Do we Have 80-Year-Old Students?

I was at a meeting earlier this month (on my campus) about developmental education.  We had a broad conversation about ‘what works’ and what we would like to do.  The person leading the meeting has a background in writing — including developmental writing courses; I’ll call him George for convenience.

George told the story of one particular student he was having trouble with.  The student was polite and all that, but could not write a coherent paper.  After grading some papers with agony, including one responding to Angela Davis, the instructor (George) had a conversation with the student.  Based on that conversation, they decided to have the student write about a different topic — the student’s own experiences in a war (World War II, in this case).  The result, according to George, was a well-written essay (far longer than required).  The lesson George took from this was … let them write about something meaningful to them.

My response to this story was:

We should look for 80-year-old students in our classes, who happen to be stuck inside a 20-year-old body.

You see, my lesson from the story is a different.  Students are complex human beings (there is no other kind).  For ‘good students’, they can focus on academics and see what we expect of them.  For ‘struggling students’, they have difficulty keeping their history and current life challenges separate from what we ask of them in a classroom.  The student in the story was 80 years old at the time, and had much to deal with; of course, writing about something personally important is meaningful.  However, society in general … and occupations in particular … demand that we communicate about ideas that we do not necessarily care about.

The lesson, for me, is this: Students need to learn how to separate and focus.  Many of our students have had challenges in their lives; sometimes, this is just a math challenge.  Other times, they have faced significant life issues and trauma.  Just being able to talk about these will help a student focus in class.  Sometimes, they do not realize that the challenges they have faced will be a benefit in a math class.  To some extent, the affective factors can prevent cognitive work; just articulating the issues behind the affective can let the brain focus on the cognitive.

It’s tempting to say that “the lesson is to show students that we ‘care’ about them as people”.  Many of us do care about our students.  However, my observations do not support this conclusion in general.  I think the lesson is more subtle than ‘we care’ or ‘make it relevant’.   Maybe the lesson is more like “give them credit for making it this far.”

Our students have faced a lot of life, whether they are 20 or 80.  For some, they have overcome more challenges in 20 years than I will have faced in 80.  We seem to be more gracious to the 80-year-old than the 20-year-old.  I think we should look for all of the 80-year-olds in our classes, especially those who are stuck in a 20-year-old body.

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Towards a Balanced Approach

When I hear somebody suggest that we take a ‘balanced approach’, my first thought is that the speaker either does not have confidence in their judgments about what is important … or does have confidence but does not want to offend the audience.  The phrase ‘balanced approach’ is often used in reference to a reform model balanced with traditional ideas.

I suggest that we think about the phrase in a new way.  Let’s begin with the assumption that the traditional curriculum has limited value and that the reform curricula have limited value.  What would we build from a blank slate?  How would we use scientific evidence in the process?

A balanced approach looks at implementing two basic properties of human learning:

1. Understanding (connected information) results in more transfer of learning and facilitates long-term retention.
2. Repetition (deliberate practice) results in efficient recall and abilities to apply information.

Some reform curricula emphasize (1) almost to the exclusion of (2).  I have taught courses like this, and talked with my students; few of them have a good report about the experience.  We all have students who approach a math course in that fashion — the students who usually are in class, and do very little ‘homework’ because they understand what they are doing (occasionally they are correct).

As mathematicians, we are drawn to ideas with power — ideas that can represent relationships among quantities, communicate the information, and help reach conclusions about some future state of those quantities.  [We are also drawn to special cases, as well as mathematics that is aesthetically pleasing.]  Our students need the ‘basic ideas with power’ so they can handle the quantitative demands of academic and social situations.  I think we can have fairly strong consensus on the mathematics that most students need.

The balance we need is about pedagogy.  Having a better ‘table of contents’ will not help if students do not learn any mathematics.

I see this issue of balance as our basic problem over the next 5 years.  We know that our courses are going to change in basic ways.  We understand what mathematics is important for all students.  Our issue is to address both the understanding and repetition in the learning process.

Currently, an ‘understanding’ method is based on students dealing with a situation and using guided questions so that they discover the basic idea.  In some cases, this works surprisingly well.  However, discovering an idea has little connection with understanding mathematics.  Here’s an example:  By looking at a set of ordered pairs (bivariate data for a situation), students are led to the idea of slope so that they can predict another value.  This forms the beginning of understanding, not the end: understanding takes extended work with diverse views of the same idea.  Students often over- or under-generalize.  In the case of slope, students think this applies to any set of values … or that it does not matter which ones ‘go on top’ … or that slope is like an ordered pair.  Understanding is a natural human process, but does not happen spontaneously with correctness.

As for ‘repetition’, we seldom get this right.  Textbooks often confuse ‘any sequence of problems’ with ‘repetition for learning’.  Much is known about properties of practice that result in different degrees of learning — a sequence can highlight the most important idea (or hide it), a sequence can reinforce good understanding (or prevent it), and a sequence can reinforce accurate recall (or prevent it).  We somehow make the mistake that good authors can design good assignments; these are vastly different sets of expertise. We also make the mistake that computer systems provide appropriate repetition.

We can (and need to) focus primarily on the big ideas in mathematics; our courses need to match the amount of material with reasonable expectations for students learning with understanding and repetition.  With a balanced approach, we can help students succeed.  With a balanced approach, we can show policy makers that we have the professional skills to solve the problems that they are concerned with.

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Hey, they did NOT ban Developmental Education! And a call to arms …

Florida.

In case you have not heard about this, Florida (meaning the legislature) passed a law which requires colleges to cease requiring courses prior to college level for many (perhaps most) students.  A summary of the bill (which has other interesting components) is at http://www.flsenate.gov/Committees/BillSummaries/2013/html/501

One way this has been reported is that ‘Florida has banned developmental courses’.  In basic ways, this seems to be the intent and the effect.  As is typical for many states in this era, the process involves an outcome designed with little professional input with a process based on no patience (or perhaps based on the attention span of the legislators).

Relative to other states, and our profession, here is the risk I see: The law changes the basic definition of ‘developmental education’.  We can have developmental education without having any courses (or credits).

Historically, ‘developmental education’ has two related meanings:

1. Remedial courses (perhaps done with more student support)
2. Student development as learners (secondary goal of student success in general)

The new definition in Florida is that “developmental education is the extra service provided to enable all students to begin in college level courses” (my paraphrasing).  Most of us would call this ‘just in time remediation’.

If you read the Florida law, and the reports of it, you will see the word ‘flexibility’ repeatedly.  I am sure that this was actually a goal in the process.  However, the new ‘developmental education’ is a risk to our students.  Flexible enrollment does not mean reasonable opportunity; access for all does not mean equality.

We could agree, I hope, that a significant portion of students referred to remediation (old developmental) do not need it — either they have no meaningful gap to fill, or the gap is small enough that they would do fine with a little bit of help (new developmental).  This is a valid criticism.

We could also agree, I hope, that we have been too quick to have more developmental courses (old developmental).  It is not reasonable that a student who passed Algebra II or AP Statistics would need 2 or 3 courses before college math.  Inefficiency was a fatal flaw in the automotive industry, and it is a fatal flaw in developmental education.

Florida has declared, in effect, that the old developmental education is bankrupt and going out of business.  No bailouts.  No loans.  No recovery.  Just gone.

That is the risk raised for all of our students.  Other states have similar pressures and political forces, and read the same reports that were read in Florida.  “We don’t need to waste all that money, and we can solve this problem at the same time.”

We need to rise up.

We need to be clear that we know of problems in our work, and that we are willing to make basic changes; further, we need to show evidence of better ideas so the ‘bankrupt and out of business’ model is not so easily taken.  Yes, we even need to become involved in the political process.

Do YOU know where your state legislator is?

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Discovery Learning in Developmental Math

Compare these two situations:

You are placed in a laboratory with 12 objects of varying shapes and matching openings on the wall.  The learning goal is for you to discover which opening goes with which object.

You are placed in a room with 3 sheets of paper, 3 pencils, and two other people.  The learning goal is for you to discover the properties of a good drawing of a rose.

Really, take a minute to see yourself in each situation.

When we design a class to depend on discovery learning, we often assume that the students will experience a good thing.  We’ve listened to the sages say ‘guide on the side’ (rhyming makes right!!), so we have stepped out of the way.  If students discover the math, they will own it and learn it better (or so the story goes).

Like other people doing reform courses, I have been using discovery learning more.  From that point of view, the most important thing to say is this:

Discovery learning is very difficult to design and implement for positive results.

Two problems routinely come up with discovery.  First, many students have a difficult time seeing the idea that we want them to discover; this is primarily a communication issue.  The second: students come with prior knowledge, some not so good; the ‘discovery’ process often activates erroneous patterns, and the student ‘discovers’ initially that they had a great thing — which then needs even more effort to re-direct to better ideas.

The instructional materials being developed for the emerging models (Math Lit, Carnegie Pathways, Dana Center Mathways, commercial texts) tend to build a discovery process in every lesson, centered around a sequence of questions.  In general, the materials insert ‘check points’ for the instructor to assess the quality of the learning.  The developers work hard to create these materials that can be used by a variety of faculty.

The research base for discovery learning has not been consistently positive, and I think we sometimes confuse the motivational impact with learning mathematics.  I did some searching for a productive analysis of the issues with discovery learning, and found something that might help us.  The article is by Kirschner, Sweller, and Clark; called “Why Unguided Learning Does not Work …”, and is available at http://www.ydae.purdue.edu/lct/hbcu/documents/Analysis_of_the_Failure_of_Discovery_PBL__Experiential_Inquiry_Learning.pdf

This article is not a quick read.  However, I think you will find useful information which will impact your work in the classroom.

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