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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Mathematical Literacy Course WITHOUT a Math Prerequisite

We did a session (Mark Chapman and I) at the recent MichMATYC conference on our newest course … a second version of Mathematical Literacy with no math prerequisite.

Here are the materials from our session:

Presentation Slides:  math-literacy-without-a-math-prerequisite-for-web

Handout 1 … Information on the course:  math-literacy-without-a-math-prerequisite-handout

Handout 2 … Math Lit Goals and Outcomes: mlcs-goals-and-outcomes-oct2013-cross-referenced

We started offering the new course this semester, so these materials describe the course design.  Data will come later!

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Can We Save “Order of Operations”??

In one recent post, I looked at some basic flaws in the mnemonic “PEMDAS” (there are several fundamental flaws). In another recent post, I talked about how unimportant a ‘correct answer’ can be in a math class.  Let’s examine the intersection of those thoughts, and deal with saving the important topic of ‘order of operations’.

The two most common statements about why “order of operations” is important are:

• “The order of operations is just an agreement so we all get the same answer.”
• “You need to follow the order of operations so that you will get the correct answer.”

Both of these miss the point; their implication is that we can change the correct answer just by changing the ‘agreement’ about order of operations … that we could declare subtraction is always done before multiplying, for example.  The order of operations is not just some coincidence of the mathematical language which will evolve to be anything fundamentally different.

The reason the ‘order of operations’ is so important is that the meaning of a mathematical statement is based on understanding the order of operations.  In natural languages, the presence of multiple verbs in a statement is unusual … in mathematics, this is commonplace.  Multiple operations in a statement with nouns and adjectives provides an efficient method of communication, which is why scientific advances increased dramatically after the use of symbolic mathematics (as opposed to the original verbal forms).

Not only does “PEMDAS” have little to do with correct order of operations, the way ‘order of operations’ is typically taught has little to do with mathematics.

When we learn a computer programming language, we face the issue directly — what is the precedence order for operations?  Although there are some minor differences in the details, almost all precedence orders are based on a fundamental mathematical idea:

The more advanced operations are done prior to simpler operations.

We teach students that exponents are repeated multiplications; what we don’t divulge is that this means that exponents are more advanced operations … and therefore are done prior to multiplying.  We cover the procedures for multiplying and dividing fractions, but do not make sure that students know that these procedures are based on the fact that multiplying and dividing are at the same level of complexity, mathematically speaking.

The fundamental idea that “more advanced operations are done first” covers the majority of what we try to do with ‘order of operations’.  The difference is this:  order of operations is treated as a memorization issue, while ‘more advanced operations first’ is calling for understanding and communication.  How students get to a ‘correct answer’ is more important than the fact that they got a correct answer.

In those computer programming languages, operations are categorized into binary and unary types, just as mathematicians do.  The ‘more advanced first’ principle handles almost all cases in both types.  Even the type some of us complain about:

-5²

Even though this ‘ambiguity’ is not encountered very often in real-world problems, this is a core issue in communication.  How do we interpret:

-x²

We certainly don’t want people to apply the opposite operation prior to squaring, and we certainly don’t want the answer to change when given in variable notation.  In both of these problems, the “-” means opposite … which is less advanced than squaring; therefore, square first, then apply the opposite.

The few places where ‘more advanced first’ fails are also places where ‘order of operations’ fails, and these are often due to our failures to maintain integrity in our language.  Our notation for trig functions is sometimes bad, or even incorrect (when it creates an inconsistency with other operations or functions).  Even if we don’t change our behavior in trig functions, students will be better off with ‘more advanced first’ than they are currently.

I’d be happier if we never used the phrase ‘order of operations’; the entire implication of this phrase is ‘memorize the rules, or else’.  Our students would have a higher quality learning experience if we just focused on ‘more advanced operations first’.  The emphasis this involves on the meaning of expressions helps novices reach a deeper understanding of our mathematical language.

Which of these is a better answer to the question “why did you multiply before you subtracted”:

• I multiplied first because the order of operations says to multiply before subtracting.
• I multiplied first because multiplying has a higher precedence because multiplying is more advanced.

As we strive to help our students understand and reason in mathematics, an ‘order of operations’ has no place in the curriculum.  Knowing a structure for operations, including ‘more advanced’, is critical.

 
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