LearningArithmeticinthe21stCenturyJonMai

LearningArithmeticinthe21stCenturyJonMai

To-Do

Develop principled arguments (backed up by: research, empirical findings, your personal experience, hypotheses about resulting cognitive developments, and the topics discussed in class) which of the four positions YOU will favor!

Principled argument which of the four positions YOU will favor!

The proposed plan for the Boulder School District would be to develop new curricula and more intuitive calculators that would simultaneously require a more in-depth knowledge of the concepts and naturally lead the students towards the correct process that would be required to solve the equation. The obvious goal would be to lead students towards a natural understanding of the concepts so that they would not be reliant on the gadget, but instead use the gadget to speed up the already understood concepts and indeed as a tool for learning the concepts. The MAPS (Memory Aiding Prompting System), developed for people with cognitive disabilities at the University of Colorado, is one of these tools designed for learning and uses several types of scaffolding with fading (Carmien). We can relate this concept of learning to calculators. Homeschoolmath.net, which assists parents who are homeschooling their children in math, says this about calculators: “Calculators? should not be used for a random trying out of all possible operations and seeing which one produces the right answer.  It is crucial that the child understands the different mathematical operations so she knows WHEN to use which one - whether the actual calculation is done mentally, on paper, or with a calculator. (Using)” By developing calculators that lead to this knowledge of when to use the different functions and how they work, we can effectively teach students both the underlying mathematical concepts and how to utilize the technology at the same time. In a paper written by Bert K. Waits, a professor of Mathematics at Ohio State University, he states “We believe what is needed in the future is a university mathematics curriculum that takes advantage of computer technology to assist students in gaining mathematical understanding, in becoming powerful and thoughtful thinkers, communicators, and problem solvers. We seek a balanced approach to the use of technology in the future.(Bert)” When we use scaffolding with fading, we can use the calculator to bridge the gap to more advanced concepts that would be difficult to understand in and of itself, and by creating understanding of the concept we can fade away the bridge, using innovative curriculum to create the understanding of the proverbial water beneath the bridge. From personal experience, knowing when an answer is correct assists the learning process by teaching when the correct formulas or procedures are used. By utilizing this power to teach the students, we're reducing dependence on technology by using the technology as a learning tool. The technology effectively makes itself obsolete. Think of it as a plane that would fly itself, but in doing so teach the pilot how to fly. Therefore, if the plane ever fails, the pilot would have the knowledge to fly it themselves, or to fly other planes that do not have the same technological capabilities. We need to utilize technology to assist learning, but the underlying concepts are still extremely important for students to learn. Why not use technology in our favor?

In their article “Creating Meaning for and with the Graphing Calculator”, Helen M. Doerr and Roxana Zangor find that graphing calculators in particular have outstanding functionality in demonstrating graphical analyses to students as well as teaching rounding concepts and alternative problem-solving techniques. Doerr and Zangor found that in the pre-calculus classes they examined, “the students began to see the calculator as a tool that should be checked based on their own understandings of mathematical results” (Doerr). In a demonstration with M&M's that was designed to display the limitations of calculators, the students would spill the candies onto a desk and remove the ones without the 'M' facing up. The students observed that they would arrive at zero candies in their experiments, but when they tried to model the experiment on the calculator, the calculator would exhibit zero as the limit, but would never reach zero itself. The article was written in the year 2000 when the TI-82 and TI-83 had become prevalent as the primary calculator in use in higher level mathematics classes (Doerr). The paradigm shift from hand-drawn and traditional calculator based analysis to the use of electronic graphical representation and analysis runs parallel to the shift we propose in our support of a position that favors the creation/invention of new curricula and technologies. Just as graphing calculators solidify traditional mathematical concepts in the minds of students, further iterations of calculator technologies and the curriculum behind them will help further solidify other mathematical concepts. In her article “Calculator Use and Problem-Solving Performance”, Charlotte L. Wheatley found similar results to Doerr and Zangor in an elementary school setting. Following the four steps in problem-solving identified by Polya in 1945 (understanding, planning, executing the plan, and evaluation), Wheatley claims that her results from groups of students allowed to use calculators for higher level multiplication and division suggest that the result of allowing more time for formulating strategies and less for computation would give the student a better understanding of the underlying concepts. She concluded that “evidence exists that calculators can facilitate mathematics performance with little risk of loss in computation proficiency” (Wheatley). Wheatley further observed that the students allowed calculators as opposed to the ones who were not were more likely to recheck their work and thus gained more from the evaluative step in problem-solving. Also, these students more frequently arrived at the correct answer despite calculator errors. As technologies develop, students can spend more and more time analyzing their problem-solving techniques instead of mucking through pages of hand-drawn computations only to abandon the problem as soon as the solution appears. Clearly, there are still some drawbacks to encouraging calculator use in school, but due to the vast benefits of calculator use, this implies the technologies and teaching methodologies should not be abandoned, but instead, improved upon.

 

Hiphophippotomi:

Jon Mai

Ariel Aguilar

Eric Holton

References:

Carmien, S., & Fischer, G. (2005) "Tools for Living and Tools for Learning." In, Proceedings of the HCI International Conference (HCII), Las Vegas, July 2005, (published on CD). http://l3d.cs.colorado.edu/~gerhard/papers/tools-hcii-2005.pdf

Helen M. Doerr and Roxana Zangor. (2000). "Creating Meaning for and with the Graphing Calculator". In, Educational Studies in Mathematics, volume 41 #2. http://www.jstor.org/stable/3483187?&Search=yes&term=calculator&list=hide&searchUri=/action/doBasicSearch?Query=calculator&wc=on&acc=on&item=2&ttl=7012&returnArticleService=showFullText

Using calculator in elementary math teaching. (2003-2009). HomeschoolMath.Net Retrieved September 6, 2010 from the World Wide Web: http://www.homeschoolmath.net/teaching/calculator-use-math-teaching.php

Bert Waits. The Merging of Calculators and Computers: A Look to the Future of Technology Enhanced Teaching and Learning of Mathematics, Proceedings of ICTMT 3, 1997

Charlotte L. Wheatley. (1980). "Calculator Use and Problem Solving Performance". In, Journal for Research in Mathematics Education, Volume 11, #5. http://www.jstor.org/stable/748623?seq=3&Search=yes&term=math&term=calculator&list=hide&searchUri=/action/doBasicSearch?Query=calculator+math&gw=jtx&acc=on&prq=calculato&hp=25&wc=on&item=3&ttl=1636&returnArticleService=showFullText&resultsServiceName=null