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.

References

• McCulloch, A. W. (2008). Insights into graphing calculator use: Methods for capturing activity and affect. International Journal for Technology in Mathematics Education, 16(2), 75–81.

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.

References

• The National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Learning and Knowledge in the 21st Century

Chapter 3: Learning and Knowledge in the Twenty-first Century

To be completely honest, I thought the article was just short of a textbook. I remember reading about different perspectives of educational psychology in another course, and much of the article was information that could have been left out. A few of the topics deviated from (what I understood to be) the original thesis of the article, which was about how technology is changing learning and teaching in this day and age.

I enjoyed the introduction of the article because it compared different beliefs and views on learning from very different time periods, while relating back to the theme of technology. It seemed as if educators of the late 19th century were preparing their students for the specific jobs they thought they would have. Whereas now, students seem to be more well-rounded. In the past, school was seen as a means to an end, but now, school is seen as preparation for more learning in the future. One thing that prepares young learners today is that they really do learn how to learn with technology. Views on learning nowadays emphasize cognitive processes like critical thinking, problem solving, and decision making over lower arithmetic and computational skills.

A few big questions did come across my mind while I was reading about this. At what point do we draw the line? Students don't need to be able to take the cube root or write the prime factorization of very large numbers anymore, so why should they need to be able to compute the limit of a rational function or find the general solution of a first order non-linear differential equation? At what point do we say, “That's enough, the calculator can do the rest.”? Why are we, as educators, so selective about what we decide students ‘should’ know?

Aside from discussing the differences due to technology of learning beliefs across time, the article also discussed different perspectives on learning. The Behavioral perspective (Skinner) focuses on external, observable responses. Drill and practice are reinforcements for learning, and educational technology can be highly effective (unless it is excessive or premature, etc.). Behaviorists state that learning is sequential and hierarchical, such as an axiomatic system.

Cognitive psychologists (Piaget) accept that learning is a result of adaptation motivated by disequilibrium. Learners apply existing schema to change what they know about new information, but also alter existing schema to fit new information. This push-and-pull balance of assimilation and accommodation is required when transferring from disequilibrium to equilibrium, thus satisfying the learner's drive. Cognitivists also support scaffolding, which requires teachers to guide and assist learners. Through scaffolding, teachers can determine what type of help to offer and when and how to offer it. Discourse is encouraged so that teachers will be able to recognize students' Zones of Proximal Development, the zone in which the transfer from disequilibrium to equilibrium is most effective, and keep them right in that zone to maximize learning. Before the ZPD, students are unchallenged and bored, while after the ZPD, students are intimidated and discouraged.

Constructivists say that teachers should create complex and realistic learning environments, encourage social interaction and communication, present multiple and diverse perspectives and representations, and facilitate student ownership in learning. Researchers today are emphasizing learning environments that take a mix of all three perspectives. Instruction should be student-centered, multisensory and multimedia-involved, collaborative as well as competitive, active and exploratory, critical, logical, and both theoretical as well as practical.

References

• Niess, M. L., Lee, J. K., & Kajder, S. B. (2008). Guiding learning with technology. Hoboken, NJ: John Wiley & Sons, Inc.

Assessment with CAS

The article on using a Computer Algebra System for teaching mathematics had several major points. The introduction was about how CAS changes teaching and learning of math. In a classroom where students are using CAS, the topics, focus within topics, and goals of lessons will change. Lessons are no longer algorithmic but focus on using operations to understand meaning. The article also mentioned two goals for teaching: to develop theory of mathematical concepts, and to use mathematical concepts in real-world models and applications. Teachers should try to construct exam questions to meet these goals. There were two schemes for analyzing exam questions.

1. The first scheme was based on testing skills and abilities of students, and took a more educational approach. It measured the educational value of problems as questions on an exam. The educational scheme had five categories.
   1. The CAS-Insensitive Questions did not make very much use of the CAS, and computation plays a minor role.
   2. The Questions Changing with Technology made a big difference once the CAS was introduced because time needed to solve the problem was greatly reduced.
   3. The Questions Devalued with CAS involved rare tricks and hard-to-remember equations to solve, so the CAS appeared insignificant.
   4. The Questions Testing Basics became trivial when using CAS because the answer would be produced immediately; however, the student needed knowledge about syntactical structure.
   5. The last category, Rediscovered Questions, are geared to support creativity, fluency, and flexibility, but are rare because of the difficulties in evaluating and grading.
2. The second scheme was based on the usefulness of CAS and took a rather technical approach. It measured the role of technology for answering a question. The technical scheme also had five categories, grouped into how significant CAS was in solving the problem, and how well the student should be familiar with CAS.
   1. In terms of significance, Primary Use needed CAS as a major activity,
   2. while Secondary Use did not facilitate CAS as strongly.
   3. Regarding familiarization, Advanced Use required in-depth knowledge of CAS,
   4. while superficial knowledge of CAS suffices for Routine Use.
   5. CAS is of very little help for questions in the last category, No CAS Use.

After discussing different types of problems in different categories, the article compared the two and made some observations about CAS. It facilitates the two teaching goals discussed above, it reveals educational value of exam questions, forces teacher to be conscious about exam questions, and revives the “forgotten” questions. When choosing exam questions, teachers need to keep the aforementioned goals in mind, but also question how they test the student. Exam questions should test general abilities rather than computational skills. In any testing environment the act of understanding and the act of overcoming an obstacle are equally important in the learning process. In addition, intellectual concentration and emotional tension are present and culminating, which creates a learning situation per se.

References

• Kokol-Voljc, V. (year unknown). Exam questions when using CAS for school mathematics teaching. publication unknown.

Lesson Structures, Methods, Solutions

When reading the article, I kept in mind that although there is a big difference in Type 1 and Type 2 lessons, there is no “right answer” in choosing a lesson type to model after. Type 1 lessons are narrower and don't elicit as much critical thinking and decision making, but they are more structured and make a classroom more organized. Type 1 lessons may be necessary depending what kind of students I will teach. If I am teaching students that are “behind the curve” and may need extra assistance with math, Type 1 lessons may be required. On the other hand, Type 2 lessons may be more beneficial for gifted students who can handle a more independent learning environment.

As I read the section describing Type 2, a lot of questions popped up in my mind: If the Type 2 lesson allows students to be independent and explore on their own, how do the goals of the lesson (and curriculum) fit in? How do they learn what they're expected to learn? What happens to classroom management if students get off track? What is my role as a teacher if technology allows students to learn on their own? Is my purpose solely that of a ‘marshall’—I'm just there to make sure students are doing what they're supposed to be doing? These questions have significant purpose not only tonight but throughout all the readings. I want to be able to use technology to guide students' learning, not to replace my teaching. My purpose as teacher is to help students investigate and understand relationships among objects, and to complexify this investigation so that the students can simplify (p. 309) it. Rather than making decisions about what and how to investigate, my job is to guide students to make those decisions, and reaffirm that they're making the right (or wrong) ones.

A miscellaneous topic: I noticed that teachers and students often throw around the words ‘lesser,’ ‘greater,’ ‘bigger,’ and ‘smaller’ a lot without realizing their meanings. ‘Bigger’ and ‘smaller’ can be quite confusing in the domain of negative numbers. (Positive numbers are easy to understand in this regard). As teachers, we should be careful when choosing these words to describe relationships and ordering of real numbers. We should use the terms ‘lesser’ and ‘greater’ to make explicit the ordering of numbers. For example, negative five is definitely less than negative three. When we ask, “Is negative five bigger or smaller than negative three?”, students will most likely get confused because they will think of absolute value. Negative three is closer to zero (also known as ‘nothing’) than negative five is, therefore it must be ‘smaller.’

One last point: Tonight's reading enforced my belief that the main goals of any lesson should be investigation and discovery. When I first transferred to the Math Ed program here at Tech, I was excited to teach in a math classroom and lecture to the students, much like many college professors do today. However over time I realized that in a high school environment, students do not want to be lectured to. They want to investigate and discover things on their own. So rather than giving out all the answers, I hope to guide students to create and find their results and own them, so that way they will learn better, remember easier, and maybe even be enthusiastic about math.

References

• McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 303–317). Reston, VA: NCTM.

Teaching Strategies for Technology

When I first watched the instructor talk about technology in the hands of students versus in the hands of teacher, I suddenly imagined a classroom full of students with open laptops working on a geometry problem together. I realized that even though technology is great to demonstrate, the students would get a more beneficial (and intriguing) experience if they could discover things for themselves. I remember the feeling of playing around with dynamic figures and devising my own conjectures. I couldn't wait to see if I was right or not, so I tried to prove them right away. I want to instill this feeling in my students. The instructor also talked a little about equity. I agree that the aid of technology helps level the playing field. It equalizes opportunities for all students.

In the reading, there was a focus on teaching strategies that should be used to implement technology in the classroom. The main focus was that technology should extend math and enhance learning. It should promote higher-order outcomes, such as reflection, reasoning, problem posing, problem solving, and decision making (NCTM, 2005, p. 1). I want my students to be able to be able to develop these processes without worrying about technical difficulties or syntax errors. I want to be able to teach students how to use technology to help them, not do math for them. Technology should also be a tool or aid for students, not their brain. One really interesting argument against the use of technology is that it does the work for the students. One really good example is prime factorization (Fundamental Theorem of Arithmetic). If the CAS can do it for the students, do they really need to know how to do it? Some might say no, but I think the students should at least learn how to do it first, and then use the calculator for more complex problems. That way, it's no magic trick. Once the students learn how to do something, they can use the CAS to do it for them afterward so they can focus their energy on the bigger picture, e.g., when should prime factorization be used? This example illustrates the white box-black box strategy. First teach the students how to do it by hand, and then allow the technology to do it for them. (For those computer science folks who know what “information hiding” is, this is a really interesting subject to talk about.)

References

• Ball, L. & Stacey, K. (2005). Teaching strategies for developing judicious technology use. In W. J. Masalski & P. C. Elliott (Eds.), Technology-supported mathematics learning environments (pp. 3–15). Reston, VA: NCTM.
• The National Council of Teachers of Mathematics. (2005). The use of technology in the learning and teaching of mathematics. In W. J. Masalski & P. C. Elliott (Eds.),Technology-supported mathematics learning environments (pp. 1–2). Reston, VA: NCTM.
• Technology in mathematics education [video file]. (2006). Retrieved from http://www.youtube.com/watch?v=W58ReRyNYp8.

The Net Gen, Continued

Chapter 4: Using Technology as a Learning Tool, Not Just the Cool New Thing
Chapter 5: The Student's Perspective

Chapter 4 emphasized further what was discussed in the first three chapters. It's interesting to read about how the Net Gen learns differently than other generations do. They use computers and other most recent technology in everyday life including work, school, play, and hobbies. But at the same time, the Net Gen is similar in work ethic and ambition to the Greatest Generation: their grandparents. NetGeners like to be creatively challenged.

I don't necessarily agree with the way the article mentions how NetGeners should study. Although a stimulating study environment is essential, I think it's up to the student to decide where and how to study. Personally, I like a quiet room with not much distraction. I find that I can get more done, and get it done faster, with a concentrated attention span.

Human interaction was also mentioned. This section really brought light to how I think about online classes. Social interaction is extremely important, and online classes need to do as much as possible to enhance it, not replace it. I get angry at people who notice trends from statistics of Math Emporium classes and think that it must be working because grades are higher. But it never occurs to them that maybe the reason grades are higher is that cheating is easier, which brings me to my next topic.

Technology is definitely helping cheating. This was made obvious, what with text messages, copy/pasting, internet sourcing, etc. However, technology is also helping teachers (and software) detect cheating. This wasn't mentioned in the article. Cheating detection software has gotten better through time, just like all technology has. Technology makes it easier to cheat, but it also makes it easier to get caught.

References

• Oblinger, D. & Oblinger, J. (2005). Educating the net generation. Retrieved from http://www.educause.edu/educatingthenetgen/.

The Net Gen

Chapter 1: Introduction
Chapter 2: Is It Age or IT: First Steps Toward Understanding the Net Generation
Chapter 3: Technology and Learning Expectations of the Net Generation

I want to start off saying that I think the term "technology" is defined very subjectively. This was shown in one of the surveys that asked what "technology is," where there were many diverse responses. Technology is relative to generation and culture. For example, we don't consider automobiles to be very technological, but to someone living in the 17th Century, they would consider it a form of technology (if not witchcraft). The article talked about how members of the Net Gen don't consider computers and cell phones technology but rather part of their everyday lives. On the other hand, members of the "Matures" generation would likely be uncomfortable with using these objects all the time.

The eBook also talked about the pros and cons of using technology in an education environment, although I believe the benefits of using these tools highly outweigh the drawbacks. No matter what form of technology a teacher uses, there are going to be problems and malfunctions that are going to have to be worked around. However, one must consider the rate at which we introduce these tools in a classroom. I'm sure that there were some skeptics back in the day when the calculator was invented, but look where we are now. It's not a matter of which technologies we use, but a matter of how fast we use them.

Another point to make is which determines the characteristics of the Net Gen: age, or experience? There were both sides of view present, but I think that no matter what generation you're talking about, whether it be a generation born between 1900-1946 or between 2100-2146, there are always going to be technological advances that the newer generations are going to be more comfortable with. There's nothing special about our generation, it's just the most recent one we can accurately study at this time.

References

• Oblinger, D. & Oblinger, J. (2005). Educating the net generation. Retrieved from http://www.educause.edu/educatingthenetgen/.