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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Can Developmental be done Online?

A challenge in our profession is being able to integrate discrete components and results to create a stable vision of our work.  Over the years, various trends have impacted us and our classrooms; in the short term, many of these trends look good.  Most produce results that are initially positive.  Few trends have created long-term change.

The question is “Can Developmental be done Online?”  I’m taking this in the broadest sense — not just courses, but online in general.  Can the purposes of developmental mathematics be met by learning in an online environment?

Like education in general, we are in danger of being reduced to a finite set of skills or competencies.  We need to keep education as a separate goal from those in training programs — training adds competencies, while education creates capabilities.  Training is what we do when success is measured by how ready people are for identified jobs or behaviors; training is critical to our well-being both individually and collectively.  However, our survival depends on our adaptability and problem solving — and this speaks to education.

This distinction between training and education is critical if we are to answer the question “Can developmental be done Online?”.  Developmental mathematics, long defined in training concepts (skills), is far more of an educational endeavor.  As long as we focus on skills, our students leave our courses with the same basic capabilities as when they entered — in other words, generally ill-equipped for education.  Within a training program, a skill-focus makes sense; in an educational program, a skill-focus tends to defeat us.

The connection between these ideas and ‘online’ is indirect; online work is capable of dealing with either training or education.  I will conjecture, however, that increasing capabilities is more difficult to achieve in an online environment; not impossible — more difficult.  The movement from novice toward expert (in other words, education) is facilitated by varied supports — modeling, discussion, non-verbal cues, individual conversations, group support — which are easier to build in a face-to-face environment.

A parallel issue is “does online work for the population of community college students?”.   The Community College Research Center (Columbia University) has just released a research study based on a large (state-wide) dataset; see http://ccrc.tc.columbia.edu/media/k2/attachments/adaptability-to-online-learning.pdf for details.  The findings are disturbing:

While all types of students in the study suffered decrements in performance in online courses, some struggled more than others to adapt: males, younger students, Black students, and students with lower grade point averages.  [abstract, pg 2]

I say disturbing because the ‘lower grade point averages’ points to a developmental population more than general, and because Black (ie, African American) students already have a statistical risk in developmental courses.

This study was limited to online courses, which might not reflect the entire nature of online learning.  However, I would point out that most non-course learning online is done in an individual-based structure — a person finds their resources, and uses them as best they can, sometimes with a little support (tutors, for example); this learning is less supported than many online courses, so I would not expect non-course learning to have better results.

There are environments that stretch the concept of online courses towards the non-course format — “MOOC”.  MOOCs offer the excitement of more equal access to educational opportunity with reduced cost, and policy makers are considering this as an alternative.  I have large concerns, however, relative to MOOCs offering any help to developmental students … either the focus is on skills (training) which won’t help students very much in education, or the focus is on education without adequate support for building capabilities (movement towards expert).

Our own local data about online courses in developmental math is not that promising; most commonly, the online courses have a lower pass rate than other methods.  It’s possible that this is part of an overall trend towards lower outcomes in online courses.

Online learning is here to stay, and will continue to evolve.  This does not mean that online courses are here to stay.  Perhaps we need to look at that format as an option for a limited group of students, perhaps even for limited purposes.  My own view is that developmental math can not be done successfully online for the population we serve; I have doubts about whether there is a significant sub-population for whom online courses is a reasonable choice for developmental math.

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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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Reasoning with Units: Correct Wrong Answers

In the world of problem solving (as an academic endeavor), we talk about non-routine problems … ill-defined problems … and we talk about problem solving strategies beyond specific content issues.  When facing these types of situations, many students find great difficulty in transferring content knowledge; as mathematicians, we sometimes see this problem solving as the core outcome of learning mathematics.

Unfortunately, the teaching of mathematics often discourages broader reasoning.

I have been running in to a consistent error in thinking about units, which has led me to think about how this happens.  Here is the situation:

A desk has an area of 5 ft².  How many square inches is that?

Two bits of knowledge (neither correct) get in the way of solving this routine problem.  First, students equate 1 foot with 12 inches, whether we are talking about length or area or volume (they get 60 square inches).  Second, students treat the exponent (square) as affecting the 5 as well as the feet (300 square inches).  The first issue was addressed in an earlier post (see https://www.devmathrevival.net/?p=1471).  How about the exponent issue?

Misapplying the exponent could be caused by an over-generalized property of exponents.  However, I think the more likely error is a combination of two practices in mathematics education:

“find the area of a 4 inch square”

“just use the numbers in formulas, and write the correct unit with the answer — area is always squared”

The first practice, extremely common in early work with area, leads students thinking that something needs to be squared when they see a square indicated (like an exponent).  The second, more to my point today, leaves students with no reasoning about units.

For example, in last week’s Math Lit class, I asked the group what the formula for distance is (and got D=rt).  I asked how we usually measured distance; once we agreed on a context (a car) we agreed ‘miles’.  The next question — how do we measure speed?  This was much tougher, even though students deal with speed limit signs every day (usually without units 🙁 )  Once we got to ‘miles per hour’, we then wrote a typical calculation showing what happens to the units.

The next step:

A car has a speed of 40 miles per hour.  How far do they go in 20 minutes?

Many students see this as a trick question, saying that we should always give the time in hours (we would say ‘consistent units’).  However, including the units in the calculation makes it more obvious that we just need to change minutes to hours (they could do that).

Back to the square feet situation, few of us show the units in calculating area.  If we consistently did include units in calculations, students would have more experience in seeing where the ‘square’ came from (in ft²), and would be less likely to apply the square to the feet.

We have another instructional practice which discourages reasoning with units: the degree sign for temperatures.  By itself, the degree sign is not the unit — the unit for temperature must include the scale involved.  When we require units for temperatures, we should not accept just the degree symbol — 40° F is much different from 40° C, and nobody wants a household temperature of 40° K.  Even the simple conversion of F to C temperatures does not make sense if the scale is not included — the process becomes a black box of non-reasoning.

It is certainly true that “reasoning with units” will slow us down.  Our work is ‘cluttered’ by non-numerical information.  However, numerical information is the easier part for our students — it is the ‘clutter’ that needs to be seen and reasoned through if our students are to have any lasting benefit from our courses.

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