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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This is Hard Work … Why It is Important!

The AMATYC 2014 conference is almost over.  Many conversations and sessions have dealt with changing the curriculum in basic ways, whether shifting to a New Life model or Dana Center or Carnegie … in part, whole, or modified.  Some of us get so enthusiastic about this change that we don’t get slowed down by worries or concerns about the amount of hard work this will take; for us, it’s like nothing will stop us from reaching  our goal.

Most of us, however, are facing constraints.  We are intimidated by the work involved — the hard work of developing an idea, getting consensus or approval in the department, and building institutional support, all of which is required before we get to develop the teaching in the new courses.  Perhaps, we think, it would be best to take a small step like replacing one chapter in the current course with a new one, and see how the ideas work out at other institutions.

Take the biggest step you think you can.  In fact, take a step a little bigger than you think is realistic.

I could justify these statements by citing policy initiatives that are coming, by showing data on how ‘bad’ things are now, or by invoking the Common Core mythology.  Today, I want to take a different approach to the rationale for why it is important to make changes a little bigger than you think is realistic.

Call it personal, or perhaps religious.  My world view, informed by my beliefs, goes something like this:

  • Do no harm to others
  • Place the needs of others ahead of (some) of your own needs

For a large portion of students placing below calculus, we are doing harm to them.  They come to us with dreams and aspirations, and we place steps in front of them that are frequently and artificially difficult.  Yes, we bear some responsibility for students giving up on their dreams.  We have been doing significant harm, even though our work is driven by a desire to help.  We must stop doing harm to such a degree.

We should be placing the needs of students above our own needs.  We will have jobs, though different, even if we eliminate half of our developmental courses; sure, it’s not comfortable … it can even be threatening.  However, continuing what we have been doing is not putting students’ needs first.  Does any student need the 100 learning outcomes of my college’s developmental mathematics courses?  Absolutely not.  Those 100 outcomes are there because we there might be a need sometime for somebody to use them in some situation; much of this ‘need’ is driven by assessments within mathematics courses.

Herb Gross, founding president of AMATYC, gave a popular keynote speech here at the conference.  At one point, Herb said that we should re-phrase the golden rule; instead of “Do unto others as you would have them do unto you” it should be “Do unto others as they would do unto themselves.”  I appreciate the intent with this re-phrasing, but it totally misses the faith justification for the original … we are called to see ourselves as the other person; the ‘do unto you’ refers to what they could do to you or for you if your roles were reversed.

So, envision yourself starting at college, and your initial mathematics course is not very ‘advanced’.  Perhaps you are the person who had AP Calculus in high school who is told that you now must take two courses before taking college calculus.  Perhaps you are the person who passed two algebra courses with weak grades who now finds herself sitting in a class reviewing grade school arithmetic with 25 other students of color.  Perhaps you are the person who did not do well before, and really needs help building a mathematical base … and you find yourself in a math course which deals strictly with procedures, with some drive-by attention to concepts, and no real applications in sight.

The issues I am talking about are definitely not just in ‘developmental’ or remedial courses.  At all levels, we tend to have a mismatch between student needs and what we provide.  The harm is done in pre-algebra and pre-calculus, in algebra and calculus, and even in statistics.

I have never met a math teacher at any level who wanted to do harm to students.  Almost all of us have a sincere desire to help students, to provide the best mathematics possible.  Change can involve a risk to do harm; the profession has enough knowledge about mathematical needs and learning to avoid much of the risk.

Do no harm, as much as possible.  We must take the biggest step possible … and a little more … to reduce the harm we do.  We should take advantage of the external forces, and create the types of change we think are best — to put our students’ needs first.

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AMATYC 2014 — Accelerate and Improve Dev Math with “New Life”

Here is the presentation file, as well as the handouts, from the November 15 session.

Presentation: New Life Accelerate and Improve Dev Math 2014 AMATYC

Main handout: References_NewLifeSession_AMATYC2014

Math Literacy Outcomes and Goals: MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

Algebraic Literacy Outcomes and Goals: Algebraic Literacy Goals and Outcomes Oct2013 cross referenced 2 by 2

Summary of 3 Emerging Models for Dev Math: Summary of Three Emerging Models for Developmental Mathematics Updated 2014

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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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