Mathematical Literacy: One third is less … or is it more than?

Our Math Lit class is dealing with a network of algebraic concepts, including some basic problem solving.  As we often find, students don’t see the point of an algebraic method for simple problems and then have great difficulty using algebra when the problem is a bit more complex.

A simple problem was something like:

The IV is supposed to deliver 50 mL/hr, and the patient is supposed to get 400 mL.  For how many hours will the patient be on the IV?

Most students ‘just divided’, even though they could not explain why that would provide the answer.  When asked for an equation, they could see why ’50n = 400′ would provide a solution; students just did not see the value of the equation.

The next problem was something like:

A house is listed as having an assessed value of $42,000. The assessed value is one-third of the true value of the home.  What is the home actually worth?

Every student started off finding one-third of 42000, with a few then adding this ‘one third’ to the 42000.  Those that added-on were doing a ‘one-third more than’ (a more complicated relationship) rather than a simple factor of 1/3.  In other words, some students thought that the answer should be less than $42000 … and some thought that the answer was 4/3 of $42000.

Students were doing these problems in groups, as they often do in this class.  In this case, however, students did not question each other about their thinking.  Hints and ‘simpler case’ finally got most people to the correct representation.  I suspect that a few students said that this made sense just to be polite.

I suspect that students are being trained to look at “one-third of” as always meaning multiply the numbers — instead of usually meaning that there is a multiplying relationship being stated.  This seemed so strongly held a belief that writing “1/3*n = 42000” did not make sense to them.  Yes, this ‘of’ means multiply — but not ‘multiply by the number stated’.  In addition, I suspect that students are having trouble with the conceptual part of using variables.  This problem is very easy if the ‘one-third of the true value’ is seen as one-third of a variable; this view was difficult for this class of students.

Some similar problems show up in traditional algebra courses, including my intermediate algebra course.  The good thing (or not so good) is that the Math Lit students are not really having that much more trouble with this than students in a ‘higher’ course.  There seems to be a larger baseline ‘desperation’ triggered when a problem involves a fractional relationship, with students reverting to ideas with little or no validity.

This particular relationship (a number is one-third of another)  is not that important within the mathematics of this course.  The more important thing, to me, is students avoiding those bad ideas in a desperate move to answer a question with fractions.  To help with that, I may approach this problem with more scaffolding next time.

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Mathematical Literacy: Understanding Algebra

Our mathematical literacy class is taking the 2nd Test today.  Over the past few weeks, we have been working on several topics but the most emphasized of these is using basic algebraic expressions.  We are seeing some real progress, and also some areas that are resistant to improvement.

A central issue in algebra is:  What objects can be combined in adding, as opposed to multiplying?  Initially, most students used the same rule for both types — if they knew to add like terms, the thought that they could only multiply like terms.  After a few cycles through this territory, most students now see the correct rule for each type.

However, we are still struggling with the details.  Even after exploring problems in small groups and in class discussions, several students in class are making these mistakes:

2n³(4n³) = 8n^9

2n³ + 4n³ = 6n^6

I think the problem with correctly learning these ideas is that it involves “2 dimensions” like a graph — students need to hold coefficients in one process and variables (exponents, really) in another process.  Humans are not naturally that good with visual learning; interpreting a bivariate graph does not happen spontaneously.  In algebra, we ask the same type of 2 dimensional thinking; instead of vertical and horizontal change, we are looking at coefficients and exponents.

The most challenging problem on this test?  This one:

Pick a negative integer and perform the following operations.

Add three.

Multiply the result by 4.

Subtract 4 from the result.

Divide the result by 2.  Write that answer.

Write the calculations for the generic number x and simplify the result.

A little more than half the students managed the basics of the operations on a constant.  Several students came up during the test to ask what the last part was talking about.  Sadly, we had spent one entire class day working on numeric patterns (seats around table arrangements) and generalizing the result for n tables.  Nobody got the algebraic version of the question correct, and only a couple of students came close.  It is true that the class this semester has students with weaker reading skills than we normally require, so it’s not surprising that those students had trouble.  Even those students who are ‘well qualified’ for the course and completed homework did not get this one.  [The problem is very similar to one in the homework.]

In a way, that problem illustrates the central theme of the material for this test — making the transition from numeric information to algebraic representation.  Clearly, work in class needs to require more attention to that transition.

Some areas seem to have worked well; issues that we struggled with earlier came out okay on the test.  Order of operations, including the pesky opposite of a square [like -6²] are definitely going better; I’d like to think that this is due to our working on ‘priorities of operations’ in class and de-emphasizing PEMDAS (I actually omitted PEMDAS from their reading).

 
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Mathematical Literacy: Have we Been Here Before?

Our Math Lit class is nearing the halfway point in the semester; our second test is coming up.  Most of the content now relates to algebraic reasoning; our class work this week dealt with building expressions to model situations and slope.

The ‘modeling’ situation involved variations on the classic problem:

We have tables arranged in a straight line, and chairs are placed (one per open side).  With 4 tables, how many chairs?  With 5 tables, how many chairs?  With n tables, how many chairs?

This problem went quite well; our book does a good job of providing some scaffolding to go from concrete cases to expressions.  The class created at least 3 different descriptions of the pattern, and we showed that all three resulted in the same final expression (2n + 2).

We then looked at different shapes (L-shape, for example … or plus-sign shape).  Because we were dealing with 3 shapes, the process did not work as well; however, the discussion was even more productive.  The book provided a hint: Look at the number of tables with 0 chairs, 1 chair, … 4 chairs (for each shape).  Students did not have much trouble counting.  The challenge came in expressing the unknown for each shape; since the variable category is not the same for each shape, this led to the conversation:

Which type of situation varies for the shape, and which types are always the same?

Several students had the very positive experience of resolving an initial confusion into something that made sense to them.  In the plus-sign shape, for example, the 2-seat table varies, with 1 ‘0 seat’ and 4 ‘1 seat’ tables, and students saw that we needed ‘n – 5’ for the 2 seat count.  We also simplified the expressions for these shapes, and these particular shapes resulted in the same total — 2n + 2.  We talked about whether this is always the case (no), and solicited suggestions for shapes that would work differently; the strangest shape we had was a shape where 2 tables had 1 seat.

The next day focused on slope, and you might think that this would go better; most students had already learned about slope, and many already could say ‘rise over run’.  Two issues got in the way.  First, students wanted to count by coordinates when they started at a point; if the points were (-3, 4) and (2, -2) they would start at (-3, 4) and go right 2 and down 2 (as if they already knew the slope).  Second, the order issue was not obvious for students; they saw every rise as positive and every run as positive.  We had to discuss this for several minutes before it started making sense.

The good thing about this slope work was that we did not start with the classic “m = —–” formula for slope from coordinates.  We have a little better understanding of what slope is measuring, and depend less on memorizing.  Since we had already talked about linear change and exponential change, we could even talk about the slope being a constant for linear situations and slope changing for exponential situations.

Overall, this was a good week in class.  Our assessment (test) will help show how effective the work was.

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Mathematical Literacy: Algebra Struggles, Building Algebraic Reasoning

One of my concerns with a traditional curriculum is that we put the content in ‘boxes’ — this week, we combine like terms … next week, we work with graphs … the following week we work with exponents & polynomials.  An average student proceeds through the course with very few opportunities to mis-apply concepts.

Our Math Lit class had a quiz today.  The first two problems are shown below:

1. Simplify the expression  -8x+2y-5x²-6y+2x

2. Simplify the expression (-8x)(2y)(-5x²)(-6y)(2x)

Most students did fine on the first problem, with combining like terms; a couple changed the exponent when adding.  The second problem caused the class to have a 15-minute discussion about what our options are.

To back up a bit, the prior class had worked on like terms (as a counting activity) and some very basic exponent patterns (multiplying with the same base, for example).  We had not formally covered the commutative property (did that today!), nor the distributive property (a start on that today).

The most common misconceptions that students brought to problem 2:

We can only operate on like things.

The numbers are connected only to the variable.

These were often presented as a package of ‘wrongness’, to create a common wrong answer:  -16x(-12y)(-5x²).  That is not a typo — students multiplied coefficients but did not change the variable (did not multiply those).  There was a general resistance to a suggestion that the constant factors could be separated from the variable factors — essentially, an over-generalization of the adding rule that we can only combine like things and the variable part stays the same.

A good outcome of this quiz is that students are more aware of some problems with their algebraic reasoning; every day, we talk about the reasoning being the important goal of this class, more important than ‘correct’ answers by themselves.  Students  partially buy in to this goal of reasoning; we did have a tense period in class when several students said ‘why do you have to make this so complicated!’.  I was honest with them that the second problem is overly complex compared to what we will need in our course.  And honest with them that the goal is knowing what our options are.

In our typical algebra course, these two problems are not addressed on the same day (except on one test day — even then, the problems are separated by space … one early on the test, one later on the test).  In our intermediate algebra course, I see the alumna of our algebra course struggle with basics — adding, multiplying, properties; the Math Lit experience sheds some light on how this might happen.  Students can pass a beginning algebra course and not understand the difference between processes for adding and multiplying.

We are early enough in the semester that I have to be cautious; just because an issue was raised does not mean that the students resolved the problem to get better understanding.  We will continue working on algebraic reasoning, so I will be looking for progress.

One thing I can say: If an issue is not raised for students, there is a very low probability that they will address the underlying problem.

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