Math Wars or Math Together

Change.

Change happens.

Change is happening faster.

Progress is different from change.

Now that I have said the obvious, what does it MEAN?  How do we promote progress, and not just change without progress?  To remind us of the basic meaning, ‘progress’ means movement towards a goal.  In the context of mathematics (including developmental mathematics), our goals reflect shared understandings and values.

And, I think that is part of the problem with ‘math wars’ … we do not focus on our shared understandings and values, and we do not articulate the core goals of our work.  I’ve been thinking about this after reading a ‘math wars’ type article (see http://betrayed-whyeducationisfailing.blogspot.com/2011/05/why-i-quit-teaching-math-at-sfcc.html) … this particular article (possibly not accurately) describes major disagreements at a community college, where the situation resulted in a faculty member resigning their position rather than ‘change’.

Too often, we leave the question of goals as a ‘given’ or a factor not requiring direct attention.  Bad idea!  If we want particular skills for our students, this implies some methodologies would be more appropriate than others.  If we want our students to experience situations like a mathematician, then different methodologies would be more appropriate.  If we want flexible problem solving (involving elements of both prior goals) for our students, an intelligent mix of methodologies would be more appropriate. 

My own guess is, as a community, we would answer “We want all of these things” but to varying degrees.  Within the framework of two courses, or perhaps three, I suspect that the capacity to reach multiple goals like this is just not there … between the resources that we can apply and the resources that our students can apply.   The New Life model, overall, tends to favor the ‘mathematician’ and problem solving goals with less on basic skills.  Other models, including the traditional framework and the redesign models (like emporium) tend to focus on basic skills (with little or no ‘mathematician’). 

We are in this together, so we should get math together.  Our conversations should start with, and focus on, the broad goals for our courses.  Too often, we begin our conversations with “Do you include factoring in beginning algebra?”  Topics are often not the end goal; topics are often secondary to the larger goals in a discipline.

Before launching a redesign project, your department (program, or whatever) should get its math together.  As a profession, we need many more conversations about the core goals; we can have areas of agreement, which will lead to shared work … and shared work can lead to progress.  Merely changing the delivery system is definitely not progress.  Sure, we want higher pass rates; however, higher pass rates just means that there are more people at the end of step n  … we would never accept an explanation that had a good conclusion without examining the quality of the steps preceding it, and this applies to our curriculum as well.   Do steps 1 to n have any connection to our core goals?   That’s my question for you.

Math wars helps nobody.  Math together can lead to progress.  Let us get our math together!!

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Redesign: Not an Either/Or Situation

NOTE: This post comes from Kathy Almy, who has worked on the New Life project, and is very involved with the profession.

There are two emerging approaches in developmental math redesign:  the emporium model and non-STEM pathways like Quantway and Statway.  These approaches are very different in their makeup so it seems that schools have to make a choice.  That doesn’t have to be the case.  Both can live together in one department and actually, that model may serve students and faculty better.

The emporium model uses online software to help students fill gaps in their understanding.  It can work for a student who needs a brush-up on skills.  It also works well for the student who plans to take college algebra or precalculus but doesn’t quite place into those courses.  For the student who is motivated and just shy of where they need to be on the college track, it can shorten the time for completing developmental coursework.  But what about the student that places into beginning algebra or below and has many more issues than just filling skill gaps? 

For these students the New Life course Mathematical Literacy for College Students (also in the Carnegie Quantway path), can be a less time-consuming model and one that serves them well.  MLCS works extremely well for the student whose college level math class will be statistics or general education math.  In one semester, students gain the mathematical maturity and college readiness they need to be successful in one or both of these college level courses.  The course integrates algebra with numeracy, functions, proportional reasoning, geometry, and statistics.  It does so using an integrated approach, touching each of these 6 topics in each unit.

How can using both approaches serve students and faculty?  The reality is that most new initiatives bring controversy and potentially resistance.  These two redesign models are no exception.  Faculty often feel very strongly, pro or con, about one or both of them.  Because of that, using one model across the board at a school may bring more challenges.  Using a variety of models helps faculty and students work and learn in ways that make sense for them.  People feel respected in terms of how they work and what they need in their future.  That’s an important facet of a successful redesign.  Because when redesign is imposed instead of invited, the effects will be short-lived and potentially less than they could have been.

Ultimately, both approaches have the same goal:  change the current one-size-fits-all mindset of developmental math to serve students better.  How we go about doing that need not be one size either.

Kathleen Almy is a math professor at Rock Valley College in Rockford, IL.  She has worked with more than 30 colleges nationwide to assist with their developmental math redesigns.  She is also implementing a pilot of the MLCS course in fall 2011.   For more information, see her blog at almydoesmath.blogspot.com. 

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Bumper Mathing

In bumper bowling, there is no ‘gutter ball’ — every attempt hits something.

I’ve been thinking about this concept related to mathematics; do our developmental courses create ‘bumper mathing’, where all students hit something most of the time?

My college offers a course that I totally love … it’s a mathematical literacy course, covering a collection of really good mathematics.  It has only a beginning algebra prerequisite.  A common theme in the course is ‘applying percents’.

All of the students in this course have ‘mastered’ percents.  They have converted percents to other forms, they have solved percent ‘problems’ (like 80% of what is 60), and applied percents to life situations; in our beginning algebra course, they ‘mastered’ mixture problems dealing with percents.

Here is a chronology of percents in my ‘math lit’ course:

10% increase from a known value, find new value … almost all are okay

Old and new values, find the relative change … almost all are okay

10% increase from original unknown amount, express new value … almost none are okay, almost all need remediation

10% increase each year, express as a function … almost none are okay, almost all need remediation

95% confidence interval dealing with survey results … half think the 95% has to be used in the computation

10% probability of A happening, probability of ‘not A’ … almost all are okay

10% increase from original amount (known or unknown), express as a function … still difficult

10% increase from a known original amount, graph the function … almost all need remediation

Notice that there are 3 times that we revisit the ‘10% increase, represent new amount’ concept.  Each time, the majority of students do not see why we get ‘1.10n’ … they’d like to see ‘0.10n’.  The problem is that they want to compute with the percent stated (10%), because that has worked almost all of time in the past.  Part of the process of ‘remediation’ is to work through concrete examples (like 6% sales tax leading to ‘1.06n’), but this is a slippery process: The prior learning keeps drawing them down to computing with the 10%.

In our pre-algebra course, we cover perecents in a very template driven way … convert % to decimal, ‘2 places left’; ‘is over of’, and others.  These templates increase the proportion of correct answers (bumper mathing), but disguise the lack of percent understanding.  Our course is not alone in this problem; our collective pre-algebra courses are supposed to prepare students for algebra, which is all about generalizing … but percents are template-taught.  There is no transfer of learning, because of bumper mathing.

To create mathemtically literate people, there needs to be a chance for ‘gutter balls’ (as in real bowling).  We see 100 correct percent answers, and conclude that there is a good understanding of percents; that is not the case most of the time.  I’ve had a lot of students over the years say “I used to be good at math, and now I am struggling”.  Perhaps we have enabled this disability by practicing bumper mathing!

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Basic Skills … what is important?

Okay, here is a quiz for you.  Which of these categories of skills is more important in today’s world?

    A) Basic skills (correct computation, solving routine problems, etc)

or

   B) Critical thinking (solving novel problems, connecting sets of learned material, metacognition, etc)

You see, I think we have a problem with “Goals”.  Philosophically, we believe that mathematics is a venue to improve critical thinking … operationally, we deliver a curriculum much more focused on basic skills.  To complicate matters, our college curriculum follows the school curriculum which is decidedly skill oriented (given the high stakes testing, understandably so).

Back in February, NPR ran a story about some reports dealing with how well students are prepared for college (see http://www.npr.org/2011/02/09/133310978/in-college-a-lack-of-rigor-leaves-students-adrift).  In their story, writing was connected more to critical thinking than math was … I would not disagree, but would hope that our math courses are part of the answer.

Our developmental mathematics textbooks are severely ‘skill bound’, and we sometimes choose solutions that exacerbate the situation — such as modularized programs that discourage integration and accumulation.

I am sure that we all share a goal of improving our students critical thinking, and the evidence indicates that the need for this work is greater than ever.    My own courses are, sadly, typical of what we are all doing.  Could we, if we wanted to, create something better?

Absolutely!  Development of critical thinking is a field with its own theories, research, and methodologies.  For starters, see the nice chapter by Diane Halpern at http://education.gsu.edu/ctl/FLC/Foundations/criticalthinking-Halpern.pdf .  Even a brief online search will provide you with more material.

We can do better!  Let’s work together so our students are better prepared for the problems they will face …inside academia and outside.

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