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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Quantway(tm) Materials, available — the Dana Center “FMR” course

The Dana Center (University of Texas – Austin) has launched an updated web site.  As part of this, the original Quantway™ materials (version 1.0) are available along with other documents — see http://www.utdanacenter.org/higher-education/new-mathways-project/new-mathways-project-curricular-materials/foundations-of-mathematical-reasoning-course/  

To get the Quantway information, you need to scroll down to the end of that page.  If you want general information on the New Mathways project, go to http://www.utdanacenter.org/higher-education/new-mathways-project/

This site is part of the New Mathways Project.  I’ve talked before about their project; the first course in their paths has now been named “Foundations of Mathematical Reasoning” (FMR).  It looks like this course will be shared by all 3 paths that will be developed — statistics, quantitative reasoning, and STEM.   The learning outcomes for FMR will be adapted from the Quantway outcomes, which were adapted from the New Life MLCS course.  The position of FMR in a math curriculum is very similar to that of MLCS; the Carnegie Foundation Quantway program is different in that students are tracked from the start … both New Mathways and New Life provide a flexible structure at this level.

The fact that 3 reform efforts share a curricular element (the first course) is part of my optimism that we can create basic change in developmental mathematics over the next several years.

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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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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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