8th Grade – Linear Relationships
Standards- and Research-Based Study of a Curricular Topic
Section and Outcome
I. Identify Adult Content Knowledge II. Consider Instructional Implications III. Identify Concepts and Specific Ideas IV. Examine Research on Student Learning V. Examine Coherency and Articulation VI. Clarify State Standards, 21st Century Skills, and District Curriculum |
Selected Sources and Readings for Study and Reflection
IA. Science for All Americans (Nature of Mathematics, Symbolic Relationships, and Mathematical Models) „ Linear relationships contain or sometimes overlap with many other concepts such as symbolic relationships, mathematical modeling, and the skill of recognizing patterns and connections. This resource stressed the importance of mathematics to the scientific endeavor while also highlighting the beauty and simplicity in mathematical reasoning rather than just the importance of applications. Specifically, recognizing connections such as the relationship of the symbolic representations of algebra to the spatial representations of geometry is an important skill. The reading suggested three phases of mathematical inquiry, which seem to me especially helpful in understanding the overall intent of the subject throughout a K-12 education: abstraction and symbolic representation, manipulating mathematical statements, and application. These are all practical uses of mathematics with which any math literate adult should feel comfortable. Furthermore, adults should be aware of the limitations of mathematical modeling. It is relatively easy to define a rule or relationship between two variables, especially when it is a linear relationship. However, over a wide range of conditions such complex circumstances may not correlate to the relationship. The example given was that of velocity = (gravity)(time) – this function fits for rocks falling from rest for a few meters, but a leaf or a feather, for instance may not correspond to such a relationship. IB. Beyond Numeracy (Algebra – Some Basic Principles, Variables and Pronouns) „ Linear relationships fall under the broader category of Algebra. Algebraic reasoning involves the “restoration and balancing” of equations or systems of equations, as the title of Al-Khowarizmi’s Al-jabr wa’l Muqabalah translates. I hadn’t quite considered it this way before, but vital to this process is the “crucial understanding that the variables stand for numbers and hence manipulations of them must be governed by the same arithmetic rules that govern numbers” (pg. 9). Without this foundation, generalizing the basic principles of algebra would be nearly impossible. Paulos draws the analogy of variables to pronouns and constants to nouns, something I had not considered before. An example was given of a rather lengthy sentence in which many pronouns were used. The sentence was re-written with pronouns replaced by general nouns (ie, his = this selfsame person’s, it = this thing, him = this person, etc). I thought this would be a great example to relate to students because it is easy to generalize variables as being too abstract and complicated. IIA: Benchmarks for Science Literacy „ As stated in NCTM Standards, during the middle grades "work with patterns should emphasize concrete situations and be informal and relatively unburdened by symbolism." However, teaching sometimes tends to focus on the skill set necessary to solve formal representations of functions. A stream of consciousness vignette was provided to illustrate this point. Given the equation s=1/2at2 and values for s and a, the student reasoned through possible values of t. This careless plug-and-chug approach indicates a lack of understanding of variables. As the resource suggests, “variables should not be approached through abstract definition but rather through real-world situations familiar to students in which they can understand, perhaps even be interested in, the multiple possibilities for value.” Another possibility for illustrating linear relationships is the use of shapes, rectangles and triangles in particular. Incorporating this aspect of geometry may aid in deeper connections between the two topics. Constancy and change presents another challenge to most middle school students’ understanding. Students first must fully understand constant rates of change before considering the increasing and decreasing rates of change that transcends the purposes of mere calculation and computation. I did not initially think to look at shapes, constancy and change to better understand linear relationships and their symbolic representation. However, this resource emphasized how these three topics are intertwined and develop in complexity from Kindergarten to twelfth grade. IIIB: NCTM Principles and Standards for School Mathematics „ Some students may make it all the way through calculus not truly understanding change. Without this foundation, students may face great difficulty understanding functions. Ideas of changes should be formed and strengthened by analyzing change in multiple contexts (pg. 40). In middle school, in particular, the focus should be on quantities that change (pg. 229). Effective questioning can serve to focus students on basic and important issues determined by the relationship represented, which in turn can teach students how to interpret and analyze a table or graph. In the example problem comparing to phone plans, questions can lead students to a formula, to observations of similarities and differences between each plan, or to particular characteristics such as the y-intercept, x-intercept, slope, etc. Observing the similarities and differences between each plan would lead into finding a point where the two plans have the same cost at a certain number of minutes used. Such an exercise can lay the groundwork for solving systems of linear equations. Students often struggle to understand variables because they have not had enough experience exploring relationships of quantities in multiple contexts. As a teacher, I recognize that there is a need for me to increase students’ familiarity with variables in various meaningful contexts. IIIA: Benchmarks for Science Literacy „ The learning goals that align well with this topic are as follows: - Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes. - An equation containing a variable may be true for just one value of the variable. - Tables, graphs, and symbols are alternate ways of representing data and relationships that can be translated from one to another. - A graph represents all the values that satisfy an equation, and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where the graphs intersect. - Geometric shapes and relationships can be described in terms of symbols and numbers and vice versa. Some of the above are simply skills, procedures, or specific ideas. However, the concept of an object’s change represented in multiple ways is at heart in these learning goals. These goals help me focus on this overarching idea of change rather than just specific skills that aid to interpret only particular representations of data. IIB: NCTM Principles and Standards for School Mathematics – What facts, concepts, principles, and broader generalizations are embedded in the standards? How do the expectations listed help clarify what the standard involves? „ Specifically, the following expectation which can similarly be a learning goal aligns well with the topic of linear relationships: “write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency – mentally or with paper and pencil in simple cases and using technology in all cases” (pg. 296). Embedded within this learning goal is the primary understanding of patterns, relations, and linear functions, and the various ways of representing them. The following standards of algebra for K-12 provide insight to the skills and understanding that should be developing (pg. 296): - to understand patterns, relations, and functions - to represent and analyze mathematical situations and structures using algebraic symbols - to use mathematical models to represent and understand quantitative relationships - to analyze change in various contexts In this sense, the Benchmark ideas and those of the NCTM Principles and Standards for School Mathematics complement each other. The broader algebra standards from NCTM encompass the specific learning goals from the Benchmarks For Scientific Literacy. However, the Benchmarks force you to consider linear relationships over a broader context than pure mathematics. More emphasis is placed on the changing type of variable than on a generalized function. The expectations clarify what understanding is to be developed for each set of grades. Looking at the progression of complexity from elementary expectations all the way to high school expectations tells me what prior knowledge my advanced 8th grade students should have. For instance, by this point in their education, my students should understand the difference between linear and nonlinear functions. IVA: Benchmarks for Science Literacy „ Generally, students grapple with the idea of change, and more specifically, rates of change. They ought to be able to interpret slope from a graph, but the research shows that unless it is explicitly illustrated, many students struggle to do this. The idea of a rate is not very well understood by students and as a consequence, rates of change seem particularly abstract. Rectangular shapes can be a powerful visual tool to demonstrate rates of change. However, before this relationship is illustrated, students must first be able to recognize specific properties of shapes, and building from that, students should be able to analyze relationships between shapes. Students ought to have this knowledge by the time they reach middle school. Algebra as a whole can first be introduced by symbols, followed by relationships represented by symbols, and finally, manipulating these relationships to find solutions or other useful information. Because it the use of symbols is more abstract, students struggle to make sense of it, especially if by the middle grades they have not had enough preliminary experience with symbols. Technology can greatly enhance student’s understanding of algebra, but it can also detract if students do not first compute and graph functions by hand. Through the research, teachers and collaborators can in some ways generalize the difficulties in understanding students face at different grade levels. Consequently, if some topic is not being intellectually digested at the associated grade level, introduction to that topic may need to be postponed to a later time whether it is pushed back further in the sequence of topics covered for that grade level or designated to a latter grade. Thus, research on student learning leads to adjustments in the Benchmarks. IVB: Research Companion „ The Research Companion suggests that although many textbooks move from the skill of isolating the variable in one-variable equations to isolating one variable in two-variable equations seamlessly, this transition is in fact very demanding for students. Furthermore, different representations (y = mx +b vs. Ax + By = C) denote different conceptions of meaning despite the fact that either form maintains equivalence. It is important for students to work with both the implicit and explicit forms of linear equations in order to gain fluency transitioning between either and more importantly, to understand the appropriateness of either form depending on context. Solving systems of equations in two variables is usually introduced in a first-year algebra course. Once students have a strong foundation of algebraic manipulation and the methods associated with it, solving systems of equations may follow more naturally. However, solving equations with one unknown results in a solution represented by a number rather than ordered pairs. Students face difficulty solving systems of equations because of this significant difference in the solution sets to equations in two variables. The concept of a variable can present a challenge to students. The resource suggests that students must first come become acquainted with variables informally through class discussion and activities, even in elementary school. By the time of middle school, students ought to begin to make their own generalizations using variables. With this incorporation of the idea of change and more specifically, variables throughout grades, students entering high school should have the basic prior knowledge necessary to understand more than linear relationships. The basics of many learning goals in the US seem to be too shallow and constricted. The popular phrase, “a mile wide and an inch deep,” is often all too true upon closer inspection of the standards. However, research can be used to develop and improve upon these educational standards. V: Atlas of Science Literacy „ I reviewed the Describing Change and Symbolic Representations concept maps from the Atlas of Science Literacy. Maps help you trace a concept or skill by laying out the foundation or application of the concept or skill in the most basic terms during the earliest grades. Following this foundation, subsequent ideas progress in complexity as you trace the connections into higher levels. From the simple beginnings in K through 2 to the concepts mastered upon completion of 12th grade, all of the underlying concepts (such as symbolic representation or change) flow in increasing complexity throughout each section of grade levels. However, many predominantly skill-related ideas are not introduced till a certain grade level. For instance, the 6-8 beginning idea that “a number expressed in the form a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b.” This varying use of notation is not directly related to any elementary concepts, and therefore this introduction is delayed till middle school grades when this notation becomes common. Throughout each concept map, it is evident that neither symbolic representation nor change is an isolated idea, but rather an underlying concept to many other subjects. The use of symbolic representation is present in mathematical models for natural phenomena, which illustrates how math is basically the language for mathematics. Similarly, symbolic representation describes change, which connects the two concept maps. Furthermore, describing change really envelops much more than the algebraic representation. Language can describe change – socially, physically, environmentally, and across many other disciplines. Therefore, whether it explicitly involves mathematics or not, describing change is not limited to the realm of mathematics. In this way, the concept map helped me to make sense of change in a broader context. The research on student learning for this grade level indicates that middle school students have difficulty solving word problems, specifically the problem context and the necessary proportional reasoning. For example, problems involving speed appear to be more complicated than problems of exchange. This map seems to help bridge that gap. Perhaps if students, too, had a broader understanding of change it would help them better understand problem contexts. Symbolic representation entails many challenges to students based on the research. The idea of equivalence on both sides of an equation is often not completely understood. For example, in the equation 3x + 4 = x + 8, many students assume that the right side will always indicate the answer. There appears to be a general lack of understanding of the meaning of a solution. It seems that students who learn how to solve for equations first by trial and error, and then formally develop more intuitive skills for solving equations or systems of equations. Some of these issues may be alleviated if students learned the meaning of equivalence through various contexts. Research continues to center around how students might come to understand solutions in a deeper way, and there seems to be no clear-cut teaching implication. However, I think perhaps creating more real-world applications where a solution or solution set is meaningful to the students may help to create a deeper understanding of solutions. 21st Century Skills Solving systems of linear equations is not an isolated, meaningless skill, but rather enhances students ability to reason and think critically, which is especially important in the 21st century. We are surrounded by choices for cell phone plans, meal plans, housing choices, investment opportunities, and so on the list continues. If we are incapable of this simple mathematical comparison, how are we to thrive in an age where poorly thought out loan repayment plans could land you in massive amounts of debt? Furthermore, manipulating algebraic equations pushes students to think beyond the scope of what simply is present, to consider and potentially better understand the concept of change mathematically. Even many low-level jobs require the basic skill of interpreting various representations of data. CTS Reflective Essay
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