Symbols as Window

Like humans in general, our students develop expectations based on experience.  Habits form, often without awareness or conscious effort.  Behaviors exhibit, which are used to measure knowledge.  In assessments, we often confuse correct behavior with correct knowledge.

Symbolic work can be difficult for novices.  We (experts) see large amounts of information in short symbolic statements.  For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.

On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):

• -3² + 5² = 2^4 = 16
• 8^6 divided by 8^2 = 1^6

The first error is a coincidental ‘right answer’ for a very wrong method.  The second one, not at all.  Both involve over-generalizations of ‘same number’ rules.  Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).

My concern is not these particular students, nor these particular errors.  My concern is our overall approach to mathematics.  We tend to take one of these approaches to symbolism in mathematics:

1. Emphasize symbolic procedures, and measure understanding by correctly completing more complicated problems.
2. Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.

Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures.  We need to learn how to balance the ‘symbols’ and ‘reasoning’ aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.

Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student’s future.  I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student’s future.

Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science).  Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.

When we assess the mastery of symbolism, we need to deal with much more than ‘correct answers’.  In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice’s understanding based on the individual learner.  Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.

Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks  ago — dealing with the adding terms error (first error above).  On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class.  Something like half the students added the exponents as well as adding the ‘terms’ (coefficients).   About 40% of these students apparently maintained this erroneous method up until the test.

Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules.  Mathematical symbolism can be a window into the houses where students keep their math knowledge.  Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.  

We need to slow down our courses.  Learning mathematics is not a fast or spontaneous activity.  Learning mathematics is hard work for both us and our students.

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When Change becomes Progress

This semester, we have more than 150 sections of Mathematical Literacy offered at colleges across the country … and these are outside of the grant-related work (such as Quantway™).  In other words, the New Life Mathematical Literacy course is now the most implemented reform math course in the United States.

Getting to this point is the result of the incredible effort of dozens of math faculty, many of whom have been members of the New Life wiki at http://dm-live.wikispaces.com/.  Our work has not involved large grants from foundations; rather, collaboration and local initiative have allowed us to create significant change.

However, change is not the same as progress.  Progress involves sustained efforts which achieve explicit goals.  We have achieved more than other efforts … but we “are not there yet”.

Where are we headed?  How will we know when we have arrived?  These are not questions for which we create singular responses and data-based conclusions; these are questions for a profession to use as standards for our work.

In the world of process and product design, one set of strategies involves having people seeing themselves in the situation that they are trying to create.  For example, we might ask 100 math faculty to imagine that the mathematics curriculum works like it is supposed to.  What does this look like?  What does it sound like?  What does it smell like?

For our work, here are some answers I would give to those questions about what we are trying to achieve:

• Students text each other about the latest exciting math problem.
• Students pass every math class unless something unexpected comes up.
• Over 10% of students major in a STEM field and over 10% of degrees are awarded in STEM fields.
• Students learn diverse mathematics, with understanding, in both pre-college and college math courses.
• Fewer students are in college-prep math classes than are in college level math classes.
• Half of the students who start in college-prep math classes change their goals to be more STEM-like.
• Math faculty are the happiest faculty on campus.

Part of our difficulty has been that we have not had a goal in mind — beyond having higher pass rates.  Higher pass rates is not a design standard; it’s a production standard (and a poor one, at that).

Progress would exist if we would judge that we are substantially closer to achieving our goals.  If we don’t articulate our goals (like the 7 statements above), we can never have progress … because we are not directing our efforts towards anything.  Change is cheap; progress is where the power is.

I started off this post thinking a next step, like getting the Algebraic Literacy course on the radar — and I still think that is very important.  Or, thinking about salvaging the college algebra and pre-calculus curriculum, which is very important.  I hope that you will be involved with one or both of those reform efforts.  Overall, however, I am concluding that we need to have more conversations about our goals.  What does progress look like?  How do we know when we are there, as opposed to where we are now?

We have created significant change.  Progress?  We’re not there yet.

Serving all students

Lately, I have been reading a lot of college schedule books.  One interesting idea I found was a college that offered “Intermediate Algebra for STEM” as well as regular intermediate algebra.

However, I saw an overall pattern that is disturbing.

Math programs tend to respond to struggling students by either creating a new course in addition to the old OR by splitting existing courses into 2 or more parts.

Let me clarify the ‘splitting’ — although I did notice a few places with modules (same semester), what I saw more was partial courses to be taken over an entire semester.  Beginning algebra might be a deliberate 2 or 3 semester experience (if all parts are passed); intermediate algebra might be another 2 or 3 semester experience.  The only way this pattern can be justified is by meeting two criteria:

1. A pass rate approaching unity (>90% in every segment)     AND
2. A retention rate approaching unity (>90% continue to the next segment)

These conditions still only allow about 50% of those starting beginning algebra to complete intermediate algebra after a 4-semester sequence (.91^7 = 51.7%).  The more typical 60% pass rate, combined with a more typical 80% retention rate, suggests a whopping 7% of students completing both algebra courses.  Most students are not done with intermediate algebra; getting to pass a college math course would happen for about 3% of those who start a beginning algebra sequence.

The issue here is not whether something needs to be done.  A significant portion (a minority, but non-trivial group) of our students have excessive struggles in spite of good effort on their part in the presence of a good classroom.  My own gut-level estimate is that approximately 20% of students need something more to help them succeed in a traditional algebra course.

Clearly, this is one of the motivations for the New Life courses, Mathematical Literacy and Algebraic Literacy.  These designs provide a more diverse curriculum, with reasoning emphasized, which is meant to help more students succeed.

However, not all institutions are able to implement the Literacy math courses at this time; some institutions will take 5 years to reach that stage.  What should the rest of us do?

I suggest this principle as being a valid guideline for our work with all students (including those who struggle):

The math curriculum should provide a one semester experience at each ‘level’ for all students.

Our current levels are beginning algebra and intermediate algebra.  Courses before beginning algebra have their own issues, and we need to justify their  existence. A beginning algebra course should be one semester, whether a student needs minimal support or maximal support.  A intermediate algebra course should be one semester, whether a student needs minimal support or maximal support.

So, instead of a two-semester beginning algebra sequence, we could offer an expanded class time version of beginning algebra.  If we think students need twice as much ‘instruction’, then we could offer a 8-class-hour per week version of a 4-credit course; at many institutions, this translates into a 4 credit course with 8 billing hours.

Most of us introduce the sequence of partial courses in response to lower pass rates for a group of struggling students.  My college did this for about 10 years.  Our rationale was that only 25% to 30% of the struggling group passed the single course, compared to 60% for the remainder of the students.  However, our best efforts only achieved 60% pass rate in the two half-courses; this resulted in about 29% of the students completing both halves — about the same rate as completed the single course.

Helping struggling students is not about providing more courses in a sequence.  Helping struggling students is much more about what we do in one course, in one semester.  Whether ‘struggling’ comes from learning disabilities (the most common reason), or historical accidents of the student (no diploma, no GED), helping students should not hurt students chances of completion.  In the political parlance, we are talking about societies most vulnerable adults.  Our work should be about catching them up, not setting them further behind.

Let’s drop those sequences of partial courses.  The design can not succeed as a strategy.  Let’s create some truly helpful solutions that fit within one semester.

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The 1st National Summit on Developmental Math — Coming to a Town Near You?

In some states, the big problem is policy makers imposing a single solution on developmental mathematics.  In most states, the big problem is that math faculty are not involved with their professional organizations (AMATYC, NADE) … which means that they lack current information on solutions and options, and that they do not have a support system outside of their own college.

The 1st National Summit on Developmental Mathematics was an initial step to deal with these problems.  Most of us could not be there.  However, you can already see most of the presentations at https://sites.google.com/site/amatycdmc/national-mathematics-summit

I hope that you will take some time to look at those presentations.

Dealing with developmental mathematics is a state and/or local issue.  We need to have state and/or regional summits on developmental mathematics.  AMATYC might suggest that each state affiliate host their summit, perhaps attached to an annual state conference, and hopefully coordinated with their NADE affiliate.

These state summits can use the online presentations as a source of information so that faculty can have conversations about solutions.  As needed, these remote resources can be supplemented by one or two live experts with broad information about the possible solutions.  The costs of these summits would be minimal; those with an interest could even apply for an AMATYC mini-grant (<$750) … see http://www.amatyc.org/?page=MiniGrants  for some information.

I hope that you will be involved as this reform effort reaches more and more states.

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