Technology in the Mathematics Classroom

Running Head: Mathematics and Technology








Technology in the Secondary Mathematics Classroom
Hai Lamb
Northern Arizona University

Abstract
This paper will discuss the need to incorporate technology into the secondary mathematics classroom, describe what a classroom should look like that implements technology using a constructivist approach, and suggest several strategies secondary mathematics teachers could use to incorporate technology into their classroom.

Technology in the Secondary Mathematics Classroom
Albert Einstein once said “education is what remains after one has forgotten everything he learned in school” (as cited on Wikiquotes, 2007). From this statement, we understand that Einstein did not view formal education positively.  He perceived education as a hindrance to creative thought because of its strict rote learning (Wikipedia, 2007).  Einstein lived between 1879 and 1955, which was over half a decade ago.  However, he may find that formal education has not changed much from when he was school aged.  Nevertheless, the National Council of Teachers of Mathematics (NCTM) tried to abandon learning strategies through mimicking and has developed a series of learning math standards that began in 1989 (Wikipedia, 2007), which focuses on conceptual learning instead of rote memorization.  Conceptual learning involves students to be creative thinkers, apply content knowledge in multiple situations, and develop effective learning methods.  Students are expected to be metacognitively aware of their own learning habits and deepen their knowledge by utilizing their curiosity to create conjectures and seek out justifications for mathematical thought.  However, change is slow and these changes are not yet seen in mathematics classrooms everywhere.
Why incorporate technology into the mathematics classroom?
With the shift in the algebra curriculum from one of rote manipulations of symbolic exercises to one of analyzing solutions (Yerushalmy & Gilead, 1997), technology plays a key role in the secondary mathematics classroom.  Symbolic manipulation is no longer emphasized because it does not require students to think critically about the results (Yerushalmy & Gilead, 1997).  Students need to be able to acquire solutions and interpret their meaning.  With the use of technology, students do not need to spend endless amounts of time manipulating mathematics, but can spend more time on analyzing it.  Mundane skills to solve problems may be left to be manipulated with the use of technology because there is a push for students to increase their higher ordered thinking skills.  Synthesizing, analyzing, communicating, conjecturing, justifying, and developing independent methods to solve problems are skills that are valued for a successful transition into the 21st century and its technology rich workplace.  Oftentimes, students are not able to transfer knowledge form the mathematics classroom to the real world.  This does not prepare students to be educated beyond the math classroom.  If students are given tools and more opportunities to think critically, they will strengthen their analytical minds. 
Another reason secondary mathematics teachers should infuse technology into their classrooms is to promote equity among gender, ability, and socioeconomic status (Picciano, 2006, p.43).  Accessibility of technology needs to be addressed when teachers integrate technology into their classroom.  Sensitivity to students’ accessibility should be considered when assigning tasks using technology.  However, when these situations are taken into consideration, and teachers reflect their awareness in their technology use with students, the outcomes are beneficial.  Regardless of race, ethnicity, gender, socioeconomic status, disability, or English language ability, all students can gain from using technology.  For instance, with a strong math background, students have better choices about higher education, which is a gateway to more careers (Herzig, 2005, p.254). 
As our need for mathematically literate workers and citizens increases, our economic security will depend on how well we have educated other individuals and incorporated them into full participation in the economy of the twenty-first century.
(Herzig, 2005, p.254)
Providing students the opportunity to learn with technology will broaden their technological understanding along with their content knowledge.  Both are important to be successful in a competitive economic world.  Teachers should level out the playing field for all students.  Therefore, it is essential for secondary math teachers to integrate technology into their daily lessons to provide students with a foundation of technology literacy and deepen their content knowledge by interpreting problems and solutions.
What does a technology rich learning classroom look like?
Constructivism and using technology effectively in the classroom setting are symbiotic.  One characteristic is a “talkative” classroom.  Students should engage in math dialogue regularly because learning and participating are inseparable (Herzig, 2005, p.255).  When students are able to communicate their thoughts, they are taking complex concepts and making them more concrete mentally and expressing them verbally (Jonassen, 2006, p.13).  Communication plays a key role in understanding what students understand.  Another characteristic is the thoughtful arrangement of the classroom.  This should reflect the task on hand.  Workplaces for groups and individuals should be separated.  Computer workstations should be available for students to access.  Resources should be displayed and readily available for students to use.  Third, students should work in groups regularly because it increases student achievement and elicits better communication skills (Walmsley & Muniz, 2003).  When working in groups, students experience working with others with diverse backgrounds and personalities, which will help them in future career interactions.  Developing an openness of to one another’s opinions is essential in the constructivist classroom because this invites students to take risks and share their thoughts.  Students will be able to make conjecture and justifications (Yerushalmy & Gilead, 1997) without fear of being embarrassed.  Teachers will be able to pose problems that require higher ordered thinking skills (Yerushalmy & Gilead, 1997) and students will be able to discuss with one another about their methods.  Cultivating a resilient personality trait for students is advantageous for students to learn to communicate with others in disagreements.  Finally, learning in a constructivist, technology rich classroom is meaningful.  Assigned tasks should be authentic and relevant to students.  The use of technology allows students to explore.  Therefore, teachers should ask more open ended questions.  Students will struggle, but their problem solving and decision making skills will increase (Erickson, 1999, p.520).  In a classroom where technology is infused effectively, students learn both content and technological skills.
What technology can be incorporated into the mathematics classroom?
Virtual manipulatives.
Dynamic representations of the real manipulative are virtual manipulatives, which are usually found on the internet.  Manipulatives offer students the opportunity to construct meaning as oppose to following an algorithm.  They can be slide, turned over, and flipped.  These may be useful in a classroom because when there is a short supply of manipulatives, virtual manipulatives can be used.  In addition, students may access these outside the classroom to help with their assignments and further explore meaning not limited to just the classroom.  Furthermore, virtual manipulatives offer students more control of their learning (Moyer, Bolyard, & Spikell, 2002, p.375).  It provides a non-threatening environment for students to make mistakes and take risks.
Computer Algebra Systems (CAS)
         With secondary mathematics curriculum changing towards conceptual learning, Heid (2002) suggests that CAS are a powerful technology to integrate into the classroom (p.662).  They are designed to display numerical, graphical, and symbolic manipulations.  They also have spreadsheet capacity.  When used effectively, CAS can allow students to “interpret algebraic expressions and use algebraic language to describe real and mathematical worlds” (Heid, 2002, p.664).  Using CAS in the classroom effectively emphasizes the focus of mathematics to analyze the results and not stress mindless algorithms.  Although CAS will derive solutions for students, these solutions do not involve higher ordered thinking but mere procedures.  Therefore, students can delve deeper into understanding the problem itself, various methods to solving the problem, and what the outcomes mean.
Laptops
         Having the capability of being wireless, laptops offer the students more accessibility to computer applications and the internet.  In Levine and Wasmuth’s (2004) case study, two algebra classrooms both taught by Wasmuth was used to investigate the effectiveness of laptops in the classroom.  One class received laptops that were assigned to them at the beginning of the experiment and the control class did not.  The control class was also taught using a traditional teaching style; whereas the laptop class had the unique opportunity for both a traditional class when the laptops were not being used and a dynamic one when they were.  Using final exam scores, Levine and Wasmuth found that the laptop class performed slightly better than the control group.  However, Wasmuth made some non-experimental comments about the laptop group such as students emailed homework questions prior to attending class, so Wasmuth was better prepared to answer questions.  Students were better prepared and spent more time in dialogue than doing algebra (p.139).  This case study is an excellent example of when content and technology can be taught simultaneously.
Websites & Webquests
Often overlooked as useful tools, websites may be used in the classroom as a resource.  Although there are many websites that may be outdated or present misinformation, teachers may research websites and develop webquests to focus students’ attention to pertaining information.  As cited in Wikipedia (2007), educators could also vary the degree of difficulty of the webquest.  Teachers could also include websites that have virtual manipulatives to use so that students may interact and engage in the activity instead of being passive.  Furthermore, webquests may be used in a collaborative group, enhancing students’ cooperation skills.
Spreadsheets
Similar to CAS, spreadsheets also derive solutions for students and display multiple representations of mathematics.  Data patterns can easily be executed, allowing students to discover the pattern themselves.  Spreadsheets are also an easy way to display and organize data.  As states previously, when students are able to organize their internal thoughts externally, a higher ordered thinking skill is being used.  Spreadsheets can be used to easily display graphs allowing students more time to interpret data.  They can also compare and contrast different graphs for more analysis.  Spreadsheets involve the user to understand the algorithm to utilize its functions.  The use of spreadsheets in the mathematics classroom is constructivist in nature, and it follows the NCTM standards moving toward conceptual learning.
Because learners off-load cognitive effort to the computer, they are able to apply more of their effort to understanding the relationships being calculated and represented graphically by the spreadsheet (Jonassen, 2006, p.121).
Geometer’s Sketchpad
The Geometer’s Sketchpad is a mathematics’ software that uses a constructivist approach to learning mathematics.  It provides students tools to create meaning.  There are no pre-programmed applications for students to run, but simply a blank document in which student create shapes and develop interpretations.  Similar to the virtual manipulatives, the Geometer’s Sketchpad offers a non-threatening environment for student to develop conjecture and test them.  Allowing students to construct meaning and develop their own definitions “does not replace basic understanding and skills” (McGehee & Griffith, 2004, p.344).  This dynamic platform offers students the opportunity to investigate relationships, cultivating a synthesizing mind.
Calculator Based Ranger (CBR)
Motion detectors that allow students to gather data involving two variables, CBRs usually measure either time and distance or time and velocity.  CBRs are a wonderful hands-on exercise that allows students to control their own experiments.  They link to the graphing calculator and data may be displayed graphically or numerically.  Students may use this information to develop a deeper understanding of how two measurements relate to one another.
TI Navigator
The TI Navigator allows the teacher to view all of the students’ calculator screens.  This is useful when instructing students how to use functions on the calculator as well as providing quick feedback to the teacher of students’ understanding.  Research shows that the TI Navigator can improve conceptual understanding, increase student achievement, improve student collaboration, and decrease amount of time it takes to complete a task (as cited in Texas Instruments’ TI Navigator Research Support, 2006).  TI Navigators also allow the teacher to monitor when students are off tasked.
Graphing Calculators
Graphing calculators have been debated on their uses in the mathematics classroom.  Teachers fear that calculator use will lead to dependence and does not increase mathematical skills.  However, as pointed out previously, mundane tasks are no longer emphasized and according to research performed on Texas Instruments graphing calculators, they can be used to increase student efficiency of completing tasks. Similarly to the research findings on the TI Navigator, the use of TI calculators showed an increase in student achievement, conceptual learning, and also an increase in student attitudes towards mathematics (as cited in Texas Instruments Graphing Calculator Research Support, 2006).  Furthermore, graphing calculators can help bridge meaning with multiple representations of symbolic, numeric, and graphical mathematics (Erbas, Leford, & Orrill, 2004, p.303.
Discussion Boards
Not only can technology benefit students, but its use can also benefit teachers.  Groth and Bergner (2007) suggest that teachers engage in more dialogue using a discussion board.  Despite location or time (p.530), teachers can communicate and share their ideas and teaching strategies with one another.  In doing so, teaching is no longer an isolated assignment (p.535), but one of collaboration and support.  Seldom are teachers able to meet with one another due to busy schedules, but online discussions promote reflection and personal growth (p.535).
Conclusion
         Albert Einstein also said “Never memorize what you can look up in books” (as cited on Wikiquotes, 2007).  In other words, Einstein realized that rote memory does not benefit the learner.  NCTM also realizes that procedural learning does not create independent learners.  Therefore, secondary mathematics classrooms need to shift from one of procedures to one of conceptual learning.  To do this successfully, math educators should integrate technology into their classroom to alleviate some of the tedious algebraic manipulations and focus on analysis of information.  Once mathematics classrooms follow this shift in curriculum and technology integration, students will benefit in both content and technological skills needed for the 21st century.
         
References
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