Apples, Hats, and Word Problems

In almost all math courses, an emphasis is placed on word problems … applications … real-world  … translation.  Is there a valid reason to include this work?  Or, should the mathematics in a course be restricted to that which is needed to deal with the contextual situations that students encounter?  Do these verbally-presented situations have a valid purpose in our math courses?

Here is the reason I am thinking about these issues — one of my ‘Facebook friends” (also a friend in ‘rl’ = ‘real life’) posted a link to a captioned picture at an online site.  The caption reads:

Everytime I see a math word problem, it looks like this:  If I have 10 ice cubes and you have 11 apples … How many pancakes will fit on the roof?
Answer: Purple because aliens don’t wear hats.

This ‘spoof’ takes its energy from the fact that we tend to have problems that are either obviously worthless (pancakes on the roof) or unreasonable (hats).  You are most likely responding the same way I did … “The problems in MY course are good and realistic problems!”  The criticism here is not what experts might see … the criticism is in what students (novices) see in this work.

First, here is a link to a short report I wrote a few years ago:  Ignore the Story  This report does not deal with how effective ‘word problems’ are — it deals more with qualitative studies.

Second, let us admit a basic fact:  The problems we can include in a course will not convince the majority of the students that those problems provide a justification for the mathematics covered.    Yes, I realize that some faculty will not agree, and hold the position that properly chosen contexts and applications will convince students.  We have a tendency to underestimate the complexity of going into a context to apply mathematics and coming back out of the context; this is hard work, and many students will seek any avenue possible to avoid dealing with the deeper relationships — sometimes working harder to avoid the process than it would be to complete it.  In other words, students will tend to map application processes to the procedures required at the shallowest level that ‘works’.

This minimalist tendency is not unique to word problems; nor is it unique to mathematics … our colleagues in other disciplines experience the same problem.  The difference is that we, in mathematics, expect students to deal with short (often cryptic) descriptions of situations in a variety of areas; students are expected to see mixing two levels of milk fat to be mathematically equivalent to mixing acid solutions of different percent concentrations even though the phrasing is often significantly different.

Elements of a course should support the instructional objectives of the course.  This implies that verbally stated problems should contribute to the mathematical outcomes for a student.  We should be using verbally stated problems to encourage and build a more complete understanding of basic mathematical ideas, and these verbal problems should also contribute to linguistic literacy for our students.  Achieving correct answers is not nearly as important as being able to paraphrase the situation, summarize it, state the known and unknowns, identify relationships between quantities that might be helpful, and write at least one mathematical statement that can be used to ‘solve’ the problem.  These abilities (which combine the linguistic and the mathematical) are much more related to a good prognosis for employment and other goals … more than basic skill accuracy, and more than algebraic manipulation by itself.

The emerging models for developmental mathematics (such as New Life, Pathways, and Mathways) tend to emphasize deeper processing of verbal problems, and de-emphasize repetitive ‘word problems’ which might look like the ‘apples, hats, and aliens’ spoof.  I encourage you to examine how verbally stated situations are used in your courses.  Do they contribute to both understanding basic mathematical concepts and linguistic abilities?

 
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Graphing and Models

One of the current trends in mathematics is ‘models’, often connected as ‘functions and models’.  What do students bring from their work on graphing in beginning algebra (often linear graphs) to this broader work?  Is this an easy transition?  Do we face challenges or hidden dangers in this work?
One thing I have noticed is that we often assume facility with basic graphing based on the linear function graphing included in a beginning algebra course.  A student can generate a table of values and use those to graph; a student can graph the y-intercept and use slope to find more points on graph to create the line.  I suggest that we face a significant gap in knowledge when we present a model to graph on their own.

This is the type of thing I am talking about:

A company finds that it costs $2.50 per glass, in addition to a basic set up cost of $80.  Write the linear function for the total cost based on the number of items (glasses).  Graph this function for a domain 0 to 100.

The typical beginning algebra class does not prepare students for this work.  Here are some of the gaps:

This is not a scientific analysis of the knowledge needed for this problem; there are details at a finer grain of analysis that would show more gaps.

Essentially, this is a problem caused by “Bumper Mathing” (see an earlier post on that).  We constrain the graphing environment to the extent that the resulting knowledge is not applicable in any realistic situation.  We can do better than this.

“Graphing”, as a collection of related concepts and procedures, is fairly complex yet very useful … and is worth doing well.  We can certainly make more room in the algebra course so that students leave with good mathematics and knowledge that transfers.

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Math – Applications for Living XIII

We are at the end of the semester, so today is “final exam day” in our quantitative reasoning course.  Here are two problems from the final — both fairly complex, though students are doing okay with them.

First problem:

A family is filling a child’s “swimming pool” – a round pool that is 6 feet in diameter (3 feet radius). They will fill the pool to a depth of 2 feet, and will be using a garden hose to fill the pool We know that there are about 7.5 gallons of water in 1 cubic foot, and the hose will deliver about 10 gallons per minute. How long will it take to fill the pool, starting from empty, to the desired volume of water?

The formulas for volume are provided.  Students need to find the volume of the pool, and then use the units to correctly convert cubic feet to minutes 

This is similar to an earlier problem shared here.

The other problem is shorter, but more complicated in reasoning:

For a new play area, a school is using 200 meters of fencing. Find the area of a square enclosure, and of a circular enclosure, using this amount of fencing.

Again, formulas are provided.  Students need to find the side of a square with perimeter 200m … then the area; the circle is more challenging … students need to find the radius given the circumference, then find the area.  I have used this problem (with different quantities of ‘fencing’) for a while, and have been pleased with the reasoning students are showing.

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Math – Applications for Living XII

In our ‘math – applications for living’ class, we are reviewing what we have learned this semester.  Some parts (like probability) are still tough for students, partially because there is some memorizing to do with new material.  Truth is … I like to cover probability mostly because the process encourages reasoning about quantities.  [For example, we had a problem to solve about the probability of having 5 children — 2 girls followed by 3 boys; some of us wanted to look at this as dependent probability: 2/5 for the first girl, 1/4 for the second, and then confusion about what to do with the boys.  Clearly, knowing that events are independent is critical.]

The best problem we worked on today was one with almost no practical value: 

We had to really work on this problem.  The intent is to have students focus on the units (we need ‘square feet’ for area; we have cubic feet and feet … how can we do this?).  When students asked how to do this problem, I would ask them “How do you measure area?” (to get them thinking about units).  Every student (individually) said “length times width”; clearly, we are still too focused on one formula, and not thinking about what we are measuring. 

Of course, we could follow up on the “length times width” idea with something more reasonable. 

S: Area is length times width.
I: Okay, for a rectangle we calculate area that way.  How do we calculate the volume of a box?
S: Multiply (writes V = LWH)
I: So, the volume is L*W times H; right?
S: Yes
I: We know that L*W is the area of a rectangle.  Think of that volume formula as “V = area * Height”.  How would we solve this for the height, which is like the depth of the lake?
S: Hmmm (thinking) … we would divide
I: Yep — divide both sides by area.  Does that give you an idea how to solve the lake problem?

Most students originally decided that they had better multiply the numbers in the problem; of course, they only dealt with the value not the units.  They did not think about getting “feet to the fourth power”, and what this might mean.  A couple of students thought that the ‘cubic’ in ‘cubic feet’ meant that that value needed to be cubed.  [More evidence of a ‘messy landscape’ of math knowledge.]

The good news from today’s class was that students actually did a reasonably good job figuring out a complicated ‘unit conversion problem’ (given dimensions of a box, the flow in gallons per minute, and rate of gallons per cubic feet … how long would it take to fill the box).  Prolonged effort on related problems with diverse settings has paid off.  We are having more difficulties with geometry (concepts) than we are with proportional reasoning.

 
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