Wednesday, November 28, 2012

MLCS: Where's the math?

MLCS is a very different course, whether you're looking at the materials, the content development, the class structure, or the homework.  Every aspect of the course is different than the traditional algebra courses that are common in all developmental math programs.  Those differences are exciting and lead to great gains in understanding and mathematical maturity for students.  I'm so impressed with what students can do when we get to this point in the semester.  But change is hard.  

Einstein said, "We can't solve problems by using the same kind of thinking we used when we created them."  To get to the point we want students to be at the end of the math literacy course, which is college readiness, we use very different approaches.  And they do work, but they do not look like what instructors expect, especially in terms of the printed page.  For example, I taught a lesson today that started with reviewing linearity and building a model, then moved into solving literal equations, and concluded with factoring the GCF.  All of this worked together in the context of building a model and then working with the most useful and simple form of it.  Everything had a purpose, but nothing was easy.  Beyond that, those topics never appear together in an algebra book.  But they need to if we want students to see connections.  And while that mix of topics may be strange to us as instructors, it can make sense when solving a bigger problem.

College readiness is more than a skill set, but possessing the ability to use that skill set.  To get students there, we use open-ended problems each unit.  This unit, we're having students work on problems framed around the question, "how big is big?"  We want students to grasp the largeness or smallness of numbers in a variety of settings.  We ask students questions like, "could you carpet the U.S. with its own debt if it was in $1 bills?"  $16 trillion is a large number but how large is it really?  Putting it in perspective helps to make sense of it.  That idea is pervasive in real life.  I just used an example:  $16 trillion instead of $16,000,000,000,000.  Writing "trillion" allows me to work with a number that's more manageable:  16.

We ask students to take the U.S. debt and find its weight in quarters, its height if it was stacked in $1 bills, its area, etc.  But the numbers are enormous, so we ask students to make a comparison that's useful and helpful.  To make this idea and goal concrete, we use the example of blue whales, the largest mammal.  But how big are they?  We could find their length or weight but those are just large numbers.  So put them in perspective.  This interactive app provides a great demonstration.  For me, seeing that a blue whale is the length of a basketball court is really illuminating.

This unit project has an amazing amount of math in it to be solved.  Students have to work with scientific notation, dimensional analysis, concepts from geometry as well as do research on quantities and find objects of comparable size to make comparisons against.  If you want a fun challenge, find the weight of the U.S. debt in quarters and then something on the earth to make sense of that number.  It's not a trivial exercise.

Like the quarter problem, the entire open-ended problem description is brief and there are no mathematical symbols in it.  But to complete the project takes pages of calculations and research.  That isn't how we typically provide "word problems" to students because it's really hard for students to complete.  We don't provide short statements and tell the students to figure it out.  As educators, we want to make math accessible so we try to find ways through problems, be it decoding word problems or finding key words or recognizing problem 'types.'  That approach is full of good intentions but the outcome is a student who can't really solve problems.  Students can often mimic their way through, but true understanding is not often gained.  It's like we've left the training wheels on the bike and then expect students to ride without them.  If they never have to ride on their own, we shouldn't be surprised when they can't.

As with the project description, there are lessons throughout the book we've written that are devoid of symbols, at first glance.  Take a look at this page from the manuscript:

It begs the questions, "where's the math?"  Take a look after students complete the lesson:

Like the national debt problem, students start with few words and a problem to solve.  But getting there requires a serious amount of work.  The mathematics is evident and abundant but it was not given to students but instead created by students.
The level of expectation is higher and so is the difficulty level.  But the understanding is deeper.  Students move past skills alone and work to use those skills without constant prompting on when or how.  Not easy, but the payoff is greater.

A colleague of mine said years ago when we began redesigning our developmental math program, "what we're doing is good, but I wish we could do more mathematics with students."  I just looked at him puzzled because what else are we doing with them in an algebra class if not mathematics?  But I understand his point now.  Doing mathematics is rich and complicated and messy and often, circuitous.  Algebra is neat and linear and simplified to a set of procedures.  Do we need those procedures?  Absolutely.  But focusing on them alone is akin to learning all the scales in music.  If we never move past the scales and into the songs, how can we say we've made music?

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