Mathematical Literacy: Assessing Progress

Our Mathematical Literacy class is taking our first test today, which creates nervous students and concerned instructors.  After 5 weeks of work (even some hard work), we get an opportunity to see how well students can apply the ideas we have been working on.

I might share later info on the kinds of scores students get on tests.  At this point, I am thinking more about the outcomes in detail — assessing the class as a whole on important abilities.

One of the early items on the test is:

The base of a triangle is 4 feet, and the height is 18 inches.  What is the ratio of height to base?

The major outcome here is knowing to have the same units when writing a ratio.  About half of the students are showing that.  Within class, this was somewhat of a minor topic; yes, we talked about it; we spent more time talking about percents requiring the units to be the same.  The feet – inches comparisons came up primarily in the homework; I’ll check later, but I think the students getting this problem correct are generally those who have been doing their homework.

Another item on the test lists a table of values and students need to identify them as being linear or exponential.  There are two tables provided, each with 4 ordered pairs; as you know, the 3rd ordered pair is sufficient to discriminate types (between these two).  We never addressed this directly in class, though we spent one class entirely immersed in creating table of values for each type.  Something like 80% of students are getting this correct (both items), which is fairly good.

A related item provides a verbal statement for linear change:

We have $200 in our savings account, and add $10 per month.  Complete a table of values.  Write a formula to find the balance in month T.

I am especially interested in whether students can create the more abstract model, as opposed to the table of values; in a beginning algebra course, we do a very similar problem — as part of a long sequence of topics related to slope & intercept.  In Math Lit, this is not the case; we have been doing different models, and not focusing on the y=mx+b symbolism. I see this problem in Math Lit as being more difficult.  It looks like about half of the students are getting the formula correct (balance = 200 + 10T), with a few having elements of the model but not all of it.

Overall, I am expecting students to do better on the concrete outcomes (numeric only) than on abstract (symbols, different representations).  Of course, this is a statement about students in general.  As the course progresses, I will watch to see if the gap remains wide — or if our course is having an impact on the gap.

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Math Lit at Achieving the Dream Conference

I am currently on my way … to Anaheim for this year’s Achieving the Dream conference.  On Wednesday, I will have a poster at the “Emerging Ideas” event (11:00am) about the Mathematical Literacy course (and the AMATYC New Life work); Thursday, I am part of  a workshop (1:45pm) on developmental math … my part is the Math Lit course, and we will have extended time for discussions and questions.  This is my first “AtD conference”, and I am really looking forward to the opportunities and dialogue.

So, I have been thinking about how progress is made in academia — about how a basic change is accepted by large numbers of faculty and implemented at their college.  The AtD “mantra” uses phrases like “move the needle”, “acceleration”, and “progression with completion”; within the official communications of AtD and related foundations (Lumina, Jobs for the Future, etc) these phrases are repeated, and much conversation centers around engaging faculty in this work.  Parallel to this, the groups provide some outstanding professional development on theory and practice related to developmental education.

My hope is that the work of the New Life project touches and excites the values and beliefs of mathematicians and math educators.  Certainly, part of this is developing a better set of vocabulary phrases to communicate about our values and beliefs; the name ‘mathematical literacy’ is one effort to develop such a phrase.  However, vocabulary alone does not produce any change of significance; many prior efforts have failed because a new phrase was layered on to an existing curriculum (like ‘basic skills’, ‘application focused’, ‘mastery learning’).

I am convinced that our survival depends upon basic changes in our curriculum — and in our ideas behind the design of the curriculum; I believe that these basic changes will only happen as we all engage in conversations and even arguments about what things mean and what is really important.  Sure, we will need some resources, which means that we need to convince foundations and grant sources that our work is important; this will mean the strategic uses of phrases like “algebraic reasoning” and others like we use in the New Life work.  However, this is much more about our profession and our work together than it is about better words.

Progress occurs after dialogue; progress will happen when we actively seek to engage all members of our profession in a deep conversation about purposes and values, goals and beliefs.  Indeed, nothing can stop progress from happening if we can do so.

If you are coming to AtD 2013, I hope we have an opportunity to have some of that conversation!

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Mathematical Literacy: Student Capabilities

In our Mathematical Literacy course, we are working through concepts from a numeric point of view with less emphasis on algebraic (symbolic) statements.  This weeks’ content dealt with ratios, scaling rates up or down, linear rate of change and exponential rate of change.  Our work might indicate what our students are capable of, in a general way.

This course is ‘at the same level’ as beginning algebra, which means that we share prerequisite settings for math, reading, and writing; the students are similar, in many ways, to a typical beginning algebra class.  The Math Lit class also has a few students who did not meet all three prerequisites (due to some system problems at the college).

It’s true that students struggled at times in class.  One of those struggles dealt with language processing; we are using nutrition labels as a context for working on rates and scaling.  When students needed to read specific questions and then extract information from the label, most students did not see what they should do.  This is not a matter of mathematical ability or skills; in fact, students who have passed our beginning algebra class often exhibit the same pattern when I see them in the applications course (Math – Applications for Living).  A few students are having trouble with the scaling ideas, which is a non-standard approach; however, since they usually know an alternate method this is not a big issue.

Although I have not done an individual assessment yet, students did not seem to have any trouble with the concepts of linear and of exponential change.  We did numeric examples in two settings, and I observed groups and individuals — no issues spotted.  Most students are having difficulty connecting a situation to a symbolic model — both linear and exponential.  In the case of linear, we did “the salary is increased by 5%” … all of them could calculate the result for a given salary, but few of them could make the transition to the symbolic model (new = 1.05S).  The same kind of thing happened with exponential models.  Since we are not emphasizing symbolic work (yet!), this gap is not a big problem (yet!).

I’ve dealt with exactly the same issue in the Applications course (symbolic models for linear and exponential change), and observed the same proportion of students having difficulty.  The traditional beginning algebra course has an insignificant impact on students’ abilities to write symbolic models for situations — except when the correct key words are used in the problem.  If the problem is stated in a way that “normal” people talk every day, students can not make the connection to symbolic forms (in general).

In some ways, this was a discouraging week.  The difficulty with language is very frustrating; my judgment is that students (and people in general) are far less skilled with the written word than in prior decades.  Basic verbal skills like parsing and paraphrasing are not normally seen.  The transition to symbolic forms seems like such a small step, so that difficulty is troubling to a mathematician.  Our course is designed to build these skills over the course by visiting similar ideas from different points of view; I can hope it gets better!  However, I find it encouraging that these students — even the ones who lack all the prerequisites — are having no more difficulty than students who passed our beginning algebra course.

This Math Lit course is a good class for a mathematician to teach; we deal with basic ideas in detail and work on transfer of knowledge, with an emphasis on problem solving (as opposed to exercises and repetition).   In that work in-depth, we can see where students really do not get the idea and work on creating better mathematical knowledge.

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Mathematical Literacy: Growing Pains?

I am sharing some of the experiences in offering our new Mathematical Literacy course here at Lansing CC (Math105); we have 2 sections of the class, and I’ll be sharing from my section in general.

A normal class day (2 hours, twice per week) involves about 50 minutes of small group work.  The text we are using (a class test of the Almy and Foes Math Lit text) organizes the lessons around this approach; the group work is well designed, and the authors even include time estimates for each activity.  We usually cover 2 lessons per day, and the pace is reasonably comfortable for students.  I experience more stress about the pace than students, because that normal class day involves 4 separate small group activities followed by sharing results and often completed by whole-class discussion or lecture.

Tuesday, we dealt with a problem that involved megabytes and gigabytes … and a conversion between those units.  Each group had people who thought that a gigabyte was exactly 1000 megabytes, and each group had somebody who checked this with their phone using an internet search to provide the correct value (1024).  I was hoping this would happen, though I did not mention the possible problem; the text did not mention a possible need to search for an answer.  We used this problem to introduce a ‘multiply or divide’ approach to converting units; simultaneously, we are building our understanding of rates so we can use the more sophisticated process later.

Yesterday, we had a salary simulation with two different plans for raises; the groups did a lot of numerical work with the two plans and several cases, and discussed how we could tell when one option would be better than the other in a given case.  We then made a transition to writing algebraic expressions as a template for the numeric work, and showed a little bit of combining like terms.  I used these expressions to create a spreadsheet for the example salaries, and also showed the process on a graphing calculator.  Most students did not have a computer to bring to class, and only a few had a graphing calculator yet;  this is one issue that we will have to deal with soon, as a phone or smart phone is not a good device for doing mathematics (especially when we need to proctor tests).

Attendance is a little strange, because it is clear that most students do actually enjoy the work in class; most days, I am only getting about 70% attendance, which is low for my classes.  Since we have a test in two weeks, the absences are a concern.  I don’t think the students in this class have a significantly different lifestyle than my beginning algebra classes, where I normally have 80% to 90% attendance.  This is a puzzle.

The largest problem so far is doing homework.  Assignments include just 6 to 10 problems in the printed textbook, and (usually) another 6 to 10 in the online homework system.  This is pretty light, and we talked in class about the importance of studying for learning … to include these steps.  Only a few students are doing anything outside of class (5 or fewer, out of 18).  This has led me to modify the participation point strategy for each class — starting with the next class, students will lose half of their ‘daily’ points if they do not complete the assignments.  I’ll check the text problems during the first group time, and the online system before class.  I’ll report on how that goes in a couple of weeks.

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