The nature of doing and learning math, as experienced by students in any math classroom, is extremely complex and multifold, influenced by many factors inside and outside of the classroom. These factors 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, 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 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 a 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.
Accordingly, in my classroom there are three, central messages I strive to communicate:
1) Math can be learned by anyone who puts in the effort;
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; and
3) Making mistakes in math is OK – as long as you learn from them so as not to make the same mistakes repeatedly.
These messages are primarily modeled by showing them how to overcome obstacles they may encounter while working on warm-up problems, homework, and classwork, 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. They are reinforced when I 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, a gentle coax helps them to do their utmost, and if they persevere they typically find their way forward; however, if they stay flummoxed, a call to their classmates to bail them out works nearly every time. In a rare moment, we need to jumpstart volunteers with a question or suggestion. On more than a few occasions, students themselves identify an error, or two, that I make at the board. Students feel comfortable pointing these out since I encourage them to do so, and thank them.
As to making mistakes, I mention frequently in class, both one on one with students, in small group sessions, and in front of the whole class that we learn best when we make mistakes. I point out that mistakes help us understand what we may have done incorrectly, misunderstood, or omitted; it also shows we are trying. 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 can learn from us. 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 by teachers.
During whole class review for an 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 specific examples in this post 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 students.
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.