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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Technology and Transfer Credit for Math Courses … the Value in Mathematics

At a certain university in my state, there is a policy which states that they will not grant transfer credit from an institution if that institution offers the course in an online format; this is applied even if they know that only 1 section is offered online and 100 are face-to-face.  The policy is applied regardless of the course’s policy on proctored tests for online courses.

At a certain university in a different state, there is a policy which states that they will not grant transfer credit from an institution if that institution allows the use of any calculator in the course; the policy is applied even if students can only use the calculator for trivial purposes (computation).  The policy is applied regardless of the course’s assessments of outcomes and regardless of the overall quality of the course.

These issues are coming up in conversations here at the AMATYC conference in Nashville.  Both policies are implemented out of negative motivation on the part of the universities … whether a lack of trust for their colleagues or a lack of understanding concerning the uses of technology to support the learning of mathematics.  Certainly, universities need to stop their use of arbitrary policies concerning technology, which amounts to a conceited attempt to impose a narrow view of what a ‘good’ math course must be like.

In other conversations, some of my colleagues suggest that we need to present arithmetic and basic skills without the use of a calculator.  One person presented a good point in this regard:  Some students confuse the input/output from a machine for the mathematics.  I agree that students need to have a personal understanding of mathematics.  However, we too often present arithmetic as the initial barrier in front of students, a barrier with little redeeming value and almost no long term benefits to students.

At the same time, I routinely see us in a general consensus of what good mathematics is … and what value it has for students.  Concepts, properties, choices … reasoning, communication, problem solving.  We generally support a ‘common core’ of properties that describe good mathematics.  How, then, can we let minor details about technology determine the transfer of credits and the nature of a student’s first “mathematics” course in college?  Are we so easily fooled by a surface feature (technology) that we do not see the value of the work going on?

This is not to say that all uses of online learning and calculators is good or valuable.  Not at all.  If we use that criteria — sometimes not used wisely — we would not grant transfer credit for any course taught in a face-to-face format because research shows that a significant portion of such classes provide no significant learning of mathematics.  No technology, no pedagogy is beneficial without regard to the quality and wisdom of usage.  Every tool can be used poorly.

It’s time for all of us to make decisions based on an evaluation of all components of a course — the outcomes, faculty, instruction, assessment, and integrity.  There is no room for prejudice in dealing with people … or with courses.  If a person feels that they are unable to evaluate the quality of a course due to the presence of a particular technology, then their professional responsibility to allow others to make the determination.  I would prefer, however, that a person with such a prejudice to seek a better understanding so that their prejudice does not exist anymore.

This is not a problem about ‘us’ and ‘them’; this is a problem about ‘we’.  A professional community, committed to providing good mathematics in service to our students and their success. This is not easy work; rich communication is required, and levels of trust. The path forward is always walked by all.

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Intermediate Algebra … the Bridge to Nowhere

Yes, I am using an emotional label being used about developmental education … yes, I am saying intermediate algebra might be such a thing.  A bit of a cheap trick, but I hope that you will continue reading anyway!

The content of our intermediate algebra courses is usually based on topics that were once covered in a second year high school algebra course.  That course, in turn, was created by companies and teams of authors (often a combination of university mathematics educators and high school math teachers).  I have not seen any documents relating to how the companies and authors determined the content; I suspect that much was based on a view “well, this topic would be good for them”.

All of this work occurred long before a general emphasis was placed on understanding, application, and cognitive science.  Procedural accuracy is the hallmark of our intermediate algebra courses — even more so than the high school algebra II course; it’s like we copied the content but limited our work to the lowest levels of learning.

We actually have some helpful stuff in there, if students can remember it later when (and if) they take more advanced courses (whether a pre-calculus/analysis course or in calculus).  The better students may do this; most do not, because the material is not usually taught in a way to create long term use.

So, here is an initial list of reasons why intermediate algebra is the biggest ‘bridge to nowhere’:

• content created over 50 years ago outside of our curricular process
• textbooks focus on procedural accuracy
• learning heavily weighted towards lowest levels of learning

Students who pass an intermediate algebra course meet the prerequisite for some college math courses; however, the intermediate algebra course did not prepare students for that course.  Nor does the intermediate algebra course contribute to mathematical understanding, nor to positive attitudes about mathematics.

Fortunately, we have a model for replacing intermediate algebra — the Algebraic Literacy course from the New Life model.  The outcomes for this course were extracted from what students need in subsequent courses, and these outcomes include both procedural and understanding emphases.   In addition, the Algebraic Literacy course includes the use of mathematics to understand the quantitative components of issues in the world — such as the spread of infectious disease.

The Dana Center work on a Stem Path is also involved in creating a replacement for intermediate algebra.  Those teams are approaching the problem from a similar viewpoint, so I expect their results to be compatible with the New Life Algebraic Literacy course even if their content has some significant differences.

To learn more about the Algebraic Literacy course, I encourage you to come to my session next week at the AMATYC Conference (Nashville); this session is at 8am on Friday (November 14).  [I am also doing a general session on the New Life model that Saturday (November 15) at 2:15pm; this session will include basic information about Algebraic Literacy.]

If you are not able to be at the AMATYC conference, take a look at the Instant Presentations page on this blog https://www.devmathrevival.net/?page_id=116 .  After the conference, I will be posted the materials from the session on that page.

Of course, if you have any questions about the Algebraic Literacy course, just contact me!

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Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)

Earlier this month, the Conference Board of Mathematical Sciences (CBMS) held a forum on mathematics in the first two years; many of the presentations are available on the web site (http://cbmsweb.org/Forum5/)

As part of one of the first plenary sessions, Eric Friedlander commented …  Students in the Biological Sciences now outnumber those in the Physical Sciences in the standard calculus 1 course.  (David Bressoud shared some specific data on those enrollment patterns.)

Historically, the developmental mathematics curriculum was all about getting students ready for pre-calculus.  Our “applications” tended to be puzzles created with physical sciences in mind — bridges, satellites, pendulums, and the like.   Few problems in our developmental courses draw the attention of those in biologically-oriented fields (including nursing).

We could include:

• Surge functions to model drug levels
• Functions to estimate the proportion of a population needed to be immunized to prevent epidemics (P_sub_c = 1 – R_sub_0)
• Models for spread of cancer … and for treatments
• Pollution prediction (simplified for closed systems)

This list is a ‘bad list’ because there is no common property (except being related to biology) … and because I do not know enough to provide a better list.  Take a look at books in applied calculus for the biological sciences; you will see applications that are perhaps better than those above.

There is a trend in the new models for developmental mathematics (AMATYC New Life, Dana Center New Mathways, and Carnegie Foundation Pathways) to include a balance of applications — including more from biology.  We need to bring in more of these applications throughout our curriculum (from the first developmental course up to calculus).

Most of us realize that the ‘applications’ in our courses and textbooks are puzzles created by somebody who knew the answer; generally, these problems do not represent the use of mathematics to solve problems and answer questions in the world around us.  Sometimes, we are not able to provide enough non-mathematical information to provide representative problems … in those cases, some reduction to the ‘puzzle state’ is acceptable.

Our puzzles should represent the diversity in the uses of mathematics, with a significant portion of applications being realistic in nature.

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