Is Calculus Instruction Changing (or Curriculum)?

In our applications for living course, we are finishing our work with statistics.  One situation involves students deciding which situation is likely to involve statistical significance:.

Rolling a die 40 times and getting 5 threes             OR

Rolling a die 400 times and getting 25 threes

This is a tricky thing, as students often focus on the sample size only.  Although this problem presents situations where the larger number is connected with significance, there is no general pattern that says ‘larger sample sizes means significance.’

Within our curriculum, developmental mathematics has dominated the news and much of our work for a long time.  There definitely is a larger sample size; is there a difference in statistical significance between developmental math and calculus courses?  In a basic way, yes, there is — developmental math serves a large group of students with multiple academic problems, while calculus serves a group of students with general academic success.  A 60% pass rate in calculus is not good, and is statistically significant given that most students in the calculus courses are expecting a high grade (while developmental math students often expect low grades).

You can try this as I did — search for ‘reform in calculus college’ or similar terms.  Most of the results of this search will be historical artifacts from the 1980’s and early 1990’s.  What’s with that??

I think we have fallen into the large number fallacy — a larger sample size indicates significance (dev math), instead of analyzing each situation separately.  We should be able to expect an 80% pass rate in calculus 1, given the academic skills of students who typically enroll in that course.  My own college gets about 60% pass, and this seems pretty normal.

For programs which require a 4-semester sequence (Calc I – II – III plus diff eq), a 60% pass rate means that a maximum of 13% will ever complete the sequence.  The likely values are far less — some portion of students are lost between courses even after passing.  I suspect that observed values will be between 5 and 10% — which, coincidentally, is the same range as a developmental math sequence.  [These low values are the result of ‘exponential attrition.]

Recently, I did learn of some work of our friends in the MAA on calculus.  It’s not on reform; rather, their focus is to analyze data to identify what we are doing and what is more successful — the Characteristics of Successful Programs in College Calculus (CSPCC) project.  The web page for their work is http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/characteristics-of-successful-programs-in-college-calculus

David Bressoud is a lead ‘PI’ on this work; he’s written a few articles about their work for the MAA Newsletter, so you may have seen those if you belong to the MAA.  The web site (above) has links to those articles, as well as papers written about their work.

One of the co-PI for the work is Vilma Mesa, who has done quite a bit of work on community college mathematics.  Dr. Mesa did a session at our recent MichMATYC conference on some of the data from community colleges for the CSPCC work, including the contrast between homework and exams in terms of assessment level … and an analysis of prompt types and expected response types (verbal, symbolic, graphic, or multiple).

I encourage you to read the material on the CSPCC web site.

The larger question is this:  Are we doing anything of substance to make basic improvements in calculus?  Or, is our ‘best shot’ using Mathematica and/or MatLab with our students?  I hope that is not the case.

If you are doing some reform in calculus, I hope that you will share your work — the good and not so good.  “Developmental Math” should not be having all the fun!!

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Math for Emerging Technologies — CRAFTY and the Vision

Perhaps we do not think enough about what mathematics is needed in emerging technologies — those fields that are relatively new and generally have good employment prospects for our students.

So, I am getting ready for presentations at the AMATYC conference and at the National Summit on Developmental Mathematics later this month (Anaheim, CA).  My presentations focus on the New Life model, which has been around for about 4 years.  Partially due to the related work of Quantway™ and Statway™, much of the attention in our field has been on the Mathematical Literacy course.  I am actually putting more energy in to the new course which follows Math Lit — Algebraic Literacy.

In the process, I am going back to the source documents that we used to develop our curriculum.  We used professional references to identify learning outcomes for the target courses our students will take.  Among those targets was ‘technology careers’ — what mathematics students need to succeed.

If you have not seen this source, I highly recommend that you study the “Vision: Mathematics for the Emerging Technologies”.  This report is the result of an intensive effort organized by AMATYC in 2000 and 2001, and involved convening dozens of experts outside of mathematics education.  Here is the link: http://c.ymcdn.com/sites/www.amatyc.org/resource/resmgr/publications/visionweb.pdf

Among the identified needs:

• Critical Thinking, Problem Solving, and Communicating Mathematically
• Algebra (ability to apply mathematics topics outside of mathematics, or in a new setting, is vital)
• Geometry
• Trigonometry
• Statistics

The list also included arithmetic (proportional reasoning, measurement), which our Math Lit course takes care of.  The report lists calculus as needed for a few technologies.

The question is … should the development of these needed skills fall entirely on the responsibility of courses in the student’s program?  Or, should ‘developmental’ mathematics provide a foundation in these areas?

If these needs were unique to emerging technologies, we might have some rationale for leaving this work to the occupational programs.  However, each area listed is also important for other reasons — other “STEM” fields need them.  These needs fit very nicely into a course preparing students for a first math analysis course (also known as pre-calculus).

Here are the current goals and learning outcomes for Algebraic Literacy: Algebraic Literacy Course Goals & Outcomes Oct2012 Algebraic Literacy Goals and Outcomes June2013

In case you are wondering, the ‘CRAFTY’ in the title of this post comes from the MAA acronym “Curriculum Reform Across the First Two Years”; the AMATYC grant involved work that supported this effort.

I’ll be talking about this “Vision” report in my Summit presentation on the Algebraic Literacy course, as well as in my New Life session at the AMATYC conference.  I hope you can make it to one of those events.  [I’ll post the presentations here later.]

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Can Developmental be done Online?

A challenge in our profession is being able to integrate discrete components and results to create a stable vision of our work.  Over the years, various trends have impacted us and our classrooms; in the short term, many of these trends look good.  Most produce results that are initially positive.  Few trends have created long-term change.

The question is “Can Developmental be done Online?”  I’m taking this in the broadest sense — not just courses, but online in general.  Can the purposes of developmental mathematics be met by learning in an online environment?

Like education in general, we are in danger of being reduced to a finite set of skills or competencies.  We need to keep education as a separate goal from those in training programs — training adds competencies, while education creates capabilities.  Training is what we do when success is measured by how ready people are for identified jobs or behaviors; training is critical to our well-being both individually and collectively.  However, our survival depends on our adaptability and problem solving — and this speaks to education.

This distinction between training and education is critical if we are to answer the question “Can developmental be done Online?”.  Developmental mathematics, long defined in training concepts (skills), is far more of an educational endeavor.  As long as we focus on skills, our students leave our courses with the same basic capabilities as when they entered — in other words, generally ill-equipped for education.  Within a training program, a skill-focus makes sense; in an educational program, a skill-focus tends to defeat us.

The connection between these ideas and ‘online’ is indirect; online work is capable of dealing with either training or education.  I will conjecture, however, that increasing capabilities is more difficult to achieve in an online environment; not impossible — more difficult.  The movement from novice toward expert (in other words, education) is facilitated by varied supports — modeling, discussion, non-verbal cues, individual conversations, group support — which are easier to build in a face-to-face environment.

A parallel issue is “does online work for the population of community college students?”.   The Community College Research Center (Columbia University) has just released a research study based on a large (state-wide) dataset; see http://ccrc.tc.columbia.edu/media/k2/attachments/adaptability-to-online-learning.pdf for details.  The findings are disturbing:

While all types of students in the study suffered decrements in performance in online courses, some struggled more than others to adapt: males, younger students, Black students, and students with lower grade point averages.  [abstract, pg 2]

I say disturbing because the ‘lower grade point averages’ points to a developmental population more than general, and because Black (ie, African American) students already have a statistical risk in developmental courses.

This study was limited to online courses, which might not reflect the entire nature of online learning.  However, I would point out that most non-course learning online is done in an individual-based structure — a person finds their resources, and uses them as best they can, sometimes with a little support (tutors, for example); this learning is less supported than many online courses, so I would not expect non-course learning to have better results.

There are environments that stretch the concept of online courses towards the non-course format — “MOOC”.  MOOCs offer the excitement of more equal access to educational opportunity with reduced cost, and policy makers are considering this as an alternative.  I have large concerns, however, relative to MOOCs offering any help to developmental students … either the focus is on skills (training) which won’t help students very much in education, or the focus is on education without adequate support for building capabilities (movement towards expert).

Our own local data about online courses in developmental math is not that promising; most commonly, the online courses have a lower pass rate than other methods.  It’s possible that this is part of an overall trend towards lower outcomes in online courses.

Online learning is here to stay, and will continue to evolve.  This does not mean that online courses are here to stay.  Perhaps we need to look at that format as an option for a limited group of students, perhaps even for limited purposes.  My own view is that developmental math can not be done successfully online for the population we serve; I have doubts about whether there is a significant sub-population for whom online courses is a reasonable choice for developmental math.

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Khan, Comfort, and the Doom of Mathematics

Perhaps you already knew this:

If students perceive instruction as clear, the result will be reinforcing existing knowledge (often not so good knowledge).

I recently ran into a reference to a fascinating item posted by Derek Muller, specifically about videos like the Khan academy; Dr Muller’s specific interest is science education (physics in particular), and you might find the presentation interesting http://www.youtube.com/watch?v=eVtCO84MDj8 (it’s just 8 minutes long).

In mathematics, even more than physics, students come to our classrooms with large amounts of prior experience with the material.  Of course, much of their existing knowledge is either incomplete or just plain wrong (whether they place into developmental math classes or not).  A ‘clear’ presentation means that the existing knowledge was not disturbed in any significant way.  Clear presentations make students even more confident in the validity of the knowledge they possess.  This is not learning.  Reinforcing wrong information is the doom of mathematics.

In Dr Muller’s study, two types of presentations were done.  The first were the ‘clear’ ones; students felt good about watching, but the result was absolutely no improvement in their learning.  The second type were ‘confusing’ ones, where the presentation deliberately stated common misunderstandings and explored them.  Students did not like watching these;  however, the result was significantly improved learning.

We see this in our classrooms.  This past Friday, a young man from my beginning algebra class came in to see me … he had left class in the middle, in a distracting way to other students.  Turns out that he left because he could not stand the confusion.  In talking to him, he believes he can do the algebra but he is getting very confused by the discussion in class about “why do that” and “here is another way to look at it”.  In fact, this student has a very low functioning level about algebra.  If he does not go through some confusion, his mathematical literacy will remain unchanged; that is to say … he won’t have any meaningful mathematical literacy.

Khan Academy videos are popular; I understand … I have watched some myself.  I consider them to be very clear and essentially useless for learning mathematics.  If a person already has good knowledge, they will not need them; if a person lacks some knowledge, they will not perceive what they lack from watching a video.  [Just like witness research in criminal justice, students perception is controlled by their understanding.]

The attraction of modules and NCAT-style redesign is often the clarity and focus.  Students do not generally see anything that might confuse them; the environment is artificially constrained to avoid as many confusing elements (inputs) as possible.  To the extent that students in these programs are ‘comfortable’ and the instruction ‘clear’, that is the extent to which existing knowledge is reinforced.  Learning can not happen if we primarily reinforce existing knowledge; confusion is an essential element in a learning environment.  [I sometimes tell my students that instead of being called a ‘teacher’ they should call me ‘confusion control expert’.]

I suspect some readers are thinking that “He has this wrong … I have data that shows that students do really learn.”  It’s true that I don’t have proof; I don’t even have my own research (though I would love to see some good cognitive research on these issues).  What I do know is that student performance on exams — especially procedural items — is a very poor measure of mathematical knowledge.  I suggest that you interview some average students that you think know their mathematics based on exam performance; have them explain why they did what they did … and have them explain the errors in another person’s work.  Based on what I have heard from students, I think that you will find that only the best students can show mathematical knowledge in an interview at a level equal to their exam performance; average students will struggle with the interview about their mathematics.

How do we avoid the doom of mathematics?  How do we prevent our classes from becoming reinforcers of existing knowledge?  I think we need to create environments for learning where every student faces some confusion on a regular basis … not overwhelming confusion, and not trivial confusion, but meaningful confusion about important mathematics.   Do we need an LCD to do that?  Must we move terms in an equation before we divide by the coefficient?  Is that distrubuting, or is that subtraction?   Confusion is where students bump into the areas of knowledge that need their attention.

Our students have a strong tendency to drive through our courses as fast as possible, without really dealing with mathematics.  They believe the myth that the experts always understand, that we are never confused.  We need to be comfortable in showing confusion to our students and model appropriate behavior to resolve it.  The appropriate response to confusion is figuring out where we went wrong … not running away for a comfortable explanation.  Confusion may call for some meta-cognitive efforts, or we may simply need to polish one particular mathematical idea.

Confusion is the fertile soil of learning.  Avoiding confusion creates a sterile environment without growth.  Comfort is fine, and we all need comfort; however, comfort never learned anything. 

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