The Myriad Representations of Math

The Myriad Representations of Math

Part 1: Analysis

The nature of doing and learning math, as perceived by students in any math classroom, is extremely complex and multifold, influenced by many factors to include: a teacher’s intrinsic beliefs established over a lifetime of experiences with math and communicated in actions, words, mannerisms, assessments, tasks, curricula, and pedagogy; each student’s individual perspective, which may vary within a class period, and from day-to-day, as shaped by their parents’, siblings’, friends’, and classmates’ own opinions, and the natural ups and downs in an adolescent’s life; the school’s emphasis, or de-emphasis, of math with respect to other subjects, its policies for assigning students to specific courses, as well as its math API score; the school district’s initiatives in math curricula, pedagogy, and professional development as well as its overall API status, such as Program Improvement (PI); the local community’s views towards math, and its significance; state and national efforts regarding math, especially as reflected in learning standards required for instruction and used to test for proficiency, as required by law; and finally, society’s depiction of math, and its role in everyday life – is it visible, important, and respected, or is it hidden, insignificant, and ridiculed?

While the complexities of sources, frequencies, and content in the multitude of messages sent to, and received by students, as corrupted by the noise present in any communications medium, might complicate the efforts of educational researchers to study, classify and model in any fashion that would yield statistically valid results, the role of the teacher, nonetheless, is fundamental as a first order indicator of what messages student’s take away from the classroom about mathematics.

Specifically, in the classroom where my cooperating teacher and I teach 35 Algebra 1 students and 40 AP Statistics students in two 58 minute periods, the two, central messages we strive to communicate are:

1) Math can be learned by anyone who puts in the effort and,

2) There are multiple approaches to solving math problems – no one way is more correct than the other – whatever works best for an individual student is best for that student, however, it might not work as effectively for another student.

Our students receive consistent messages, from both my CT and me, that math can be learned if students put in the effort.  This message is primarily modeled by showing them how to overcome obstacles they may encounter while working on warm-up problems, homework, and class-work, mostly by select questions meant to guide them to a way forward, so they develop the confidence to challenge similar problems and not give up, whether for the moment or the duration.  This message is reinforced when we occasionally call student groups to the front of the class to work problems.  When students reach a point where they are confused, do not know what to do next, or are simply overwhelmed with shyness, we gently coax them to do their utmost, and if they persevere they typically find their way forward; however, even if they stay flummoxed, we call in their classmates to bail them out, which works nearly every time.  In a rare moment, we need to jumpstart volunteers with a question or suggestion.  On a few occasions, students themselves identify an error, or two, that either my CT or I made at the board.  Students in both periods feel comfortable pointing those out to us since we encourage them to do so, and thank them such as in the following situation.

Antonio points out a mistake when solving a problem with not carrying the decimal 0.6 down for 2.6, the CT had written 2.0.  The CT compliments him “Thank you, Antonio for watching for my mistakes.  Sometimes I forget to copy something correctly even if I know how to do the problem.”

My CT and I also spend considerable time walking among the groups of students in the classroom, checking for understanding with a question or two, answering questions as they arise, perhaps with questions of our own, shepherding students back on task, and suggesting different ways to approach problems if students have difficulty with a specific method.  Example questions and statements include:

–        “Call me or Mr. Math Teacher over if you need help.

–        “Are you going to add or subtract?”

–        “So, you should be talking about distributions, their outliers, center, spread, etc.”

–        “Can you tell me what you got for #4?”

While spending most of my time among students, I do not hesitate to kneel on the floor next to a student, showing my commitment to helping them learn math as long as they give it effort, and commending them on their tenacity and progress, however much it may be at that moment.  During these one on one moments, I often show students multiple approaches to problems to see if they have a preference for one or the other, or if it helps them better understand how to approach future problems.  I believe many appreciate the approach I take, where I show how problems can be approached in different ways, rather than giving up if they cannot recall one specific procedure.

Additionally, in the structuring, implementation and assessment of lessons, class-work and homework, my CT and I signal to students that we value the topics being taught, the students, and the subject, as a whole.  We do not create problems that obfuscate the essential points in the lesson at hand, which mostly frustrates students and harms the learning process.  Instead, we create problems that enable them to build a foundation and through scaffolding, improve the depth and breadth of their understanding, at least that is our intent and goal.

Although it may sound as if we have a superbly functioning classroom, we are far from perfect.  We could have supportive math-focused posters, messages, and other media surrounding students in the class, providing: examples of the myriad of math’s applications in everyday life; reminders of key formula, shapes, acronyms, and problem solving approaches; and encouraging messages such as Attitude, Persistence, Effort with associated graphics or pictures.

There is room for improvement in the ratio of time a teacher is speaking to the time a student is speaking about math.  It is too natural for a teacher to feel as if they must impart wisdom to the class, especially as these pearls flow into our mind in a stream of consciousness while standing at the front of the class.  Letting go of that need, and turning the floor over to students more frequently provides them the opportunity to reinforce what they may not have completely understood initially, to expand on specific elements, to request a repeat of certain portions of instruction, to pose unique questions to the teacher and/or class, and to take ownership of their own destiny.

Even though my CT and I are highly aligned in many ways, we do differ in certain areas owing to differences in our lives, mathematical background, educational experience, as well as classroom management and pedagogical philosophies.  As an example, I have a very low threshold for tolerating students who speak among themselves when I am speaking to the class, unless I sense that it is math talk.  While my CT states she does not tolerate talking when she is talking, my observation is that she has a much higher threshold for detecting when such talking might be occurring.  I have the utmost respect for, and appreciation of, my CT.  It is apparently clear that she is a great, caring, encouraging, and knowledgeable teacher.  Nonetheless, I struggle with how to interpret her subconscious tolerance for higher levels of student talking than myself, and more importantly, to what extent either of ours is best for students’ learning about math.

On the pedagogical front, we differ in how we show Algebra 1 students how to solve certain problems.  From what I have observed to date, she appears to prefer approaches that emphasize specific procedures, such as cross multiplying, even if it’s not necessary, and minimizes addressing the placement and factored forms of numbers in a problem.  The following excerpt from my blog captures my current dilemma regarding the different approaches to teaching math for the most effective learning.

Is it best to show students procedures that are easy to remember in hopes they recall the procedure, apply it correctly, and hopefully, more often than not, respond correctly on the various assessments placed before them?  Even if they may not know how to apply what they’ve learned in a problem that deviates slightly in context, content or construction from what they may have grown accustomed to seeing?  Is it OK if they simply show rudimentary facility or is comprehensive transference required?  What is best for the future of those students and our nation?

It is still not clear to me what is truly best for students; I do believe my CT’s experience has shaped her approach and sensing her commitment to student learning, it must be the most effective one for her.

Part 2: Reflection

As mentioned above, and reprinted below, the following two messages are central and key to the nature of mathematics and mathematical activity.

1) Math can be learned by anyone who puts in the effort and,

2) There are multiple approaches to solving math problems – no one way is more correct than the other – whatever works best for an individual student is best for that student, however, it might not work as effectively for another student.

Several examples of each of those two messages are provided in Part 1: Analysis.

Additionally, I believe the following is a key message for students of math.

Making mistakes in math is OK – as long as you learn from them and do not repeat the same mistake often

I communicate this message frequently in class, both one on one with students, in small group sessions, and in front of the whole class.   I point out that mistakes help us understand what we may have done incorrectly, misunderstood, or omitted.  I also mention that the mistakes we make are likely to be made by others so we should strive to share them with others so they do not make the same ones.  Also, when mistakes are made at the board, they provide excellent opportunities to illustrate common errors, such as the following example, and even those made occasionally by instructors, as mentioned in Part 1.

During whole class review for this week’s end of marking period final in Algebra 1, students were called up in groups, of four, to the board to show how they solved a problem assigned to them.  One student, walking to the board, whispered to me that he did not know how to do the problem.  I whispered back not to worry, that I would help him do the problem.  As we worked the order of operations problem, a typical error was made where a negative sign was improperly addressed, resulting in an incorrect result.  I did not notice the error when the student wrote the problem so as we worked the remainder of the problem out, he and I were stumped when we arrived at a solution that was not one of the multiple choice options.  I quickly worked back through the problem, using it as an opportunity to show students how to retrace your steps through a problem to detect where an error was introduced, corrected the mistake and we revised the solution, which was now correct.

I also believe it is important to dispel the following messages that might have been communicated to students previously, and may have been internalized.  I have heard these throughout my life, from many people, in multiple places.

  • Math is:  (with common examples or student thoughts)
    • Boring (lecture, skill & drill worksheets, homework, quizzes/tests)
    • Difficult (hard to follow, understand, remember, apply, transfer, etc)
    • Complex (too many things to remember and apply without mistakes)
    • Confusing (differences for operations, order, rules, theorems, signs, etc)
    • Unnecessary (don’t need it for most jobs, especially if “blue collar”)
    • For “smart” people (and I am not smart)

While I do not have examples for how to rectify each of these, I believe many are addressed with curricula specifically designed with these common perceptions in mind, so as not to let them take hold with the curricula and its associated pedagogy.

I firmly believe that the longstanding approach to mathematics instruction, as a one-size shoe fits all approach, disenfranchises many to the subject; the math field is shooting itself in the foot, so to speak.  So being keenly aware of the various learning preferences students have, as well as the multiple pedagogical approaches available to be matched to students’ preferences, along with passionate instructors, holds hope that the preponderance of these thoughts will diminish.

Lastly, the increasing emphasis on standardized tests and how they determine the success or failure of students, teachers, schools, districts, etcetera is an ominous force crowding out many attempts to introduce experiential, group work oriented, or other novel approaches which may maintain students’ attention, interest, and focus on math, as well as expand on their ability to handle more complex situations life will bring their way, unlike tidy, boring, and lifeless equations, expressions and such.  This hangs over my head as I consider how to send messages about math to my future students.  I hope that I am able to obtain greater clarity soon.


About Dave aka Mr. Math Teacher

Secondary math teacher teaching math intervention, algebra 1, honors precalculus, and AP Calculus AB. I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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