Include ‘statistics’ in a math course? Maybe not :(

The new prep curriculum in mathematics uses a course like Mathematical Literacy as the starting point, where the course is not just about algebra.  The ‘average’ math lit course includes a little bit of statistics, though it’s not clear that this is helping students.

A recent test in my Math Lit course included this question:

According to a study of player salaries back in 1998, here are three bits of information:

Mean Salary: $2.2 million       Median: $1.3 million              Mode: $272,000

One source made a claim that “most NBA players made more than $2 million” that year.
This claim is false.  Explain why; use a complete sentence.

This was question #2; question #1 was to find the 3 ‘averages’ for a given set of values.  As usual, the class did very well on those computational questions.  If somebody asks them for a median, there is a pretty good chance that the students will do fine.

However, when asked to use that same information to support an argument … well, let’s just say that my students did pretty lousy.  One student out of 2 classes (47 students) did a decent job — and all students would have encountered this type of question in their ‘homework’.

The word “most” was the key challenge, because students automatically connected that word with the mode — which is not an answer for this question at all. A good portion of the students actually presented an argument that the claim was true — based on the mean.

We are teaching more statistics (integrated in other courses, specifically) because we are led to believe that everybody will be ‘using’ statistics.  I certainly agree that we are all subjected to good and bad uses of statistics in the everyday navigation of western society.  Does having computational fluency with ‘averages’ contribute to statistical education?  No, not for the majority of our students.

I think we add statistics (and/or probability) to our classes because doing so allows us to believe that we are providing something of value to our students.  A little bit of statistics might very well be worse than no statistics at all.

The Bad Part of Dev Math

This past weekend, I was at our state affiliate conference.  MichMATYC has a long history (relatively), and we have had a number of AMATYC leaders from our state (including three AMATYC presidents).  We’ve been heavily involved with the AMATYC standards (all 3 of them).  However, you can still see some bad stuff among our practitioners.

One of the sessions I attended focused on lower levels of dev math — pre-algebra and beginning algebra.  The presenter shared some strategies which had resulted in improved results for students; those improved results were (1) correct answers and (2) understanding.  That sounded good.

However, the algebra portion was pretty bad.  The context was solving simple linear equations, and the presenter showed this sequence:

• one step equations (adding/subtracting; dividing)
• two step equations (two terms on one side, one on other)
• equations with parentheses, resulting in equations already seen

All equations were designed to have integer answers; the presenter’s rationale was that students (and instructor) would know that a messy answer meant there had been a mistake.  All equations were solved with one series of steps (simplify, move terms, divide) — even if there was an easier solution in a different order.

When asked about the prescriptive nature of the work, the presenter responded that students understood that it was reversing PEMDAS (which, of course, makes it even worse for me).

The BAD PART of dev math is:

1. Locking down procedures to one sequence
2. Building on memorized incomplete information (like PEMDAS)

As soon as students move from linear equations taught in this way to any other type (quadratic, exponential, rational) they have no way to connect prior knowledge to new situations.  In other words, the student will seem to ‘not know anything’ in a subsequent class.

To the extent that this type of teaching is common practice, developmental mathematics DESERVES to be eliminated.  Causing damage is worse than not having the opportunity to help students.  When we offer a class on arithmetic (even pre-algebra), the course is very likely to suffer from the BAD PART; offering Math Literacy to meet the needs in ‘pre-algebra’ and ‘basic algebra’ will tend to avoid the problem — but is no guarantee.

All of us have course syllabi with learning outcomes.  Those outcomes need to focus on learning that helps students, not learning that harms students.  Reasoning and applying need to be emphasized, so that students seldom experience the BAD PART.

 
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Students at the Center of Learning

“Teaching and Learning” … a phrase often used in professional development for us teachers, as well as in titles of articles and books.  Perhaps a better phrase would be “Learning and Teacher Behaviors”, or “Learning … Teaching without getting in the way!”

I am thinking about how well our Math Literacy course is doing in the Math Lab format.  The Math Lab format creates a learning environment by establishing assignments and a structure for students to work through those assignments.  The instructor ‘stays out of the way’ as long as learning is successful.  This format has been used with very traditional content, and is now being used with a modern developmental course — Math Literacy.

Although some students struggle longer, and do not initially ‘get’ new ideas, the vast majority of students in the Math Lab Math Literacy course have been successful with:

• identifying linear and exponential patterns in sequence
• using dimensional analysis for unit conversions
• identifying the type of calculation for geometry (perimeter, area, volume)
• writing expressions for verbal statements

What’s been tougher?  Anything dealing with percents — applications, simple & compound interest, etc.  Of course, these are weak spots for students in any math class; over the years, I have not seen anything that ‘fixes’ these in the short term; the fix involves unlearning bad or incomplete ideas, and this takes time and long-term ‘exposure’ to errors (along with support from an expert).  Direct instruction or group activities have limited effectiveness against the force of pre-existing bad knowledge.

The instructional materials form the basis for the learning in this Math Lab format.  If the ‘textbook’ is focused on problems to do, contexts to explore, with the expectation that the instructor will provide ‘the mathematics’, then the learner centered approach requires that we use specialized processes in the classroom.  The classroom becomes the focus, and we spend resources & energy on tactical decisions such as ‘homogeneous groupings’ or ‘group responsibilities’ or ‘flipping the classroom’.  The materials we use in this course are well crafted to support learning; the authors ‘expected’ the classroom to be the focus, though our Math Lab ‘classroom’ is working quite well with the materials.

What if we could offer a true “student at the center of learning” design?  Seems to me that this goal would lead us to use methods like our Math Lab, where students interact with the learning materials without an instructor mediating (as much as possible).  Students in our Math Literacy course have been successful in learning new mathematics with decent reasoning skills in this format.  Although initially confusing to students, the classroom is lower stress than a ‘regular’ classroom; there are no artificial social processes used to ‘facilitate’ the learning.  Think of it as being more like a student as an apprentice, where direct engagement with the objects of the occupation is the key for learning.

Of course, we are not normally able to offer all math courses in this format of active learning.  For me, the approach is to design my ‘lecture’ classes to be more like workshops.  In a 2-hour class, I might deliver 45 minutes of very focused presentations (direct instruction) distributed in a deliberate manner through the class time.  The length of ‘lecturing’ is varied according to the course and somewhat according to the needs of the students in a given class.

The point of this post is …

Stay out of the way of learning.

Students can learn by interacting directly with the learning environment.

We want students who are independent, and able to learn without a special structure.  Prepare your students for the real world by creating learning environments where they develop those skills while they are learning mathematics.

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Math Literacy: What do Students Struggle With? (part I)

In our “Math Lab” sections of Math Literacy, all tests are given individually … and graded while the student watches.  This is done by having 3 alternate forms of each of the 6 tests, with an answer sheet.

The process results in deeper knowledge of what students struggle with as well as what is going well.  For example, the little bit of algebra on the first test went well for all students.

On the other hand, two ‘estimating’ problems are struggle zones.  One question involves angle sizes:

The choices are provided to make this less stressful for students.  Quite a few students select sizes that are obviously too big for the image (choice A or B in this case). Very few select a ‘too small’ option (C).

The other estimating looks like this:

Our answer key allows about 10% leeway around the expected answer (390 miles for this one).  Again, students who miss this usually estimate too high (sometimes way too high).  A rare student went low in their estimate.

The issues seem different in each problem.  Estimating angles seems to be a perceptual challenge, where the eyes look at the distance between the rays instead of the opening size (or ratio of distances).  The map problem appears to be a simpler challenge — not using the measuring device provided (the scale at the bottom).

This test has a third estimating problem:

Students are missing this one for an odd reason:  instead of writing “-82”, they write “82”.  They knew that they were on the left side (it’s not like they said ’78’) but did not connect the sign with the estimate (even though it’s on the graph).  I don’t look at this as a struggle as much as ‘attention to detail’ … an issue for many of our students at all levels.

All of theses problems have similar exercises in the homework.  [We also have a Practice Test in the online system, which also has the problems.]  For most topics, those exercises are sufficient.  The first two listed above ‘not so much’.  These fit in the category “learning how to learn” … noticing a problem, seeking help, reasoning about it, practicing, etc.

Overall, the Math Lab method is working well for this course.  We will see other ‘struggle points’ for other topics as we go through the material.

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