September 29, 2010

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.


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

September 15, 2010

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.


September 8, 2010

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.


  • McGraw, R., & Grant, M. (). 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.

September 1, 2010

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.)


  • Ball, L. & Stacey, K. (). 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. (). 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]. (). Retrieved from