TMI in the GCF and LCD of MATH: TTYL, PEMDAS!!

If we could tweet and text math, we would say things like “Need LCD, remove GCF, remember PEMDAS”.  Wait a minute, we already say those things.  Seems like math classes are ahead of the curve on not communicating well.  Let’s look a little deeper.

The human brain has some limitations that impact how well acronyms work in communication; as a teacher, I would say that communicating with acronyms tends to keep the information processing at the surface level — translating the acronym in to the words — rather than connecting ideas to important concepts.  We say “you need an LCD to add fractions” and have to remind students that an LCD is not needed for multiplying fractions.  Perhaps we would improve our instruction if we banned acronyms.

I’ve tried taking a compromise approach in my intermediate algebra class, where we are currently taking the test on rational expressions.  We use the label “LCD” after we’ve shown that terms need something in common before adding and subtracting.  I’d like to say that the approach improves student learning, but that is not likely to be measurable — partly because it is such a challenge to get students to reason mathematically instead of memorizing rules for getting answers.

Within five to ten years, we will have a different curriculum for most of our students (STEM and not-STEM) where we provide a better mathematical experience for all students.  In the meantime, many of us will struggle with better understanding in our students.  It might help to say “TTYL” (talk to you later) to all acronyms, as much as we can manage.

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Math: Applications for Living XVII

One of the big story lines in our ‘Math  — Applications for Living’ course is percent change. The first level is comparing absolute and relative changes in quantities, along with percent increase and decrease expressions.  We began to work on writing a mathematical model for percent increase or decrease as (1+r)^n, and saw compound interest as a variation on this.  The course ends soon after we formalize this work by looking at exponential models for growth and decay.

Recently, I saw an article in our local newspaper that illustrated exponential growth in the world of internet traffic.  With an accompanying story, the following graphic was used:

The original data is part of the Cisco IP modeling report (2011 to 2016); they have a report wizard at http://ciscovni.com/vni_forecast/index.htm

As you can see, the first and third graphs in the article are great examples of showing exponential change; the mobile data chart has the largest rate, but both graphs are delightfully exponential in nature.  The problem is the middle graph — for corporate accounts.  That graph is labeled “21% growth per year”, when the pattern is clearly linear; the data shows a slight decrease in growth rate later in the forecast period.

When we get to the exponential models in class, I plan to show this set of graphs in class and ask … ‘where is the error in this chart’.  I think it is interesting that a journalist writing about internet trends does not understand exponential change enough to clearly communicate about two different patterns.

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Do the Math! What does that mean?

I was at a conference this past week, when a keynote speaker used the phrase “do the math!”.  A redesign methodology states that one of the benefits is ‘students spend more time doing math’.  If we ever needed evidence that the mathematics curriculum is mis-directed, these comments would seem to be conclusive evidence of a problem — they are quotes from fellow mathematics faculty.

Perhaps we have lost track of what mathematics is.  According to a dictionary (Merriam-Webster in this case, though they are all similar), mathematics is

the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

Open any developmental ‘mathematics’ textbook, and you will see something that resembles mathematics … a bit like a scary Halloween costume, where the outside looks different than the true character underneath.  We have gone far from the path of mathematics, much to the detriment of our students.

In particular, we have lost all elements of science within mathematics (especially in developmental math, but also in gateway college courses).  Science is a ‘system of knowledge’.  If it were not for the undeserved special treatment of mathematics, our science colleagues would have long ago challenged our mathematics courses as being a ritualistic mis-education of the masses.  Two hundred types of problems with remembered procedures to manipulate values and symbols to acquire a ‘correct’ answer does not represent knowledge; the resemblance is stronger with uninformed rituals performed with no redeeming value (practical or intellectual).

The emerging models (New Life, Dana Center Mathways, Carnegie Pathways) are all movements towards mathematics.  We can, and must, reform our mathematics courses so that students learn mathematics more than rituals.  As mathematicians, we have knowledge systems that help people understand the world around them … and a knowledge domain that is enjoyable just for the learning.

The person who said “Do the math!” was simply saying “you need to agree with me, because my view of the data says you should”.  The person who said “students spend more time doing math” really meant that students spend more time in some activity that resembles mathematics … but was most likely engagement in the rituals that have taken the place of mathematics.  The fact that the majority of American students believe that they are bad at ‘mathematics’ says more about our curriculum than it does about them.

I still spend a large portion of my teaching time in courses where the content is still traditional; change is not instantaneous.  However, whatever the course, we can take a more mathematical approach by focusing on concepts and connections even as we get students to accurately perform the rituals.  We each need to start on this path towards teaching mathematics so that we are ready for larger changes; our curriculum in 10 years will have little resemblance to that of 5 years ago.  The good news is that we will be truly teaching mathematics when that change comes.

I hope that you will begin your personal journey towards teaching mathematics; perhaps you can even contribute to the professional work that will lead to the change that is coming.

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Remediation as Cheese, Remediation as Fishing

You may have noticed that the emphasis on completion, combined with a high priority on getting a job right away, has resulted in pressures on colleges to provide training and skills development … with less emphasis on the subtler goals of intellectual development, curiosity, and liberating arts.  In both developmental and gateway mathematics courses, we have become the epitome of the completion/job methodology; to the extent that this is true we have failed as educators and mathematicians.

My thinking on this got a boost from a short piece on “Habits of Mind” by Dan Berrett (see http://chronicle.com/article/Habits-of-Mind-Lessons-for/134868/).  Dan’s main point is that the current focus on measurable outcomes applied to a college ‘education’ results in using simplistic measures, where these measures miss the most powerful advantages of an education.

Earlier this year, I was in a conversation with a group discussing placement tests and diagnostic tests.  The predominant approach was summarized by a food metaphor:  Our goal is to fill in the ‘swiss cheese holes’ for our students, so that they do not have any gaps.  The ‘remediation as cheese’ metaphor is very much the common approach, whether a college uses modules or emporium or traditional classes; we measure success by counting holes (or lack thereof).  I’d like to think that education in general and mathematics in particular is more than the absence of holes.

Compare the cheese metaphor with this:  Remediation as fishing.  According to a quote, often cited as a Chinese proverb:

Give a man a fish and he will eat for a day. Teach a man to fish and he will eat for the rest of his life.

“To fish” is the “remediation”; we are not about holes … we are about building capacity as well as building ability … we are about attitudes about learning as well as learning about attitudes … we are about enabling students to become more than they intended at the start of our course.  Remediation succeeds when students are fundamentally different when they leave our classrooms; the ‘lack of holes’ with arithmetic and algebra does not improve a student’s preparation for education or for employment as much as the habits of mind that can be developed.

Let’s help our students learn how to fish.  The broader goals of education are just as important as discrete skills and immediate performance measures.  We can, and should, contribute to our student’s capabilities within our developmental mathematics classes. 

Nobody makes a greater mistake then he who does nothing because he could only do a little.  [Edmund Burke]

 
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