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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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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Is Mathematics a Science?

My college has recently completed a ‘reorganization’ of programs and departments.  As a result of this change, mathematics is now in the same administrative unit as science. Is this a good fit?

Although we share much, I have seen some interesting differences.  One striking difference is this:

Mathematics faculty are expected to be flexible generalists.

Science faculty are expected to be specialists.

We are likely to be posting one full-time position in mathematics, and at least 2 full-time positions in science.  As the programs talked about requirements for the positions, mathematics consistently kept flexibility as a top priority — to be able to teach a variety of courses.  Science faculty, on the other hand, consistently listed specific backgrounds — micro-biology versus biology, physics versus geology, etc.  I have asked about why this is the case, irritating a few of my friends along the way; the rationale basically boils down to ‘we need a specialist to teach x’.

In mathematics, we sometimes seek a specialist — like a math for elementary teachers course, or statistics.  The vast majority of math faculty (full-time) are qualified (in our view) to teach any of a dozen courses.  Science faculty seem to keep themselves in a box, where they may have 3 to 5 courses that they can teach.  I am not sure which approach is superior, but I do know that the situation is related to the other observation about math & science.

Science, in general, does not do developmental.

Students in K-12 have had a variety of science.  When students arrive at college, the college-level science courses they take are determined by their program — not by ‘deficiencies’.  Certainly, students who have struggled in science select programs that will provide them with lower-level science courses.  Every student begins chemistry with a college-level chemistry class; every student begins biology with a college-level biology class.  [My college had, at one time, a developmental science course — never a large population.]

Part of this is the acceptance of ‘science’ as a set of (almost) independent disciplines (sometimes competing disciplines).  Students will generally take courses in 2 science disciplines.

Mathematics is seen by policy makers as a single, continuous strand.  At the bottom is arithmetic; at the top, calculus … in between, lots of algebra, a little geometry, and some trigonometry.  There is “one mathematics”; there are “multiple sciences”.

Of course, this ‘one mathematics’ is an incorrect view.  First of all, that image confuses a sequence of prerequisites for a content structure; only parts of algebra are needed for calculus, as is the case for geometry and trigonometry.  Students in occupational programs are the ones who might get to experience the other parts of these mathematical disciplines.  We, the faculty, reinforce this incorrect view by testing and placing all students along this single continuum (including the requirement for remediation of arithmetic and algebra).

Secondly, there are mathematical disciplines that are relatively unrelated to calculus preparation … disciplines that are used extensively in the modern world.  Students are more likely to interact with network problems than they are common denominators.

As we talk with career experts and other programs about what their students need, what topics do we ask them about?  I suspect that 99% of the discussion focuses on the ‘calculus continuum’ (arithmetic to calculus, via algebra).  Do we ask about topics that are not in developmental math courses?  Topics that are not in introductory college courses?  I’ve not seen that done.

Could we envision a world where there really was no need for developmental mathematics (in the sense of repeating school mathematics)?  Unless students need calculus for their program, would it be possible to start with “basic quantitative reasoning” or “introductory statistics” or “math for electronics” for students less prepared?  Better prepared students, perhaps, could take “applied calculus” or “diverse mathematics for college” or “statistics and probability”.   Students needing calculus could take “general calculus” as a preparation for a calculus sequence. These questions, perhaps, are related to the nudge that some state legislators are giving us when they limit developmental education.

Although mathematics is the “Queen of the Sciences” (historically), our practice of mathematics is not so much a science.  A science is based on a collection  of methods applied to related sets of objects (like chemistry does); mathematics does consist of several disciplines.  However, we do not function like a science, nor do we provide students with preparation for scientific thinking within our math classes.

Mathematics in college is not a science.  Would we serve our students better if it were?  What would that look like?

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Are Math Courses Worthwhile?

Colleges usually require students to take math courses, whether for an associates degree or a bachelor degree.  The common approach is to identify a level on the “developmental math to calculus” ladder that seems like the best fit.  I believe that this approach is bound to failure, partially because these math courses are normally not worth taking … from the students’ point of view.

Much of the current conversation focuses on developmental mathematics, and whether those courses are worth the investment of institutional resources.  This approach hides the assumption that the gateway college courses are worthwhile for the institution and its students.

During a meeting about general education last week, a comment was made that we all know what college algebra is; to be fair, the speaker meant college algebra as opposed to intermediate algebra.  It’s true that math faculty can tell when they see a college algebra course — because it matches a generic description of college algebra.  The suggestion was also made that a college algebra course includes more demanding problem solving than intermediate algebra.

A student perspective on courses is naturally simpler than ours, but perhaps we need to attend to that perspective to solve our deep-rooted curricular problems.  A course is worthwhile for students when one or more of these conditions is met:

• The content of the course is naturally appealing to a curious mind.
• The abilities developed in the course enable success in other courses (easily seen as such).
• The process of learning in the course is stimulating and/or rewarding (innately).

I’m not describing students who think a course is worthwhile because it was easy, nor those who see primarily value in the social relationships.  I am thinking of students who are looking for an academic reason for taking a course.

A course such as college algebra is doomed to fail all student criteria, at least for most students.  It seems like we, as mathematicians, want students to take these courses so that we can spot the unusual students for whom such an artificial set of content appeals to students via the third condition (the learning is innately stimulating or rewarding).  We seem to take pride in the tidy logic and coherence of the traditional content, forgetting that students might need something different for their needs.

In other disciplines, a gateway course is often seen as an attractor for students — show students how wonderful the discipline is so that they want to see more.  Sociology and french inspire students, sometimes, because wise faculty design such courses to be potentially inspiring to a broad cross-section of students.  When was the last time your college algebra course inspired somebody who was not already STEM-bound?

We would like to have more math and STEM students, but we put courses in front of students that have a strong track record of discouraging student interest in our discipline.  Whether we call it college algebra or pre-calculus, a central goal of the course should be student inspiration.  I do not think our typical courses serve students sufficiently well to be worthwhile.

My concern also applies to other courses besides college algebra or pre-calculus.  We often use statistics as an easier path for students, a sort of “we can’t win anyway, so let’s make it easier” approach.  It’s time for us to re-build our gateway math courses so that they are appropriate introductions into the science called mathematics.  The emerging models of developmental mathematics — AMATYC New Life, Dana Center Mathways, Carnegie Foundations Pathways — can form a foundation for these new gateway math courses.

Our gateway math courses are not usually worthwhile for students, but they can be … and we should make them be worthwhile.

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