Students at the Center of Learning

“Teaching and Learning” … a phrase often used in professional development for us teachers, as well as in titles of articles and books.  Perhaps a better phrase would be “Learning and Teacher Behaviors”, or “Learning … Teaching without getting in the way!”

I am thinking about how well our Math Literacy course is doing in the Math Lab format.  The Math Lab format creates a learning environment by establishing assignments and a structure for students to work through those assignments.  The instructor ‘stays out of the way’ as long as learning is successful.  This format has been used with very traditional content, and is now being used with a modern developmental course — Math Literacy.

Although some students struggle longer, and do not initially ‘get’ new ideas, the vast majority of students in the Math Lab Math Literacy course have been successful with:

  • identifying linear and exponential patterns in sequence
  • using dimensional analysis for unit conversions
  • identifying the type of calculation for geometry (perimeter, area, volume)
  • writing expressions for verbal statements

What’s been tougher?  Anything dealing with percents — applications, simple & compound interest, etc.  Of course, these are weak spots for students in any math class; over the years, I have not seen anything that ‘fixes’ these in the short term; the fix involves unlearning bad or incomplete ideas, and this takes time and long-term ‘exposure’ to errors (along with support from an expert).  Direct instruction or group activities have limited effectiveness against the force of pre-existing bad knowledge.

The instructional materials form the basis for the learning in this Math Lab format.  If the ‘textbook’ is focused on problems to do, contexts to explore, with the expectation that the instructor will provide ‘the mathematics’, then the learner centered approach requires that we use specialized processes in the classroom.  The classroom becomes the focus, and we spend resources & energy on tactical decisions such as ‘homogeneous groupings’ or ‘group responsibilities’ or ‘flipping the classroom’.  The materials we use in this course are well crafted to support learning; the authors ‘expected’ the classroom to be the focus, though our Math Lab ‘classroom’ is working quite well with the materials.

What if we could offer a true “student at the center of learning” design?  Seems to me that this goal would lead us to use methods like our Math Lab, where students interact with the learning materials without an instructor mediating (as much as possible).  Students in our Math Literacy course have been successful in learning new mathematics with decent reasoning skills in this format.  Although initially confusing to students, the classroom is lower stress than a ‘regular’ classroom; there are no artificial social processes used to ‘facilitate’ the learning.  Think of it as being more like a student as an apprentice, where direct engagement with the objects of the occupation is the key for learning.

Of course, we are not normally able to offer all math courses in this format of active learning.  For me, the approach is to design my ‘lecture’ classes to be more like workshops.  In a 2-hour class, I might deliver 45 minutes of very focused presentations (direct instruction) distributed in a deliberate manner through the class time.  The length of ‘lecturing’ is varied according to the course and somewhat according to the needs of the students in a given class.

The point of this post is …

Stay out of the way of learning.

Students can learn by interacting directly with the learning environment.

We want students who are independent, and able to learn without a special structure.  Prepare your students for the real world by creating learning environments where they develop those skills while they are learning mathematics.

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What Does the Future Look Like? College Mathematics that Works!

We live in a transition time, for college mathematics.  Developmental Mathematics is shifting away from the traditional curriculum, with an over-use of “prerequisite remediation” in the short term.  At the same time, both of the primary professional organizations for our work (MAA and AMATYC) have been calling for basic shifts in both what we teach and how we teach within the ‘standard’ college level mathematics courses.  What does our eventual ‘target’ look like?  Can we anticipate where we will end up?

In a basic way, the answer to the last question is ‘yes’, due to the fact that all of the forces shaping the future are known at the present time.  We don’t know precisely which forces will have a larger influence, and that is fundamental since the forces are not operating in the same direction.  Imagine yourself in an n-dimensional force field where you can see the vectors around you.  Although the wind varies over time, some types of vectors dominate your environment.

These vectors around us originate from power sources.  Professional standards (MAA, AMATYC, etc) send out vectors in the direction of higher levels of reasoning, modern content, more diverse content, and more sophisticated instructional methodologies.  The K-12 educational system, the Common Core in particular, send out vectors in very similar directions.  Policy influencers, higher education provosts and chancellors, and state legislators send out vectors representing forces in different directions from those in the prior lists.

In the short term, this latter set of forces will dominate … because some of the individuals involved have sufficient decision making power that they can impose a set of practices on portions of our work.  However, these practices will not survive long term except to the extent that they support the prevailing set of forces around us.  As the people in authority change faces, the practices will tend to revert … either to the pre-existing conditions (bad) or to a condition making progress in the direction of the prevailing forces.

Here is a description, a picture, of where we will be in 10 to 15 years.

  • Remediation will be smaller than in the past, but still normally discrete (not combined with college courses as in co-requisite models).  Arithmetic will be ‘taught’ but never as a separate course and never will be a barrier to a college education.  Content will focus on the primary domains of basic mathematical reasoning — algebra, geometry, trigonometry, statistics, and modeling.  No more than two remedial courses will ever be required of students, regardless of their ‘starting condition’.
  • “College Algebra” will not be used as a course title.  Similar courses for non-STEM majors will have titles such as “Functions and Modeling in a Modern World.  The content of this course, never used as a prerequisite to standard calculus, will be from the same domains as remedial mathematics — algebra, geometry, trigonometry, statistics, and modeling.
  • “Pre-calculus” courses will be replaced by a one-semester “Intro to Math Analysis” course which focuses on the primary issue for success in calculus: reasoning with flexibility supported by procedural understanding.  This course will have a very strategic focus in terms of objects and skills involved, with a shorter topic list than prior courses … taught in a way which results in a true readiness for calculus.
  • “Calculus” courses will be re-structured to focus on a combination of symbolic and numeric work.  The first semester of the two-semester sequence will include derivatives and integration for basic forms, as well as an introduction to scientific modeling using matrices such as those encountered in the client disciplines; this eliminates the need for our client disciplines to teach basic quantitative methods, and provides modern content to serve those disciplines.  The second (and final) semester calculus course focuses on multi-variable processes combined with a more complete approach to scientific modeling — appropriate for students who may eventually conduct their own research in a client discipline
  • “Liberal Arts Math” and “Quantitative Reasoning”  will have merged in to a new QR course at most institutions.  At some institutions, these courses are replaced by the “Functions and Modeling” course (which is fundamentally a QR course).  Where QR exists as a separate course, the ‘practical’ content will be de-emphasized relative to today’s courses, with an increase in symbolic mathematics. The primary distinction between QR and Functions and Modeling is that QR does not include as much trigonometry.
  • “Intro Statistics” will exist with similar content to the best of today’s courses.  The primary change will be a relative decrease in the number of students taking a Stat course to meet a degree requirement, as program planners realize that their mathematical needs are more diverse than statistics … and that requiring statistics should not be based on just a desire to avoid college algebra (which does not exist in this ‘now’).
  • Students will become inspired to consider a major in mathematical sciences by the diverse quality content along with the effective methods used within the courses.  Instead of a focus on weeding out students not ‘worthy’ of majoring in mathematics, we will focus on including all students on the mathematical road to maximize the distance covered.

I see an exiting future, once we get past the relatively short-term impacts of changes imposed from outside.  In the long-term, nothing can stop us from achieving a desired goal … except for our own doubts and lack of clarity.

My hope is that you see something in this image of the future to get excited about, something that plays the role of a beautiful sunrise in the forest.  If you can SEE where you want to go, you can get there … and it is a lot easier to survive temporary struggles along the way.

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The Selfishness of the Corequisite Model

One of the major ‘things’ right now is “corequisite remediation”, referring specifically to the practice of placing (most or all) students in a college mathematics course with a requirement that certain students take a support class.  Over time, we will discover that this has sufficient promise to justify further exploration and use.  The problem is … the practice is very selfish on our part in many of the common implementations.

Most data on this practice comes from two college math courses — introductory statistics and liberal arts math.  For most offerings in those courses, few prior skills are needed for success; in both cases, the most common need is for expressing fractions as percents and some proportional reasoning.  Algebraic reasoning is seldom needed.  In both cases, the legacy prerequisite (usually intermediate algebra) was an artifact more related to establishing transfer than to course needs.  Few co-requisites structures have been done in college algebra nor in quantitative reasoning with an algebraic emphasis

My contention is that using co-requisite methods for a non-STEM math course amounts to a selfish decision on our part.  We place a higher priority on improving our ‘measures’ of completing that one math course … at the expense of preparing students for other quantitative needs in college.  This is especially an issue for our friends in science, who often depend on a variety of algebraic concepts in their courses (as they should).  The co-requisite model focuses almost totally on “What do they need for THIS terminal math class?” (which is a small set); ignored is the larger set “What do students need for college quantitative work?”.

Now, it is true that the traditional developmental mathematics courses do not deliver on that larger set — at least, not in an efficient manner or with good results.  Replacing developmental math courses with the co-requisite model (as is being suggested) is placing students at risk … just so a math course can ‘look better’.  Our response should be:  “How can we make fundamental improvements in the course content and design so that developmental mathematics works for almost all students across their college program?”

Our reason for existence in developmental mathematics is the whole student for their whole college experience.  We can achieve short term ‘better results’ for ourselves with co-requisite remediation.  This comes at the expense of leveling the playing field, equity, and student success in general.  Can we be that selfish?

I realize that I am attributing a personal motivation to a practice in the profession.  I’m okay with that … most of us are in this profession because of personal motivations.  I think large numbers of ‘us’ have a deep commitment to equity, as well as upward mobility.   Co-requisite remediation creates a system focused on the short-term ‘results’, often involving minority-heavy support classes, with few long-term benefits (if any).

Our responsibilities involve much more than “one math course”.  Let’s do our job.  Instead of taking the easy ‘out’ of co-requisite remediation, we should replace the traditional sequence of developmental math courses with a very short structure to serve all college students and all of their needs … Mathematical Literacy and Algebraic Literacy (or similar courses).

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Math Literacy: What do Students Struggle With? (part I)

In our “Math Lab” sections of Math Literacy, all tests are given individually … and graded while the student watches.  This is done by having 3 alternate forms of each of the 6 tests, with an answer sheet.

The process results in deeper knowledge of what students struggle with as well as what is going well.  For example, the little bit of algebra on the first test went well for all students.

On the other hand, two ‘estimating’ problems are struggle zones.  One question involves angle sizes:


The choices are provided to make this less stressful for students.  Quite a few students select sizes that are obviously too big for the image (choice A or B in this case). Very few select a ‘too small’ option (C).

The other estimating looks like this:


Our answer key allows about 10% leeway around the expected answer (390 miles for this one).  Again, students who miss this usually estimate too high (sometimes way too high).  A rare student went low in their estimate.

The issues seem different in each problem.  Estimating angles seems to be a perceptual challenge, where the eyes look at the distance between the rays instead of the opening size (or ratio of distances).  The map problem appears to be a simpler challenge — not using the measuring device provided (the scale at the bottom).

This test has a third estimating problem:


Students are missing this one for an odd reason:  instead of writing “-82”, they write “82”.  They knew that they were on the left side (it’s not like they said ’78’) but did not connect the sign with the estimate (even though it’s on the graph).  I don’t look at this as a struggle as much as ‘attention to detail’ … an issue for many of our students at all levels.

All of theses problems have similar exercises in the homework.  [We also have a Practice Test in the online system, which also has the problems.]  For most topics, those exercises are sufficient.  The first two listed above ‘not so much’.  These fit in the category “learning how to learn” … noticing a problem, seeking help, reasoning about it, practicing, etc.

Overall, the Math Lab method is working well for this course.  We will see other ‘struggle points’ for other topics as we go through the material.

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