Webinar Registration is open — Treisman & Rotman, June 6 (AMATYC)

Registration is now open for AMATYC’s next webinar: Issues in Implementing Reform in Developmental and Gateway Mathematics.  Details appear below; registration is currently open to AMATYC members and there is no cost to register for the webinar.

Webinar Details:

Presenters: Uri Treisman and Jack Rotman

Date: Wednesday, June 6, 2012

Time: 4:00pm EDT / 3:00pm CDT / 2:00pm MDT / 1:00pm PDT

Description: Uri Treisman and Jack Rotman will discuss issues that should be considered in implementing reform efforts in early college mathematics starting with a comprehensive theoretical framework to guide the work and narrowing down to key principles for the work on-the-ground.

Sponsoring Committee: Developmental Mathematics

To Register, Click Here

Registration is limited to AMATYC members; however, the webinar will be recorded and posted for general viewing later.  I’ll post a notice when that recording is ready.

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Apples, Hats, and Word Problems

In almost all math courses, an emphasis is placed on word problems … applications … real-world  … translation.  Is there a valid reason to include this work?  Or, should the mathematics in a course be restricted to that which is needed to deal with the contextual situations that students encounter?  Do these verbally-presented situations have a valid purpose in our math courses?

Here is the reason I am thinking about these issues — one of my ‘Facebook friends” (also a friend in ‘rl’ = ‘real life’) posted a link to a captioned picture at an online site.  The caption reads:

Everytime I see a math word problem, it looks like this:  If I have 10 ice cubes and you have 11 apples … How many pancakes will fit on the roof?
Answer: Purple because aliens don’t wear hats.

This ‘spoof’ takes its energy from the fact that we tend to have problems that are either obviously worthless (pancakes on the roof) or unreasonable (hats).  You are most likely responding the same way I did … “The problems in MY course are good and realistic problems!”  The criticism here is not what experts might see … the criticism is in what students (novices) see in this work.

First, here is a link to a short report I wrote a few years ago:  Ignore the Story  This report does not deal with how effective ‘word problems’ are — it deals more with qualitative studies.

Second, let us admit a basic fact:  The problems we can include in a course will not convince the majority of the students that those problems provide a justification for the mathematics covered.    Yes, I realize that some faculty will not agree, and hold the position that properly chosen contexts and applications will convince students.  We have a tendency to underestimate the complexity of going into a context to apply mathematics and coming back out of the context; this is hard work, and many students will seek any avenue possible to avoid dealing with the deeper relationships — sometimes working harder to avoid the process than it would be to complete it.  In other words, students will tend to map application processes to the procedures required at the shallowest level that ‘works’.

This minimalist tendency is not unique to word problems; nor is it unique to mathematics … our colleagues in other disciplines experience the same problem.  The difference is that we, in mathematics, expect students to deal with short (often cryptic) descriptions of situations in a variety of areas; students are expected to see mixing two levels of milk fat to be mathematically equivalent to mixing acid solutions of different percent concentrations even though the phrasing is often significantly different.

Elements of a course should support the instructional objectives of the course.  This implies that verbally stated problems should contribute to the mathematical outcomes for a student.  We should be using verbally stated problems to encourage and build a more complete understanding of basic mathematical ideas, and these verbal problems should also contribute to linguistic literacy for our students.  Achieving correct answers is not nearly as important as being able to paraphrase the situation, summarize it, state the known and unknowns, identify relationships between quantities that might be helpful, and write at least one mathematical statement that can be used to ‘solve’ the problem.  These abilities (which combine the linguistic and the mathematical) are much more related to a good prognosis for employment and other goals … more than basic skill accuracy, and more than algebraic manipulation by itself.

The emerging models for developmental mathematics (such as New Life, Pathways, and Mathways) tend to emphasize deeper processing of verbal problems, and de-emphasize repetitive ‘word problems’ which might look like the ‘apples, hats, and aliens’ spoof.  I encourage you to examine how verbally stated situations are used in your courses.  Do they contribute to both understanding basic mathematical concepts and linguistic abilities?

 
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Our Success — What does it look like?

Perhaps you have been involved with a process which includes classic design principles.  One of the basic design principles basically says “Imagine what success looks like … what it feels like … what it smells like.”  Ideally, this process is done by a group in a relaxed environment; no particular outcome is expected (besides a description).  After this description of what success is (based on perceptual characteristics), the process is designed to lead up to that outcome.

For us in developmental mathematics, what would our description be?  How would we describe success based on what our senses could directly perceive?  Would we even be able to describe success without the use of tests or assessments?

My concern is that we have described our work so much by learning outcomes and by tests (placement tests in particular) that we have very little thoughtful design in our work.  I worry that ‘success’ in developmental mathematics is mostly measured by correct responses to a predictable set of questions.

If developmental mathematics is about ‘getting ready’ for success, then our success imagination should reflect this concept.  Getting ready is not a description that can be used for design — we need to make ‘is ready’ concrete.  Descriptions like “articulate in quantitative issues”, “flexible with basic symbolic procedures”, and “responds positively to novel problem situations” are a start.  What descriptions would you add?

In the emerging models for developmental mathematics (New Life, Pathways, Mathways), some thought has been given to answering this basic question of what success would look like.  However, design is not a universal process; we can not just copy what some smart people have done.  Designing for success is a local process … what does success look like for your students?

I suggest to you that sustainable change in developmental mathematics will only be possible if we apply a deliberate design that considers a larger picture than categories and sets of learning outcomes.  The emerging models provide a necessary component, but not a sufficient one.

I invite you to initiate a ‘design for success’ process at your college.

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Good Algebra

A recent comment on this blog basically asked the blunt question:  Basic math seems fine, but WHY did I have to learn algebra?  Mathematicians know that the word ‘algebra’ has multiple meanings.  In developmental math courses, the ‘algebra’ is usually various procedures relating to polynomials with integer exponents, with a collection of procedures for rational exponents.  The traditional algebra course packages material that is either (A) thought to be important for pre-calculus and/or calculus, or (B) what students should have had in high school.

Given this, my honest answer to the question is “There is no good reason for you to learn that algebra.”  If you need calculus, we probably are not building your understanding deeply enough; we certainly are not developing your reasoning in the way you will need in calculus.  If you do not need calculus, what you experience in ‘algebra’ is unrelated to any mathematical need you might have (such as science classes, technical careers, or life in general).

A reasonable follow-up question would be: “If this algebra is sort-of okay for calculus bound students (and could be improved), and this algebra is not helpful to most students in the course, WHY does the profession maintain these courses built around an amazingly consistent content package?”

I believe that we, as a profession, are committed to helping students … that we want to provide the mathematics they need.  We seem to be ignoring a logical analysis of the situation; there must be a strong reason for us continuing the traditional ‘dev math’ package.  I believe that there are two processes which combine to create this reason (an illusion of a valid reason):

Myth 1: Algebraic manipulation is evidence of either understanding or mathematical reasoning; quick and correct execution are evidence of better understanding and/or mathematical reasoning.

Myth 2: Developmental students can not be expected to deal directly with abstractions (core mathematical ideas); the best we can do is provide basic skills.

For my college, we use a common departmental final exam for these courses … a practice which I support.  However, the final exam for our intermediate algebra course is a set of 40 problems to be completed in 2 hours; the 40 problems represent 40 learning ‘objectives’ in the course … no item on the final involves applying synthesis or learning based on multiple objectives.  Good algebra seems to be seen as quick algebra … good algebra seems to be seen as repetitive algebra.

Every day, people make mathematical claims.  Whether it is economics, environmental, or political … somebody says “this is growing exponentially”.  Do our algebra courses help students understand this phrase?  Would students have any idea what conditions allow truly exponential growth … could students tell when the phrase is being used as a rhetorical tactic?  Does the phrase “we expect 150000 jobs per month to be added to the economy” imply an equation for our students … could they estimate when we will have replaced the number of jobs lost in the recession?  Given a graphical representation of either an equation or data, can our students determine if the representation is accurate or if it is distorted (by inappropriate scales, for example)?

Yes, we have good algebra we can and should provide to our students.  Good algebra is not quick algebra (except for experts like us); good algebra involves abstractions and reasoning, and can be messy.  We need to have faith that our students are capable of doing good algebra; if we do not have this faith and act on that, we are enabling students to be ‘bad at math’ as a way of life. 

It’s time for us to step out of our constraints created by history and myths … step out of that cage, and build a new experience centered on good algebra for our students.

 
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