Symbols as Window

Like humans in general, our students develop expectations based on experience.  Habits form, often without awareness or conscious effort.  Behaviors exhibit, which are used to measure knowledge.  In assessments, we often confuse correct behavior with correct knowledge.

Symbolic work can be difficult for novices.  We (experts) see large amounts of information in short symbolic statements.  For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.

On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):

• -3² + 5² = 2^4 = 16
• 8^6 divided by 8^2 = 1^6

The first error is a coincidental ‘right answer’ for a very wrong method.  The second one, not at all.  Both involve over-generalizations of ‘same number’ rules.  Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).

My concern is not these particular students, nor these particular errors.  My concern is our overall approach to mathematics.  We tend to take one of these approaches to symbolism in mathematics:

1. Emphasize symbolic procedures, and measure understanding by correctly completing more complicated problems.
2. Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.

Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures.  We need to learn how to balance the ‘symbols’ and ‘reasoning’ aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.

Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student’s future.  I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student’s future.

Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science).  Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.

When we assess the mastery of symbolism, we need to deal with much more than ‘correct answers’.  In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice’s understanding based on the individual learner.  Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.

Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks  ago — dealing with the adding terms error (first error above).  On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class.  Something like half the students added the exponents as well as adding the ‘terms’ (coefficients).   About 40% of these students apparently maintained this erroneous method up until the test.

Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules.  Mathematical symbolism can be a window into the houses where students keep their math knowledge.  Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.  

We need to slow down our courses.  Learning mathematics is not a fast or spontaneous activity.  Learning mathematics is hard work for both us and our students.

 Join Dev Math Revival on Facebook:

Product As Sum: The Language of Algebra

I’ve been puzzling over some types of errors that seem both common and resistant to correction.  Essentially, the errors involve a disconnect between meaning and symbols especially in the two basic structures of quantities — adding and multiplying.

Here is a brief catalog of the errors:

• 3x²+5x² = 8x^4
• 4a(2b) = 8b + 4ab  (or some other ‘distributing’)
• (5y²)^3=15y^6  or 125y^8
• (3n +2) + (5n + 4) = 15n² +22n + 10
• sqrt(4x^9) = 2x^3
• sqrt(-50) = 5i + sqrt(2)

I’ve been seeing these types of errors for many years; however, it seems like the first 4 are becoming more common.  The radical context is not that important by itself for most of my students — except as a window into the same fragile knowledge about mathematical notation and meaning.  The errors appear with both new-to-college students and students who have ‘passed’ an algebra course.

In talking to students about these patterns, I’ve concluded that quite a bit of the problem is based on procedures removed from meaning.  Students usually know the phrase “like terms”, but seldom talk about counting when we have them; they know to combine the numbers in front but are often unsure about the exponents.  A focus on the meaning of the expression would make it clear what should be done.

The fourth error (‘foiling a sum’ or ‘distributing when adding’) is triggered by the “distributing is great” attitude; students really like to distribute, and we talk about distributing all the time.  In exploring this error (which shows temporary improvement) students say that they did not “see” the operation between the parentheses; what they mean is that they thought that parentheses means a product.

It’s likely that experienced teachers are not surprised by any item on the list above.  The issue for us is this: If these are important enough, how do we change our curriculum to decrease the frequency of such errors of meaning?  My own view is that the basic errors (the first 4) are very important, and I want to address them in all courses (whether traditional algebra or a math literacy course).

One strategy that I plan to use is more “unblocked practice and assessment”.  Much of a traditional developmental math course is severely blocked: the problems deal with a small set of procedures, separated from other types that might trigger an error.  We need to provide opportunities for these errors to be shown during the learning process.  Instead of trying to include quite so many types of each procedure, I will include some competing types from earlier work.  A student who can complete 50 ‘foil’ problems with 90% accuracy may not understand much at all, and may mis-apply the procedure … if we’ve never given them a chance to develop skills in discriminating types of problems.  This unblocked approach needs to be in all stages of learning (initial, practice, assessment, cumulative, etc).

Another method I use in my beginning algebra course is based on language learning concepts.  The idea is not complicated: Present students with either the symbolic statement or a verbal equivalent and ask them to identify the other.  Usually, this is done in a ‘multiple-select’ format: more than one correct choice is possible.  Students need to know that there is more than one verbal statement for a symbolic statement, and that there are sometimes equivalent symbolic statements.

For years, I have included some vocabulary or concept questions on daily quizzes.  I am concluding that I need to expand this to other assessments including tests, and to include perhaps more types.  Some of the online homework systems we use have these types of items, and the students who need them the most tend to skip  them … putting more emphasis on these in assessments will encourage students to take them more seriously in the homework.

I called this post “product as sum” because I am seeing students not being able to consistently treat them accurately.  This is such a fundamental concept that such errors bother me, especially when they occur in students who have passed an algebra course last semester.  Perhaps this is more evidence that:

1. We are trying to ‘cover’ too much (not enough time to understand and connect knowledge)
2. We focus on procedure too much (removes meaning as a critical feature to deal with)
3. We compartmentalize content too much (problems tend to be blocked, sometimes severely)

Meaning, connections, and concepts are important.  Procedures by themselves?  Not so much!

 Join Dev Math Revival on Facebook:

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).

 Join Dev Math Revival on Facebook: