Workshop at AMATYC 2012

At the AMATYC 2012 conference (Jacksonville, FL — November 8 to 12) I will be making a general presentation on New Life … AND a workshop on the two courses in the New Life model (Mathematical Literacy for College Students – MLCS, and Transitions).  For general conference information, see http://www.amatyc.org/Events/conferences/2012Jacksonville/index.html 

I am thrilled to be able to provide both the general session and the workshop on the courses.  We are collecting ideas for the workshop over on the wiki for New Life (see http://dm-live.wikispaces.com/AMATYC+2012).  If you are not a member there yet, just follow the directions for joining the wiki; it is fairly easy to join, and membership is open to anybody with an interest.

Tentatively, the general New Life session is scheduled for November 8 (Thursday) from 9:00 to 9:50; the Workshop on New Life courses (MLCS and Transitions) is scheduled for November 9 (Friday) from 1:45 to 3:45.  Both sessions are being held in a larger room — feel free to pass along this information!

Hope to see you there.

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Ignore Common Core?

Can college math faculty ignore the Common Core?  Specifically, can those of us working in developmental mathematics ignore the Common Core?

If you need to read more about the Common Core Math Standards, take a look here http://www.corestandards.org/the-standards/mathematics. The Standards are listed for each grade K to 8, and then high school by area of mathematics.

As you might know, a primary motivation for the Common Core was that of alignment … getting K-12 outcomes to align with expectations, especially for college readiness.  This alignment is connected to the standardized tests used for ‘No Child Left Behind’ (NCLB) as well as some teacher evaluations.  [A current theme in teacher evaluations is the use of ‘value added models’ (VAM), which is a statistical methodology to estimate the impact of individual teachers; I may address VAM in a future post.]

A logical approach might be to think that … if a student places in to developmental mathematics … there is no reason that we need to be especially aware of the Common Core.  If this placement is accurate, we might conclude that the Common Core ‘did not work’ for whatever reason, so our work is independent.

Look at the situation in a different ‘frame’:  Because the Common Core is closely tied to standardized testing and NCLB, the mathematics assessed is often discrete skills with a focus on procedures and simple applications.  This emphasis in K-12 will, therefore, tend to produce students in college — whether ‘developmental’ or not — who have a less complex package of mathematical proficiency.   

I have been suspecting something like this happening in the last few years (even before Common Core, though the Common Core will expand the impact) … students obtain about the same average scores on placement test even though their functioning, mathematically, is more limited.  Solving a linear inequality might go okay for them, and then difficulty emerges when there is a discussion about how to represent the solutions in a different way.  Finding slope from two ordered pairs might be okay, and then confusion appears when slope needs to be interpreted in words or a context.

Recently, I did a post on “Lockhart’s Lament”; in that essay, an observation is that a sure way to ruin a subject is to require all students to ‘take it’.  With the Common Core, we have a movement to make all students take the same subject for almost all of their K-12 experience.  Since this ‘subject’ is almost always tied to standardized tests and sometimes to teacher evaluations, the forces operate on the subject to reduce all topics to operational steps.  (I’m reminded of the “paint by numbers” analogy in Lockhart’s Lament.)

Policy makers are often looking for simple solutions, which makes the Common Core look very attractive as well as standardized tests.  If only we could present ‘understanding and reasoning’ as simple solutions for the mathematical needs of K-12 students.  Are not those the central enablers of success for students  in our college courses?

We ignore the Common Core at our own peril.  Some college faculty actively support the use of the Common Core mathematics standards, and there is a real danger that this wish will be granted.  There is no single mathematical standard in the Common Core that I object to; the tragedy is that the summation (or integration in the mathematical sense, if you will) of the Common Core is a worsening of the mathematics problem in colleges … starting with developmental, but including all college mathematics in the first two years.
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Math – Applications for Living VIII

In our class (Math – Applications for Living) we are investigating linear and exponential functions.  One of the assessments in the class is a quiz which covers linear functions connected to contexts.  This quiz had 3 items, which gave almost all students quite a bit of difficulty.  Every student in class has passed our introductory algebra class where we make a big deal of slope, graphing with slope, the linear function form, finding equations of lines, and a few related outcomes.  Perhaps the difficulty was due to ‘normal’ forgetting … I am more inclined to attribute the difficulty to a surface-level knowledge that inhibits transfer to new situations.

The first item was not too bad: 

The cost of renting a car is a flat $26, plus an additional 23 cents per mile that you drive.  Write the linear function for this situation.

The class day before the quiz, we had done quite a bit of work with y=mx + b in context like this.  The majority got this one right (though there were some truly strange answers).

The second item got a little tougher:

At 1pm, 2.5 inches of snow had fallen.  At 5pm, 3.5 inches had fallen.  Find the slope.

This item involved two related (connected!) ideas — the independent variable (input) ‘results’ in the change in the dependent (output) variable, and slope is the change in dependent divided by the change in the independent.  This ‘sieve of knowledge’ filtered out about 2/3 of the class — a third missed the fact that time is usually an independent variable, and another third lost the idea of slope as a division.  This is the outcome that I was most concerned by.

The last item was the ‘capstone’:

A child was 40 inches tall at age 8, and 54 inches tall at age 10.  Write a linear function to find the height based on the age.

A quick read of this problem might make one think that it is problem 2 (two issues) with a third issue piled on.  However, the problem said which variable was independent (age); the intent was to combine the ‘what is slope’ issue with knowing  how to find a y-intercept.  Essentially, nobody got this problem correct.  Some missed the independent variable stated in the problem … some could not find slope … the majority found some slope-type number but had no clue what to do with the problem from there.  If we strip the problem of context, it becomes this classic exercise:

Find the equation of the line through the points (8, 40) and (10, 54).  Write the answer in slope-intercept form.

Every student in class had survived doing at least a dozen of these problems in the prior math class; this item is pretty common on all of our tests in introductory algebra … and often is on the final exam for that course.

We spent about 10 minutes going over this one problem after the quiz. The questions from class were really good — and indicated how weak their knowledge was (I can only hope that their knowledge is getting deeper!).  Some students found a ‘slope’ and just used that (“y = 7x”); several felt compelled to use one of the given values in the equation (y=7x + 40 and y=7x +54 both were seen).  One common theme that came out was that students forgot that the ‘b’ in the function was the y-intercept; however, it was more than that … they were mystified by my statement that we could find the y-intercept from the given information.  I showed the symbolic method; not much luck with that.  I showed a graphical method … that helped a little more.  On this one item, I am guessing that we went from about 25% correct knowledge to about 60% knowledge.

Behind all of this difficulty is the manner of learning normally seen in a basic algebra class — 40 topics (sections), containing a few types of problems each, lots of repetition but few real problems (as opposed to exercises), and almost no connections between topics.  The mental map resulting from this is ‘not pretty’; an open-ended and unusual problem like on my quiz shows a number of gaps and misunderstandings.

In a separate post, I have called for “depth and breadth” (mile wide and mile deep!).  If we need to error on one of these two dimensions, let us error on the side of depth … wide exposure without depth is often worse than no exposure at all.  My students are having a difficult time unlearning what they ‘learned’ before; it is easier to extend good knowledge to a new area.

We all have these experiences — where we see the basic problems with student’s knowledge of basic mathematical objects like linear functions.   It helps to know that we share this process.  Perhaps together we can build a mathematics curriculum that does a much better job of building mathematical proficiency.

 
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Rotman’s Race to the Top

A few year’s ago, Paul Lockhart wrote an essay entitled “A Mathematician’s Lament”, sometimes known as “Lockhart’s Lament”.  I find this essay to be incredibly inspiring.

If you have not read it, here is a copy: lockhartslament

As far as I know, Lockhart never published this essay. I actually know very little about Paul Lockhart, besides the information that he went to teaching in a K-12 school after being a university professor for a time.  Some brief bio stuff is at http://www.maa.org/devlin/devlin_03_08.html

Okay, back to the essay.  Paul Lockhart is presenting a vision of mathematics as an art, not as a science, where learning is very personal and where inspiration is a basic component.  A fundamental process in Lockhart’s vision is the personal need to understand ideas, often ideas separate from reality.  The ‘lament’ aspect of the title for the article refers to the view that school mathematics has made mathematics into ‘paint by numbers’ instead of an exciting personal adventure.  Where we might focus on insight and creativity, we instead deliver an assembly line where students receive upgrade modules to their skill sets; Lockhart presents the view that these skill sets have nothing to do with mathematics.

I have two reasons for highly appreciating this essay.  First of all, people often assume that I believe that mathematics should be presented in contexts that students can relate to (probably because I advocate for “appropriate mathematics”).  Contexts are fun, and we can develop mathematics from context.  However, contexts tend to hide mathematics because they are so complicated; mathematics is an attempt to simplify measurement of reality, so complications are ‘the wrong direction’.  When I speak about appropriate mathematics, I am describing basic mathematical concepts and relationships which reflect an uncluttered view of mathematics; applicability might be now, might be later … or may never come for a given student.  Education is not about situations that students can understand at the time; education is about building the capacities to understand more situations in the future.

The second reason for appreciating Lockhart’s Lament is the immediate usefulness in teaching.  If we believe that students want to understand, then we can engage them with us in exploring ideas and reaching a deeper understanding.  Here is an example … in my intermediate algebra class this week, we introduced complex numbers starting with the ‘imaginary unit’.  Often, we would start this by pointing out that ‘the square root of a negative number is not a real number’ and then saying “the imaginary unit i resolves this problem.”

Think about this way of developing the idea instead (and it worked beautifully in class). 

1. Squaring any real number results in a positive … double check student’s understanding (because quite a few believe that this is not true).
2. The real numbers constitute the real number line, where squaring a number results in a positive.
3. Think of (‘imagine”) a different number line where squaring a number results in a negative.  Perhaps this line is perpendicular to the real number line.
4. We need to tell which number line a given number is on; how about ‘i‘ to label the number?
5. Let’s square an imaginary number and it’s opposite (like 2i and -2i).
6. Squaring an imaginary number produces i²; since the imaginary numbers are those we square to get a negative, i² must be -1

Any context involving imaginary or complex numbers is well beyond anything my students would understand.  I would always want to explore imaginary numbers; first, it provides a situation to further understand real numbers … second, the reasoning behind imaginary numbers is accessible to pretty much all students.

You might be wondering why I gave this post the title “Rotman’s Race to the Top”.  One reason is just not very important … I wanted a alliteration comparable to “Lockhart’s Lament”; well, I tried :).  The important reason is this:  The “Race to the Top” process is based on  some questionable components that do not connect directly with the learning process (common standards [a redundant phrase], standardized testing, and outcomes.  The world we live in, as teachers of mathematics, is a personal one: We interact with our students, we create situations that encourage learning, and we want our students to value learning for its own sake.  We also want our students to understand some mathematics.  Inspiration and reasoning have a lot more to do with this environment than any law or governmental policy.

Rotman’s Race to the Top is the process of teachers showing our enthusiasm, modeling the excitement of understanding ideas, and helping each student reach a higher level of functioning.  Building capacity and a maintaining a love for learning are our goals.  Rotman’s Race to the Top is about our students and their future.

Will you join this Race to the Top?

 
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