Proofs and Counterexamples about Continuous Functions

This study focused on reasoning and proof from undergraduate math majors in calculus. Proofs of truths and counterexamples of false facts have been shown to help students understand them. Not only do they help explain why a statement is true or false, but students will also be able to see how. However, reading proofs and constructing them are different processes. In my personal experience, constructing proofs is very difficult especially when I don't know what direction in which I need to head.

Before constructing proofs/counterexamples, students and teachers need to decide which one to produce (to prove a truth or to disprove a falsity?), which means they must decide on the validity of the proposition (is it true or false?), which means they must have a deep understanding of the concepts involved. In fact, many students and teachers have difficulty deciding whether a proposition is true or false because of their inadequate understanding of the mathematical content. Still, many mathematicians often struggle with formal proofs. Thus more effective ways of teaching these are needed, at least in the undergraduate math major domain.

Ko and Knuth (2009, p. 69) pose two research questions for this study:

1. How well do undergraduates construct proofs and generate counterexamples in the domain of continuous functions?
2. What problems appear in the proofs students construct or the counterexamples they generate?

One thing that the authors stated that I found striking was the fact that proof serves as a means of communication among mathematicians. This makes sense, because I noticed when learners can explain something in each others' terms, they gain a better understanding of the idea. Informally, they are proving it to one another.

We usually think of counterexamples, as disputes to generalizing statements. For example, the number $2$ would be a counterexample (in fact, the only one) to the false proposition, All prime numbers are odd. Some counterexamples can actually be ‘examples’, if we think of them as a proof to the generalizing statement's negation (e.g., $2$ is an example that proves the conjecture, There exists an even prime number).

The authors also discussed different types of proof schemes and counterexamples and the properties of each. They created a scale or spectrum of proofs, and one for counterexamples, that ranged by robustness. After receiving calculus students' work on a problem set, they rated it using their own spectra.

Since constructing a proof requires sufficient background knowledge and the decision of whether to prove or disprove, Ko and Knuth (2009) suggest that math teachers should provide a learning environment in which students can easily decide the veracity of statements. There are several methods of doing this. Especially in secondary school, formal proof is not even in the question. Therefore teachers must foster students' informal proof and reasoning as often as possible. Fishbein (1982) states, Ignoring students' intuitive representations of proof does not promote their mathematical understanding (as cited in Ko and Knuth, 2009, p. 75). In addition to drawing attention to students' intuitive thinking, teachers should also explicitly identify students' logical flaws and misconceptions so that they will have a stronger understanding of the processes in constructing (even informal) proofs.

References

• Ko, Y., & Knuth, E. (2009). Undergraduate mathematics majors' writing performance producing proofs and counterexamples about continuous functions. Journal of Mathematical Behavior, 28, 68–77.

How Students View Assessment

Assessment seems to have a completely different effect on students' attitudes based on whether they frequently pass or frequently fail.

Students who demonstrate winning streaks are confident and hopeful, demonstrate continual evidence of success, and are excited to learn. These students are likely to seek more feedback and more challenges. On the other hand, students who demonstrate losing streaks are frequently hopeless and give up easily. They do not feel safe at school and feel that they are always being evaluated. They see feedback as criticism and do not seek challenges or new ideas. Stiggins tries to eliminate this gap.

Rather than using assessment to sort students into winners and losers based on performance, educators should use assessment to help student learn. Stiggins offers an alternative notion, Assessment For Learning (as opposed to assessment of learning). This alternative encourages teachers to turn assessment into a process that involves sharing goals and targets with students, provide frequent and continual feedback in student-friendly language, and provide examples of outstanding student work. This causes students to self-assess and notice trends in their own achievement. Students become more aware of their academic progress. They begin to understand what is expected and make decisions on how to become better. They also generate their own feedback and set their own goals. The hope, Stiggins says, is not to eliminate failure but to eliminate losing streaks. This helps boost student confidence and motivates them to try more.

References

• Stiggins, R. (2007). Assessment through the student's eyes. Educational Leadership, 64(8), 22–26.

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.

Teachers and the Internet

I think it's a bit unfair that teachers are rated based on online activity. One's personal life need not interfere with one's professional life. We are all entitled to do as we please outside of the work place (provided it's legal). If the administration hadn't seen the picture, Sydner shouldn't have been fired. Personally, I have my online profile set to private so that only friends can see it. I have also not made any efforts to befriend my subordinates (be they future students or the swimmers I coach). To the people who argue that employers could hack the privacy settings of social networks, I would counter that if they're using illegal methods to see my profile, they have no room to talk. The moral of the article is that current and future teachers (or any role models) should be careful of what they upload to the Web. But the fact of the matter is, we are unable to control what our friends post.

The results of the Facebook study conducted conflicted with my beliefs about online networking. Reading the material on the Net Generation earlier this semester, I was convinced that social networking and online communication enhanced human interaction. Even when we're physically alone, we're not ‘alone’ due to instant messages, text messages, video/audio chat, direct messages, emails, and status updates. It's hard to believe that these kinds of phenomenon are linked to anxiety, anger, and depression.

References

• Michels, S. (2008, May 6). Teachers' virtual lives conflict with classroom. ABC News. Retrieved from http://abcnews.go.com/TheLaw/story?id=4791295.
• College Facebook users have lower GPA's, more neurotic behavior. (2009, September 29). News Channel 9. Retrieved from http://www.newschannel9.com/news/facebook-985115-college-students.html.

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/.