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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