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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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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FOIL in a Box (algebra!)

Some of us have a ‘thing’ about FOIL as a topic in an algebra class; there are concerns about emphasizing the FOIL process as it can submerge the real algebra going on.  Some (perhaps the majority) are not significantly handicapped by being “FOILed”.  This post is not about FOIL itself … it’s “FOIL in a Box”.

Okay, so this is what I am talking about.  The problem given to the student is to multiply two binomials, such as (2x – 3) and (3x +4).  Here is the “FOIL in a Box”:

Some students like this approach, and I think this is because the box lets them focus on one small part of the problem.  The overall process is submerged, and the format does all of the work.  Of course, this is exactly what many procedures in arithmetic do.  The FOIL in a Box method is much like column multiplication, where partial products are arranged in a mechanical way to produce the correct place value.  If correct answers to multiplying were the primary goal, there would be nothing wrong with either FOIL in a Box or partial products in arithmetic.

My observation has been that almost all students who use FOIL in a Box are handicapped in working with polynomials.  Students have trouble integrating the Box into longer problems.  And, though they may have some ‘right answers’ for factoring trinomials, the transition to other types is more difficult. 

What should we do instead?  My own conclusion is that we need to keep emphasizing the entire idea involved.  FOIL is used for “distributing when both factors have two terms”, and “distributing is used to multiply when one factor has two or more terms”.  We too often assume that students will keep information connected to the correct context … they don’t automatically know that distributing does not apply to 3 monomial factors [3(2y)4z ≠72yz], nor to a power of a binomial  [(x + 3)² ≠x² + 9]. 

The achievement of correct answers in the short-term should not come at the price of handicapping the student’s future learning.  All learning should be connected to good prior learning, and imbedded within the basic ideas of the discipline.  We need to be comfortable articulating the full name of what we are doing (multiplying two factors each with two terms), and not use a mnemonic such as FOIL as a container for knowledge of mathematics.

 
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Getting Into Statway or Quantway

In case you missed the webinar on January 24 (by the Carnegie Foundation for the Advancement of Teaching) … institutions can begin a process that may lead to being included in either Statway or Quantway.

Here is the slide from the webinar dealing with ‘getting into Statway or Quantway’:

Just to review … Statway is a two-semester sequence designed for students who place into beginning algebra, which takes them ‘to and thru’ a college-credit statistics course; statistical topics are the focus of the course, with developmental math topics integrated in both semesters.  
Quantway is a one-semester course designed for students who would take a ‘non-STEM’ course the next semester; it is also designed for students who place into beginning algebra, with a focus on numerical reasoning, proportional reasoning, algebraic reasoning, and functions & models.  After Quantway, students would take a college-credit math course such as quantitative reasoning, statistics, or math for liberal arts.

“Getting into Statway or Quantway” means more than just offering the course; joining a Pathway means being part of the Networked Improvement Community coordinated by the Carnegie Foundation … that is why institutions need to meet the criteria listed above.

If you are ready for this type of change, send your letter of interest to the Carnegie Foundation (email address on the image above).

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