Saving College Mathematics

The problem with reality is that it tends to get in the way of where we want to go.

I’m thinking of two recent communications.  One, a comment in response to a post here, suggested that the Common Vision will have the same fate as Calculus Reform al a 1990 … in other words, ‘n.s.d.’ (no significant difference), no impact, nowhere.  The other, a presentation by a leader of the Common Vision work who suggested that we have reach a critical mass for modernizing college mathematics.

Both speakers are experienced professionals with a strong mathematical background.  Both can site ‘data’ to support their conclusion, and both can be wrong.  [No surprise to either of them!]

Before continuing, let us consider the three types of college mathematics courses:  Developmental; Freshman/sophomore level mathematics; and upper division mathematics.  Each of these types has a unique set of forces acting on it to either change or remain the same.  The Common Vision report is directly related to the freshman/sophomore mathematics in particular.

Attempts to revolutionize freshman/sophomore mathematics have focused on part of a system.  Both the ‘lean and lively’ calculus and college algebra ‘right stuff’ dealt with content, primarily.  The AMATYC Standards (Beyond Crossroads) maintained a focus on processes (such as instruction or assessment), though “BC” was hardly calling for revolutionizing college mathematics.

We should consider what has led to a fundamental change in developmental mathematics.  The process that is leading to long-term basic change (a good revolution) is driven by three compatible projects which focus first on the content and second on process.  [These efforts are the Carnegie Pathways, the Dana Center Mathematics Pathways, and the AMATYC New Life project.]  The three projects collaborate in basic ways, even though they could be seen as ‘competing solutions’.

For this purpose, I will ignore the co-requisite movement, which seeks to displace developmental mathematics without impacting freshman/sophomore mathematics in any significant manner.  Such an effort has a low probability of long-term survival, though it certainly will create some unintended changes.

The developmental mathematics revolution is working (though it is not yet complete) because the work appeals to mathematicians and because the modern content encourages active learning methods.  There is also a continuity with prior professional work, and the engagement of diverse stakeholders in the process.

If we seek to save college mathematics, the core of our work is the freshman/sophomore curriculum.  A variety of forces are acting on this work to make ‘revolution’ difficult; even modest reforms seem to be too much of a challenge.  However, I think the largest sets of forces in this matrix have their origins in us … the mathematics faculty of colleges.

We worry about ‘transfer’, and we sorry about ‘prerequisite material’.  The transfer worry means that we don’t change because our sister institutions might decline the transfer … the prerequisite worry means that we don’t change because it might disrupt a mythical sequence of necessary steps.  In many ways, the transfer worry feeds off of the prerequisite worry.

In many states, the transfer worry is managed by a state system.  In most of these systems, the decisions are made by ‘us’ (college math faculty).  Therefore, the transfer worry is a self-imposed set of forces to resist change.  Clearly, the solution is to develop a consensus that change is needed … which means to support colleges who are willing to begin the revolution in mathematics.

Earlier I mentioned the ‘critical mass’ comment.  This observation was based on evidence of process changes (mostly, in active learning and some social psychology) primarily in R1 institutions (research universities).  Although these changes are welcome and help students, I don’t think the long-term impact will be anywhere near large enough compared to the problems we seek to solve.

College mathematics, especially freshman/sophomore level, is defined by content and structure defined by the needs of a 1965 education for a 1955 occupation (engineers especially).  Any long-term solution has to address the known needs of today’s education for 2010 occupations (a more diverse list).  Modern teaching methods are not enough.

Saving college mathematics requires that we change the mathematics.  Sufficient information exists to develop a new system of courses.  Instead of two college algebra or pre-calculus courses followed by three semesters of symbolic calculus … perhaps we can design a system with one pre-calculus course followed by two semesters of calculus which combines symbolic and numeric methods.  People more experienced would know how to structure this work, so that both content and process are modernized.

We need to stop the pattern of ‘solving part of the problem’.  Solving part of a problem is failing to solve the problem.

It’s time to build a college mathematics system that solves problems and serves our students.  We can’t let “reality” prevent us, because often we are that reality.

 
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Are Our Students Changing?

In most ways that matter, college mathematics has not changed in the past five decades.  Whether we are looking at developmental math, college algebra, or calculus, the mathematics has not changed … the changes have been in the mediating tools (computers), not in matters of substance.

Of course, that assessment is too harsh with respect to developmental mathematics.  At this writing, perhaps greater than 10% of students in remedial mathematics are enrolled in a modern course (Math Literacy, Foundations of Mathematical Reasoning, Quantway, or Statway).  However, those modern courses are too often implemented around the edges … only students needing non-STEM math courses are allowed to take the improved dev math course.

At the same time, our students have changed in basic ways.  One shift is the high school math they have experienced.  When our current remedial courses were designed, the median high school math experience ended in Algebra I.  Currently, the median experience includes Algebra II … and more, in many cases.

The lighter bars represent the graduating class of 2009; 76% of them completed Algebra 2 … and 35% completed something like pre-calculus in high school.  [This data is based on a detailed study of a sample of transcripts.]

Note that the high school courses have changed in basic ways, in response to the NCTM standards and even the Common Core State Standards.   Our college courses have held on to the abandoned property at the corner of 1965 and Elm Street.

Student intended majors have also shifted.  Using data from 4 year colleges, this is the pattern over an extended period.

[From “Insights and Recommendations”, MAA Calculus Project  http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf ]

That chart is not especially clear.  Notice the curve upward for one of the trends?  The steepest increase is in biological sciences, which used to be less than half the size of engineering majors … though now the bio sciences majors outnumber all other groups.  Our college math courses continue to emphasize the needs of 1965 engineering programs, with a fixation on ‘the calculus’.

I am not as concerned with whether students have ‘more skills’ now; they likely do, based on the long-term trends in national assessments.  However, talking about ‘more skills’ often limits our discussion to particular subsets of either high school or college mathematics.  My point is that we, in college mathematics, are significantly blinded by our viewpoint in the traditions of college mathematics … and that we would not notice changes in student mathematical knowledge because we are looking in the wrong places.

It’s time for ALL college students to experience a modernized mathematics curriculum, one which reflects student backgrounds and goals while providing content based on professional college standards.    Take a look at the guiding principles in the Common Vision document … http://www.maa.org/sites/default/files/pdf/common-vision/cv_white_paper.pdf

The status quo is not just unacceptable.  The status quo is a professional failure on our part.  We can fix that, and help both our students and society thrive.

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Modern Dev Math

Let’s pretend that we don’t have external groups and policy makers directing or demanding that we make fundamental changes in developmental mathematics.  Instead, let us examine the level of ‘fit’ between the traditional developmental mathematics curriculum and the majority of students arriving at our colleges this fall.

I want to start with a little bit of data.  This chart shows the typical high school math taking patterns for two cohorts of students.  [See  http://www.bls.gov/opub/ted/2012/ted_20121016.htm ]

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There has been a fundamental shift in the mathematics that our students have been exposed to, and we have reason to expect that the trends will continue.  We know that this increased level of math courses in high school does not translate directly into increased mathematical competence.  I am more interested in structural factors.

Intermediate algebra has been the capstone of developmental mathematics for fifty years.  At that time, the majority of students did not take algebra 2 in high school … so it was logical to have intermediate algebra be ‘sort of developmental’ and ‘sort of college level’.   By about 2000, this had shifted so that the majority of students had taken algebra 2 or beyond.

The first lesson is:

Intermediate algebra is remedial for the majority of our students, and should be considered developmental math in college.

This seems to be one lesson that policy makers and influencers have ignored.  We still have entire states that define intermediate algebra as ‘college math’, and a number that count intermediate algebra for general education requirements.

At the lower levels of developmental mathematics, the median of our curriculum includes a pre-algebra course … and may also include arithmetic.  Fifty years ago, some of this made sense.  When the students highest math was algebra 1 in most cases, providing remediation one level below that was appropriate.  By fifteen years ago, the majority of students had taken algebra 2 or beyond.  The second lesson is:

Providing and requiring remediation two or more levels below the highest math class taken is inappropriate given the median student experience.

At some point, this mismatch is going to be noticed by regulators and/or policy influencers.  We offer courses in arithmetic and pre-algebra without being able to demonstrate significant benefits to students, when the majority of students completed significantly higher math courses in high school.

In addition to the changes in course taking, there have also been fundamental shifts in the nature of the mathematics being learned in high school.  Our typical developmental math classes still resemble an average high school (or middle school) math class from 1970, in terms of content.  This period emphasized procedural skills and limited ‘applications’ (focusing on stylized problems requiring the use of the procedural skills).  Since then, we have had the NCTM standards and the Common Core State Standards.

Whatever we may think of those standards, the K-12 math experience has changed.  The emphasis on standardized tests creates a minor force that might shift the K-12 curriculum towards procedures … except that the standardized tests general place a higher premium on mathematical reasoning.  Our college math courses are making a similar shift towards reasoning.  Another historical lesson is:

Developmental mathematics is out-of-date with high schools, and also emphasizes the wrong things in preparing students for college mathematics.

We will never abandon procedures in our math courses.  It is clear, however, that procedural skill is insufficient.  Our traditional developmental mathematics curriculum focuses on correcting skill gaps in procedures aligned with grade levels from fifty years ago.  We appear to start with an unquestioned premise that remediation needs to walk through each grade’s math content from 5 decades ago … grade 8 before grade 9, etc.  This is a K-12 paradigm with no basis in current collegiate needs.

The 3- or 4-course sequence of remedial mathematics is, and always will be, dysfunctional as a model for college developmental education.

There is no need to spend a semester on grade 8 mathematics, nor a need to spend a semester on grade 9 mathematics.  When students lack the mathematical abilities needed for college mathematics, the needs are almost always a combination of reasoning and procedural skills.  If we can not envision a one-semester solution for this problem, connected to general education mathematics, we have not used the creativity and imagination that mathematicians are known for.  Take a look at the Mathematical Literacy course MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2 .  If students are preparing for pre-calculus or college algebra, take a look at the Algebraic Literacy course  Algebraic Literacy Goals and Outcomes Oct2013 cross referenced

Pretending that the policy influencers and external forces are absent is not possible.  However, it is possible for us to advocate for a better mathematical solution that addresses the needs of our students in an efficient model reflecting the mathematics required.

More on the Evils of PEMDAS!

The most common course for me to teach is ‘intermediate algebra’, and I’ve been thinking of the many issues with that course as part of the college curriculum.  However, my interest today is in poking at PEMDAS … and the poor way we often teach the order of operations.  As you know, understanding the order of operations concept is one key part of understanding basic algebraic notation.

An easy poke at PEMDAS is the “P” (parentheses for us, ‘B’ bracket in some other countries).  The problem below is actually from our beginning algebra curriculum:

16÷(4)(2)

Operator precedence usually places products and quotients at the same level, with the normal parsing from left to right (answer: 8).  Of course this ‘tie breaking’ rule is arbitrary; however, a convention about this is necessary for all machine calculation … and our students interact with these machines.

I’ve seen people say that this is a silly point, without merit … and they suggest including sufficient grouping to avoid any “ambiguity” from the expression.  I’ve also seen people say that there is no such thing as implicit multiplication (as in the problem above, or as in an algebraic term like -3x).  What they mean is that implicit multiplication has the same priority as explicit multiplication; some programming environments do not allow implicit products in order to avoid issues with that precedence.

If we state the problem algebraically, it might be:

16÷4k, where k=2

We, of course, prefer fraction notation for quotients due to the ‘confusion’ created by the divided by symbol (which our students write as a slash):

16/4k

One discussion site has a comment that we should use those grouping symbols to be clear, and concludes with a comment that the answer changes when we use algebraic notation for the same quotient & product expression.  (see http://math.stackexchange.com/questions/33215/what-is-48%C3%B7293  )  This ‘changing answer’ feature should bother all of us!

In the original problem above, the product involves parentheses … so our PEMDAS-based students always calculate that product first.  They have no idea that there is an issue with implied products when variables are involved; I’m okay with that at the time (we get to it later).  In all of my years of reviewing missed problems like that one, I’ve never heard a student justify their answer by ‘implied products have a higher priority’.  They always say “parentheses first”.

If we could say “GEMDAS” (for “grouping”) we would be more honest.  I’m not sure what “G” means for my poor aunt Sally … but, then, having a sentence for an mnemonic with no connected meaning is likely to be a bad thing.  When we continually talk about ‘remember my dear aunt Sally’, we encourage students to process information at the lowest possible level — instead of a beginning understanding, all they get is a memorized rule which is fundamentally flawed.

The role of mnemonics in ‘remembering’ has been studied.  The book Cognitive Psychology and Instruction, 4th edition Bruning et al has a review of research on this on pages 72-73 (it’s also in their 5th edition though I don’t have that page reference).  The basic conclusion was that mnemonics help students remember when mnemonics help students remember … and can interfere with remembering when the student does not find them helpful.  That means the some students can use them to remember, some students get confused … and (in my view) all students have negative consequences for using poor aunt sally.

I think the emphasis on PEMDAS also creates a mental ‘twist’ in our students’ minds.  They take expressions which do not have stated grouping and insert parentheses so that the basic meaning is changed:

5x²  is mistakenly processed as (5x)²

In the intermediate algebra course, some strange things happen relative to parentheses.

(3x² – 5) + (4x + 3) is treated as a product

A good portion of my class time is spent on un-learning PEMDAS and building some understanding of notation with order of operations.  The biggest problem … grouping that is done with other symbols besides parentheses (fraction bars, radical symbols, absolute value, etc).

Because I’ve been teaching so long, I’m occasionally asked about any changes I notice.  Folks expect me to report that students are less prepared now compared to 30 or 40 years ago.  Actually, there have been improvements in the mathematics preparation of our students.  However, these improvements are not uniformly distributed both in terms of students and in terms of mathematics.  In particular, students struggle more now with order of operations; some of that degradation seems to be due to the over-use of PEMDAS.

We should avoid books that build in PEMDAS, and we should avoid the mnemonic in our classes.  Understanding something is much better than memorizing an erroneous rule.

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