Towards Success in Developmental Mathematics — A Toolkit

Have you been looking for practices that encourage and support student success, especially in developmental mathematics?  I will share a toolkit of practices that seem to be effective with students in beginning algebra (often the most challenging course for us and for students).

Communication is the key.  To start with, write your syllabus (first day handout) with the reader in mind.  My own syllabus is conversational, and full of textboxes and a few graphic elements.  A traditional syllabus discourages reading; I think they also discourage engagement.  We encourage engagement by the design and tone of our syllabus.

It’s all about the learning.  In the syllabus and your attitude, emphasize that the top priority is to learn to the  best of each student’s abilities.  Homework is not just about practice … homework is a learning endeavor.  In my case, I emphasize the ‘learning cycles’ (see https://www.devmathrevival.net/?p=1229) AND reinforce these ideas by comments and actions every day in class.  repetition and action count; saying ‘it’ once does not matter.

Provide a reward for seeking help.  This is really important as a step to change behavior.  Most students, especially those in developmental mathematics, are reluctant to seek help.  At the same time, help is what makes the difference between passing and failing.  My method for this is to give an assignment (8 to 10 points, out of 1000 for the course) for students who seek help within the first two weeks of the semester.  Not only do students get more help, they feel more connected to the college experience.

Dig deep and build; don’t assume ‘they get it’.  Many of my students could combine like terms … as long as the sum was not zero; they could use exponents … as long as the problems were limited.  This is my most recent change; one of my students this semester wrote me an email (in the first week) that this was the first time she understood algebra.  Even some of the ‘high-performing’ students found some gaps.  Specifically in beginning algebra, I am using language concepts (see https://www.devmathrevival.net/?p=1253) and building processes in great detail: we started from “x + x + x = 3x” and “x·x=x²” … we went through zeros in sums [2x + (-2x) – 8 = -8]. 

Assessment as a routine activity, with instructor feedback.  If a student can go 2 or 3 weeks before getting feedback from me, I am assuming that they are ready for college work (and can make their own judgments about learning).  Everyday we have a quiz or a worksheet; I’ve even run a class where we do both (all classes are 2 hours, twice a week).  Obviously, the more assessment activity the more work we have.  My assessments fit in to the “It’s all about the learning” concept; daily assessments are 5 or 6 points, and I ‘drop’ 3 or 4 over the semester so random absences don’t hurt students.

We are a community of learners in this class.  You might call this ‘group work’, or ‘learning together’.  However, it’s not good enough to have 2 people in the class that help a student … we can all help each other.  If I can create an environment where each student is comfortable asking almost anybody in the class … and where every student is willing to help others, this is a powerful tool for motivation and ‘connecting’.  In my case, I model this behavior during class, and build opportunities for students to work with different people; seldom do I arrange the groups.  (Sometimes I will direct them to work with somebody they have not yet worked with.)

My goals for these practices focus on student engagement and learning capabilities; if the practices ‘work’, I will see better learning this semester and the student will be better prepared for other courses — even if they never use the mathematics we study.   Obviously, these practices are just a part of what I do … I hope you find some ideas within them.

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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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Flip? Reverse? Teaching?

For some reasons, math teachers seem to change their classroom behavior because of popular media … perhaps more than we do based on cognitive psychology, learning theory, and research.  It’ as if we think “a million ‘likes’ can’t be wrong” … and , as if we think that there is something new about the structures of ‘flip teaching’ or ‘reverse teaching’.

In a typical flip-teaching structure, students get their lectures outside of class time.  Class time becomes mostly individual work and small group time.  People who use this model report that it is popular with their students, and I have no doubt that this is true.  The anecdotes also tend to report that students are more engaged; I find this part a bit humorous — the structure pretty much demands more student activity.  I would bet that we could generally increase student ‘activity’ just by not talking. 

There are some questions about ‘flip teaching’ that are really questions about what we see as the goals of learning in our classroom.  Are we about basic skills?  Mastering procedures?  Reasoning?  Application?  Seeing a coherent whole?  I find it sad that we are drawn to structures that direct students towards the smallest aspects of mathematics instead of the largest.  We tend to worry a great deal about whether a student will be able to perform a known procedure.  (And, yes, I know that there are such things as high-stakes testing which tend to reinforce this bias.)

Flip teaching is a new name for an old idea.  My first teaching experience was in a program where developmental classes were run like that … students used materials before class time for instruction, and we focused on questions during class.  Originally, this program was individual classes in this format and eventually we blended all classes in to one large program.  I generally observed that students would mimic procedures, often without understanding — the same problem that we find in other classes.

Flip teaching is not a solution for learning mathematics.  If you need a new structure for motivational purposes, flip teaching can work for a while.  While you have this ‘break’, think seriously about real solutions that help the learning of mathematics.  You won’t find these solutions on YouTube or KhanAcademy … you will find them in AMATYC and NCTM and MAA and other professional groups.

I will point out that the emerging models — AMATYC New Life, Carnegie Pathways, and Dana Center Mathways — all have a focus on the learning environment in the classroom with the faculty “facing forward” (not reversed).  We use resources outside of class time, and we also emphasize directed activities during class to build understanding of mathematics.  Faculty are professional designers of learning experiences.

 
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Change in Developmental Mathematics

Most of us involved with developmental mathematics understand that change is coming; to some extent, we welcome this — though we also have concerns.  How should we conceptualize this change?  How will we even know when this change represents progress, and not just change?

Part of this conceptualization depends on having a concise vocabulary to describe what is changing and what would be progress.  I have heard one phrase that is not helpful; I’d like to explain what I see as being so bad about “change the culture of teaching” and suggest a better vocabulary.

To change a culture, there needs to be a culture.  Culture, in the formal sense understood by anthropologists, refers to shared symbolisms and understandings by a large group of people communicated through generations.  Teaching in developmental mathematics does have some shared norms, such as developing concise work habits and reasoning in students; this does not make it a ‘culture’.  (See http://www.tamu.edu/faculty/choudhury/culture.html for some definitions of ‘culture’.)  What we have is a partially shared set of norms and values out of a larger framework of understanding our settings; we lack the ‘completeness’ of natural cultures as part of a society.  The phrase “change the culture of teaching” is an oversimplification of our problems, often meant to dismiss concerns about change.

How should we talk about change in developmental mathematics?  I suggest that we focus on some central goals and beliefs, not as cultural artifacts but as deliberate and thoughtful statements about our work.

First:

Developmental mathematics deals with increasing student’s capacity for dealing with quantitative situations.

Our central goal is not preparing students for pre-calculus or calculus.  We focus on basic ideas of mathematics, understood deeply, and able to be employed as needed.  We serve all mathematics, not just algebra of polynomials.

Second:

Developmental mathematics contributes to general education.

Our students are preparing for introductory college courses; therefore, specialization is not appropriate.  The design and delivery of developmental mathematics should contribute to the goals of general education, as a priority over specialization.

Third:

Developmental mathematics allows for the possibility of inspiration and discovery of mathematicians in unlikely places.

We have the opportunity to open doors, to allow students to see beauty in mathematics, whether through specific artifacts from a discipline or by the rich connections between aspects of mathematics. 

Fourth, and most importantly:

Teaching in developmental mathematics involves deep understandings of what it means to learn mathematics combined with a broad and varied collection of tools to help students learn and the professional judgment to apply appropriate methods.

Faculty who have accepted the challenge and honor of working in developmental mathematics are advanced professionals who build individual and collective expertise by sharing and learning with others.  We are not there yet, and are not even close; we achieve as much as we do now primarily due to an amazing willingness to work very hard for our students.  Faculty can not be replaced by computers, nor by Khan videos (as good as they are); we use technology as one part of our tool set, not the entire tool set. 

Up until recently, developmental mathematics has lacked a model and mission; most people used the term to describe remedial mathematics, meaning a repeat of school mathematics.  We have not articulated our goals and beliefs, distinct from the school mathematics situation.  Saying that we are doing ‘school mathematics differently and better’ is a very weak justification for our existence.  We can do much better; we can articulate positive statements about our goals and beliefs.

We need to be able to tell when we have made progress, and not just change.  A higher passing rate is only a partial measure if our design is valid; I suggest that it is not.  We need to keep our eyes on the big picture, on the strong and unique justification for developmental mathematics as part of our country’s promise of upward mobility and work ethic.

We can, and must, do a better job of maintaining a focus on mathematics in college to prepare our students for success.

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