Assessment Rubric: General Education Mathematics

Assessment Rubric:  General Education Mathematics

	3	2	1	0
Interpretation Ability to articulate a problem and explain information presented in mathematical forms (e.g., equations, graphs, diagrams, tables, words)	Provides accurate explanations of information presented in mathematical forms. For instance, accurately explains the trend data shown in a graph.	Provides somewhat accurate explanations of information presented in mathematical forms, but occasionally makes minor errors related to computations or units. For instance, accurately explains trend data shown in a graph, but may miscalculate the slope of the trend line.	Attempts to explain information presented in mathematical forms, but draws incorrect conclusions about what the information means.  For example, attempts to explain the trend data shown in a graph, but will frequently misinterpret the nature of that trend, perhaps by confusing positive and negative trends.	No meaningful work done.
Representation Ability to convert relevant information into various mathematical forms (e.g., equations, graphs, diagrams, tables, words)	Competently converts relevant information into an appropriate and desired mathematical portrayal.	Completes conversion of information but resulting mathematical portrayal is only partially appropriate or accurate.	Completes conversion of information but resulting mathematical portrayal is inappropriate or inaccurate.	No meaningful work done.
Calculation Use the tools of mathematics.	Calculations attempted are essentially all successful and sufficiently comprehensive to solve the problem. For example, the student decides to use the quadratic formula to solve and problem and substitutes and simplifies appropriately.	Calculations attempted are either unsuccessful or represents only a portion of the calculations required to comprehensively solve the problem. For example, the student decides to use the quadratic formula to solve and problem and substitutes incorrectly or makes a minor simplification error.	Calculations are attempted but are both unsuccessful and are not comprehensive. For example, the student decides to use the quadratic formula to solve and problem and substitutes incorrectly and has simplification errors.	No meaningful work done.
Application Ability to apply mathematical generalizations, principles, theories, or rules to real world problems.	Select and apply the appropriate mathematical principles to correctly solve a real world application problem taking into account important assumptions. (In calculating the area of an irregular polygon, student correctly divides the area into simple shapes and correctly uses known formulae to calculate the areas.)	Chooses appropriate mathematical principles but has errors in applying principles to solve real world problem. (In calculating the area of an irregular polygon, student correctly subdivides the area into simple shapes and improperly uses formulae to calculate the areas.)	Attempts to solve application problem but is unsuccessful. (Student knows some of the technique for subdivision of an irregular polygon but incorrectly divides the area; knows some simple area formulae, but cannot put all the steps together to get a correct result.)	No meaningful work done.
Analysis Ability to make judgments and draw appropriate conclusions based on the quantitative analysis of data, while recognizing the limits of this analysis	Uses the quantitative analysis of data as the basis for competent judgments, drawing reasonable and appropriately qualified conclusions from this work. (Interpolates or extrapolates data from a graph or table to calculate information not specifically given; creates a formula from information to predict results for future events.)	Uses the quantitative analysis of data as the basis for workmanlike (without inspiration or nuance, ordinary) judgments, drawing plausible conclusions from this work. (Given a problem statement, the correct relationship can be identified and known values used to calculate the desired unknown.)	Uses the quantitative analysis of data as the basis for tentative, basic judgments, although is hesitant or uncertain about drawing conclusions from this work. (Given a problem statement, known values can be correctly identified, however, the appropriate relationship is not found or applied correctly and the desired result is not found.)	No meaningful work done.
Communication Expressing quantitative evidence in support of the argument or purpose of the work (in terms of what evidence is used and how it is formatted, presented, and contextualized)	Uses quantitative information in connection with the argument or purpose of the work, though data may be presented in a less than completely effective format or some parts of the explication may be uneven. For instance, effectively uses verbal and/or written skills to explain the quantitative evidence.	Uses quantitative information, but does not effectively connect it to the argument or purpose of the work. For instance, the quantitative evidence may be correct, but verbal and/or written skills are weak.	Presents an argument for which quantitative evidence is pertinent, but does not provide adequate explicit numerical support. (May use quasi-quantitative words such as "many," "few," "increasing," "small," and the like in place of actual quantities.) For instance, does not effectively use verbal and/or written skills to explain quantitative evidence.	No meaningful work done.

A lens for assessment

Recently Stephen J. Dubner in a blog entry entitled Are You Smarter Than an Eighth Grader? Published a copy of document from 1895 that had been published in the Salina (Kansas) Journal. By all appearances, this document looks to be a final exam for eighth graders though Snopes.com casts some doubt on this. As with so many historical questions, we may never know the full truth here, but I would still like to use this as an opportunity to discuss assessment by using this document as a lens.

Whenever I give a test, I am assessing two things. One of these things is how well each of the students understand the material. Their scores reflect how well they have learned. This can either be done as simply an abtract exercise in discerning what percentage of the material has been absorbed or, as is more often the case, against a set of standards. Did the student pass the exam? Did they learn 70 percent of the material and get a gentlemanly C? Did they learn 90 percent and earn an A? The other thing I am assessing is how well I taught the subject. Which questions did they know the answer to? Did they do so well that I could spend less time there and more on an area that they didn't do as well on?

For this exam, we don't have this sort of information. We don't know how many people passed if any or whether anyone even took it. As of this writing, I just don't know.

I recall a conversation at a local mathematics confernence a few years back with a man who had been teaching Intermediate Algebra for many years. He had carefully tracked his how his students had done on an impressive number of student learning outcomes. One of those was arithmetic with imaginary numbers. He'd tried a number of different methods of approaching the topic. He'd spent more time, given more emphasis, but he could never the the student success above a certain threshold. And he had years of data to prove this.

Others there had questions to ask about things he'd tried. The question I wanted to ask was "You are teaching the arithmatic of complex numbers to Intermediate Algebra students? Why?" Intermediate Algebra is taught in college to students who've come to college unprepared to take College Algebra. One need not know much about the arithmetic of complex numbers to meet the challenge of College Algebra, and, indeed, the time might be better spent elsewhere.

The topic is not appropriate for the students in the class. I find the arithmetic of complex numbers to be enjoyable and fascinating; it can be a useful intellectual exercise; and there are areas in which it is actually useful. If I were this teacher, I would remove the topic from the course to spend more time on other topics. I open myself up to the charges of dumbing down the course by doing this, but quite frankly the students who need it will learn about it in Differential Equations or the Theory of Functions of a Complex Variable. The students who are trying to prepare themselves for College Algebra will be able to learn what they need to be ready for that course.

As I look at the document from 1895, I see a lot of things that I don't know about. Putting the other sections aside and concentrating on the math, which is my area of expertise, I see some things there that are good, but I don't see any algebra. We teach a little algebra in the eighth grade now. This has none. There is nothing about computers, of course.

This exam proves that things change. We as teachers can use assessment methods as a tool to attempt to change for the better.