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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Proportional Reasoning

What is proportional reasoning?  Are “proportions” truly a different topic than “functions”?  If our students master linear functions, will that enable them to reason with proportional quantities?   Stated another way: Does knowing the mathematical object imply that a person can reason with that object to solve problems?

I think we, as mathematicians, tend to see proportionality as a special case of functions … or as a particular type of equation involving two ratios or rates.  Why does a person need to be able to reason with proportions … or proportionality?

Take a look at this problem, which is paraphrased from my “Math – Applications for Living” course:

A car is driven at an average speed of 44 miles per hour.  At this speed, the car averages 33 miles per gallon.  How much gasoline does the car use per hour?  Follow-up question:  If the tank is full with 14 gallons of gasoline, how long could the car be driven (in hours)?

I’ve seen students struggle with this problem, which I use to highlight proportional reasoning.  Each rate is a statement of proportionality … miles driven is proportional to the time, the gallons of gasoline are proportional to the miles driven, etc.  Further, each statement of proportionality (rate) is equally true in two forms — as stated, and the inverted rate.

By looking at the units we need (like gal per hr in the first question), we set up the rates to provide that answer:

To the extent that our students take basic science classes, this proportional reasoning is very valuable … and has no direct connections to the concepts of functions.  Bringing up concepts of input and output only complicates these problems … because each statement of a rate allows any of the quantities to be the output; identifying an ‘output’ quantity is done by looking at the nature of the output needed for the question at hand.

Instead of addressing proportional reasoning, we often ‘help’ students by teaching them keywords to indicate multiplication or division (if it says “how many pieces can you get” it is division, etc).  You may have noticed how limited this approach is, because problems are phrased differently.  If we look at proportional reasoning with the rates, it becomes much easier.

Proportional reasoning comes up, in a natural way, when we start studying probability.  Take the classic type of problem to introduce probability:

A container has 6 red marbles, 3 blue marbles, and 1 white marble.  What is the probability that a marble, chosen randomly from this container, will be blue?

The concept here is something like “the probability of something happening is proportional to the number of those ‘things’ that are in the entire group”.    Many simple probabilities are based on this, as is the question of ‘drawing two blue marbles’ from this container.

Unfortunately, what many students remember about proportions has limited value (and is often mis-applied) … “cross products”.  Given a proportional situation, the important thing is being able to write two rates or ratios which make the same comparison (they follow the proportionality involved); as described above, it is also important to be able to write products of rates to produce the desired units.

Although a slope is a rate, which might suggest proportionality, the use of linear functions is not proportional reasoning (especially as experienced by novice learners).  The connection between proportionality and linear functions is not an equivalence; it is more of an issue of ‘shared concepts’.  We should not assume that knowledge of linear functions has much to do with proportional reasoning.

 
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Do we need developmental education?

You may have seen the news story about Connecticut considering a law that would eliminate all developmental education in the state … except for imbedded remediation and an ‘intensive college readiness program’. General story: http://communitycollegespotlight.org/content/connecticut-may-end-remedial-requirement_8674/ and more details http://www.insidehighered.com/news/2012/04/04/connecticut-legislature-mulls-elimination-remedial-courses

I see two basic issues raised by this.  An obvious issue is a statement about the perceived value of developmental education.  In the case of mathematics, some developmental programs have 4 courses before the first college-level math class; a logical analysis of this system can easily show that there is a basic design flaw … a two-year ‘getting ready for college’ track is enough credits for a major, but these are courses that do not have value in themselves.  A rejection of this design is basic in our development of the New Life model, where we reduce that pre-college work to 1 or 2 courses, depending on the student’s program.  Does a rejection of the 2-year developmental math program imply that it can be replaced by a ‘just in time’ remediation model, combined with a boot-camp experience?

The other basic issue raised is the change process.  We appear to be in a period when politicians are policy makers in broad areas of education.  It’s not like the state said ‘We are spending way too much money … and not getting enough benefit; we are appointing a task force of experts who are charged with creating a model that meets the needs of our communities in a process that is much more efficient’.  Whatever the process was, the lawmakers believe that they have a solution that can be legislated.  Have we done such a poor job of articulating the power of a good developmental program that lawmakers believe that this is a solution? 

I have no doubt that some students — even many students — would be well served by the ‘imbedded + boot camp’ model; historically, we have underestimated the capabilities of students to cope with challenges … if they have a little more support.  However, I believe that this model will leave many students defeated; these will be the types of students for whom community colleges were created — the ‘first generation college’, the un-empowered and vulnerable, and those for whom the K-12 system did not ‘work’ … as well as the returning adult. 

We need to do a better job of articulating the power of what we do everyday.  Our courses are not just about some collection of basic skills, that our goals include developing learning and thinking in our students; we need to tell people in authority that we have expertise and methods that produce results.

We also need to be willing to ‘take the criticism’ … that our developmental programs have become entrenched and stagnant systems that do not serve enough students nor well enough for all students, that we can develop models that better serve our students with dramatically reduced credits and costs.  If we continue to insist on the same-old programs, or even if we fail to recognize this problem, then we deserve to have others (like politicians) determine a better system.  I believe that we are wise enough to do the right thing.

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