Suggestions for Making Vertical Alignment Between Courses Tangible Throughout the Math Curriculum Using Dynamic Solution Exercises Abstract: This paper justifies the need for, and offers some suggestions on the selection and implementation of mathematical problems known as Dynamic Solution Exercises (DSE s). The intent of the manuscript is to help provide insight into how mathematics teachers can go about making vertical articulation a cooperative and tangible part of the mathematics curriculum. A sample Dynamic Solution Exercise is provided based on research at Metropolitan Community College in Omaha NE. Some strategies for selecting and building a DSE instructional environment are included in the narrative. Introduction: The term articulation is often expressed as the interrelatedness of various aspects of the didactic content. Academic articulation can be conceptualized in a number of ways but the most common representations of articulation relating to instruction are the substructures of vertical and horizontal articulation. Vertical articulation addresses sequential issues of lessons, topics, or courses while horizontal articulation focuses on the association between or among components of the curriculum that occur simultaneously (Ornstein & Hunkins, 1998). In an applied environment such as a math department, curriculum designers would vertically focus the important ideas of one course to support the critical ideas of the next sequential course in a given discipline, and horizontally focus the key ideas of one course to support the primary topics of concurrent coursework in other areas. Despite the need to logically sequence the topics within a curriculum, the importance of articulation as an applied practice can still be easily overlooked. This manuscript will focus on vertical articulation and provide suggestions for selecting mathematical tasks that are best suited to vertical articulation. In addition, a classroom focused Dynamic Solution Exercise (DSE) will be provided as an example of tangible vertical articulation using a mathematical topic that spans the scope of instruction from basic mathematics to College Algebra. Questions that often surface are why pay attention to vertical articulation why expend effort to build DSEs? The answer is because with thoughtful development, curriculum can support learning as much as engaging instruction does. Romberg (2000) suggests that with appropriate guidance from teachers, students can build a coherent understanding of mathematics, and that their understanding about the symbolic processes of mathematics can evolve into increasingly abstract and scientific reasoning. This happens through opportunities to participate in unique kinds of mathematical tasks. Unfortunately, a coherent understanding of anything does not happen over the course of a single academic term. The evolution in thinking that allows students to demonstrate a transition from informal to formal semiotics presumably happens over an extended period. It follows then that developing the kinds of appropriate mathematical tasks, the kinds that allow for this transition to take place, can most appropriately be done by a team of mathematics educators, each considering the nuances of what the others do, and then documenting each teachers part in the process through thoughtful curriculum (i.e. DSE).
To some degree, a broad effort in this area has already been initiated by the National Council of Teachers of Mathematics (NCTM) with the release of the Curriculum Focal Points for Pre-Kindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006). Though the document does not specifically address vertical articulation, it is presented for consideration as a part of a larger idea. The NCTM manuscript presents three primary characteristics for a topic to be considered a point of focus: Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school? Does it fit with what is known about learning mathematics? Does it connect logically with the mathematics in earlier and later grade levels? (p. 5) The third bullet point addresses the notion of vertical articulation, and is of particular relevance when considering how easily the idea of mathematical fluency can be misinterpreted. While most instructors would agree that it is important for their students to be able to master difficult forms of textbook problems; most would perhaps also agree that this kind of mathematical processing does not necessarily guarantee mathematical fluency. Understanding why mathematical processes work is preferable to simply being proficient at them. It is then important to be able to maintain and extend fluency through specifically designed mathematical tasks that use mathematical ideas to build mathematical ideas. If we operate on the assumption that understanding mathematics at any level, and within any instructional paradigm is a result of the mental processes that include adaptive and strategic reasoning, teachers must provide the opportunity for students to experience these processes. Cobb (1994) suggests that the informal mathematical activities in which students are initially engaged should create a basis from which they can create increasingly sophisticated mathematical conceptions. The Dynamic Solution Exercises as described below provide an opportunity for this kind of instructional environment. Suggestions for Implementing Vertically Aligned Tasks: For vertical articulation to be implemented into a department s curriculum, it should be concrete in some fashion beyond what is innate in the course descriptions or instructional texts. Mathematics instructors at different levels should work together to develop a critical mass of problems that represent a broad scope of learning, and that have dynamic solutions which become increasingly more sophisticated as the students progress through successively higher level coursework. In this format, students see the same basic problems term after term, but their familiarity with the problems will provide background and context for their current coursework. These problems will hereafter be referred to as Dynamic Solution Exercises (DSE). DSEs are standard mathematics problems appearing in textbooks, but are adapted to focus instruction on using students current understanding of mathematics to build new understanding of mathematics. For this reason, the dynamics of the solutions, not simply correct answers, must be the focus of teachers instruction and assessment. The effective implementation of a tangible, vertically articulated process using a DSE approach, hinges on three primary factors: Staying focused on common assessment
tools; making DSEs a priority in curriculum planning; and developing specific strategies for selecting and building the DSEs that can best be vertically articulated through many levels of mathematics coursework. 1. Staying focused on common assessment tools: If a curriculum team sees merit in the use of standardized assessments, it increases the likelihood that important ideas can be drawn from standardized tests and focused on updating curriculum and instruction to service the specific needs of students related to interpreting mathematical questions in standardized formats. 2. Making DSEs a priority in curriculum planning. Dynamic Solution Exercises can consist of just about any topic from a traditional mathematics curriculum. The strategy for selecting materials to support the DSE approach can be best thought of as a series of questions that keep the instruction and assessment focused on the ideas that are determined to be the most important common threads. Some ideas to consider in developing DSEs are as follows: Did you specify Pre-requisite and Post-requisite knowledge? Does each stage of the DSE prepare student for the next stage? Have you made it possible for students to draw together ideas from previous understandings and justify how they will lead to new understandings? Can you distinguish DSEs from other types of practice problems as your own brand of Curriculum Focal Points? To some extent, the suggestions provided here were inspired by those in professional development related lesson studies (e.g., Lewis, 2002) where teachers ask questions similar to those stated above. In this model however, the attempt is to make vertical articulation tangible, and in so doing, teachers must distinguish the DSEs from other types of problems as outlined in the last question. 3. Developing specific strategies for selecting and building the DSEs that can best be vertically articulated through many levels of mathematics coursework. DSEs are essentially mathematical models which re-teach the central concepts of a specific unit over time. The solutions for each stage of the tasks must be dynamic as a topic is explored in greater detail; and must use increasingly sophisticated mathematics to achieve similar outcomes as were completed in earlier coursework. In short, a DSE (the tangible product of vertical articulation) demonstrates what it is to build new mathematical knowledge from existing mathematical knowledge. Basically, a DSE consists of a single topic that is taught in successive classes. Each course offering from basic mathematics to calculus could represent one stage of a DSE. Some DSEs may have two or three such stages, while others can have as many as seven. It is important at this point to note that although the DESs are taught from the bottom up, they must be developed from the top down. Ultimately, this means the calculus teachers developing a DSE may want to identify a central concept or theme they deem to be a critical element of the curriculum. The DSEs must then make efforts to support that cause, each at a level that feeds the next.
Introduction Results from a study done at Metropolitan Community College in Omaha NE were used to develop the DSE presented here in an attempt to improve conceptual understanding for our students of a key concept in intermediate algebra. A common final was administered to selected sections of students taking Intermediate Algebra classes during three consecutive quarters beginning with spring quarter 2009. It was administered to students in both traditional and completely online sections. The results were analyzed to identify concepts where students struggled. Organization of the Study The test included twenty four multiple choice items. The items were picked to be representative of concepts that should be learned and mastered in Intermediate Algebra after studying linear equations, systems of equations, polynomials, factoring, rational expressions, radicals, and complex numbers. Results Individual items on the test were analyzed and one of the most commonly missed items was setting up a two by two system of equations for a basic mixture example. This article will present three levels of a DSE to illustrate how the problem of solving a mixture example could be developed and solved using progressively more sophisticated methods as a student progresses from basic mathematics to intermediate algebra. A Classroom Focused DSE We have assumed that the evolution of understanding about a given topic can be structured through thoughtfully considered DSE s. In point of fact, nearly any problem that is repeated throughout a school s mathematics curriculum can ultimately develop into a DSE. The derivation of process for solving a two by two system of equations can be incorporated into a Dynamic Solution Exercise that appears repeatedly in a sequenced curriculum. It is helpful to think of the DSE as an approach where one problem acts as an instructional medium, a context if you will, for multiple levels of instruction. It is important to note that knowledge about the DSE topic is less important than understanding the evolutionary dynamics of the mathematical solutions being shown. A basic DSE approach using the topic of solving systems of two by two systems of equations follows. Instructions and solutions for instructors, worksheets for students, and video explanations for the instructor for each level can be found at the following website: http://resource.mccneb.edu/math. To locate the DSE - Level 1, find the link: Select A Course, and choose: Math 0910; to locate DSE Leve1 2, choose: Math 0960; for DSE Level 3, choose: Math 1310. DSE Stage 1: Guess and Test Process (General Mathematics Example) Solving a system of two equations with two unknowns could be explored and solved in a class where fractions, decimals, ratios, and percents are the main topics. This could be done by having the students investigate the different possibilities that will occur with varying amounts of each solution being mixed to produce a weed killer of a required strength. By substituting values for amount 1 and amount 2 we can see how the various combinations affect the final strength of the weed killer. The students will perform the
calculations and then make the conclusions based on the questions asked. This can lead to questions about what combination yields the strongest solution verses the combination that yields the weakest. If students can understand the concept of how to weaken or strengthen the potency of the solution, they will develop the background to see the logic and reason for the algebraic solution. Given Information and Task: You are working for a lawn service and have been asked to mix two different strength weed killers together to obtain 10 liters of weed killer that has a 61% strength. This is to meet new government environmental standards. On hand you have one barrel of weed killer that is 40% strength and another and another barrel that is 70%. How many liters of each should be mixed to obtain 10 liters that is 61% weed killer? To help understand the problem, keep in mind that a solution that is 40% weed killer means that 40% of the liquid in Amount 1 is weed killer. In the same way, Amount 1 + Amount 2 = 10 Liters of Total Liquid Percent of Weed killer in the mixture to be 61% of the total Liquid, so 70%*Amount 1 + 40%*Amount 2 = 61%*10 Liters 0.70*Amount 1 + 0.40*Amount 2 = 0.61*10 = 6.1 Liters of Weed killer Fill in the following table and use the results to find your solution. Amount 1 + Amount 2 = 10 Liters of Liquid Amount 1 Amount 2 70% of Amount 1 + 40% of Amount 2 = # of Liters of Weed killer % of Weed killer = # of Liters Weed killer/10 Liters 0 10 0.70*(0) + 0.40*(10) = 0 + 4 = 4 Liters of Weed killer in the Mixture 1 9 0.70*(1) + 0.40*(9) = 2 8 0.70*(2) + 0.40*(8) = 3 0.70*( ) + 0.40*( ) = Pattern continued for 4 to 8. 9 0.70*( ) + 0.40*( ) = 10 0 0.70*(10) + 0.40*(0) = 4/10 =.40 = 40% Weed killer Questions for extension: At what combination of Amount 1 and Amount 2 was the percent of weed killer the least?
At what combination of Amount 1 and Amount 2 was the percent of weed killer the most? At what combination of Amount 1 and Amount 2 was the percent of weed killer 61%? Based amounts used and the percent of weed killer in each, what generalizations might you make? Solution: With some explanation by the instructor, the students should be able to complete the table. After the student has successfully completed the table, they should formulate answers to the questions posed. This DSE could be assigned in a variety of ways. Use as a class activity with students working in groups or as an independent project to be completed outside of class. At the introduction and completion of the DSE, time for class discussion would be beneficial to bring conceptual understanding to the students. DSE Stage 2: Manipulation of formulas (Introductory Algebra Example) We will solve the same problem. We will use a similar table to the DSE in Stage 1, but note that stage 2 we introduce the variable x to represent Amount 1 and (10 x) to represent Amount 2. This allows us to write an equation in terms of the variable x. We will still fill in the table to find the solution, but in DSE Stage 2 we will follow it with the question of how to formulate a linear equation that will allow us to end up with a 61% final solution. We then let the students move to Part 2 where they will formulate their own algebraic equation to solve the problem. We have moved to an algebraic solution that can be adjusted based on the strength of the final solution required. The skills drawn from this stage of the exercise focuses on developing an algebraic solution that can be applied without the need for a guess and test process. Given Information and Task: Part 1) Statement of the problem is identical to DSE Stage 1. The student worksheet begins with the following explanation. Amount 1 + Amount 2 = 10 Liters of Total Liquid Amount 1 = x, and if we need 10 liters of the final mixture, then: Amount 2 = 10 - x Percent of Weed killer in the mixture to be 61% of the total Liquid, so 0.70*x + 0.40*(10 x) = 0.61*10 = 6.1 Liters of Weed killer Fill in the following table and use the results to find your solution. Amount 1 Amount 2 Amount 1 + Amount 2 = 10 Liters of Liquid x + (10 - x) = 10 Liters of Liquid
x 10 - x 70%*x + 40%*(10 -x) Equal the Amount of Weed killer in the 10 Liter Mixture Percent of Weed killer = Liters of Weed killer/10 Liters 0 10 0.70*x + 0.40*(10-x) = 0.70*(0) +0.40*(10) = 4 Liters Weed killer 1 0.70*x + 0.40*(10 - x) = 0.70*( ) + 0.40*( ) = Pattern continued for 2 to 9. 10 0.70*x + 0.40*(10 - x) = 0.70*( ) + 0.40*( ) = 4/10 =.40 = 40% Weed killer Solution: Part 1) The student should fill in the values in column 2 knowing each one is (10 x), then substitute into column 3 and perform the computations to find the amount of weed killer in the final 10 liters of solution. They will find the correct combination at 7 liters of the 70% solutions and 3 liters of the 30% solution. Now is the time to help students formulate an equation that can be solved from above. 0.70*x + 0.40*(10 x) = 0.61*10 = 6.1 Liters of Weed killer Part 2) After the first application, the strength of the weed killer was found to be too strong and it was killing the other plants. You have been directed to put together 10 liters of a mixture of the two different weed killers that will have a final strength of 49%. Can you write one equation in terms of x using the model from above that will allow you to find the correct combinations of the two liquids? Solution: Part 2) They will see that the only thing that has changed in out computation is that we now want a 49% solution and that in order to get there all we need to do is multiply the 0.49 strength times the total amount of the mixture needed. Using substitution to solve yields the equation: 0.70*x + 0.40*(10 - x) = 0.49 * 10 = 4.9 liters of weed killer Students can solve algebraically to find x = 3 and y = 7. DSE Stage 3: Using a system of equations & graphing (Intermediate Algebra Example) This stage of the model introduces a second variable allowing us to set up a system of two equations and unknowns and find a solution by either an algebraic or geometric process. We need to introduce the second variable in order to look at a two dimensional solution by graphing. It allows us to write the two constraints as linear equations and find the solution by graphing the two constraints and finding the intersection. Given Information and Task: Part 1) Statement of the problem is identical to DSE Stage 1. The student worksheet begins with the following explanation.
Amount 1 + Amount 2 = 10 Liters of Total Liquid Amount 1 = x, and if we need 10 liters of the final mixture, then: Amount 2 = (10 x) = y and so: x + y = 10 Percent of Weed killer in the mixture to be 61% of the total Liquid, so 0.70*x + 0.40*(10 x) = 0.61*10 = 6.1 Liters of Weed killer 0.70*x + 0.40*(y) = 0.61*10 = 6.1 Liters of Weed killer Fill in the following table and use the results to find your solution. You will also be asked to solve the system algebraically and graphically. Amount 1 Amount 2 x y = 10 - x Amount 1 + Amount 2 = 10 Liters of Liquid x = Amount 1 and y = Amount 2 y = 10 -x x + y = 10 Liters of Liquid 70% of x + 40% of y = Amount of Weed killer in mixture Percent of Weed killer = Amount of Weed killer/10 0 10 0.70*x + 0.40*y = 0.80*(0) + 0.40*(10) = 4 Liters 4/10 =.40 = 40% Weed killer 1 0.70*x + 0.40*y = 0.70*( ) + 0.40*( ) = Pattern continued for 2 to 9. 10 0.70*x + 0.40*y = 0.70*( ) + 0.40*( ) = x + y = 10 and 0.70*x + 0.40*y = 0.61*10 = 6.1 Notice we have a system of two equations and two unknowns. Solve this system by substitution or elimination. To solve the system by graphing we could solve both equations for y. y = -1.75x + 15.25 and y = -x + 10 Plot these equations and find the intersection. How does the graph of the two equations relate to the final solution to our problem? Solution: Part 1) The student should complete the table as in the earlier DSE. For the algebraic process, the student should also solve by the elimination or substitution method. After they can substitute 10 x for y into the second equation, they will solve in a similar fashion as in the last DSE.
For the graphical solution, the student should graph the two equations that are solved for y. They will identify the ordered pair (7, 3) as the solution and explain that the intersection of the two lines is where both conditions are met at the same time. Part 2) Suppose that after the first application, the strength of the weed killer was found to be too strong and it was killing the other plants. You have been directed to put together 10 liters of a mixture of the two different weed killers that will have a final strength of 52%. Without having to construct another table, can you construct a system of two equations and two unknowns using the model from above that will allow you to find the correct combination of the two liquids? Solve the system algebraically and by graphing. Part 2) The two equations are: x + y = 10 and 0.70x + 0.40y = 0.52*10 = 5.2 The algebraic solution can be accomplished with the substitution of y = -x + 10 into the other equation to obtain x = 4 and y = 6. The geometric solution can be found by solving each equation for y and graphing. y = -x + 10 and 0.70x + 0.40y = 5.2 y = (-.0.70x)/0.40 + 5.2/0.40 so y = -7/4*x + 13 The solution is the intersection of the following two lines which is observed to be the ordered pair (4, 6) on our graph.
Follow up Thoughts: Why spend time teaching knowledge about the setting up a system of equations? Don t. Instead, teach for understanding about how the use of mathematics is strategic and can vary depending on a mathematical task is approached. The system of equations task is nothing but a convenience to provide context. If you look closely at the underlying tasks assigned for each stage of the DSE illustrated above, you can see an obvious progression of mathematical expectations. Stage 1: Computation and mathematical reasoning Stage 2: Algebraic solution to a system of equations Stage 3: Geometric solution to a system of equations Additionally, at each stage of the DSE, students are encouraged to construct coherent explanations of the mathematical processes they are demonstrating. This is an important part of the DSE instructional approach, and may include other such techniques as indepth exploration, teacher-student dialogue, and peer reviewing. Conclusion: Unfortunately there is no effortless way to develop good DSEs, but the suggestions reviewed here may act as a starting point for math departments interested in vertical articulation activities. The development of DSEs is a time consuming task, and requires unprecedented cooperation between teachers at different levels, but there is a silver lining. Well-constructed DSEs will naturally connect important topics at each level, so a departmental assessment may need only ten or fewer DSEs to help students revisit the most critical elements of their entire mathematics curriculum. Problems related to optimization, particularly with sundials and such make excellent DSEs because they require an evolving understanding of numeric processes, algebra, geometry, advanced algebra, trigonometry, and calculus. In other words, a good DSE can offend your intuition at many levels! At the very least the process of developing DSEs may help instructors from the developmental level to calculus start talking to each other about what
constitutes important mathematics, and help students to understand how mathematics naturally evolves. References: Cobb, P. (1994). Theories of mathematical learning and constructivism: A personal view. Paper presented at the Symposium on Trends and Perspectives in Mathematics Education, Institute for Mathematics, University of Klagenfurt, Austria. Lewis, C. (2002). Lesson study: Handbook of teacher-led instructional change. Philadelphia: Research for Better Schools. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author Ornstein, A., & Hunkins, F. (1998). Curriculum: Foundations, principles, and issues. 3 rd Ed. Boston: Allyn & Bacon Romberg, T. (2000). Changing the teaching and learning of mathematics. Australian Mathematics Teacher, Vol. 56, No. 4, 6-9. Watanabe, T. (2007). In pursuit of a focused and coherent school mathematics curriculum. The Mathematics Educator. Vol. 17, No. 1, 2-6.