Assessment Rubric: General Education Mathematics

Assessment Rubric:  General Education Mathematics

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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.

New Year’s and Assessment

During the New Year's season, which stretches from the day after Christmas until the last resolution is broken, i.e. sometime around the Feast of Epiphany, we take time to look at what we've done during the previous year and make plans for what we will do during the coming year. Our plans for the coming year are called resolutions, and as I mentioned above, our resolutions typically have a short self-life. One year I resolved to give up French fries, which lasted approximately to the next time I smelled French fries.

While this is assessment in a sense, it is assessment that is done badly. Before I go any further, I must remind everyone of the importance of context in the practice of assessment. The type of assessment I am talking about here is assessment of personal behavior as opposed to the institutional assessment as it is practiced by universities, but the general principles are the same. Our New Year's resolutions are goals by another name. One of my goals is typically to lose weight. I choose this for a number of reasons one of which would be pressure from my publics, i.e. my wife. Her reason for this pressure is that she desires that I be happy and healthy. These are desires I should, and I do, have for myself, but because of my love of good food I have internal pressures to overeat.

I choose strategies or objectives in order to aid in accomplishing that goal. The particular one I mentioned was cutting out French fries. One advantage of this goal is that it is easy to measure. Either you are successful in cutting out French Fries or you are not.

But, I did say that I had failed in this particular endeavor. Why?

One thing is that I chose a poor strategy, i.e. cutting out a particular food. There were reasons for this. One of which was that French fries are a particular weakness of mine that pull me into overeating. The problem is that the measure is an all or none instrument. Eating a few French fries is measured as a failure in this system, even if it doesn't result in the intake of a large number of calories in the end. The failure is discouraging and results in a premature abandonment of the goal.

It strikes me that the real problem here occurred with the choice of the goal. The goal itself should have been to live a lifestyle conducive to health. Objectives created in the aid of this could have been eating a healthy diet and exercising an appropriate amount. One can then address the healthy diet through means of portion control and a regular exercise schedule. While there is no set of goals that is so good that a human being such as myself can't find a way to wiggle out of it by Epiphany, this goal and these objectives are at least amenable to long to planning, i.e. I can lay myself out a plan for portion control and regular exercise and can put in some more effective measures of house closely I am meeting my objectives.

At the same time, this is a goal which, while pleasing my publics, also is a good in and of itself to me.

This is something we need to keep in mind when we do institution-level assessment at the university. While the impetus is often pressure from a particular public, those applying the pressure are doing so because they care about education…as do we. Our challenge is to articulate the desires of those applying the pressure into an achievable plan. In attempting to do this, it is easy to set up ourselves to fail, so we need to take care in the early stages.

Happy New Year!

Being killed by perfection

I recently talked to one of my cousins, who is a retired engineer. He was interested in what the world of higher education is like these days, and, as a part of our conversation, I talked to him about assessment. When I explained the basic principles to him, his response was, "Ah, TQM."

TQM, for those who don't know, stands for Total Quality Management. My understanding of the history may be faulty, but this was one of those ideas that came into American industry from Japan when we were falling behind them in terms of quality. The idea isn't new, of course. The general idea of set a goal, take an action, and measure how far the outcome is from the goal is as old as civilization. I suppose the novelty of the notion lies in the use of modern analytic tools and institutionalizing it to such a large degree.

My cousin said the process had run afoul in his company when it was transmuted by the slogan "Perfect the first time."

There is no such thing as perfection, so arriving at it the first time is nonsense. And, of course, this is not what a culture of continual improvement about. I don't even like the phrase "continual improvement" as it doesn't seem to allow even for normal statistical variation. The idea is to organize our activities and create structures so that improvement is a natural outcome of the process. We need to recognize that we can improve and to institute processes so this will, over the course of time, happen.

Part of this process might include setting up a "perfect" ideal of what our desired outcomes would be. But, I believe it was Voltaire who said, the perfect is not the enemy of the good. We can be good, but we can always strive to be better. Using the tools of assessment is a natural part of this process.

Math Task Force: On the Road

I look at my previous entries in this blog and see that I'd written about preparing to begin with the Math Task Force on September 16. It is now early November and we've had our second weekly meeting. Those outside of Higher Education might think that there has been some sort of a delay. The truth of the matter is that it has taken this long with everything going as well as possible. A while back I was in a meeting with a faculty member who was in the process of starting a new program. I told her that one I'd been involved in starting took 3 years to begin. She asked me what went wrong. I told her that nothing went wrong; it took that long with everything going perfectly.

There are two constraints that come together that make academic processes go so slowly. The first of these is inclusivity. We operate under a system of shared governance which means that any program or policy that is created will have to be reviewed and approved by representatives of the entire faculty. Including the appropriate representatives from the various sectors of the university is essential. The second of these is scheduling around classes. So many members of the university community teach that meetings must be scheduled around classes; the classes cannot be moved because students come first. Along with this is the fact that most of the folks who you would want on a committee are already on several committees because they are good workers, so we have to schedule around those committees too. The upshot of this is that it takes a while to schedule that first meeting and sometimes all subsequent meetings.

But, as I said earlier, we've already met twice and are meeting on a weekly basis. The conversation has begun. We say that a lot on the university campus, because so much of what we do is based on conversations among thoughtful people who care very much and have their own opinions. Meetings are a good site for these conversations as ours opinions can be given and the opinions of others can be heard. We then leave the meeting and consider what we've heard and whether we want to make those words our words. By this process a group opinion is born which is informed by the knowledge of all those present.

I've been thinking of the issue of mathematics as general education for a considerable time now. It is very affirming to hear many of the thoughts I've had coming independently from the mouths of others. It is also good to hear perspectives I've not considered before. It is a beautiful process.

On standardized tests

This article is from the New York times.

Math Task Force: The Beginning

I am interested in mathematics for a number of reasons. One is that it is my discipline. Another might be that it's simply an inherently interesting subject, but then, given my previous statement, some might consider me biased. The reason that I am interested in mathematics as a subject at this point is that it is listed as a basic skill in my university's general education program and we are currently in the processes of developing a structure of assessment for it.

I speak with no fear of rebuttal when I say that mathematics is a very technical subject, but I will add to this that mathematics has its own beauty. Within its jungle of formulas and theorems, there are nuggets of beauty that await a prepared mind. I've seen them and I know. This sort of beauty is the stuff envisioned by those who have the vision of idealized, liberal-arts-type general education courses. My university doesn't currently offer such a course. Ought we? I don't know, and I don't know whether it's even something we should think about, but I will toss it to the family dog to see if he will chase it.

Beyond that sort of vision of general education math, there is also mathematics as a life skill. It is knowledge that people can actually use in daily life. Mathematics can help you live and it can help you to prosper.

But mathematics is classified as a basic skill in our general education program because it is foundational. There are disciples—the sciences, the social sciences, business, technology, and so forth—that use mathematics. Currently the general education offerings on our campus are structured in order to feed into these various disciplines.

We are currently in the process of putting together a Math Task Force which will explore how well these needs are being met. In the past, the Department of Mathematics has paid attention to these needs without the aid of any larger structure, i.e. it has been done informally which met the needs of the time. As the character of the university changes, regularizing these processes which were previously informal and opportunistic, will help aid the university in demonstrating our commitment to educating our students in this basic skill.