It’s Time for Algebra Class … Do You Know Where Your Linguist Is?

We’ve heard … and many of us say … “math is a language” or “algebra is like a foreign language”.  In our classrooms, these statements are often intended to motivate students to pay attention to vocabulary and syntax.  In general, I think the net result is neutral or even negative.  [Students are told to attend to something that they do not understand, and also lack a structure for learning.]

Twenty-five years ago, the Center for Applied Linguistics (http://www.cal.org/) published a pair of books on “English Skills for Algebra”, authored by Joann Crandall et al (Crandall, Dale, Rhodes, and Spanos).  One book was a student workbook … the other a tutor guide; the goals were:

“provide practice in manipulating the specialized language of mathematics and algebra through listening, speaking, reading, and writing activities in English; and

“provide practice in using language as a vehicle through which they can think about and discuss the processes used to perform basic operations in beginning algebra.”

I notice that the authors (linguists) include four modes of langauge usage (active — speaking & writing; passive — listening & reading).  I suspect that this is obvious to linguists … but not to mathematicians … that fluency depends on prolonged and deliberate efforts in all four modes.  Our math classes tend to focus on the passive modes; we consider ourselves progressive if we include talking in small groups. 

You probably will have difficulty finding the books mentioned.  I am adopting some of the content for my beginning algebra classes, and can provide a sample of one activity.  This is a worksheet, delivered through our course management system, with the purpose being to understand both one correct meaning for an algebraic statement AND to identify a correct paraphrasing.  Here is an image (you may need to right click on it, and open separately so you can enlarge it):

This activity has 7 questions, and I have a series of 3 for students to use. 

As you can see, this still works on the passive modes.  For active modes, here are two things I do in class:

Speaking — I ‘cold-call’ on students to have them explain how to do problems (they have had a couple of minutes to work on the problem, which is related to an example I have worked with verbal explanations).  I am able to encourage correct spoken language, as well as identify gaps in language or understanding.

Writing — I use “no-talk quizzes’, where students review other student’s work and provide feedback in writing phrases or sentences.  The focus is on explanations; feedback must be verbal (can not be symbolic).” (pg iv)

I encourage you to think more deeply about ‘algebra is a language’.  If you are fortunate enough to have a linguist nearby (which I was for a few years), talk to them; you might need to draw an analogy to learning a foreign language.  [Most linguists actually have some background in applied mathematics, but not so much in learning issues in mathematics.]  My own work in this regard is unfinished … I am most concerned about getting a process for spoken algebra with feedback, and I want to add more writing with feedback.

Writing across the curriculum is wonderful; however, the language within mathematics is more fundamental to our work.  If we conceptualize algebra as a language, we should have a deliberate plan for developing the fluency of our students in all modes of usage.  Just saying “it is a language” is a bit like saying “don’t you understand this yet?”.  The langauge learning process is not just a matter of a label like that, or motivation; language learning has its own processes.

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Don’t Do Your Homework!!

Within days, close to a million students will attend their first class in some developmental math course; some have already started.  The vast majority of their teachers want every student to succeed, though this may not be the students’ perception.  Therefore, one phrase is likely to be heard by close to a million students in a short period of time:  “Do your homework!”

I do not tell my students to do their homework.  Why?  Well, think of this metaphor:

My cousin Alfred walks in to his doctor’s office; being astute, the doctor notices that Alfred is obese (the only question is ‘how obese’).
After letting Alfred share his health concerns, the doctor makes two statements to him:
1) Do you think that you are overweight or obese?
    Alfred’s answer: Well, yes … that is kind of obvious.
2) Okay, your first step is to eat right and exercise?
    Alfred’s response: Yes, I’d like that.

What do you think Alfred is going to do?  Will he eat right?  Will he exercise?

Our students, especially in developmental math courses, do not know how to operationalize “do your homework”.  The vast majority of students believe that ‘doing homework’ means completing the assigned exercises, whether online or on paper.  We sometimes reinforce this perception by “collecting homework”, where we make sure that the student has ‘done it’.  However, the basic purpose of homework is to learn the most possible for that content for that student. 

Instead of telling my students to ‘do homework’, I tell them to follow the learning cycles.  These ‘learning cycles’ are simply stated components of doing homework with a focus on the purpose (learning).  Here are the phrases I am using for the 3 cycles I talk about with my students:

1. Study and Learn
   Read the explanations and information, study the examples, re-work the examples.
2. Practice
   Try every problem assigned, and check your answer.  Look at what is going well for you, and look for areas that you did not understand yet; figure those out.
3. Get Help
   After you examine areas you did not understand, get help on anything you still don’t get. 

You can probably come up with different phrases and a different set of ‘cycles’.  I like to use the word ‘cycles’ because of the implications that the process is repeated and that cycles are related.  My intent is to create an impression that learning involves deliberate work, as well as an impression that answers (right or wrong) are just a step in the process.  It is likely that my students do not see most of what I am trying to say — though indications are that listing these learning cycles helps most students do a better job.

I never collect ‘homework’, because homework is something that happens in the brain while doing the learning.  I would love to be able to directly measure all aspects of learning at the biological level; the world is probably a better place since I can not do so.  Instead, I use assessments in class (like a simple quiz on about half of the class days) along with discussions with students.

Students should not ‘do homework’.  Students should learn math, which involves discrete activities that work together to help that student do the best they can do.

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Towards Effective Remediation

Do we have a vision of effective remediation … a model which minimizes the pre-college level work for students, in total, while providing an opportunity for all adults to be included in the process of completing credentials leading to better employment and quality of life?  Based on some 39 years in developmental education, what would I suggest?

I have been thinking, as hard as I can, lately on the problems caused by policy makers looking for a simple solution.  Often, the policy makers’ interest in remediation has been prompted by reports issued by groups like Complete College America (CCA); the CCA “Bridge to Nowhere” report is excellent use of rhetorical tools, but is not a good foundation for building policies in support of effective remediation.  The simple solutions involved are usually crafted by groups that do not include people with expertise in developmental education.  Somehow, the viewpoint that we present, as experts, is difficult to understand by non-experts; perhaps some policy makers are worried that experts will only want to preserve the current system, or that we will suggest that even more courses be provided in our field.

Effective remediation involves providing the appropriate learning opportunities for each learner so that the learner reaches college courses with adequate preparation.  Traditionally, we establish remediation in discrete content areas (reading, writing, math), with an independent decision in each area based on a placement test.  Some promising practices have evolved recently with efforts to link developmental content courses, and efforts to include learning skills.   Especially within mathematics, considerable effort has been invested in creating a modularized approach; modularization is a topic of its own.  However, two observations might help us:

1. Each student is considered for 0 to 4 developmental courses in each of the 3 content areas, usually based on one placement test in each area.
2. The content is the developmental courses is often severely constrained by the historical roots of the system; especially in mathematics (though still true in reading and writing), the focus is on mechanics and procedure, with less emphasis on reasoning and analysis.

For us to develop a vision of effective remediation, we need to understand the deeper problems with the existing system such as those suggested by these observations.  In order to provide appropriate learning opportunities (whether courses, workshops, or other experiences), we need a more advanced conceptualization of remediation itself.  We need to more beyond a simple binary choice independently made in discrete content areas based on one test in each.

I suggest that we consider the following framework:

• Students roughly within a standard error of placing in to college work in a content area be provided just-in-time remediation and register for the college course.
• Students over one standard error away are placed into one of two populations based on other measures (such as high school GPA).  Some might be placed into the ‘just-in-time’ remediation group.
• The low-intensity developmental students are placed into a one-semester ‘get ready for college work’ course in one or more content areas.
• The high-intensity developmental students are provided a year of connected course work which blends reading, writing, math, and learning skills designed to address content and thinking needs.

The first two categories involve a significant portion of our developmental students, who have less intense needs; their remediation can be quicker than we often provide now.  For those who ‘almost place’ into college work, the ‘discontinuity’ research on placement tests suggests that we might be able to avoid any developmental enrollments in that content area.   The low-intensity developmental students are those who are not predicted to succeed in college but have limited needs; within mathematics, this group would include those who can review areas of algebra and quantitative reasoning in one course with minimal support outside of the class.

The high-intensity developmental group would include students with broad needs across multiple content areas.  These are the students who now struggle to complete developmental courses.  However, their educational needs are not limited to the content area skills; reasoning skills and study skills are a problem for many students in this group.  I am envisioning a two-semester package of courses (three or four each semester) with intentional overlap of cognitive skills being addressed … the math course, the reading course, and the writing course would all address issues of inference and concise language use.  This high-intensity group would also have a student-success type course to prepare them for the academic demands of the college course that lie outside of a content area.

Here is an image of this model:

A goal of this model is to make a better match between student needs and the remediation that they receive.  Our traditional system is designed for the ‘low-intensity’ type of student, and I believe that these students are well served on average.  The just-in-time remediation group is the source of our current problems from the policy makers; because these people exist in our current developmental program when the evidence raises questions about this practice, policy makers generalize the conclusion to all developmental students. 

The biggest change, and our largest opportunity, comes from the high-intensity group of students.  In our society, these are often called ‘low-skilled’ adults; they might be functionally literate (or perhaps not), and generally have few options in the economy.  Our traditional developmental program tends to be either limited in helpfulness or a problem for these students.  In a mathematics class, the high-intensity students have difficulty with both the mathematical ideas and the language factors in the work.  We tend to expect some magical cognitive growth in these students, as if working on discrete content areas will generate spontaneous global changes in the brain; I have no doubt that this does, in fact, happen to some students … I have seen it.  However, we do not create conditions for the larger cognitive changes.

Colleges might create a one-semester option for the less intense of the high-intensity group — those who can accomplish the goals with a one-semester package.  Smaller colleges might have difficulty with the logistics of this, while larger colleges would probably benefit from having two categories of high-intensity students.  Part of the rationale for the design for high-intensity need students is that preparation for them, is a more complex challenge.  Some will have had special services in the K-12 schools, and some will have significant learning disabilities.  This is the group most at risk; if community colleges are to serve all adults, then our remediation design needs to provide an appropriate pathway through to college work.  The alternative is to have a significant group of adults who will always be economically and socially vulnerable.  This high-intensity group are the ones that we need to educate policy makers about, so that they can understand the needs better — both the student needs, and our needs if we are to truly help them.

If you would like to do some reading on research related to this model, much of what I am thinking of resides in reports from the Community College Research Center (CCRC) at Columbia (http://ccrc.tc.columbia.edu/). Two specific articles: placement tests in general (http://ccrc.tc.columbia.edu/Publication.asp?uid=1033) and skipping developmental based on discontinuity analysis (http://ccrc.tc.columbia.edu/Publication.asp?uid=1035).    An article of interest by Tom Bailey and others on state policy appears at http://articles.courant.com/2012-05-18/news/hc-op-bailey-college-remedial-education-bill-too-r-20120518_1_remedial-classes-community-college-research-center-remedial-education.

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Policies that Encourage … Policies that Inhibit … Social Mobility and Equity

Recently, I heard that Ohio is the latest state to officially declare that Intermediate Algebra is the minimum prerequisite to college credit bearing math courses.  The results of such policies are seldom positive for students (and these policies do not help us in mathematics education), and they reflect archaic notions about college mathematics.

I suggest that this ‘intermediate algebra’ policy is a regressive practice which disproportionately impacts students from under-represented groups and those from social groups with lower levels of resources.  Stated another way: These policies prevent community colleges from properly serving specifically those groups for whom community colleges are the institutions of choice.  These groups, collectively seen as “low power social groups”, are critical to both the community college mission and our country’s future.

Most data that I have seen suggests two separate factors that make this policy (and its consequences) so bad:

1. Low power groups (underrepresented, or low resources) are placed into developmental math at disproportionate rates and at the lower levels of math at disproportionate rates.
2. Low power groups tend to have even lower rates of success in developmental mathematics (compared to majority/high power groups).

An “intermediate algebra is a gatekeeper” policy reinforces existing inequities in our society, as the students with the fewest options are placed in lower levels of math with more courses to complete but with a lower probability of doing so.

The emerging models (New Life, Carnegie Pathways, Dana Center Mathways) have a basic strategy of creating appropriate mathematics courses for all of our students with a deliberate reduction in the length of the math sequence; instead of 3 or 4 math courses, the new models plan on 2 as a typical sequence.  The “intermediate algebra is gatekeeper” policy conflicts with quicker access to college work, and will limit college completion initiatives; such a policy creates a 72-credit associate degree (counting the required math prerequisites), which means that students using financial aid will ‘run out’ of resources. 

Policy makers are likely to be creating these rules without information on their impact for our students and for the success of our programs.  The AMATYC Developmental Mathematics Committee (https://groups.google.com/forum/?fromgroups#!forum/amatyc-dmc) has a small team currently working on a position statement which might help inform those involved with such policies in the future.

 
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