Literature+Review

=Literature Review=

math \[\sum\limits_{k = 1}^\infty {\frac{1}} \] math

math \[\int_0^\pi {\sin \left( x \right)} \,dx\] math  Mathematics is a core subject in all school curriculums. It starts in early elementary school where children learn arithmetic as their first introduction to the subject. Once they have moved on to high school, algebra is introduced as an entirely separate entity. Most junior highs however, offer a pre-algebra class as a transition between these two seemingly different worlds of mathematics. Historically Algebra was very slow to develop. It might come as a surprise to find out that modern notation actually was not used until the sixteenth century (Sfard 1995). Before this, these things we expressed in words or pictures. This meant that it was very difficult to build on ideas since expressing them took so much room/time. Since there was an inability to condense ideas and organize them it meant that is was extremely hard for people to make these ideas something objective that they could build off of (Sfard 1995). Historically Mathematicians were very slow to accept things like imaginary numbers and other abstract ideas, due to the very operational nature of their notations (Sfard 1995), “These kinds of historical phenomena remind us that the process of symbolization in school mathematics has a deeply transformative effect on what is known” (Kaput, Blanton, & Moreno 2008). Concepts that were the hardest to gain acceptance in the mathematical community historically are often the same ones that give students the most trouble (Sfard 1995). New ideas were brought into the mathematical community with a great deal of mistrust, yet when introducing these concepts to students, we often expect them to accept it as truth and to understand it easily (Sfard 1995).

Due to older children’s struggles with Algebra it is often said that younger children are not capable of learning algebra (Schliemann, Carraher & Brizuela 2007). This assumption is the reason given for teaching arithmetic completely separate from, and before algebra. However, Schliemann, Carraher & Brizuela performed a series of experiments in which they showed that children as young as first grade can understand basic algebra concepts (2007). Contrary to the way that it is taught in schools arithmetic is a proper subset of algebra, and therefore algebra should be integrated into the curriculum much earlier. By adding a generality to the arithmetic a seamless switch to algebra can be made (Sfard 1995). By considering arithmetic as a part of algebra encourages us to view isolated examples and topics as instances of more abstract ideas and concepts (Schliemann, Carraher & Brizuela 2007). According to Kaput this generalizing of arithmetic operations is the first strand and primary role of the transition to algebra (2008). When this generalizing is placed on a more specific context such as functions the algebra becomes more dependent on patterns and on the domains of its specificity, modeled in Kaput’s second strand. When this second strand is used to come up with a specific solution it leads to an expression; this is Kaput’s third strand (2008). In other words once a student can express arithmetic in a generalized way, using symbols and their own expressions, they have become able to reason algebraically.

According to Kaput there are two main aspects of algebraic reasoning that are shared with most branches of mathematics (2008). These two aspects being the ability to generalize and use generalized expressions, and the ability to use specialized systems of symbols (Kaput 2008). Kaput describes them as, “Algebra as systematically symbolizing generalizations of regularities and constraints,” And, “Algebra as syntactically guided reasoning and actions on generalizations expressed in conventional symbol systems” (2008). Graham and Thomas also refer to this idea calling it object and process (2000). In other words, owning the concept as something to be built off of, versus the way in which a concept is gained and the act of using it as a process. When a new concept is introduced three levels of understanding are used by the student to grasp this new concept and make it their own (Heid & Blume 2008). The first two are easily relatable to the concept of process. They are having an understanding at the “Action-level” and at the “Process-level,” which means being able to go through the actions of a concept, and then being able to view the process as a whole (Heid & Blume 2008). The third level is “Object-level” which is being able to use the process as an object, to “act on it without carrying it out in actuality or mentally” (Heid & Blume 2008). Reaching this “Object-level” of understanding is important for a student. If they never get there they will be unable to build off of that concept, and will have trouble understanding higher math.

To build off the idea of process versus object, Tall introduced the idea of a concept image in regards to a concept definition. The concept image is all cognitive structures including objects and mental pictures that are connected with the concept (Tall 1981). The concept definition, which can be both formal and personal, are the words that specify the concept. The definition can lead to its own concept image, or it can influence the image already formed by the student. These images can be built from informal usage but can also be shaped after a formal definition is given. The image can also be shaped by experiences students encounter in the classroom (Tall 1981). For example, when new symbols are introduced they affect the student’s perceptions of the world. (Then into represented and representing?) The represented world is the “actual world” whereas; the representing world is a system of symbols. This process of using symbols allows us to express and reason with generality and further symbolization (Kaput 2008). Once we have generalized an expression, it is possible to then further build upon this expression. This now gives a new view of the problem. This in turn changes their concept image and/or definition and shifts their way of thinking. This is easily represented by Kaput’s model of the continuing process of symbolization (Figure 1) (Kaput 2008).

Kaput uses the idea of the represented world and the representing world to describe the process of symbolization. Figure 1 shows this process and its continuous nature (Kaput 2008). As shown in this model experiences are analyzed and deciphered forming representations that together produce a new way of thinking. This new idea then has its own experiences and representations of these experiences that form another way of thinking. This process continually shapes the ideas and concepts of a student. Likewise, in Kutzler’s learning spiral (figure 2), concepts undergo three phases that help develop new ideas into building blocks for further knowledge (Kutzler 2008). In the experimentation phase students can find examples and observe patterns to form conjectures. These are then proven and later are implemented into a new concept in the exactification phase. In the final stage of application, the new concept is used to produce new examples and patterns. Due to the continuous nature of this process many new concepts are developed and a way of organizing them needs to be found. In order to organize and simplify thoughts and ideas symbols are often developed.

However, using symbols can sometimes be a complicated process for a student. Kaput compares it to using the windshield while driving, because when driving it’s impossible to look at a spot on the windshield and look through the windshield at the road, all at once. This is analogous to understanding the concept of symbols and variables. Students have a hard time seeing the symbol as a set object, and trying to look through it and see it as a variable at the same time. They miss the generality of the symbol when they see it as a set object. To be able to understand how the variable changes they have to look through what they already know (Kaput 2008). For example sometimes students have a hard time translating word problems into algebraic expressions. In one case when asked to represent 6 times as many students as professors many students wrote the equation as. This of course is incorrect, as that would mean there are six times as many professors as students. This is not the only translation issue however. It has also been shown that students have trouble using letters, as they tend to relate them to their position in the alphabet, for example because y comes after x in the alphabet (Wagner 1983). These translation issues can be problematic; however they are not the only ones.

Another type of issue many students have with symbols is not being able to “unpack” them (source or was that all Dr. Lapp?). Once a concept is understood and placed into a symbol a student needs to be able to see what that symbol means every time they look at it. Sometimes though a student has trouble with this and is seeing nothing but the symbol. This inability to call up the concept that the symbol is based in causes a rift in the student’s understanding and can be very confusing. This problem arises from not fully understanding a concept before packing it into a symbol. The concept needs to be fully understood, so that it can be both packed and unpacked as needed. Being able to use symbolization is very important to algebraic reasoning. According to Kaput “The heart of algebraic reasoning is comprised of complex symbolization processes that serve purposeful generalizations and reasoning with them” (2008). So understanding of a concept is vitally important to using symbolization properly, and to the ability to reason algebraically.

A students understanding of a concept is dependent on past knowledge. In order to build new knowledge and concepts, a student needs something to build on. In the “onion model” (figure3) created by Pirie and Kieren, they emphasize a progression through concepts in which students can “fold back” to previous layers of knowledge so that they may deepen their understanding of the concept in its entirety. They suggest that “The model has a fractal-like quality: inspection of any particular primitive knowing will reveal the layers of inner knowings” (1994). Kutzler related this to building a house. Before building the second level, a first level needs to be completed (2008). This poses an issue in a classroom because not all students progress at the same rate. Due to this, some students do not have a complete level of old knowledge to base their new level on. Kutzler suggests the use of a computer algebra system (CAS) to serve as a scaffolding of sorts. This allows a student to learn higher level concepts without making lower level mistakes that may hinder their understanding. CAS also allows for exploration in new ways. Heid and Blume performed a study in which there were two classes one with calculators and one without. In the “calculator class” the students were less likely to over generalize about specific functions (2008). These students were able make connections without prior knowledge of these functions and their characteristics. A similar study was done focusing on variables and the concept of variables. The students were able to reach a deeper understanding of specific concepts when only taught on a broader scale (Graham & Thomas 2000).

Due to its powerful nature it is important for the CAS system to be used appropriately. There are three different roles that technology can play in the classroom; tutor, tutee, and tool. As a tutor the technology is used to teach the child. If the technology is being “taught” through programming by the child it is being used as a tutee. Finally the tool mode is where the technology is being used for application programs (McCoy 1989). The CAS can play these roles in the classroom, but it is imperative that they are not used improperly. Such as (not using as a crutch). Kutzler uses the analogy of technology being compared to different modes of transportation, a bike, a car and walking. People usually don’t drive to the mail box or walk across the country. “Despite the possible misuse of [transportation], we do not demand its abolition. Likewise, we should not banish calculators and computers simply because some students use them improperly” (Kutzler 2008).

One of the most useful pedagogical features of the CAS is its ability to offer a student immediate feedback. In studies involving kinematics labs in a physics classroom, immediate feedback and student control were found to have a profound effect on the depth of understanding a student had of the connection between the physical event and the graph that was produced (Beichner 1990). “Microcomputer-based laboratory (MBL) experiences are useful in helping students understand the relationships between physical events and graphs representing those events” (Beichner, 1990). The ability to be involved in the process was extremely helpful to the students’ understanding. In addition Brasell did a study in a kinematics lab where the graph was shown only 20 seconds after the event, the students in this group did no better than the students who did not see the event at all. However those students who saw the graph immediately after the event had significant improvement (1987). Immediate feedback is also important as it allows for a student to make abstract graphs and ideas into concrete connections in their head (Hale 2000). It can also offer a way to break common misconceptions by being able to see many examples (Hale 2000). The immediacy of the connection is extremely important to the building of concepts and connections.

The CAS offers this same type of involvement, control and immediacy in an algebraic setting. By giving the students the ability to control the representations, it allows them to find a deeper understanding of those concepts. In addition it offers immediate feedback to any change the student may make. Using the CAS and having control over the representation, as well as it’s immediacy in response allows a student to find different avenues of exploration. They can feel free to experiment with different representations, and not fear miscalculations. Referring back to Figure 2, experimentation is a very important part of the learning process as it is the building block of new conceptual ideas (Kutzler 2008). The CAS offers the ability to experiment and therefore has the potential to deepen students understanding of concepts, and their connections between different representations.

The CAS is a very powerful tool however and must be used appropriately. The teacher’s approach at using this tool in the classroom is very influential to the positive or negative effects it can have in the student’s understanding. A teacher should give great care to if and how they use this tool (Kutzler 2008). There are a few things that teachers can implement while working with the CAS to help increase student understanding. One of which is asking open-ended questions while using the CAS to get the students really thinking about what is going on (Kutzler 2008). This also helps the students reach an understanding level when they can begin to express what they know. Pirie and Kieren say that “A lack of ‘expressing’ activity seems to inhibit the students from moving beyond their previous image” (1994). Open-ended questions therefore are an important compliment to the use of CAS.  Another thing a teacher can do while working with the CAS is to make sure there is a reflection component in what the class is doing. Reflection on the practice is another way to get students to think. One way of reflecting is having class and group discussions not only with the purpose of deepening understanding but of preventing groups from “converging on misconceptions” (Lapp 2000). Technology can often lend misconceptions of ideas so when using technology it becomes important to repair any misconceptions that may develop (Hale 2000). Teachers have to take extra care not to use the technology or give examples in a way that helps to confirm these misconceptions (Lapp 2000). Teachers can use cognitive conflict to motivate “sense-making.” Often times students over generalize a concept, and therefore form misconceptions. This means that a teacher should provide an example that contradicts over generalizations, this provides conflict to the student, and allows them to fix their misconceptions. Hale brings this up in his 2000 article when he states that “Students cannot repair their misconceptions until they are confronted by them” (p.417). (Tall and Vinner p.2-3; cognative conflict factors)  Hied and Blume also stress that it is not only doing and seeing the actions but the reflection on those actions and the results of them that is important (2008). In Pirie and Kieren’s onion model (figure 3), it is shown how important being able to express ideas is to the student’s learning. Expressing the concepts is a necessary step to fully understanding them, and being able to build off of them (1994). As Hied puts it, “…learning occurs through student’s deliberate action and their reflection on that action [this] suggests that students should generate instances of a pattern and also reflect on the meaning of those results” (2003). A really good way to ensure reflection in a classroom is to utilize group work. By getting a class to discuss in groups they are given an opportunity to try and express what they know to their peers. Hied and Blume found in their study that mathematical insight comes mostly from discussion with peers first (2008). Group work is a great way to get students interacting and it helps to solidify expression of different representations (McCoy, Baker & Little 1996). Group work allows student to discover things on their own, and to reach a point where they can express what they are learning.