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?

 
Join Dev Math Revival on Facebook:

Teachers as Resource … Teachers as Problem

Once in a while (more often than you would know from posts here), I read something about education that shows how poorly some people outside of teaching understand what we do, and how we develop.  Today, I read a post on an effort (New Jersey) to remove  ‘bad teachers’ … based in part on standardized test scores.  I have posted about the value-added models that use standardized test scores; if you want to see a critique of value-added models from a mathematician’s viewpoint, see http://www.ams.org/notices/201105/rtx110500667p.pdf .

The article is at Test scores add value to teacher review, which is a blog post.  (I realize that one should be skeptical about the voracity of anything posted on a blog … you never know :).)  We could get distracted by the value-added component, and miss the more central error in such efforts:

Most good teachers were bad teachers at one time.

Personally, I began teaching (like most of us) with good training but bad real preparation.  How can a teacher be prepared to be a good teacher in the first two or three years?  I believe that some people have such an unusual background that they are actually good teachers from the first day; I believe that this is not a reasonable expectation for the group of all new teachers, regardless of the particulars in their training.  Yes, we can improve preparation of teachers at all levels — even college teaching.  Yes, we should have high expectations of teachers … with commensurate high rewards.

At the college level, our ‘standardized measure’ of outcomes is the set of grades we assign to students.  If we analyze these at the level of a specific college, the grade measurements are likely to be valid and reliable enough that some meaningful analysis is possible, always supplemented by insight and wisdom.  At the level of an individual instructor, grades are not so good; depending on the institution, there may not be any standardization at all in this measure.

To me, an obvious approach to the developmental mathematics problem is this:

Faculty are the most powerful resource available.

Instead of saying ‘bad teacher’, we should say “That does not look so good; what did you see happening?”  Instead of saying ‘bad teacher’, we should say “Are there conditions which negatively impacted your students that we could, together, improve?”  Instead of saying ‘bad teacher’, we should say “Can we identify what barriers exist in the learning for specific groups of students … and what development is needed for us to help all students succeed?”

Some of us might think that we do not need to worry about ‘removing bad teachers’ coming to higher education.  Especially in community colleges, we certainly do need to be concerned … whether it is at your institution yet, many colleges have become aware that they have opportunities to prevent new faculty from continuing by critiquing their teaching in the first two years.  Some states are implementing performance-based funding, where colleges get points … and $$ money $$ … for students ‘completing’ developmental mathematics as shown by grades; colleges in these states will have a vested interest in removing ‘bad teachers’ who fail too many students.  [I would like to believe that, in most cases, this removal will not be based just on the grades.]

In the emerging models (New Life, Pathways, Mathways), faculty are seen as the most powerful resource.  Professional development is intended to be be continuous and purposeful; expertise is gained both by the deliberate professional development and by the involvement in network of faculty.  In my view, a ‘bad teacher’ is a temporary condition … and one most of us have in our history.  Like mathematics for our students, becoming a good teacher is basically working hard with appropriate strategies.

Teachers are the most powerful resource; faculty are at the core of all solutions to the developmental mathematics problem.

 
Join Dev Math Revival on Facebook:

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.

Join Dev Math Revival on Facebook:

Fractions Lament

I am a fraction, and woe is me.  Nobody seems to understand my talents.  People have a prejudice against fractions, you know.  [I hesitate to say that there is ratial profiling, but it’s not pretty.]

People on the street say they hate me, without ever bothering to really look at me.  Is it my fault that they only remember their own difficulties with mechanics and procedures to generate answers involving things that resemble me?  If only they would look at the beauty and usefulness of fractions like me … they would find that fractions can be their friend.  [Look for me on Facebook!]   Fractions are much better than those elitist snobs, the integers; integers think that they are the greatest thing since sliced bread … but let me tell you that there are an infinite number of fractions for every integer.  The world could live without integers, but take away fractions and people would be back in their caves — without their iPads and Blackberry.

Then there are the math teachers!  You would think that math teachers would be enthusiastic about showing people how great fractions can be.  What do they do instead?  They tell students that you have to learn about greatest common factors and least common multiples, before you can ‘work’ with fractions.  Don’t they know that the GCF and the LCM are part of the integer conspiracy?  They make a big deal of mixed numbers; hey, if I want to hang around with an integer I will let you know — until then, I am happy being a fraction, thank you very much.  And ‘reducing’ fractions?  You (teachers) should talk; have you looked in the mirror lately?  I don’t think I am the one that needs reducing.

I am a fraction.  I can show much more than how many cups of flour you need to make 3/4 of a recipe; that’s boring stuff.  The good stuff is when a fraction lets you compare the rate of different groups of students to make sure that they are all getting the benefit of passing that math class.  A fraction lets you communicate about a rate.  [Did I tell you that my first cousin Marcel is a second derivative? He is one beautiful fraction!]  That reminds me … fractions make it pretty easy to convert one measurement to different units; you’ve just got to line up the units to get rid of versus the ones you need in the answer.  A fraction can also tell you what the chances are for having 2 boys and 2 girls in that family you want; that’s a beautiful thing by itself, isn’t it?

I am a fraction.  Don’t show me that cute picture with 8 parts and 5 shaded; even if I was 5/8 I would not like that picture.  Sure, I can show how many parts are there, just like certain integers can show how many pieces of candy are in the bag.  How would you like it if somebody showed a stick figure and said “this is Jennifer”?  You are more than a stick figure; I am much more than parts shaded in a drawing.

I am a fraction.  It’s time you saw me, and understand me.  I can’t make you love me, or even respect me.  However, I promise that I will do my part if you give me a chance/

 
Join Dev Math Revival on Facebook: