Fragile Understanding … Building a Foundation

Our beginning algebra class is taking a test on ‘exponents and polynomials’ today; this chapter is about as popular as a math chapter can be for my students.  The processes are fairly easy, and with some extra effort in class, most students do well on this test.  All is not good, however.

Students tend to have a fragile understanding.  For whatever reasons, the symbols in front of them do not have full meaning.  Here are two examples of what I am talking about.

Subtraction versus “FOIL”:
Seeing a problem like (5x + 4) – (2x – 3), many students convert it into (5x + 4)(-2x+3) and multiply.  They know the rule about ‘changing the signs’ when we subtract, but lose the little ‘plus’ between the groups.

Negative exponents versus polynomials:
Seeing a problem like (6x² – 9x)/(3x²), many students convert (2 – 3/x) into (2- 3)/x to get -1/x.

As teachers, we feel good when students show a process that fits with a good understanding.  Showing a process does not depend on a good understanding.  The relationship works one way cause and effect (understanding leads to good processes); a good process does not lead to, nor is evidence of, good understanding.

So, we give assessments to students and say “they know exponents” because of the processes and answers.  In the extreme form, we have a module on exponents and polynomials and certify “mastery” because of a high score on the module assessment.  We do not do enough assessments that do a compare and contrast — opportunities for us to see if a student has a fragile understanding, identify the weakness, and then build up a stronger understanding.

I continue to work on this problem.  In the case of ‘subtraction versus FOIL’, I use problems like the one shown on assessments early in the semester, during our first class on ‘FOIL’, and later in the chapter.  That helps; no magic, but the opportunity to discuss with an individual student is powerful.

I believe we need to work on two components of our instruction if we have any hope of building a strong understanding in place of fragile understanding.

• Combination of active and direct instruction on the concepts, with a focus on “what choices do we have?”
• Assessments that determine the presence of confusion of concepts (aka ‘fragile understanding’)

Our professional expertise is needed, since we can not assess for the presence of specific confusions unless we know what the common types are.  To make this even more challenging, we have no assurance that the confusions are global versus local — do students in beginning algebra courses tend to have the same confusion regardless of locality?

The best resource we have is the students in our classes.  Having purposeful conversations (oral assessments) is a critical source of information about both a specific student and zones of confusion.  These conversations provide insights, and form a way to validate our more convenient forms of assessment (paper & pencil, or computer test).  When I grade today’s test on this chapter, I will be comparing what I thought they understood to what I see being shown on the test; just like my students, there should not be any surprises to me on the test.

Of course, there is a good question … does it matter at all?  We have a pride in our work and profession, so we respond with an automatic ‘yes’.  We should be able to articulate to other audiences why it does matter.  Does a fragile understanding enable or prevent a student from completing a math course?  How about a science course?  Can we develop quantitative reasoning in the presence of fragile understanding?  Does a modular design support sufficiently strong understanding?  Do online homework systems provide any benefits for understanding concepts?

The issue of fragile understanding is critical to the first two years of college mathematics, whether in a developmental math class college level.  I have heard colleagues suggest that the prerequisite for a certain class be raised to calculus II, not because any calculus is needed but only because students have a stronger understanding after passing (surviving) calculus II.  We often cover this problem with a vague label “mathematical maturity”.

In response to a recent post, Herb Gross (AMATYC founding president) wrote a comment, in which he emphasized the “WHY” in the math classes he taught.  I totally agree with his comment, in which he said that students want the why — they want to understand.  Although a human brain can learn with and without understanding, there is a natural preference to learn with understanding.

A fragile understanding, lacking the ‘why’, leads to both short term and long term problems for students.  I think we waste their time in a math class if we accept correct answers for the majority (70%) of problems as a proxy for ‘knowing’.  Determining that a student knows mathematics is a complicated challenge, and forms a core purpose for having a strong faculty professional development.

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A Natural Approach to Negative Exponents

For today’s class in our beginning algebra course, we took a different approach to negative exponents.  The decision to do something different is partially rooted in my conviction that most of our textbooks are wrong about what negative exponents mean.

To set the stage, the first thing we did was a little activity on basic properties of exponents.   The activity is based on this document Class 22 Group Activity Exponents

This activity uses the type of approach many of us use for a more active learning classroom.  I suspect mine is not as polished as many; several students found the ‘long way’ a bit confusing.  As usual, I did not present any of the ideas before students got the activity and worked in their small groups.

One of the problems on this activity is the problem shown here.  In the ‘long way’ method, students easily wrote out the factors and found the answer.  Quite a few of them used the subtraction method to create a negative exponent.  In a natural way, we noticed that m^-4 is the same as having m^4 in the denominator.

Negative exponents indicate division!

We did not create negative exponents in order to write reciprocals.  We started using negative exponents in order to report that we divided by some factors.  I find it troubling that we have focused on a secondary use for the notation, when the primary use makes more sense to students.

If you want to see what is so important about this, give a problem like this to your students.

Direction: Write without negative exponents.

Almost every student focusing on the reciprocal meaning will invert the fraction — making the 4 a multiplier instead of the divisor it really is.  Most students focusing on the division meaning will see that the m cubed needs to be in the denominator.

In part of this activity, students also dealt with a zero power.  In doing the long way (write it all out), quite a few students wrote that variable in their work; it made sense, though, to omit that factor because it said “zero factors” … and then we can talk about what value that ‘zero factor’ has in a product (one).

As we shift towards more work with exponential functions, it becomes critical that students understand the meaning of all kinds of powers.  A core understanding of negative exponents is part of this; fractional exponents are important too (though we tend not to cover these in either our Math Lit course or beginning algebra).

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Teacher as Confusion Manager — The Key for Student Learning

Early in my career, I focused on being clear — as close to perfectly clear as I could manage.  Class time was easy for students to follow.  Eventually, I realized that my students were not learning very much and decided that I was part of the problem.

Since then, I have seen my role as “confusion manager”.  In planning for what to do in class, I would look for a sequence of activities or problems that were likely to lead to some confusion.  In a basic way, confusion is the brain’s assessment that there is a gap between existing knowledge and needed knowledge.  Without confusion, learning new material is limited for most people.

There is a recent article in The Chronicle  called “Confuse Students to Help Them Learn” by Steve Kolowich  (http://chronicle.com/article/Confuse-Students-to-Help-Them/148385/?cid=cc&utm_source=cc&utm_medium=en)    The initial part of the article covers the experience of another teacher noticing this ‘confusion to help’ property.  Later, the article brings in some experts in psychology.  Certainly, the points in this article are consistent with what I know as a teacher and as a reader of cognitive psychology.

So … here is what I think is so important about the concept of “confusion to help learning”:  There is a great pressure to utilize digital resources, such as short videos.  These videos are almost always created with an emphasis on BEING CLEAR and avoiding confusion.  To check this out, just do a search for videos on your favorite math topics and watch.

Of course, it is possible to create short videos made with productive confusion in mind.  The article listed above suggests that it is not possible; however, I believe that I could create a video that would confuse my students just enough to be helpful.  The problem is that it is much easier to avoid confusion entirely OR to create a lot of confusion.  Productive confusion videos would take the type of planing that goes in to commercial production or movies.  Who has the resources to invest 20 hours in order to create a 5 minute video?

I think students who watch clear videos (or clear lectures) see the experience as validating their existing knowledge — even when their knowledge conflicts with what they just experienced.  A little additional information might survive through short-term memorization, or limited amounts of disconnected long-term information.  If all we are after is recall of facts or replication of simple procedures, clear videos might be sufficient.

As mathematicians, however, we are more interested in goals of reasoning … of application … and connecting knowledge.  This type of learning needs deliberate effort by students, and that means that we need to create and manage confusion.  Confusion, like anxiety, is a natural state for humans; too little of either leads to less learning just as too much of either impedes learning.  Our professional judgment and connection to every student is needed to provide productive confusion.

Have you confused any students yet today?  I hope so!

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Students Don’t Do Optional … or Options

In the Achieving the Dream (AtD) ‘world’, the phrase “Students do not do optional” is used as a message to colleges that policy and program decisions need to reflect what we believe students ought to do — if it’s a helpful thing, making it optional often means that the students who need it the most will not do it.  I tried something in my class that suggests a slightly different idea.

For the past two years, I have ‘required’ (assigned points) students to connect with a help location at the college.  The idea was that students need to know — before they think they need it — where they can get help for their math class.  I allow days for this — usually, until the 4th class day.

Until this semester, I provided students with options for how to complete this required activity.

• my office hours
• the college’s “Learning Commons” (tutoring center)
• the college’s library tutoring (also staffed by the tutoring center)
• special programs tutoring (like TRIO)

Typically, I would have about 70% of students complete this ‘connect with help’ activity; most of the struggling students were in the 30% who did not.  Some of these students eventually found the help.

This semester, I tried a revision to this connect with help activity.  I provided students the following choice(s):

1. the college’s “Learning Commons” (tutoring center)

The result?  I have 100% completion for this activity.  All active students have completed the activity, and most of these did it right away.

This is summer semester, and “summer is different” (though it’s difficult to quantify how different).  However, the results suggest that the existence of options creates barriers for some of our students.  We have evidence that this problem exists within the content of a mathematics class — when we tell students that we are covering multiple methods (or concepts) for the same type of problems, some students struggle due to the existence of a choice.  [For those who are curious, you may wonder if students are not coming to my office hour — so far, I actually have more students coming to my office hours.  No apparent loss there.]

I think the basic question is this:

Given that choices (options or optional) creates some risk for some students, WHEN are there sufficient advantages to justify this risk?

If dealing with a choice has the potential for improving mathematical understanding, I will continue to place choices in front of my students.  We should resist the temptation to provide simple answers when students struggle with mathematics; the process working (learning) depends upon the learner navigating through choices and dealing with some ambiguity. On the other hand, when the choices deal with something non-mathematical, we should be very careful before imposing the choice on students.

Some people might be thinking “So, it’s okay for us to be rigid and not-flexible” in dealing with students.  That is NOT what I am saying.  If one of my students gave me a valid rationale for why they could not do the ‘one option’, I would offer them an equivalent process.  Our rigidity needs to be invested in what is important to us; I would hope that the important stuff is something related to “understanding mathematics” (though we don’t all agree on what that means).

I would suggest that the AtD phrase be modified slightly:

Options will cause difficulties for some students.  Allow options when this provides enough advantages to students.

We usually try to be helpful to students, and part of this is a tendency to provide students with options. Putting choices in front of students is not always a good thing, so we need to be selective about when we put options in to our courses and procedures.

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