Assessment in Mathematics Classrooms

I’ve been doing some thinking, and writing, lately on the roles of assessment in mathematics classrooms.  For many of us, assessment is a means to assign grades to students; we see assessment as following the learning.  Certainly, we need to use assessments for grading purposes.  However, learning can not be separated from assessment … learning without assessment is simply wandering in a city hoping to find that nice hotel or restaurant; little good is likely to result, and some damage is likely.

For general reading on assessments for learning mathematics, try one of these sources:

• CUPM Guidelines for Assessment of Student Learning (at http://www.maa.org/saum/maanotes49/279.html)
• Assessment and the Process of Learning Statistics (at http://www.amstat.org/publications/jse/v5n1/hubbard.html)
• The Assessment Principle (NCTM)  at https://www.usi.edu/science/math/sallyk/Standards/document/chapter2/assess.htm
• IMAGINE: Mathematics Assessment For Learning (at http://www.gatesfoundation.org/edextranet/Pages/ImagineConvening2009.aspx)

The ideal assessment in my view would be a series of interviews with each student, where their work and comments are prompts for questions and discussion; an expert talking to the learner can identify problems and reinforce partially correct understanding, which are difficult goals for mass assessments.  I have not managed to design a class to achieve this, although I have managed to create tools that enable some interviewing to support learning.

I’ll describe some of the assessment tools I am currently using to support student learning (and student motivation):

QUICK QUIZ
This is a traditional quiz, though very short (4 questions), with half of the items being concept or ‘no work’ items.  The quiz is given at the start of a class, covering the learning that ‘should’ have occured prior to class (prior to any homework questions).  Ten minutes is plenty of time, and then we review the quiz — by students explaining each item, and I reinforce & correct as needed.  Since the Quick Quiz uses ideas and simpler problems, the process encourages students to attempt the learning; in class, we often say the “quiz is part of the learning process” to support good learning attitudes.

NO-TALK QUIZ
As a variation, a No-Talk Quiz involves students working 2 or 3 problems focusing on key processes or ideas.  Their quiz is then reviewed by two other students, who can only write comments on the quiz.  The student then has an opportunity to re-do any problem where they think the feedback suggests that they were wrong.  As an assessment, this process involves every student doing two necessary steps:  Critically reviewing work for accuracy and completeness, and explaining.  In addition, each student has to judge their own understanding compared to the feedback they get; provides a little ‘meta-cognition’.

TEST DRIVES
During class, we develop ideas and reasoning as well as master procedures.  After the large group discussion (5 to 10 minutes), I have every student try a Test Drive … their chance to try out what we have been learning.  During a Test Drive, I talk to individual students about their work; this mini-interview does not involve every student for every Test Drive, but involves most students on any given class day.  We review the Test Drive as a class, based on students explaining what we should do.

FOCUSED WORKSHEETS
My classes know these simply as ‘worksheets’ — they include material from all sections since the last test, and normally involve 5 or 6 items.  My goal during worksheet time (20 to 35 minutes at the end of class), is to talk to every student at least once about their learning.  Over a semester, I will ‘mini-interview’ every student enough to understand some of their learning needs ; this helps to inform my teaching decisions.  As part of the worksheet process, students have an opportunity to work in groups; for my own purposes, I do not structure this nor require group work … though it is strongly encouraged.

Among the assessments I no longer use regularly, ‘writing’ is the primary category.  I’ve tried different methods, such as explaining steps or sentence completion.  I am sure that other people have developed a pattern with these so that they support learning; my own attempts seem to either frustrate students without benefit or actually reinforce learning wrong ideas.

For the curious, every class involves multiple “Test Drives”.  Each class day involves either a quiz or worksheet; I have tried doing both a quiz and a worksheet in the same two-hour class, but the price is a considerable amount of stress for students.  Overall, I try to do about 40 minutes of assessment activity in every 110-minute class period — not counting ‘test days’.  I don’t label this time as ‘assessment’ for students, because they, too, view assessment as ‘grades’.  Instead, I talk about improving our learning (which is exactly the goal of assessment).

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Innovations in Developmental Mathematics — Getting Past Go

Innovations — large, small, and between — are common in developmental mathematics.  Most of us do not establish ‘Being innovative’ as a goal; rather, we figure out something to do that promises to solve a problem, and this creates the innovation.  Some innovations are very context-dependent, while others are transferable and scalable.  The New Life project (AMATYC Developmental Math Committee) describes innovations that can be locally adaptable and scaled.

The Next Dev Challenge (from Getting Past Go) seeks to gather information on innovations in developmental education, and have the larger community rate the ideas.  The web site is http://gettingpastgo.edthemes.org/ , and we are now in the rating stage of their work.

I encourage you to rate innovations at  the Next Dev Challenge link (http://gettingpastgo.edthemes.org/) . You will need to register in order to rate ideas.  Once you are registered at that site, you can submit your ratings of some innovations (they have far too many for you to rate all of them 🙂 ).  To find entries of interest to you, open the Next Dev Challenge link and then read submissions. (Be sure that you are logged in.)  You can search for phrases, and narrow the results down to one of four categories (assessment & placement; instruction & delivery; continuous improvement; student supports.  Note that you need to click on “View” for each innovation in order to rate it. In addition, you need to choose each rating (adaptability, evidence, and overall) at the end of each innovation’s page.

We have a professional responsibility to participate in projects like the Next Dev Challenge.  I hope you will be able to share some ratings of innovations.  Clearly, I hope that you will support the innovations related to the New Life project in particular; whether you do this or not, please participate in the rating process!

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Formulae As A Disguise

How do we know what a student knows?  More often than not, the use of formulae (such as perimeter or area) serve as a disguise for the lack of knowledge … a disguise which allows a person to achieve a preponderance of ‘correct answers’ in spite of having no relational or procedural knowledge. 

My motivation, sadly, is personal therapy.  Our beginning algebra classes took a test today, dealing with polynomials.  This is a traditional class, though our work together has focused on meaning and understanding.  One problem on this test is a contrived operation question:

Find the polynomial that represents the perimeter of the figure. [Figure shows a triangle with sides 3a+2, 2a+1, and 6]

A minority of students added the sides.  Two responses predominated the incorrect work — P = 2L + 2W, and A = LW.  Students retrieved these formulae in spite of the visual stimulus indicated that this was not a rectangle.  It is likely that most students had achieved ‘success’ by using these formulae in prior math courses, perhaps where the material was ‘blocked’ (all problems of a similar type, not mixed).

This thought led me to question something at the heart of our current work in this course:  ‘rules’ for operations with exponents.  The formulae for this work have been stated verbally, not symbolically; our class time has been focused on the reasonableness of our rules.  Based on the types of mistakes I see on other items, I suspect that students are storing some of their knowledge in those “formula files” just like the geometry ones.

I am suspecting that a formula in the hands of a novice math student is dangerous, just like some power tools in the hands of novice craftsmen (like myself).  Perhaps we would be better served by avoiding rules in most cases, and avoiding formulae as long as possible, so that all work is done based on some understanding.  Perhaps a student stops learning as soon as there is a rule or formula to remember.  This concern with formulae is related to concerns with PEMDAS:  The presence of a rule which provides sufficient correct answers stops the learning process, and may prevent deeper understanding.

If we are talking about finance formulae involving 6 input variables, I do not see a problem with the formula stopping the learning process.  However, when there is a key mathematical concept involved — whether perimeter or exponents — I think the formulae create enough problems to approach them with reservations.  If anybody knows of related scientific research on the impact of formulae on learning, I would love to hear about it.

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Mythical Course Holds Key for Reforming Mathematics

The concern is not that “those who ignore history are bound to repeat it”.  No, the concern is that those who ignore history create conditions that hurt students.  In the case of mathematics, the mythical course called intermediate algebra holds the key for reforming mathematics … and policy makers often fall in to self-defeating behavior because they ignore history.

Here is the core question:

Is it possible for a course to be ‘college level mathematics’ if it does not have an intermediate algebra prerequisite?

In an opinion (Sacramento Bee), Katie Hern writes of the challenges facing the dozens of California community colleges who have implemented an alternative statistics pathway (http://www.sacbee.com/2012/11/10/4974786/new-approach-to-remedial-math.html).  These colleges are either in the Statway network, or they are doing a stat path as part of the California Acceleration Project (http://cap.3csn.org/).  Policy makers in the state, along with math faculty unaware of history, may block this work.

Many problems exist within the current mathematics curriculum, and intermediate algebra is a core contributor to these problems.  With just a bit of cynicism, here are statements that define intermediate algebra in the 21st century landscape in this country:

1. Intermediate algebra is the course that protects faculty teaching ‘college math’ courses from students who might need extra help.
2. Intermediate algebra is a distorted version of a high school algebra II course from 1965.
3. Intermediate algebra is the perfect course to show students that mathematics can be totally without redeeming value.
4. Intermediate algebra is the last math course to employ technology in intelligent ways.
5. Intermediate algebra is the final course that you can assign to a high school math teacher with the directions “just do what you do in the day … the students are not likely to succeed anyway”.
6. Intermediate algebra is the course used to kill any dreams of being in a STEM field.

I do not know of any high school which offers a math course as mind-numbing as our intermediate algebra courses.  We have this belief that our intermediate algebra course is roughly equivalent to a second year algebra course in high school; even before Common Core … even before the NCTM standards … this was not true.  Back when community colleges were being born and growing rapidly in developmental math work (roughly 1965 to 1975), the curricular materials for our intermediate algebra courses were based on the general framework of an algebra II course that existed for a short time.  The high school curriculum changed — and we did not.

We might believe that our intermediate algebra course is still a good thing; after all, matching (or not matching) a high school course does not have anything to do with the merits of a course in college.  What good does an intermediate algebra course do our students?  Most readers will think something like “get students ready for college algebra or pre-calculus”; this would mean that the learning in intermediate algebra prepares students for the learning in those courses.  We confuse ‘covering the right topics’ with ‘preparing students’; college algebra and pre-calculus are more than finite sets of procedures to symbolically derive answers.  A college mathematics course is all about understanding mathematics as a science so that students both see the intellectual beauty and can apply their mathematics.  Does factoring the sum of cubes, or rationalizing a denominator, have anything to do with preparing students for that?

As a profession, we need to recognize the false nature of our beliefs about intermediate algebra.  Until we do, our students will continue to face artificially long sequences of math courses without any basic value.  If we can embrace a shared vision of college mathematics … ‘understanding mathematics as a science, can see the intellectual beauty and can apply it’ … we will open the doors to a better future.  Imagine a math curriculum where we emphasize good mathematics, the joy of learning mathematics, and developing reasoning abilities; perhaps we can build a curriculum which inspires students to consider STEM fields.

The mythical course (intermediate algebra) has been used as an artificial and false measure of ‘college mathematics’.  Our shared professional judgment, involving compromise as all shared work does, forms a reliable means to measure ‘college mathematics’.

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