Tech Analysis

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The app that I chose for this analysis is called Factor Race.  It cost $0.99 through Apple’s App store and can is compatible with iPhones, iPads, and iPod Touch.

According to the app description “Factor Race is a game where the player must identify the binomial factors of trinomial equations.”  Students have to correctly factor different trinomial equations in order to make it around the track.  As students complete each level, they earn better race cars.

I really like the idea of this app.  It’s a fun, interactive way for students to practice factoring without it being another worksheet in class.  I actually stumbled upon this app when I was looking for engaging activities for my students to practice factoring.  However, I spent days trying to get the app to work.  Once it goes through the opening “credits”, you can only see half of the app.  Not being able to see the full screen makes it really difficult to even complete one factoring problem, let alone complete an entire level.  I would really have liked to use this with my students, but I didn’t feel comfortable expending class time with an app that I’m really unsure of how it works beyond its description.  I’m really hoping that the developers fix the bug because I think it would be a great app for algebra students.

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Curriculum Analysis

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Kristen and I shared similar views on how each of three textbooks she looked at approached factoring.  We both agreed that having the repetition-style practice.  With something like factoring, having that kind of practice allows students plenty of familiarity with the process.

We also agreed that it was nice that the Core Connections book outlines in detail what teachers need to address, especially as first year teachers.  It’s easy to get lost in the details, and the CPM curriculum really aims to have an integrated story, so having a detail teachers guide is very beneficial.  We also agreed that it was nice that the Core Connections book had activities that broke up the direct instruction.  I think it’s beneficial, not just because factoring is something that students need to work at on their own to gain mastery, but also because it allows for avenues for students to connect and see the meaningful background for a concept like factoring.

We also agreed that the common errors alert was great.  It’s easy to forget what those errors are because you have so much experience with the concept, but that isn’t true for students.  Knowing where they might make mistakes can help both you and your students, both in terms of being less frustrated by the concept and in ensuring that the students are really understanding what they’re doing.

Dynamic Geometry

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There have been a number of studies that show that students have difficulty with mathematical reasoning.  I’ve seen it in my own classroom (both math and science).  Students can take an equation or formula, plug in the numbers, and get the correct answer, but they can’t explain why what they did was correct or have difficulty solving a slightly different type of problem using that same equation or formula.  The article points out that that “teaching approaches often tend to concentrate on verification and devalue or omit exploration and explanation.”  I try to incorporate different avenues to encourage students to explore and explain their work, but they are hesitant or resistant to doing anything but show you that they got the right answer.  They don’t think beyond “I’ve used the correct formula and gotten the right answer, I’m done.”  This was particularly prevalent with a unit in Chemistry about Gas Laws (a topic that is very heavy in algebraic manipulation).  As a class, we would walk through how to solve the problem and I would ask “How do I know my answer makes sense?”  The smart-aleck comment I got was “well you’re the teacher so of course it’s right”.  The point I was trying to make was to go back and look at the particular law and what the concept says.  The math should follow along from that and if they have done the work correctly, knowledge of the larger concept will allow them to check their own work (without having me there to say “yes” or “no”).

Interactive geometry software are computer programs that allow students to explore geometry concepts.  It allows them to see how the relationships in geometry work and how they all interrelate.  It allows students to see that we aren’t lying when we say that it’s all connected because the evidence to support that statement is right in front of them.  Students look at why the geometrical shapes and constructions are the way that the are, and what elements need to be there for the whole thing to work.  The main issue is with student interaction with the software.  Students have a hard time ensuring that the shape they use is correct for the program to recognize and for them to move on and actually work with the geometry.  I think if the interface was less finicky, along with clear instruction on how the program worked and its limitations, I think this could be a powerful tool in the classroom.

 

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.