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!

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Developmental Math — Summary of Three Models

This is an update to a prior post, with new information … it compares AMATYC New Life, Dana Center Mathways, and Carnegie Foundations Pathways.

If you want a download of this file, click on the link below:

Summary of Three Emerging Models for Developmental Mathematics Updated March 22, 2013

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Math: Applications for Living — The Chance of That

Our Math-Applications for Living course is finishing up our work with statistics and probability.  A couple of students commented that they realized that their thinking had to change when we talked about probability — what seems natural for experts is not natural at all for novices.   This was an issue for combining probabilities — either a sequence (multiply) or options for one event (add).

On today’s test, one item seems to be really confusing:

Determine the probability of meeting someone whose phone number ends in the same digit as yours.

Students are asking “what about the rest of the digits” and “how many digits are we including for the phone number”.  From my point of view, this question was an attempt to measure the basic rule about probability — for random events, the probability is the ratio of “yes” to “total”.  We had dealt with this in other contexts, but I wanted something new on the test to see if they would apply the idea.

Probability continues to be one of the more challenging parts of the course.  One of the ironic results is that students appreciate the algebra that we will do next, because they seem to ‘get that’ a little better.

Of course, many of us who deal with probability do so within the context of a statistics course (as opposed to my survey-type course).  I wonder if you find that students have similar difficulties in ‘probabilistic thinking’.

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Hidden Treasure in Math Class

A course design can facilitate learning, and a course design can hinder learning.  I suspect that we get so focused on the details of our math courses that we may not notice whether our course is facilitating or hindering.

In our Math Literacy class, we have been working on algebraic reasoning.  On the surface, the class looks like we are not ‘covering’ very much because we don’t include some typical algebraic (developmental) topics.  We found some hidden treasure this week in class.

As we often do, part of class is based on groups figuring out problems with some guidance and reflection.  Today this meant that we had each group do an equation ‘tag-team’ style — each student could either do the next step, or erase the last step.  Students had a little trouble playing by the rules, and wanted to switch to ‘their’ method to solve the equation.  The payoff came when we talked about the different choices, as more students figured out that they have options for linear equations.

The hidden treasure came next, not that students saw it as totally good.  We looked at how we could solve equations of a type never seen before, starting with a simple rational equation (namely, 5 = 200/x).  Students could see the solution (40) though not always obtained formally, so we talked about doing ‘opposite’ operations to solve.  We followed this with a radical equation (the pendulum model), which is not normally seen in this level of math course.  To solve for the length inside the radical, we listed the calculation steps if we knew the length and wanted to calculate the period.  Then, we reversed — the opposite operations in the reverse order.

To me, the hidden treasure in this is that students get to think about both types of skills that we use in mathematics — we have routine procedures (often based on properties) and we have reasoning about statements (often based on relationships unique to the problem).  Wouldn’t it be wonderful if students developed both strategies, instead of just using routine procedures (often memorized)?

It’s clear that my hidden treasure was not perfectly clear to students; after this discussion, we had a worksheet which included an equation of related design.  They generally understood the reverse order idea, but thought they should do them in a different order — a choice which requires applying properties of expressions.  Our conversation was more satisfying than normal because we had used the reasoning approach, and talked about choices.

Students may still ‘want’ a recipe for solving equations and simplifying expressions.  Giving students a recipe hides the math treasure; emphasizing choices and reasoning allows for the possibility of students finding our hidden treasure.

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