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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Towards Effective Remediation: Culture of Learning

Two events are fairly common in my classes: (1) A student says “just tell us” when we are doing group work, and (2) A student says “I don’t get that at all” when I am doing a (mini) lecture.  I suspect that these are also common in other classes, and wonder what meaning we each see in these statements.

Both statements deal with confusion and frustration.  One occurs when a student is struggling to find the idea in their own work, and the other occurs when a student struggles to understand the idea in my work.  Both are normal, and both are part of a learning process.

A culture of learning would be shown by an acceptance of these frustrations, combined with a determination to learn in spite of (or because of) that frustration.  Learning is rewarding just for its own sake as we see how ideas connect and build on each other.  A focus on comfort defeats a learning attitude.  Perhaps a focus on the learner raises the same risk.

We tend to see the phrase “student centered” as a positive goal usually implying a process whereby students find ideas about mathematics.  For some of us, this means that we seek to minimize frustration and/or confusion.  I think a better goal is to manage the frustration and confusion to maximize learning and build a culture of learning.  I want my students to see learning mathematics as a set of goals which are attainable given effort and attitude.

We can also see ‘student centered’ as an idea leading to a focus on context and applications, perhaps to the extent that we only cover mathematics that can be applied to problems of interest to students.  As much as I am enthusiastic about applications (I teach a course 100% ‘applications’) I think it is a mistake to construct a curriculum around problems that students can understand and care about — these must be included, but a culture of learning means that we look at extending beyond the immediately practical to the larger ideas and even the artistic beauty of the subject.

In every course, I seek to present some beautiful and useless mathematics.

I know that few of my students achieve this culture of learning, even though my goal is to get them so motivated to learn that nothing will stop them from learning more mathematics.  I know that most of my students will stop taking mathematics as soon as that becomes an option, even though my goal is to inspire them to take at least one more math course than they are required to take.

Students seldom achieve more than our goals and expectations, so I have this culture of learning as a goal in my classes.  Rather than a limited range of ‘student centered’ ideas, I am looking at the largest possible picture of what that means — including how we deal with frustrations and confusions.  Learning, as in life, mostly is determined by how we deal with such problems; learning, as in life, is damaged by attempts to avoid confusion and frustration.

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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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Towards Effective Remediation: Quantity and Pacing

I’ve been having conversations about arithmetic and similar topics (at the Achieving the Dream conference, and online at MATHEDCC).  In some cases, the conversation was the result of telling people about the New Life model (see https://www.devmathrevival.net/?p=1401).  In other cases, we had been discussing other issues.

So, I have been thinking about related issues.  As a result, I have some ideas of how to frame a conversation about developmental mathematics that might help us make progress.  To start with, we often describe developmental mathematics by using names of courses or by listing topics with implied outcomes.  In our New Life work, we actually started by examining the types of mathematics that students need to succeed, and dealt later with course names and lists of topics.

Our curricular designs are based on our assumptions and goals, which are often unstated.  One of the most problematic assumptions is:

It is effective to deal with 8 general topics, with 10 to 15 outcomes within each, in one course.

There are two fundamental problems with this.  First, the courses we design cover so much ‘material’ that we prevent the learners’ brains from dealing with the associations and connections that are part of learning; this results in most students focusing on just remembering what to do, rather than making sense of it.  Second, the design is based on the absence of prior learning (good and bad) in the learner; this is obviously not true in almost all cases.  Time is needed for us and students to identify where there are conflicts between prior learning and current need, and time is needed to deal with these conflicts.  The result is that we add another layer of ‘learning’, one that is weaker than prior learning; students after our courses are notorious for returning to wrong methods and ideas after our course is done … because we do not provide a method of correction.

We need to slow down; learning is much more complex than having a list of 80 to 120 outcomes.  Since we need to go slower, we must be strategic about what areas to focus on .. trying to do it all (or even most of it) means that we are willing to accomplish little of significance.

This strategic work should be based on our judgments as mathematicians about which mathematical ideas are most important in particular cases.  Do we want work with fractions, or do we want work with proportional reasoning?  If we want both, what are we willing to give up … percents or linear models (as examples)?

We need to do some critical thinking about our goals and purposes, and apply our problem solving skills so that our courses are effective learning experiences for our students.

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