The traditional mathematics curriculum focuses the majority of its time on calculation. Conrad Wolfram, the strategic director of Wolfram Research, the developers of mathematical computational software, identifies four key steps to solving real world mathematics problems. 1. Posing the right question, 2. Taking the real world problem and developing it into a mathematical formula, 3. Mathematical computation, 4. Taking the mathematical formulation into a real world understanding. The societal emphasis on comprehending mathematics has been focused nearly all on computation. The model has been one of compute to comprehend, this process forces students to learn the dynamic concepts of math through the tedious process of learning their formula's to understand the connections. In other words, student's understanding comes only after they process an equation with various inputs to learn how those inputs come together to produce different results. Due to the time consuming nature of calculation by hand, the few results students are able to witness over the course of a standard class period does little to build an overarching understanding of the math topic. With little time available the focus on calculation provides little time to examine the other three key steps to solving real world math problems.
The issue of developing a mathematically literate citizenry is in modifying the way the subject is taught. A study conducted by members of the University of Georgia, Detroit Public Schools, and University of Michigan Ann Arbor, studied inner city school math and science classes and identified, "in a typical class session... students were asked to copy data and a formula from an instruction sheet into a particular spreadsheet cells in order to reproduce a teacher-designed product... inability to deal with more flexible and independent modes of computer use can only contribute to the feeling of powerlessness and dependence, which is already too prevalent in urban, predominantly minority communities." (Jackson, Edwards and Berger, 1993) This focuses students on the application of computation, ignoring the dynamics involved in truly understanding math dynamics. While this was a general study of only one city school district, not to mention it was taken twenty years ago, the statistics on math education in America have not changed much.
"Math is much broader than calculation." Wolfram
There are a few companies and research organizations that have begun developing "microworlds". They are simple to understand computer programs which allow a user to examine a math concept by being able to visually manipulate the problem. One of the more dynamic microworld solutions has been developed by Wolfram Research, titled Mathematica. What the creators of Mathematica suggest is that there needs to be a shift in curriculum to help students develop logical thinking. Their software aids students in developing just that, utilizing visual means to convey understanding. (Wolfram, 2010) Mathematica assists in solving that problem; it is an impressive visual platform that provides countless types of math problems and allows the learner to change the inputs to produce new outcomes. It is through this interactive interface that students are able to develop an understanding of how the math works through a visual analysis of the problem.
As Conrad Wolfram explains in his TED Talk, "Teaching kids real math with computers", calculation is just one portion of training students how to understand and utilize Mathematica. The majority of students being accepted to post-secondary institutions will rarely ever have to complete a challenging computational problem, as computers and calculators fulfill that requirement. Those students who never make it to a four-year institution or those who never graduate will typically not need to complete complex mathematics. What all individuals do need is a solid footing in the other three keys to solving real world math problems. The 21st century is filled with truthiness, those near truths are always backed with some type of figure, statistic, or graph that all Americans need to be able to identify, understand, and weed out the seeming implausible statistics and be able to identify the factual ones.
Parse the Practice
Mathematica's design is simple as it allows teachers to select from a plethora of different interactive demonstrations or to create their own utilizing the Mathematica software. Once created or identified, a teacher is able to distribute that singular demonstration or create a do-it-yourself module, in a PowerPoint like format, with the demonstration built-in. It is the interactive demonstration or "microworlds", that makes Mathematica so unique and desirable for teachers and students. In the example below, students are able to learn about finding the distance between two points. In this example there are 5 options that may be manipulated to aid learning. First, students may change the location of either point on the line, view the answer, view the grid lines on the graph, and even receive a hint. This is accompanied by an explanation below that guides the student through the demonstration. This learning ecology allows students to immerse themselves in the process of math without being immediately bogged down in the computation of the equation, allowing them to develop logical thinking.
The below link is to a YouTube video that demonstrated modifying multiple inputs to show the effect on a visual demonstration.
http://www.youtube.com/watch?v=dQfI1D5PeaA
Mathematica is designed to be used in a way that facilitates communication in several different ways: as a resource of demonstration during a lecture, through a blended learning environment, for student led discovery either supervised by a teacher or individually led, and as an assessment tool for academically advanced students.
Within a traditional classroom setting, math teachers are often tasked with explaining topics that are hard to visualize. For example, within a lecture discussing the curve of a line when the slope is 1/X, it is challenging to impart the understanding to students that the line will never touch the X-axis. Mathematica allows the teacher to provide a demonstration by modifying the variables to that end. Within a classroom discussion, Mathematica allows a teacher to approach group assessments in a number of ways, starting with the example of the visual outcome and have the students identify what the inputs must be, or provide them the inputs and expect them to develop the outcome. This tool, in the hands of a trained educator can bring nearly any mathematic topic to life. (High School Math Teacher, 2013)
Just as a teacher could integrate this software within a lecture, a teacher who prefers a blended learning, given a 1:1 student to device ration, may take the same lesson and modify it for individualized computer to student instruction, transitioning a standard math class to one that is more representative of a science class. Students are able to hypothesize what certain inputs could produce and identifying the results aiding in the development of more logic based thinking. With the presentation mode of Mathematica, teachers are able to build entire modules on any topic they wish, they may pull in previously created interactive demonstrations or they may create their own. These demonstrations may be integrated within these presentations to aid student learning. While using a blended curriculum, teachers may move freely to aid students who are struggling and if needed are able to bring the entire class together to work on a singular demonstration with the class. Within this role teachers span the gamut of creators, tutors, and possibly instructors.
Fairly similarly to the blended learning instruction, teachers may develop independent modules for students to complete in a distance learning model. While the software doesn't facilitate reporting and assessment, it can very easily play the role of instructor. Additional software is required to assess student performance. To supplement the materials, the teacher may wish to produce or pair the module with already created instructional videos in order to communicate the basics of a concept to their students. (An additional program is required to produce videos) Many simple modules have already been created and are posted on the Wolfram site for no additional charge. The large amount of resources already available allows learners who wish to gain a desired skill set or additional knowledge on their own to pursue their those mathematic based interests.
Mathematica allows for new demonstrations to be created by students. This process allows for nearly an unlimited amount of options for teachers to challenge students. While the user interface is simple, understanding a math concept, and the calculations required, may prove to be a unique challenge to students learning a concept. This software does not utilize peer-peer communication, but within a complex project creating a Mathematica demonstration, a group of students could be challenged with the development of their own microworld.
Underlying Learning Theory
While the use of any visual to assist in learning brings about a discussion of learning theory, the aspects and techniques of the microworld environments created in Mathematica pull ideas, and pedagogies from an even broader range. Prior to a discussion of how Mathematic improves understanding it is important to define understanding within a deeper context. "To be said to have an understanding, you have to be able to put this preposition through its paces, explaining and predicting novel cases. To have an understanding is to be in a state of readiness to perform in a certain way taking the representation as a point of departure." (Perkins and Under, 1994) While this idea of being able to "explain and predict novel cases", may seem simple, it requires a learner to not only internalize a concept fully but to be able to apply the principles of that concept to other "novel" situations and examples. While the microworlds of Mathematica are most likely not going to take a student through to this advanced level of understanding on it's own, it will indeed assist in the learning process.
Microworlds and Learning Theory
The Van Hiele Theory provides a conceptual framework for which to better understand where a visual representation may fit into the learning process. Dina van Hiele-Geldof "had the objective 'to investigate the improvement of learning performance by a change in the learning method.'" This is exactly what microworlds such as Mathematica endeavor to accomplish, modify the traditional form of calculation and by hand manipulation of the world, to one that is computer assisted. Van Hiele's model is broken down into three levels "Visual-Holistic Reasoning", "Analytic-Componential Reasoning", and "Relational-Inferential Property-Based Reasoning".
| Dina Van Hiele-Geldof - Levels of Learning Theory | |
| Level 1: Visual-Holistic Reasoning | "Students identify, describe, and reason about shapes and other geometric configurations according to their appearance as visual wholes, […] they may refer to visual prototypes. […]. Orientation on figures may strongly affect students’ shape identifications” (p.851). |
| Level 2: Analytic-Componential Reasoning | "Students [acquire through instruction] a) an increasing ability and inclination to account for the spatial structure of shapes by analyzing their parts and how their parts are related and b) an increasing ability to understand and apply formal geometric concepts in analyzing relationships between parts of shapes”. This level incorporates “visual-informal componential reasoning, informal and insufficient-formal componential reasoning, and sufficient formal property-based reasoning” (pp.851-852). |
| Level 3: Relational-Inferential Property-Based Reasoning | "Students explicitly interrelate and make inferences about geometric properties of shapes.[…] The verbally-stated properties themselves are interiorized so that they can be meaningfully decomposed, analyzed, and applied to various shapes”. This level incorporates “empirical relations, componential analysis, logical inference, hierarchical shape, classif ication based on logical inference” (pp.852-853) |
(Patsiomitou, Barkatsas and Emvalotis, 2010)
Within Mathematica's dynamic interactive demonstrations/representations, learners are able to manipulate the graphic. It is through he visualization and manipulation that learners are constantly stretching across all three levels of van Hiele's learning levels, from the ability to change the perspective of the original object within level one, to slight modifications of the object's properties in level two understanding the relationships between the different variables, through to being able to make inferences about the object based off now more intuitive understanding of how the variables affect the object. (ibid) The ability to reach across multiple levels of lower to higher order thinking needs but a guide whether in the form of a teacher or learning module to guide the attention of a student to specific portions of a microworld to assist in the scaffolding of the learner's understanding, or in a case of a highly motivated learner potentially gaining a higher level of comprehension on their own.
In the idea of a highly motivated and capable individual learner gaining a high level understanding purely through individual manipulation of a microworld could be a stretch of this writer's imagination. The goal of teaching should not be to facilitate a learner who relies on a teacher, but to develop a learner who can learn on one's own. In order to develop the autodidact, one must be very cognizant of training the learner to monitor their cognitive load. If a student feels overwhelmed they are more likely to prematurely quit, assuming that the challenge they are facing is beyond their grasp, while that is not the case. "The general idea is that, as learners mature and automatize skills, available working memory increase, allowing the encoding of more complex cognitive structures." (Perkins and Unger, 1994) It is finding ways to free memory through mastery of simpler concepts that learners are able to reach higher levels of understanding. As this constructivist approach develops learners are able to use more mental power to process the challenge at hand. Mathematica provides an extremely unique path to aid in the learning process. By purely focusing on the changes that occur from different inputs to mathematical challenges aid in the learning process; temporarily students are able to focus completely on gaining a logical understanding of the forces at work. Freeing the mind of the calculation reduces the cognitive load allowing the learner to make even bigger leaps in higher order thinking.
The Tools and Methods of Microworlds
While Mathematica's approach works within higher level learning theory, there are specific tools and techniques of the software that contribute to making this software truly impressive; Among those are the interactive controls, the variety of graphical interfaces, and the ability to use with real world problems. These functions and opportunities together aid in producing an impressive opportunity for teachers and learners.
The multimodal learning opportunity created by the microworlds of Mathematica allow for more than just an animation. As has previously been mentioned the ability to personally alter the inputs allows a learner to better understand the concept. it is with microworld software that students are able to take the helm of the inputs as opposed to simply watching an animation. While a constructivist approach suggests that learners are developing their understanding of the world through a process of assimilation and accommodation. when looking at a single image or graphical example of a math concept a learner must work that example into their understanding of the world. This process is very limiting; if the learner is exposed to a different image of the same math concept, it may look so different that the process of accommodation may be beyond the learner's grasp. The same issue may occur with an animation of the extremes of a math concept. Being a passive witness to the modification of inputs which are then also demonstrated in the output example may be challenging for a learner to grasp. By making the learner an active contributor in the simulation, they are able to see the situation within their own perspective better. They are able to modify the microworld, in this case, to fulfill their own requirements to accommodate this new information within their understanding of the world. (Rieber, 1992)
It is within those microworlds that a new opportunity arises for students to gain improved understanding. In many of Piaget's later writings he discussed a topic he coined as reflective abstraction, the process of gaining understanding "new syntheses in midst of which particular laws acquire new meaning" (Tall and Dubinsky, 1991). This complex concept plays third fiddle to two necessary steps that will in some cases inevitably precede it at some point. This is intentionally a confusing statement as the steps work together to provide higher level understanding of all mathematical topics as an individual develops. Much like the idea of developing a schema set leads to greater understanding within a social science framework. The process of reflective abstraction develops learner's understandings of the stricter world of mathematical law. This learning process will typically begin with empirical abstraction, "this kind of abstraction leads to the extraction of common properties of objects and extensional generalizations". As a learner first examines a cube within the realm of a mathematics class, they will gather and internalize the basic features of six faces that have the same length on each of the twelve edges. As that information is gathered it is internalized within the framework of that individuals experiences. (Ibid)
As the process of empirical abstraction takes place, pseudo-empirical abstraction will most likely occur as well. This process includes assumptions that the learner may make during their experience that may or may not be accurate. For example, the learner may identify the six sided object as a cube and may assume that other six sided objects that are longer than they are wider are also a cube. This broad generalization was selected to simplify the concept and may grow to be far more complex as they grow older. The process of understanding continues with reflective abstraction. As the learner begins to assimilate other shapes that are not cubes they begin to reformulate their schemas, and understand additional laws of cubes.(Ibid)
The Approach in Practice
Mathematica, is at its essence what is known in math education as a "microworld" where specific concepts may be manipulated and tested within computer software. These microworlds are intended to "free students form the labors of compass and ruler manipulation... and allow them to freely and flexibly explore geometric conjectures, thus restoring an element of discovery to the usually entirely formal deductive study." (Perkins and Unger, 1994) While have been no studies conducted on Mathematica there have been on other similar microworlds. The following are studies and quotes from Geometer's Sketchpad, similar software.
Due to the focus of this section on a different piece of software it is key to understand their similarities and differences. Both software allows students to manipulate shapes, figures, and angles. Mathematica approaches the topic from providing students with a completed demonstration, while Geometer's Sketchpad provides a blank canvas for students to create and manipulate their own figures, angles, and shapes. Due to the differences and similarities the following studies have been selected due to their focus on the similar aspects of the two microworlds.
Reflective Abstraction in High School Geometry Class
A study was conducted on tenth grade Honors Geometry students enrolled in a high school near a large university. Out of the sixteen students in the class four were selected to participate in the study "who reflected a range of mathematical abilities and technological familiarity." The study lasted a total of seven weeks. The focus for the study was to "engage students in thinking about transformations as functions as they used technology and they encouraged students to provide mathematical explanations and justifications for the phenomena they observed. This enabled the researcher to examine students’ mathematical explanations for the purpose of characterizing students’ understandings of transformational geometry they were developing." (Ibid) This study conducted on a small group of students allowed the researcher to focus on how students interacted with and understood the material.
The research questions for this study asked, "What is the nature of Geometry students’ understandings of geometric transformations (i.e., translations, reflections, rotations, and dilations) when instruction capitalizes on the technological tool, Geometer's Sketchpad?"(Ibid) The researcher was very detailed in their approach to understanding what misconceptions student's developed and how they developed new understandings, whether it was through the software or through other means. As the students worked more heavily with the software the researcher identified that certain pseudo-empirical abstractions were made that challenged the student's understanding of tasks being learned. The conclusion was made that possibly the use of the computer could have led the students to those errors. Though by the end of the unit 3 out of the 4 reached a level of reflective abstraction that corrected their misconception. In the end the study clearly identifies that this study was not enough to identify the strength of utilizing microworlds to improve math education and that while it did show promise more study is required. Since, the topics covered in this study may be executed with both software it may be unreasonable to simply assume that similar results could have been achieved through Mathematica. Although it is a fairly safe bet that no software is perfect and that pseudo-empirical abstractions will be made by learners utilizing both applications, and due to the similarities of the programs and their design to assist learners in developing reflective abstractions, learners would develop proper comprehension of these topics.
Utilizing Visual Active Representations
Within a study conducted at a public high school in Athens Greece utilizing Geometer's Sketchpad environment, a similar software to Mathematica, they wished to answer two research questions:
1. How does the building of LVAR (linking visual active representations) modes impact on students’ transformation of verbal statements with regard to the construction of meanings, conjecturing and the proving process?
2. Does the building and transforming of LVAR modes lead students to structure mental transformations that correspond to the development of their van Hiele level? (Patsiomitou, Barkatsas and Emvalotis, 2010)
Much different from the previous study discussed, this one focuses on the development of internal abstraction from actual visual representations made from Geometer's Sketchpad. The authors of this study chose to include the following as a key piece to understand their perspective. "In the field of Mathematics the term representation has also been defined by Pape & Tchoshanov (2001) 'as internal—abstractions of mathematical ideas or cognitive schemata that are developed by a learner through experience. […] external manifestations of mathematical concepts that ‘act as stimuli on the senses’ and help us understand these concepts, […] also refers to the act of externalizing an internal, mental abstraction'". (Ibid) The researchers within this study focus on building understanding of the property of triangles. Prior to starting the study on the high school students they gave them an initial assessment on the level of geometric thought. While at the end of the study they do not clearly identify figures on gains or losses of participants they do take time to identify that students who demonstrated the most basic level of geometric thought were able to articulate much higher levels of logical mathematical thinking at the end of the study. (Ibid)
Critical Reflection
The interactive demonstrations provided within the microworlds of Mathematica provide plenty of pedagogical and theoretical approaches that show major signs of promise. With those exciting promises the software's costs, challenges of use, and limitations do leave something to be desired. The understanding of how theory and practice applies to Mathematica has been slightly skewed due to the lack of research on it and the various studies that have been identified from the similar software Geometer's Sketchpad.
The outstanding visuals that are produced by Mathematica, and which are very similar to that of Geometer's sketchpad were highly touted within the LVAR study as being key to the development of conceptual framework, that along with the study based around reflective abstraction, it would be logical to assume that the software is capable of assisting students in making large leaps in their comprehension of at least geometry. Additional points were given via testimonials by secondary teachers using Geometer's Sketchpad. the comments provided were similarly related to actions that Mathematica is capable of completing. Geometer's sketchpad provides the opportunity for "a much more memorable and substantive lesson." "Sketchpad allows students to discover Geometery, I don't have to tell them." "It changed the level of question that we had in the classroom instead of what is we could talk about how does this behave when you move something in this kind of way." (Edsoft, 2011)
Mathematica enables teachers to develop their own interactive demonstrations for any possible equation or concept they wish to teach, covering algebra to calculus. Most demonstrations empower the user with sliders that enable the learner to try countless different options to understand the range of the topic. As teachers provide the demonstrations, they are able to pair them with a suite of how-to instructions that enable students to quickly learn how to use the software. If an a-synchronous lesson is not enough for students, the teacher is just as easily able to interact with Mathematica in front of a class with a smart enables teachers to work on the fly, providing examples that will best help a class understand a concept.
While there are many positives to Mathematica there are also several drawbacks. The software does require some learning for teachers to be able to create their own demonstrations. This learning curve can provide a serious challenge to some teachers who may struggle to develop their own classes. A smart board is required to maximize the freedom of use while lecturing. While a teacher may use a projector but would require a mouse to interface with the software. There are no additional video or audio options to allow the teacher to create in depth explanations or additional visuals to help students understand.
While students work on their demonstrations, there is no form of communication built-in for students to share information with other students or to the teacher. The application remains an informative tool to be used on a single computer. Students are capable of creating their own demonstrations, but there is no "free" workspace for students to simply create geometric shapes and play with them to learn their properties. Each demonstration must be completed prior to class and downloaded to each workstation for students to work with them. Lastly, there is a fee for licensing to schools.
Conclusion and Recommendations
The diagram below provides a great overview of what a microworld like Mathematica is capable of assisting with. In the past, students have been limited to just the real and mathematical world. If they wanted to attempt to visualize information it would be limited to images in a textbook, personal drawings, or predesigned animations in video. Micro worlds allow for actual experimentation within a classroom, and that brings with it the ability to reason about geometry. That reasoning allows students to develop a higher order of thinking that they have not had the opportunity to do in the past.
Throughout this study there is one certainty. Providing computer representations for students learning math, is grounded in several pedagogical theories and has shown serious signs of being highly effective within studies. The fact that Mathematica's visual representation provides students the ability to reach higher levels of thinking through just the ability to see the range of possibilities, along with the ability for teachers to use the demonstrations within their own direct instruction provides an impressive combination that should not be easily overlooked, but the two major challenges with Mathematica are its cost and ease of use for both teachers and students. An additional weakness of the software is its inability to communicate information to fellow students and teachers, but since other similar software also lacks the option to facilitate communication it can hardly be held against this software. Due to a challenging format to create new demonstrations and a lack of a free space for students Mathematica may not be the best source for all schools. But if the teachers are willing to put in the work it takes to produce demonstrations to get results from students it will most likely provide a major payoff for learners.
References
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