Do the Math! What does that mean?

I was at a conference this past week, when a keynote speaker used the phrase “do the math!”.  A redesign methodology states that one of the benefits is ‘students spend more time doing math’.  If we ever needed evidence that the mathematics curriculum is mis-directed, these comments would seem to be conclusive evidence of a problem — they are quotes from fellow mathematics faculty.

Perhaps we have lost track of what mathematics is.  According to a dictionary (Merriam-Webster in this case, though they are all similar), mathematics is

the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

Open any developmental ‘mathematics’ textbook, and you will see something that resembles mathematics … a bit like a scary Halloween costume, where the outside looks different than the true character underneath.  We have gone far from the path of mathematics, much to the detriment of our students.

In particular, we have lost all elements of science within mathematics (especially in developmental math, but also in gateway college courses).  Science is a ‘system of knowledge’.  If it were not for the undeserved special treatment of mathematics, our science colleagues would have long ago challenged our mathematics courses as being a ritualistic mis-education of the masses.  Two hundred types of problems with remembered procedures to manipulate values and symbols to acquire a ‘correct’ answer does not represent knowledge; the resemblance is stronger with uninformed rituals performed with no redeeming value (practical or intellectual).

The emerging models (New Life, Dana Center Mathways, Carnegie Pathways) are all movements towards mathematics.  We can, and must, reform our mathematics courses so that students learn mathematics more than rituals.  As mathematicians, we have knowledge systems that help people understand the world around them … and a knowledge domain that is enjoyable just for the learning.

The person who said “Do the math!” was simply saying “you need to agree with me, because my view of the data says you should”.  The person who said “students spend more time doing math” really meant that students spend more time in some activity that resembles mathematics … but was most likely engagement in the rituals that have taken the place of mathematics.  The fact that the majority of American students believe that they are bad at ‘mathematics’ says more about our curriculum than it does about them.

I still spend a large portion of my teaching time in courses where the content is still traditional; change is not instantaneous.  However, whatever the course, we can take a more mathematical approach by focusing on concepts and connections even as we get students to accurately perform the rituals.  We each need to start on this path towards teaching mathematics so that we are ready for larger changes; our curriculum in 10 years will have little resemblance to that of 5 years ago.  The good news is that we will be truly teaching mathematics when that change comes.

I hope that you will begin your personal journey towards teaching mathematics; perhaps you can even contribute to the professional work that will lead to the change that is coming.

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Remediation as Cheese, Remediation as Fishing

You may have noticed that the emphasis on completion, combined with a high priority on getting a job right away, has resulted in pressures on colleges to provide training and skills development … with less emphasis on the subtler goals of intellectual development, curiosity, and liberating arts.  In both developmental and gateway mathematics courses, we have become the epitome of the completion/job methodology; to the extent that this is true we have failed as educators and mathematicians.

My thinking on this got a boost from a short piece on “Habits of Mind” by Dan Berrett (see http://chronicle.com/article/Habits-of-Mind-Lessons-for/134868/).  Dan’s main point is that the current focus on measurable outcomes applied to a college ‘education’ results in using simplistic measures, where these measures miss the most powerful advantages of an education.

Earlier this year, I was in a conversation with a group discussing placement tests and diagnostic tests.  The predominant approach was summarized by a food metaphor:  Our goal is to fill in the ‘swiss cheese holes’ for our students, so that they do not have any gaps.  The ‘remediation as cheese’ metaphor is very much the common approach, whether a college uses modules or emporium or traditional classes; we measure success by counting holes (or lack thereof).  I’d like to think that education in general and mathematics in particular is more than the absence of holes.

Compare the cheese metaphor with this:  Remediation as fishing.  According to a quote, often cited as a Chinese proverb:

Give a man a fish and he will eat for a day. Teach a man to fish and he will eat for the rest of his life.

“To fish” is the “remediation”; we are not about holes … we are about building capacity as well as building ability … we are about attitudes about learning as well as learning about attitudes … we are about enabling students to become more than they intended at the start of our course.  Remediation succeeds when students are fundamentally different when they leave our classrooms; the ‘lack of holes’ with arithmetic and algebra does not improve a student’s preparation for education or for employment as much as the habits of mind that can be developed.

Let’s help our students learn how to fish.  The broader goals of education are just as important as discrete skills and immediate performance measures.  We can, and should, contribute to our student’s capabilities within our developmental mathematics classes. 

Nobody makes a greater mistake then he who does nothing because he could only do a little.  [Edmund Burke]

 
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Summary and Comparison of the Pathways, New Life, and Mathways

Three emerging models of developmental mathematics — AMATYC New Life, Carnegie Pathways, and Dana Center Mathways — have enough similarities that people can mistake one for another.  The basic genetic codes of these models does look very similar.  I’d like to provide some comments of comparison in order to highlight the differences; most of these differences are operational or matters of implementation.  You might want to read this Summary:  Summary of Three Emerging Models for Developmental Mathematics

One difference between the models is that the Pathways are always targeted towards specific groups of students — namely, students who only need an introductory statistics course (Statway) or only need a quantitative reasoning course (Quantway).  The New Life model can be implemented this way, which is what my own college is doing; however, the New Life model describes a curriculum that can completely replace a traditional developmental mathematics program.  Although still under development, the Dana Center Mathways is likely to be flexible in the same ways as the New Life model with some differences within the design.

All three models incorporate student success design factors.  The Pathways materials have both imbedded and supplemental work on issues such as productive struggle, deliberate practice, productive persistence, and growth mind-set.  The New Life model, because it is a professional framework, does not prescribe how these factors are incorporated in the local implementation; some instructional materials for the New Life model embed the design and others are strictly supplemental.  In the case of the Dana Center Mathways, the model includes a required student success course as a co-requisite; the details are still being developed.

Placement is handled differently in the models.  The Pathways (statway & quantway) are two-semester packages; the design assumes that all students begin in the first semester and continue into the second semester.  The New Life model could be implemented this way; however, the basic design suggests a normal process where some students could begin at the second-semester level without taking the first semester.  The preliminary descriptions of the Dana Center Mathways suggests that they will also provide flexibility concerning where students begin.

None of the models assume a change in the actual placement tests at this time.  In all three cases, the first course uses ‘placement into beginning algebra’ as the benchmark; local colleges adapt this guideline to their environment.  Since all three models focus on concepts of mathematical literacy, they collectively suggest that our placement tests need to have a measure of basic quantitative reasoning at a pre-college level.

The largest difference in implementation is in the domain of ‘institutional committment’.  The Pathways model presumes (and requires at this time) that the institution will sign a multi-year agreement to develop, implement, and participate in the activities of a shared network; individual faculty can not ‘join’ statway or quantway.  [However, the instructional materials will be publicly available later this month for open-use as version 1.0 (first classroom version) under a license like Creative Commons.]  The New Life model allows for individual faculty to pursue incremental changes at their institution, as well as allowing for institutions to make a committment for a larger change.  My current interpretation of the Dana Center Mathways model is that they will seek some level of institutional committment to a change, though the change might not be total reform of the curriculum.

Mathematically, the models differ in how they address the old intermediate algebra course. The Pathways model does not address this course in any way, since the student populations are selected to avoid intermediate algebra — the Pathways include mathematical literacy and the ‘terminal’ math course only.  The New Life model provides a replacement course (currently called “Transitions”) that can be used instead of intermediate algebra; the outcomes for Transitions include a fair amount of the topics of intermediate algebra — the approach is more balanced between symbolic and numeric work, and the content is not limited by tradition (exponential change is consider a basic skill in Transitions).  Although still being developed, I anticipate that the Dana Center Mathways model will also provide a replacement for intermediate algebra; this will provide the profession with two strong alternatives to the old ‘non-functional’ intermediate algebra.  [‘non-functional’ has two meanings — does not work well, and does not integrate function work]

I have been fortunate to have (1) led the New Life project, (2) been deeply involved with Pathways, and (3) involved with Dana Center Mathways [with the hope of doing more].  My statements above do not represent an official narrative from the 3 sources; rather, this is my professional evaluation offering a comparison.  To the extent that I have information or knowledge, I would be glad to answer questions about how the 3 models are similar or different.

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Pre-Algebra Just In Time

In my beginning algebra classes, I made a number of changes for this year.  Some are related to pre-algebra as a prerequisite for success in beginning algebra.  For quite a while, I have concluded that a pre-algebra (or basic math) class is an inappropriate prerequisite to beginning algebra based on analyzing data; my experience this semester might provide an alternative.  Students need not spend an entire semester ‘getting ready’ for algebra.

At the start of our course, we have a chapter which reviews operations on signed numbers with a minor emphasis on very basic algebraic expressions (like terms, distributing, etc).  This chapter is essentially a typical pre-algebra course contracted down to one chapter; historically, I concluded that both experiences (a course, a chapter) did very little to enhance the readiness of students for algebra.

Instead of spending any class time on signed number operations, we spent every minute of class time on algebraic language, syntax, and concepts.  We talked about adding changing coefficients but never exponents, and about multiplying changing both (depending) … and followed this up with a variety of problems for students to struggle with.  Much time was spent on translations, but not just in to algebra: we did a bit of work on translating algebra into words; even when students remember the ‘right thing to do’ (procedure) there is often a misunderstanding about what the expression meant.

Given the equation concepts we will be studying, we covered zeros in adding and subtracting.  Take some time to interview your students about a simple problem like this:

5y – 5y = ??

When this  would come up later in the course, something like 20% of my students would report that the answer is ‘y’ — the fives cancel out, leaving the y.  Curiously, textbooks do not have problems involving zero for combining like terms, even though this is critical for later work.

Class spent a lot of time (very frustrating for students) dealing with the different uses of the ‘-‘ symbol: opposite, negative, minus.    Some of this was imbedded in the translation work, and others in procedural work.  As instructors, we are incredibly careless about reading the ‘-‘ symbol, tending to say ‘negative’ when it is ‘opposite’ (like ‘-x’).  The central issue here is often “do we have any options about how to treat this particular ‘-‘?”

To assess this change, I used the same test from prior years with some added ‘difficulty’ — 3 added questions on expressions (including the zero in adding).  The initial assessment is that the new emphasis (pre-algebra just in time) helped students with algebraic proficiency without harming numeric skills.  The average score on the somewhat harder test was almost identical to the average on the prior (easier test).  Obviously, this is not enough to assess the merits of the new approach:  if the change does not help students later in the course, then the new process is not good enough yet.  I am especially eager to see if the ‘zero’ in adding has been improved.

My belief is that we could improve the outcomes of developmental mathematics by a fairly simple change:  do not require any ‘math’ or ‘pre-algebra’ before beginning algebra, just some basic numeracy is good enough.  Some students have a direct need to know arithmetic skills for an occupation, but this is a different need than ‘algebra’.  By placing almost all students directly into beginning algebra, we eliminate a math course for a large group of students — without producing harm.  [From general data I’ve seen, the chance of success in algebra AFTER a pre-algebra course is statistically equivalent to the chance of those below the placement cutoff … and the numbers are not good for either group.]

However, I am also sure that our current algebra courses need to do much more about basic literacy issues — translations, syntax, paraphrasing, etc — as well as a more complete treatment of procedures (zero in particular, but also adding vs multiplying).  We tend to move too quickly into applications of literacy (procedures for solving equations, for example) without building the conceptual foundation required for understanding.    We need “Pre-Algebra Just In Time”!

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