses.  In other subjects, such as language arts or science, students were essentially equal whether they had been given homework or not.  It raises interesting questions.  Should we only being giving out math homework?  Is there another type of homework we could assign in other subjects that would be equally as successful?

Overall, I think as a teacher I will assign homework.  I think a certain amount of individual work is necessary, but I don’t think you need to drown kids in work, hoping that something sticks.  I think the Fischer article resonates most with me.  I think it is important that the homework you do assign has a purpose (and is purposefully planned) and adds to the learning.

# PLCs and Lesson Studies

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PLCs, or Professional Learning Communities, is an extended learning opportunity for teachers, encouraging collaborative learning and feedback within a school.  Implemented correctly, PLCs have the ability to aid all members of the education community: teachers, students, administrators, and parents.

Lesson studies, principally used within the Japanese schools system, work in the same way as PLCs.  Both focus on student success and creating a successful educational environment.  One of the positives of PLCs and Lesson Studies are the collaborative assessment of the teacher.  It introduces new teaching methods and perspectives, a supportive environment for both teachers and administrators, and a way to improve lessons and lesson practices.

The collaboration inherent in PLCs and Lesson Studies is applicable to all facets of education and is key to creating a successful academic environment.  However, it is important for PLCs to be organized and structured to ensure their success.  For greater teacher success, teachers should be involved in a variety of PLCs, not just subject/grade specific ones.  By being involved in a variety of PLCs, teachers get a greater perspective of the educational environment of their schools, as well as learning different techniques to approach problems.

Currently, schools operate under a “teacher as an island” mentality.  The teacher has their classroom, they teach the material in the way they think best, and there is no feedback from their colleagues.  This results in students with varying understandings of the set curriculum, many of whom aren’t adequately prepared to move on to the next level of their education.  PLCs allow teachers to work together, determining what material is being covered and what isn’t and working to best achieve their goals and the goals of their students.

While it might seem obvious that teachers should be communicating and collaborating, it really hasn’t been the case.  Just as students have different learning styles, teachers have different teaching styles.  It is incredibly important for teachers to balance their own individuality with the collaborative environment in order to create a classroom that can best prepare students for the next stage of their education, as well for the future.