Response to: Learning Algebra in a Computer Algebra Environment

Do you ever read an article and then wonder what you just read?  That's how I felt after reading Learning Algebra in a Computer Algebra Environment, by Paul Drijvers of Utrecht University, which summarizes his doctoral thesis.  Often I find that reading articles like the above mentioned are not a good use of my time as the article is written for the purpose of building a resume and publishing for the sake of publishing. Additionally, in my opinion one of the most important considerations of being a preservice teacher is the ability as an educator to explain material at a lower level, a skill that I feel differentiates a satisfactory teacher from an above average educator.  With that said, this article is difficult to read and comprehend and to be blunt, is written to sound smart and not actually deliver valuable insight to the average reader. However, if the target audience includes other doctoral candidates and researchers in the field then it may be a very well written article...to that I cannot speak.

As a requirement of this blog post I must find a point the author makes and elaborate and voice my opinion. I was able to decipher a few key take aways from the article that I feel synthesize well with other readings regarding CAS and the umbrella subject of incorporating technology in the classroom.

In the article, Drijvers discusses the instrumental approach to using computer algebra and arrives at the same conclusion presented by Conrad Wolfram in his Ted talk, "By freeing the students from the algebraic calculations, computer algebra would offer opportunities to concentrate on concept development and on problem-solving strategies" (Drijvers, 80).  However, Drijvers also discusses the dangers with incorporating technology into the classroom, "Because the CAS already contains all the algebra, the computer algebra tool might somewhat abstract and formal top-down character, and might turn out to be inflexible with respect to informal notation and syntax" and goes on to add "Furthermore, the CAS might be a black box for students, as it carries out complex procedures in a way that is not transparent to them" (Drijvers, 80).

I believe Drijvers hits the nail on the head here. By handing a student a calculator with CAS built in, you risk the possibility of requiring minimal cognitive effort to complete a task.  Often this "minimal cognitive effort" comes in the form of using the "Black Box" Drijvers mentions above, where students are only concerned with arriving at the final solution, regardless of the method or procedure used. Further synthesizing, the judicious use of technology in the classroom is required by secondary mathematics teachers to ensure the incorporation of CAS does not lead to the "black box approach" for students.

Response to: Using Technology to Promote Access to Mathematics

The chapter Using Technology to Promote Access to Mathematics from the book Teaching Mathematics Meaningfully by Allsopp, Kyger, and Lovin reiterates the message that technology in mathematics education is only appropriate when it helps "facilitate mathematical understanding" (Allsopp, Kyger & Lovin, 2007, p. 195).  The chapter touches briefly on "types of common mathematics-related technology and guidelines for selection" including mathematics software, wireless technology, and on-line resources (Allsopp, Kyger & Lovin, 2007, p. 195).

I found the topic of mathematics software to be particularly interesting, mainly the point the author makes about the equality of software packages:

As a teacher, one must keep in mind that not all software packages are equal in quality. Many software packages at first glance seem to provie students with real-life contexts and are colorful and engaging, but at second glance, there is not much substance behind the glitter and flashing lights (Allsopp, Kyger & Lovin, 2007, p. 196).

Synthesizing previous readings, I believe this is particularly applicable when incorporating new technology (i.e. the iPad) into the classroom environment, as their are hundreds of thousands of exciting and flashy applications in the "app-store" but determining which can "enhance my students' understanding of the target mathematical concept?" is the key take away  (Allsopp, Kyger & Lovin, 2007, p. 198).

The authors place a particular emphasis on Core Curriculums' Geometers Sketchpad and the use of the application to further students understanding of geometry through the creation and exploration of geometry related conjectures.  Through my own experience with Geometers Sketchpad, I have first hand seen the value of the ease with which one can manipulate geometric figures to develop understanding of a particular subject. However, I have also seen the danger in using such tool as focus can easily shift from the task of the subject matter at hand, to the challenges of using such an unfamiliar program.

Allsopp, Kyger & Lovin also touch on the use of calculators to "facilitate higher order thinking by circumventing basic skill difficulties" (Allsopp, Kyger & Lovin, 2007, p. 209).  I believe they make a good argument for the use of calculators in the classroom.

Some educators might believe that students should never be allowed to use calculators until they have mastered their basic facts and computational algorithms. One perspective is to evaluate the purpose of the activity. If the purpose is to develop computational proficiency, then calculators may not be suitable. However, the purpose is to engage students in using basic facts to learn about and do higher order mathematics, then the use of calculators is suitable (Allsopp, Kyger & Lovin, 2007, p. 209).

This reiterates the message in Teaching Strategies for Developing Judicious Technology Use by Lynda Ball and Kaye in the 67th NCTM yearbook (2005).   

References:

Allsopp, D. H., Kyger, M. M., & Lovin, L. H. (2007).Teaching mathematics meaningfully. (pp. 195-219). Baltimore, MA: Paul H. Brookes.

Ball, L., & Stacey, K. (2005 Teaching strategies for developing judicious technology use. In W. J. Malaski & P. Elliot (Eds.), Technology-Supportoted Mathematics Learning Environments 67th Yearbook.

Response to: Using Data-Collection Devices to Enhance Students' Understanding

The article Using Data-Collection Devices to Enhance Students' Understanding by Douglas Lapp and Vivian Cyrus examines the use of data collection to reduce graph-shape-and-path-of-motion confusion. Microcomputer-Based Laboratory (MBL), Calculator-Based Laboratory (CBL), and Calculator-Based Ranger are all data-collection devices which can be used with a calculator or computer to analyze motion.

Graph-shape-and-path-of-motion confusion is a common misconception that students often have about the relationship between the path of motion of an object and a graphical representation of its characteristics (i.e. position, velocity, and acceleration).  The article discusses one major advantage of an MBL is the ability to create live graphs, or plot data in real time.  A reference in the paper, Braswell, believed that "the immediacy of graph production is probably the most important feature of MBL activities" (Lapp & Cyrus, 2000).  As an undergraduate in engineering where a large portion of the curriculum is focused on motion of objects, labs where data collection systems were used to create real time plots we're very beneficial as it allowed the user (me) to formulate their own conclusions and immediately investigate these theories.

Two interesting examples are presented in the article where students often encounter graph-shape-and-path-of-motion confusion. The first, includes a ball on a frictionless flat table where "the student expects the graph of position versus time to also be horizontal rather than a straight line with nonzero slope" (Lapp & Cyrus, 2000). The importance of this example is as an educator, teachers need to use discretion when choosing examples to present material and ensure the examples presented do not support or reinforce these common misconceptions.  The second example of "a graph showing the speed of a bicycle when the bicycle ascended a hill and then descended the hill.  Prior to instruction, many students assumed that when the graph increased, the bicyclist was going up the hill"  (Lapp & Cyrus, 2000).  As a cyclist, it is very intuitive that your velocity decreases as you climb a hill, and increases as you descend.  However, as a student it is difficult to translate these characteristics to a graphical representation.  A fantastic way to present this information would be to use a clip of the Tour de France, as climb information (i.e. grade, elevation gain, peak elevation) are usually presented as the peloton approaches the base and students would be able to see the cyclists struggle on the climb as the reach the peak, and then the death-defying-descent that proceeds with speeds of 100kmh.  This example would help connect the position-velocity-acceleration of a cyclist to a graphical representation.

I believe the point the author makes about creating hypothesis prior to conducting the experiment to promote "connecting graphs with physical events" (Lapp & Cyrus, 2000) is essential to developing an understanding between position/velocity/acceleration of an object.

Access to a resources including MBL, CBL, and CBR is invaluable to educators, but like all technology an instructor must carefully, or shall I say judiciously, decide whether technology is appropriate.

References:

Lapp, D. A., & Cyrus, V. F. (2000). Using data-collection devices to enhance students' understanding.Mathematics Teacher, 93(6),

Response to: NCTM Position Paper on Technology

The article Teaching Strategies for Developing Judicious Technology Use by Lynda Ball and Kaye in the 67th NCTM yearbook (2005) discusses the judicious use of CAS and technology in the classroom. The article begins by taking the position that technology, when used effectively, can "extend the mathematics that can be taught and enhances students' learning" (Ball & Stacey, 2005, p. 1)  The article further states that "banning technology is not necessary" (Ball & Stacey, 2005, p. 3) and is in response to the "frequent calls to ban the use of calculators and computer software that "do mathematics (especially arithmetic or algebra) in schools or a particular grade levels."

One particular point that I found interesting in the article was the brief summary of a recent work by Heid - 2002, who is an "opponent of the use of CAS in American schools.  There are practical arguments about cost, ease of use, and inadequate preparation for unchanged college courses and tests." which I touched on during my last response to CAS in the classroom.  I stated this exact point, that although the adaptation in secondary school is beneficial, and may allow more time for examining underlying concepts rather than methodical manipulation, that we are in fact doing a disservice by delivering students to a VERY teacher centered university environment.

Continuing on with the current article of discussion, a teacher "Lucy" of "fourth-year high school students who were taking a college preparatory"in Australia was surveyed about her judicious use of technology in the classroom. The classroom had access to CAS calculators full-time during their final two years including assessments. Lucy's view on technology for the classroom is summarized well by the following, "she often stated that her main goal for the use of technology was improving the mathematical understanding of students, not the learning of technology."

Lucy employed a "white box-black box" approach which involves presenting material by pencil and paper and then later incorporating technology for more difficult problems.  The CAS allows the students to focus on the topic of concern and not difficult mathematical manipulation.  I find that this approach closely aligns with my current teaching philosophy, where students will learn underlying concepts before they are permitted to utilize technology to perform more difficult operations.  Lucy reflected on her own changes in teaching style as a result of using CAS in the classroom:

Lucy emphasized when technology use is an efficient and effective method for solving problems and when it is unnecessary. In fact, Lucy commented for solving problems that this emphasis was perhaps the greatest change in her teaching that had resulted from having constant access to CAS technology over two years (Ball & Stacey, 2005, p. 7).

I can however, see particular exercises where students may utilize technology to discover and form their own conjectures about new concepts and material, then through reinforced lecture and formal presentation of material students will begin making connections about their recent discoveries. The article further examines how teachers can produce judicious user of technology by creating "An environment where students are usually given the responsibility to make their own selections of mental, pencil and paper, and technology approaches" (Ball & Stacey, 2005, p. 12).

The main take away for me of the article is that as educators we have the ability to "confidently deliver programs that result in students using technology to increase their mathematical understanding instead of randomly pressing buttons in the hope that an answer may appear by magic" (Ball & Stacey, 2005, p. 14).

References:

Ball, L., & Stacey, K. (2005 Teaching strategies for developing judicious technology use. In W. J. Malaski & P. Elliot (Eds.), Technology-Supportoted Mathematics Learning Environments 67th Yearbook.

Response to: Computer Algebra System (CAS)

The introduction of CAS handhelds into a classroom environment necessitates caution, for a computer is only as intelligent as its user. The article Exam questions when using CAS for school mathematics teaching by Vlasta Kokol-Voljc summarizes the appropriate application of CAS on exams. For me, the main take away form the article is the shift away from algorithmically centered education.

Kokol-Voljc identifies three-steps that are required for the successful adaptation of CAS into the classroom, foremost a change of teaching styles is required. This is further broken down by Kokol-Voljc as "from teacher centered to student centered group work and from deductive to inductive learning."  The second step is identified as "refocusing on the teaching goals" and finally the third step is to "change the mathematical problems" including in-class activities, homework, and exam questions.

The article then goes on to identify the two types of exam questions: "theory oriented and application oriented questions."  These two exam question types can be further reduced into  "CAS insensitive questions and questions changing with technology" or in other words questions where "CAS - or any other calculation tool is of very limited help" and questions where the focus will change drastically due to the use of CAS respectively.

I believe the Kokol-Voljc really hits home the point here:

Without CAS, the solving of such problems was (is) strenuous because they involve a lot of mechanical calculations, or because they require “complicated” multi-step solving strategies. For many students, such craftsmanship activities outshine the actual point (goal) of the problem, hence they often forget to answer the original question after successfully completing the necessary calculations. 

In a traditional paper and pencil environment tasks such as solving equations, finding derivatives, and computing integrals require most of the time used for answering these questions. Therefore for both teachers and students the feedback centers on testing the performing of operations. When using CAS, the time needed for such questions is reduced drastically, hence the significance of these questions for providing feedback for teachers and students changes as well. 

 I wonder however, as educators embracing new technology and changing the way mathematics is taught at a secondary level, if we are doing a disservice to the next generation? We are delivering them to a university environment where, speaking in generalizations, technology is not openly embraced and the teaching environment is still VERY teacher centered. Are we setting up our students to fail by allowing them to rely on a device, foregoing these strenuous and time consuming "mechanical calculations" which have governed mathematics educations for years? In some ways, yes.

Great, so we've created a new generation of critical thinkers! The problem is students are inclined to use the minimal brain activity required to achieve the desired result. As educators, WE are responsible for creating activities which challenge students to learn and develop a better understanding of the underlying mathematical concepts while....emphasis on WHILE, ensuring out students are intelligently utilizing technological aids. So, am I sold on the CAS for the classroom environment? No. Do I believe there is potential for something great? Yes. But we have to remember, a computer is only as intelligent as its user - Garbage in, garbage out.

References:

Kokol-Voljc, V. Exam questions when using CAS for school mathematics teaching.