Neat Knowledge? Messy Landscape?

We all spend quite a bit of time talking with students, and we also look at massive amounts of student work.  Sometimes, we get in to “homework system mode” where we only provide feedback on the answer.  The answer, by itself, is very weak as a communicator of the knowledge a student possesses. 

I have been thinking about how messy student knowledge is (about mathematics); perhaps this is true for all domains.  In my classes, I try to create an atmosphere where people feel safe asking questions and participating; of course, sometimes the questions asked indicate significant misunderstandings … which can temporarily cause some damage to the overall learning in the class.  In this post, I want to focus more on the implications of errors in student work.

Okay, in our intermediate algebra class we just had a test on ‘quadratics’.  The material is a mixture of procedural and conceptual, with a few ‘applications’ included.  One of the applications involves compound interest (annual) for 2 years — so it creates a quadratic equation like this:

 Most students managed to write this (based on the verbal description and the provided formula).  The most common error?  Subtracting 4000 from each side, a disturbing error.  Since this problem was on a test, I did not have an oppportunity to discuss the error with the students to identify the source.  My primary suspect:  An early rule (early in education) to the effect that “large and small numbers mean divide, similar size numbers mean subtract”.  Every  one of these students can solve linear equations requiring division (like -4x=30), and most solved a quadratic involving a common factor (like 2x²=100).  What triggers  the “we must subtract” response?

In another class (the quantitative reasoning course), we have been doing geometry this week.  As for other topics, the formulas are provided — we are much more interested in the reasoning involved.  One of the problems dealt with finding the perimeter of this shape:

Two consistent problems came up.  First, students thought that ALL perimeter problems dealt with “P = 2L + 2W” … causing them to count the widths of the rectangle (which are interior, therefore not part of ‘perimeter’).  Second, even when convinced that some perimeter problems dealt with a different relationship, they did not see  why we should omit the perimeter (they still wanted to include the interior dimension).  Since I was able to discuss these issues, I have some idea of what is behind them. 

My point here is not to complain about what students do or don’t do, nor to suggest that they have had bad teaching  [I mean, beyond having ME as a teacher :)].  Rather, perhaps we need to think more about the root cause for many student difficulties: 

The basic problem is an oversimplification of a connected set of ideas down to a single ‘if-then’ clause.

Students are sometimes desperate to learn math, and we want to help them.  Too often, this results in only dealing with one thing (or one thing at a time) — which simplifies the learning, but which avoids building connections (contrast, similar).    The geometry instance of this is easily described:  by only dealing with one shape for ‘perimeter’ we encourage students to connect just one formula with that work — and to disconnect this from the concept involved (‘distance around’).   Our procedural work in algebra is even more prone to the oversimplification, because the correct learning is abstract about a combination of ‘properties’, ‘undoing’, and ‘maintaining value OR balance’. 

I’ve got to anticipate what some readers are thinking (due to that last sentence) … are manipulatives the solution for this problem in algebra?  Not really.  The research I have read on ‘kinetic learning’ suggests that this might provide some of the scaffolding to abstract ideas — but is counter-productive if the manipulative remains a focus.  This is a very difficult task, to shift from manipulative to concept; the stages vary with each learner.  If you can do manipulatives on an individual basis with continual feedback (conversation) with a highly skilled faculty, then I believe this can support better learning.   The skills involved are complex, and most faculty will not have them — including me; this is combining the skills of a cognitive researcher, qualitative researcher, and math faculty.  The problem with most manipulatives in learning is that the entire process is over-simplified to be practical in a group setting; a group of 3 students is too large, and possible 2 students is too large.

What is the answer?  We need to clearly articulate the few critically important learning outcomes in a math class (perhaps a dozen for a course), and provide diverse learning experiences around those outcomes.  “Simple” is not the solution; simple is part of the problem.  The solution is a planned sequence of activities to deeply explore the ‘good stuff’ of mathematics.

 
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Math – Applications for Living IX

In our Math119 course, we are studying models — linear (repeated adding) and exponential (repeated multiplying).  Although some of the details we are including are not very practical, some are practical … and helpful in understanding everyday numbers like ‘inflation’.

Here is a situation we looked at:

If prices increase at a monthly rate of 1.5%, by what percentage do they increase in a year?

Much of our work in class has been on translating from a “percent change” statement to a “multiplying statement”.  Most students saw that this 1.5% increase meant that the multiplier was 1.015.  To answer this question, we just evaluated

We did have a little struggle about using the resulting value (1.1956 …); with a little nudging, we agreed that the annual increase was 19.6%.  Even though we have done quite a few finance applications, this result was a little surprising … students thought we would multiply 0.015 by 12 (18%).

While we were working on models, we also introduced using a calculator procedure to find answers to ‘difficult’ questions [meaning that we used a numeric approach to solving exponential equations].  Take a look at this problem:

Fifty mg of a drug are administered at 2pm, and 20% of the drug is eliminated each hour.  When will it reach 10 mg in the body (the minimum effective level)?

We’ve got that percent change going on; students are generally getting that — this is a multiplier of 0.80.  [This problem is much tougher when I give them drug levels for consecutive 1 hour intervals … like after 3 hours and after 4 hours.].  We set up this equation

To solve this problem, we used a graphing calculator ‘intersect’ process … placing this function on ‘y1’ and the output we needed (10) on ‘y2’.   Our solution (about 7.2 hours)  is useful in understanding the frequency for some prescriptions (3 times per day in this case).  In class, we also approach this same problem as a ‘half-life’ situation; conceptually, that is more complex … and specialized, so we do not emphasize the half-life method.  [Half-life is mostly there to help students if they take a science course which uses half-life concepts.]

We also point out that the intersect process used here is very flexible; it may be one of the most practical things they get out of the course.

 
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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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Math – Applications for Living VII

Our class, Math119 (Math – Applications for Living) is in the middle of our work on statistics.  The last class included finding a margin of error and a confidence interval for a poll … like those pesky political polls we are constantly hearing about. 

So, here is the situation.  This month’s poll showed 63% of respondents supported one candidate, based on results from 384 people; last month, the same poll reported 58% supported that candidate.  The article stated that the candidate is enjoying the increased support … is that a valid conclusion?

As you know, this relates to two issues.  First, the standard error for a proportion like this is found with the statistical formula:  

Tests of significance are then based on z values for a normal distribution; the most common reference is z = 1.96 … creating a margin of error representing a 95% confidence interval.

In our class (Math119), we use a quick rule of thumb to combine these two ideas into one statement which just uses the sample size — and this rule of thumb works pretty good for the types of proportions normally seen in polls (p values between 10% and 90%).  The rule of thumb for the margin of error is just the reciprocal of the square root of the sample size     

For the poll data, the sample sizes are both about 400.  The rule of thumb gives an estimate of 5%, which is very close to the actual value (approximately 4.7%.  In our class, we make a reference to the presence of the more accurate formula, but we use only this rule of thumb.

In this poll example, we create the confidence interval … and conclude that there is no significant difference between the polls.  The confidence intervals overlap; even though the new poll has a larger number, it is not enough of an increase to be significant (with this sample size).

We also have talked about selection bias and other potential problems with polls, and have begun the process of thinking about the impact of sample sizes on things being ‘significant’ (whether they are meaningful or not).

 
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