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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Quantway(tm) Materials, available — the Dana Center “FMR” course

The Dana Center (University of Texas – Austin) has launched an updated web site.  As part of this, the original Quantway™ materials (version 1.0) are available along with other documents — see http://www.utdanacenter.org/higher-education/new-mathways-project/new-mathways-project-curricular-materials/foundations-of-mathematical-reasoning-course/  

To get the Quantway information, you need to scroll down to the end of that page.  If you want general information on the New Mathways project, go to http://www.utdanacenter.org/higher-education/new-mathways-project/

This site is part of the New Mathways Project.  I’ve talked before about their project; the first course in their paths has now been named “Foundations of Mathematical Reasoning” (FMR).  It looks like this course will be shared by all 3 paths that will be developed — statistics, quantitative reasoning, and STEM.   The learning outcomes for FMR will be adapted from the Quantway outcomes, which were adapted from the New Life MLCS course.  The position of FMR in a math curriculum is very similar to that of MLCS; the Carnegie Foundation Quantway program is different in that students are tracked from the start … both New Mathways and New Life provide a flexible structure at this level.

The fact that 3 reform efforts share a curricular element (the first course) is part of my optimism that we can create basic change in developmental mathematics over the next several years.

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Towards Effective Remediation: Quantity and Pacing

I’ve been having conversations about arithmetic and similar topics (at the Achieving the Dream conference, and online at MATHEDCC).  In some cases, the conversation was the result of telling people about the New Life model (see https://www.devmathrevival.net/?p=1401).  In other cases, we had been discussing other issues.

So, I have been thinking about related issues.  As a result, I have some ideas of how to frame a conversation about developmental mathematics that might help us make progress.  To start with, we often describe developmental mathematics by using names of courses or by listing topics with implied outcomes.  In our New Life work, we actually started by examining the types of mathematics that students need to succeed, and dealt later with course names and lists of topics.

Our curricular designs are based on our assumptions and goals, which are often unstated.  One of the most problematic assumptions is:

It is effective to deal with 8 general topics, with 10 to 15 outcomes within each, in one course.

There are two fundamental problems with this.  First, the courses we design cover so much ‘material’ that we prevent the learners’ brains from dealing with the associations and connections that are part of learning; this results in most students focusing on just remembering what to do, rather than making sense of it.  Second, the design is based on the absence of prior learning (good and bad) in the learner; this is obviously not true in almost all cases.  Time is needed for us and students to identify where there are conflicts between prior learning and current need, and time is needed to deal with these conflicts.  The result is that we add another layer of ‘learning’, one that is weaker than prior learning; students after our courses are notorious for returning to wrong methods and ideas after our course is done … because we do not provide a method of correction.

We need to slow down; learning is much more complex than having a list of 80 to 120 outcomes.  Since we need to go slower, we must be strategic about what areas to focus on .. trying to do it all (or even most of it) means that we are willing to accomplish little of significance.

This strategic work should be based on our judgments as mathematicians about which mathematical ideas are most important in particular cases.  Do we want work with fractions, or do we want work with proportional reasoning?  If we want both, what are we willing to give up … percents or linear models (as examples)?

We need to do some critical thinking about our goals and purposes, and apply our problem solving skills so that our courses are effective learning experiences for our students.

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The Logic of Change: Do the same, expect different results

You have likely heard the quick definition of insanity:  Doing the same thing, and expecting different results.  Presenters often apply this statement to teaching, frequently stated as “Why should we teach it the same way as they had before, when that obviously did not work?”.  The interesting thought in this logic is the ‘same’ descriptor can be applied to many aspects of the current environment; in spite of this, most discussions focus on the pedagogy and on the teacher behavior in particular.

What about the content?  Perhaps we can improve our results if we first improve the content.  The appropriateness of the current content is questionable, and some have argued that the current content is damaging.  You might take a look at the New Life course outcomes (MLCS Course Goals & Outcomes Oct2012  and Algebraic Literacy Course Goals & Outcomes Oct2012).

However, perhaps we are tragically over simplifying the conversation.  What do we mean by content?  “Algebra” does not always refer to the same content, nor do we use it to refer to the same assessment standards.  We also ignore, I believe, the issue of student perceptions of content.  If you want to trigger a uniform reaction to content, put a simple problem that involves fractions and variables in front of students; in developmental courses, most students will perceive this type of problem as a threat and as something they can not understand.    We could improve our courses tremendously if we would invest time in improving the accuracy of student perceptions of content.  Yes, this takes time, and we would have to give up something … look at it this way:  Most students do not achieve deep understanding of most topics anyway; perhaps the net result would be better if we went a lot slower, with fewer ‘topics’.

You might try this experiment:  After you have covered a topic in a class like you have usually done, where the class went as well as you normally see, ask your students to write their answer to this question: “What are we learning about?”  [I ask individual students this question, and suspect that your students will struggle to provide a good answer just as mine do.]

I see still another over simplification in this conversation: is our content described by objects and procedures, or is our content better described by concepts and relationships?  We do not share perceptions of content, which makes it harder on students.  My hope is that we can, through many professional discussions over an extended period, involving all parts of the country, develop shared language to communicate our perceptions of content.  Of course, I would like us to emphasize reasoning and mathematical problem solving (beyond ‘real world’ problems).  In any case, our students would benefit from our accurate use of a shared language for content.

In many cases, speakers who use quotes like ‘do the same, expect different results’ are using this as a rhetorical device in their efforts to convince us to adopt their solution.  Our profession needs us to take a deeper look at the situations and problems.  If simple statements could solve the problems, they would have been solved long ago.  Making progress at scale (in location and in time) depends on broadly shared conceptualizations and collaboration on solutions.  We each are part of the solutions.

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