Assessing Students Proficiency Strands for Elementary Algebra - Assignment Example

Assessment Task and Criteria (Student Name) (Institutional Affiliation) (Subject) (Instructor) May 12, 2013 Assessing Students Proficiency strands for Elementary Algebra Functions and Algebra So as to assess something, one must have certain concepts of that something. For my case, algebra refers to various things in present day schools. Particularly, the study of algebra is usually mixed together with the study of functions. Even though it is a fact that the concept of function is trailing much of the elementary algebra, and there are aspects of algebra of several of the tasks that we intend our students to perform with functions, the blending of functions and algebra has resulted to a significantly confusing in algebraic teaching (Olson, 2009). Thus, I would like to take some moment to explain own stance prior to discussion proficiency assessment. In the course, whereas appreciating other potential applications, I will utilize the world algebra to refer to elementary algebra, meaning the study of algebraic equations and expressions whereby numbers are replaced with letters. Goldsmith and Mark (2009) state that in the development of concepts from functions, to algebra to arithmetic, there is an abstraction increase at every stage, and the abstraction increase at the second stage is more or less bigger that the initial stage. Within the stage of algebra to arithmetic, students learn the representation of numbers with letters as well as the calculation using numbers with algebraic expressions. Within the stage of algebra to functions students learn are learning the new sort of function, object and represent functions by the use of letters. In algebraic teaching, two complementary dangers exist in which case each of them can make learners to miss the scale of this stage (Driscoll and Bryant, 2008). The initial danger being that function notations and functions seem to the learners to be majorly a concept of applying some form of auxiliary notation, in that for instance, if a (f) function is expressed with f(y) = y2- 2y-3, hence, f(y) is would simply appear as a short-hand notation to express y2-2y-3. This creates a confusion between the expressions and functions that represent them, which subsequently results in a wide area of confusion regarding the concepts of transforming expressions and equivalent expressions. For instance, intend learners to understand that (y-1)2-4 and (y + 1) (x-3) are equivalent expressions, where each one of them reveals different concepts of a similar function. However, without a well-built concept of function being an object different from an expression that defines it, the importance of transformation and equivalence would be lost in the flurry of equal signs that follows the f(y). Learners are not able to see forests for the trees (Popham, 2007). The other complementary danger being that algebra seems as some form of auxiliary to the functions’ study. It is highly to apply multiple functions representations so as to develop a concrete concept of a function to be an object in its own form. The representation of functions can be done using verbal descriptions, tables and graphs, a part from the algebraic equations. In viewing the same object in different perspective, one strives to develop the concept of its independent being. On this view, algebra offers one method of understanding functions, nevertheless functions are not treated as explicitly algebraic objects. According to Jeremy, Jane and Bradford (2007) this method helps solve the danger of learners not able to “see the forest to the trees”, however it risks the students not able to see the trees for the forest. Overstressing functions during algebraic lessons could obscure the structure of algebra and, and excessive attention to numerical and graphical approaches could sabotage the main aim: thinking or reasoning regarding symbolic representation (Lappan, Fey, Fitzgerald, Friel and Phillips, 2006). Algebraic Proficiency Look at the below common errors in algebra, that almost every teacher who teaches calculus like myself have come across at one time or the other. = =  This student has lost a 2 that he factored from the denominator. Learners usually take these kinds of errors to be negligible slips, just like a wobble when riding his bicycle, and the teacher normally treat this view point with partial mark. The remediation suggested for learners who commit this type of error is usually, numerous drills and in the case of riding the bicycle lots of practice. This error is treated as a procedural fluency matter; it is either that the student has no knowledge of the algebra rules or is inadequately practiced during their execution. In fact, with no further information, it would not be possible to precisely diagnose the error. Nevertheless, it merits recalling on experience to establish possible diagnoses, which fall within other proficiency strands. For instance, it could be possible that it is a conceptual understanding error. Experience through discussion with students regarding their mistakes point out the likelihood that it could not be a procedural unintentional slip, however a confusion regarding the procedures which are allowed in which circumstances (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006). Assessing Proficiency A lot of tasks of algebra focus on evaluating the procedural fluency. In fact the widespread notion among students on algebra is that it is completely a procedural fluency. The concept that there are thoughts in algebra surprises many students. Then, how do I go about assessing other strands of algebra? One approach would be through asking word problems and an intensive multi-stage question, which entails all algebraic strands at one go (Goldsmith, Mark and Kantrov, 2000). However, these kinds of questions hardly find their way through to the standardized assessments tasks and that is why this paper has chosen these types of assessment tasks. In assessing the non-procedural elements of algebraic proficiency there is need to ask simple tasks or questions. Below are some of the assessment tasks. Assessment Task: In this assessment task, the solution to this algebraic equation depends on the b constant. Take that b is positive, what would be the impact of increasing b on the solution’s value? Does the solution remain unchanged, decrease or increase? Provide a reason to your answer that would be understood without necessarily working out the equation. a) d/b= 1 b) b x=b c) bx = 1 d) x- b = 0 a) Increases: The bigger b is, the bigger x has to be in order to have a ration of 1 b) Remains unchanged: When b changes, both the sides of the equations remain equal and change together c) Decreases: The bigger b is, the smaller x has to be in order to get a product of 1 d) Increases: The bigger b is, the bigger x has to be in order to get 0 when b is deducted from it. Conceptual Understanding and Equity Madaus and O’Dwyer (2009) argue that completing the above assessment task requires the student to utilize what they have been taught and hence would immensely inform regarding the depth of student’s understanding of algebraic concepts as well as their skills flexibility. Based on the context deployed, this type of assessment could also provide students of differing strengths, interests and backgrounds different approaches to demonstrate their understandings and skills (Goldsmith, Mark, & Kantrov, 2000). The equations tasks are all trivially simple to work out. However, students find such a question hard since they have not being asked any demonstration of a procedure, but to construct an explanation in respect to what is implies for a number to be an equation’s solution. Students can possibly learn the mechanics of working out an equation but fails to understand the rather challenging aspect of the task as an equality statement between two equations whose fact is dependent on the variables values and the procedure of working out an expression as a sequence of logical deductions comprising such statements (Lappan, Fey, Fitzgerald, Friel and Phillips, 2006). Requiring learners to reason over equations while they do not solve them would assess this aspect of proficiency. According to Shepard and Bleim (2005) to tackle reliability issues, the guides for scoring commonly referred to as rubrics would be used. Rubrics outline the specific understandings, knowledge and skills which are being assessed, and indicate the various quality levels for each. The proficiency of algebra lies in its symbolic representations. The assessment task testing the proficiency of algebra has to include the symbolic manipulation fluency, as well as other four elements of mathematical competence, such as productive disposition, adaptive reasoning, strategic aspects and procedural fluency. However, this type of richer test is normally holdup until functions are introduced, because the various approaches of representing functions, as well as their different application within the contexts of real-world, offer a better ground for the deigning of more strategic and conceptual questions. At elementary levels of algebra simple assessment tasks rarely come by, due to the fact that it is normally taught purely as a procedural skill, and more strategic and conceptual components either veiled or ignored making the function to be more abstract. The Rubric and criteria for elementary algebraic proficiency Content Score Point Demonstration of proficiency -the student gives a satisfactory response that has clear, plausible explanations and is reasonably right. Or the student applies appropriate descriptions that exhibits the understanding of the equations, and offers sensible arguments to support his answers, with little or no flaws 3 Demonstration of minimal proficiency Student gives an almost satisfactory answers that has some flaws, for instance starts to correctly answer the question, however fails to provide answers of all the parts of the questions or excludes appropriate explanation, wrongly uses algebraic expressions, or applies wrong strategies in responding to the question 2 Demonstration of lack of proficiency Student gives below a satisfactory response, which merely starts to answer the problem, however fails to completely answer the equation, for example offers zero or little explanation that are not clear, manifests zero or little understanding of the equation that has been asked, or makes huge flaws in responding to the question 1 Demonstration of lack of algebraic proficiency Student offers an unsatisfactory explanation which inappropriately answers the question, for instance the student uses algorithms that do not indicate any understanding of the equation, gives a copy of the equation with no appropriate response, or the student fails to give any material fact that is appropriate to the equation or completely fails to answer the equation 0 References Goldsmith, L., Mark, J., & Kantrov, I. (2000). Choosing a Standards-Based Mathematics Curriculum. Portsmouth, NH: Heinemann. Goldsmith, L. & Mark, J. (2009). What Is a Standards-Based Mathematics Curriculum? Educational Leadership, 57(3), 40. Shepard, L. & Bleim, C. (2005). Parents’ Thinking about Standardized Tests and Performance Assessments. Educational Researcher, 24(8), 28. Olson, L. (2009, Jan. 11). Making Every Test Count.Quality Counts: Rewarding Results, Punishing Failure. Education Week Special Report, 11. Popham, W. J. (2007). What’s Wrong—and What’s Right—with Rubrics. Educational Leadership, 55(2), 72. Madaus, G. & O’Dwyer, L. (2009). A Short History of Performance Assessment: Lessons Learned. Phi Delta Kappan, 80(9), 693. Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Phillips, E. (2006). Getting to Know Connected Mathematics: A Guide to the Connected Mathematics Curriculum. White Plains, NY: Dale Seymour Publications, 57. Jeremy, K., Jane., S, and Bradford., F (2007). Adding it up: Helping children learn mathematics, National Academy Press,Washington, DC. Driscoll, M. & Bryant, D. (2008). Learning About Assessment, Learning Through Assessment. A report of the National Research Council, Mathematical Sciences Education Board. Washington, DC: National Academy Press, 21. Read More

Learners are not able to see forests for the trees (Popham, 2007). The other complementary danger being that algebra seems as some form of auxiliary to the functions’ study. It is highly to apply multiple functions representations so as to develop a concrete concept of a function to be an object in its own form. The representation of functions can be done using verbal descriptions, tables and graphs, a part from the algebraic equations. In viewing the same object in different perspective, one strives to develop the concept of its independent being.

On this view, algebra offers one method of understanding functions, nevertheless functions are not treated as explicitly algebraic objects. According to Jeremy, Jane and Bradford (2007) this method helps solve the danger of learners not able to “see the forest to the trees”, however it risks the students not able to see the trees for the forest. Overstressing functions during algebraic lessons could obscure the structure of algebra and, and excessive attention to numerical and graphical approaches could sabotage the main aim: thinking or reasoning regarding symbolic representation (Lappan, Fey, Fitzgerald, Friel and Phillips, 2006).

Algebraic Proficiency Look at the below common errors in algebra, that almost every teacher who teaches calculus like myself have come across at one time or the other. = =  This student has lost a 2 that he factored from the denominator. Learners usually take these kinds of errors to be negligible slips, just like a wobble when riding his bicycle, and the teacher normally treat this view point with partial mark. The remediation suggested for learners who commit this type of error is usually, numerous drills and in the case of riding the bicycle lots of practice.

This error is treated as a procedural fluency matter; it is either that the student has no knowledge of the algebra rules or is inadequately practiced during their execution. In fact, with no further information, it would not be possible to precisely diagnose the error. Nevertheless, it merits recalling on experience to establish possible diagnoses, which fall within other proficiency strands. For instance, it could be possible that it is a conceptual understanding error. Experience through discussion with students regarding their mistakes point out the likelihood that it could not be a procedural unintentional slip, however a confusion regarding the procedures which are allowed in which circumstances (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006).

Assessing Proficiency A lot of tasks of algebra focus on evaluating the procedural fluency. In fact the widespread notion among students on algebra is that it is completely a procedural fluency. The concept that there are thoughts in algebra surprises many students. Then, how do I go about assessing other strands of algebra? One approach would be through asking word problems and an intensive multi-stage question, which entails all algebraic strands at one go (Goldsmith, Mark and Kantrov, 2000).

However, these kinds of questions hardly find their way through to the standardized assessments tasks and that is why this paper has chosen these types of assessment tasks. In assessing the non-procedural elements of algebraic proficiency there is need to ask simple tasks or questions. Below are some of the assessment tasks. Assessment Task: In this assessment task, the solution to this algebraic equation depends on the b constant. Take that b is positive, what would be the impact of increasing b on the solution’s value?

Does the solution remain unchanged, decrease or increase? Provide a reason to your answer that would be understood without necessarily working out the equation. a) d/b= 1 b) b x=b c) bx = 1 d) x- b = 0 a) Increases: The bigger b is, the bigger x has to be in order to have a ration of 1 b) Remains unchanged: When b changes, both the sides of the equations remain equal and change together c) Decreases: The bigger b is, the smaller x has to be in order to get a product of 1 d) Increases: The bigger b is, the bigger x has to be in order to get 0 when b is deducted from it.

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