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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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.