January 27, 2011

Graphing Calculator Use: Activity and Affect

The thesis of this article provides an effective method that captures data on how and why individual students choose to use graphing calculators—henceforth, HGT—outside of whole class situations. I believe McCulloch's purpose is clear and valid. Most recent research on HGT focuses on its effect on student achievement, or else focuses on the teaching and learning of a specific topic in content. This past research drives her interests in this paper:

  1. When students work independently rather than in a large group, what are they actually doing with HGT?
  2. What aspects of HGT do they attend to?
  3. How do emotions, values, beliefs affect their HGT activity decisions?

McCulloch mentions several difficulties and drawbacks about traditional methods of data collection, so offering a new method and determining its effectiveness has well-founded intentions.

To address the problem solving strategies and decision making in detail, a task-based interview is typically implemented. Basically, this means the problem solver—the subject—will think out loud in front of the interviewer. However this design is unsuitable if it is used to collect data about emotions and values. McCulloch's study combines the task-based interview with a video-SR (video stimulated response, a procedure in which videotaped behavior is played back to the subject in hopes that he/she will recall his/her activity). With this combination of designs, the subject will recall both cognitive and affective activity. In addition to capturing the subject's activity, a video capture card was used to capture the screen activity on the HGT. When these three components are put together, the event is recreated.

I think this method should be used for an intensive examination of a subject, but would not be appropriate as an extensive study for a large group or class. The resources of time, money, and equipment are simply not available. The article also mentions other limitations.

McCulloch makes several conclusions from her study:

  1. The methodology used is very effective regarding capturing data from individual students, and opens many doors to future research.
  2. The methodology captures information that was unattainable in the past.
  3. The methodology allows students to reflect and recap their activities with the HGT, providing a learning experience.

I'd like to make one more point about the outcomes of this study. In the article, one of the limitations identified was that the subject's emotions played an important role when reviewing the video-SR interview. therefore… the student will associate [them] with that event in the future (p. 80). I do not see this as a drawback though. In my experience, the concepts with which I associated certain emotions were the ones that I remembered the most in school. Teachers should always promote positive emotions in students and never negative ones, but I cannot ignore the fact that I achieved well when I was tested on the concepts for which I had negative feelings.


  • McCulloch, A. W. (). Insights into graphing calculator use: Methods for capturing activity and affect. International Journal for Technology in Mathematics Education, 16(2), 75–81.

January 25, 2011

Teacher as Researcher

One of the benefits of teachers taking an active part in the field of research is the community formed among educators. Research allows teachers to reflect on their own pedagogical knowledge and to share and revise it with each other. Lankford (2003) provides a step-by-step sequence for novice researchers interested in becoming more active.

Thinking About Research involves simply motivating oneself to read more relavant literature and to stay informed on current trends and recommendations. Research is to be considered as a tool that provides direction when it comes to trying new approaches and techniques in the classroom.

When Reading and Discussing Research, educators should read a wide range of literature and discuss, using personal experience, the implications it has in the classroom. In this way, teachers are able to bounce ideas off of one another and gain broader perspectives.

Designing and Critiquing Classroom Investigations: At a certain point, teacher teams are comfortable enough to discuss suggestions and critiques for new implementation. It is at this point that the research theory becomes practical.

In addition to Lankford's approach, she provides specific examples of different types of informal investigations conducted in her team's classrooms. These illustrations make clear how important the teacher's role is in research.

After reading the chapter, I had a few questions:

  1. How would more conservative teachers feel about new ideas involving research and new teaching methods?
  2. In terms of grading, educators focus on the processes involved. However, in the ‘real world,’ only the results matter. How can teachers reduce these conflicting pressures in students? Is it fair for teachers to grade processes and results equally?
  3. In what ways does research adversely affect students? When testing new techniques, are the students acting as ‘guinea pigs?’ For example, if two different methods were used on two different classes, and later results showed that one method was significantly more effective, is it fair for the students in the opposite class?


  • Lankford, N. K. (). Teacher as researcher: What does it really mean? In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 279–289). New York, NY: Macmillan Publishing Company.

January 20, 2011

NCTM Principles Overview

After reading NCTM's six Principles for School Mathematics, I obtained a better understanding of the features required to produce a high-quality educational environment for mathematics students. NCTM emphasizes that the Principles are not mutually exclusive, i.e. they address overlapping themes. I believe the Principles are also collectively exhaustive, i.e. they try to encompass all of the features necessary in the math classroom.

I agree that all students are capable of learning mathematics. However I believe some students are more advanced than others. Like sports, some athletes are good at baseball while others are good at hockey. In the academic setting, there are going to be students ahead of the learning curve and behind the learning curve, and these students will vary from math to English. Teachers should be able to accommodate for this aspect of diversity (among others) while keeping expectations high.

Mathematics is a subject that well exhibits the Curriculum Principle. Math itself is cumulative, so I find it easy to make connections between content areas within the subject. As the NCTM says, the strands are highly interconnected (2000, p. 15). The more easily students can see and realize this, the more connections they will make. The NCTM recommendations for curriculum will help narrow in on the tasks I will need to accomplish while constructing lesson plans.

I never realized the impact teachers make on their students until I heard it. One school year might not seem like a long time, but in the minds of the students it can make all the difference. Looking back on my own experiences in high school and reading about the effects of a single teacher on an entire class enforces this view. Realizing that one can change the lives of children forever may be a scary thought but it can also be a good one if teaching is done effectively. I think an effective teacher has a good balance of content knowledge (knowledge about what the students know and what they need to learn) and pedagogical knowledge (knowledge about how to teach). Teachers also need to continually seek improvement on their own part. As a teacher I plan to continue my exposure to research on mathematics and education so that my teaching practices will be continually improving. This is furthermore in the best interest of the students because as generations change, pedagogical methods (along with content) will change.

I believe the best way to learn is to be in one's Zone of Proximal Development. Students need to be challenged and supported. When a task is challenging enough to overcome boredom, but not too challenging as to promote anxiety, the student is in the ZPD and will learn with understanding. Students need to be able to build on previous knowledge, elaborate on new concepts, and organize concepts in a way that helps them remember them the easiest.

New opinions about assessment open up doors that I had not realized existed in the past. Assessment is foremost used as a tool to detect what students learned and how well they learned it, but it is also used as feedback for the students. Students should easily distinguish their place in the curriculum so they know exactly what their strengths and weaknesses are. Assessment should be used as an intermediary to help students learn, not as the end to a unit where students will never have to use that information again. On the other side, assessment can be used to aid teachers. For example, teachers will know which students are excelling or falling behind in certain areas, and will be able to make decisions for future instruction.

The aid of technology helps level the playing field. This goes back to the Equity Principle. Technology helps students focus on the bigger problems at hand, such as those involving decision-making and problem solving. The increase in technology yields changes in curriculum and changes in views on which concepts are essential in the classroom. Technology shifts the students’ attention from thoughtless algorithms to more complex thinking.

There are a few Principles that stand out when regarding mathematics: the Curriculum and Technology Principles. I favorite these two Principles because I believe they are crucial in the mathematics classroom. As I’ve stated above, math is one of the subjects that is a continuous field with overlapping grey areas rather than a collection of discrete facts or figures. There are more connections that can be made in math than any other subject in secondary school (in my opinion), so building on these connections is essential for students to learn effectively. Technology is the other Principle that sticks out to me. It is true that technology can be used in other courses, but I believe it can be most appreciated in a math course. Not only does technology act as a tool that enhances the learning of math, it is also determines the behavior of students and teachers with regards to content. The more (and better) technology is available, the more students will be able to make decisions, think critically, and focus on meaning.


  • The National Council of Teachers of Mathematics. (). Principles and standards for school mathematics. Reston, VA: NCTM.