By Elena Galaktionova
Elena Galaktionova sent us this article shortly before she passed away earlier this year.
Foreword by Cornelius Pillen
Elena Galaktionova received her first introduction to mathematics from her favorite middle school teacher in Minsk, Belarus, her hometown. After she had finished her education at the Belarusian State University she went on to receive a Ph.D. from the University of Massachusetts in Amherst. Her area of research was representation theory. She taught mathematics for many years at the University of South Alabama, after some earlier stints at the Alabama School of Math and Science and the School of Computing at USA. In Mobile, Alabama, she was one of the organizers and teachers of the Mobile Mathematics Circle. The Circle has been going strong for 20 years. Later she recruited local teachers and a middle school principal to participate as a team at an AIM workshop on Math Teacher Circles. Upon return to Mobile she founded the Mobile Math Teachers’ Circle. Twice she gave presentations at the Circle on the Road conferences. Her work with local middle schools and her interests in home schooling were motivated by her love for mathematics. She cared deeply about math education. Sadly, Elena passed away earlier this year after a long battle with cancer.
In all my classes I try to teach reasoning, writing and problem-solving skills. I noticed that if a class is heavy on computations and dense in content, such as Calculus, the result of this effort is barely noticeable if at all. I recall a memorable moment in a multi-variable calculus class. The topic was optimization. My students knew just fine how to use the Lagrange multiplier method given a function and a constraint, thank you much. But it turned out they were helpless in the face of even the simplest application problems. Some of these students were studying Calculus with me for almost three semesters and their grades were good and I tried so hard to teach them what matters in mathematics the most. I remember a chilling realization at the moment, that we — the students and I — wasted three semesters.
A very different experience comes from another course. At our university it is called “Foundations of mathematics”. Unlike other math classes it does not have a lot of content. A bit of logic, set theory, relations, maybe some number theory. It is the first class where students are learning to write proofs. This is a writing-intense class. There are essentially no calculations. I collect the homework every class period and grade the same way one would grade an essay. My first requirement is writing in grammatically correct meaningful English sentences. This is not an easy task for most. A lot of students by the time they start this class learned to perceive math as number and symbol manipulations. At the beginning of the course I often see in students’ work words that are strung together in rather random fashion. We go together over some of the responses asking questions like : “Is this an English sentence? What is the meaning? Are all the terms defined? How could it be misinterpreted?”
By the end of the semester I observe a turn-around: there is a palpable effort from even the weakest students to put their ideas into words. The change is most noticeable in weak students. The struggle for finding the right words and writing in grammatically correct sentences may be still there. While they did not suddenly became great at math, their mental activity and learning efforts are much more productive, since they are consciously directed towards comprehension and expressing their ideas verbally with a degree of precision.
I wondered if my students noticed this change themselves; that was until I was approached at the end of the semester by two of my “Foundations” students who emphatically told me how this course entirely changed the way they view and approach math. This is reflected in their grades in other math classes as well. For example, one of my Calculus II students was taking “Foundations” concurrently. Her grade in Calculus II changed from a D at the start of the semester to a B towards the middle. Most notably, she enthusiastically confirmed and told other students how much taking “Foundations” helps with Calculus II, despite having no content in common.
What is most interesting to me is the quantum character of this change and that it was especially noticeable in weaker students.
Young children come to school as a blank slate. Yet they have the innate ability for reasoning, they have curiosity, they are eager to play and explore. Over the years their teachers influence their perception of what math is about. Two of the possibilities are:
- math is a manipulation of numbers and symbols according to a predetermined set of rules;
- math is communicated through meaningful statements.
The students who do not do well in mathematics typically view math as a manipulation of symbols. The “making sense” switch changes this so the students begin to read and communicate mathematics as meaningful, logically connected statements.
To summarize, here is what I observed:
- Both exclusively formal processing of math tasks and making sense of math tasks are learned, eventually habitual, behaviors. Either one becomes a mental process which is practiced and reinforced in every math class.
- Effective learning of mathematics does not happen until mathematical communication is perceived as meaningful statements.
- Students who view math as a formal manipulation of numbers or symbols will habitually direct their effort and mental energy toward this in a math class, unless they are given problems which naturally invite reasoning and stay away from using formulae and rules. In a class with a computational component, such as pre-calculus or calculus, even if a teacher tries to teach reasoning and making sense, it has relatively little consequence: under stress, such as homework due the next day or a test, such students revert to their habits. Some of them spend a fair amount of time studying and reinforcing these habits, often getting frustrated because of the little return for their efforts.
- A dedicated computation-free and writing-intensive class which stays away from problems that may suggest formal manipulation can turn on the “making sense” switch. Students start to perceive mathematics as meaningful statements. They look for logical connections between the statements. Their verbal skills are productively challenged.
The important qualities for such class, assuming the main purpose is to turn on the “making sense” switch:
- The class should be writing intensive.
- The tasks are such that students can rely on their existing reasoning skills, common sense, intuition. They should not be too abstract. For example, it is easier to find appropriate problems in logic, set theory, elementary geometry or combinatorics than in abstract algebra. This allows students to scrutinize math statements using their own “sensometer” and keep working with them until they can make sense of them.
- The course should be light on content and big on thought, allowing sufficient time to think and write about problems.
- The class should not include tasks which could tempt students into formal manipulation.
- Feedback on writing is continuously provided by the teacher: students’ attention is brought to details of their writing, the meaning of what is written, and how the writing could be improved.
- The time required for the “switch” to turn on in such class is less than a semester. This is my experience with undergraduates.
- This works even if this writing intensive class is taken in parallel with other, computationally intensive math classes.
- Once the switch is turned on it stays on in other classes, even in those with computational components, as long as teachers pay attention to making sense and reasoning.
- A practical aside on grading: a class where non-routine and sometimes difficult problems are part of homework presents certain challenges for grading. I told my students that if they could not solve the problem, they should write down their attempts, for example, how they used problem-solving strategies discussed in class, such as looking at related simpler problems or generating examples and trying to find a pattern and showing why it did not work. Adequate effort and quality writing would earn almost full credit. Of course, it is important to also include easier problems which are within reach for nearly everyone. If I did not have sufficient time to grade the full homework, I selectively graded 2 or 3 problems.
Unfortunately, the “Foundations” class is a sophomore level for math and math education majors. In fact, no prerequisites are needed for it. So we started to encourage students to take this class as early as possible, when noticing that it helps them in other math classes. There is no reason why a class with similar characteristics and goals is not taught to seventh graders. It would improve their learning of mathematics for years to come. As an example, in Russia, the class that perfectly fits the bill is Geometry class. Systematic study of Geometry starts in 7th grade and continues through 11th grade. The Geometry class meets 2 or 3 times per week. All statements and theorems are proven based on what is already known. Thus, Geometry is presented as a unified theory and not a random collection of facts. The students are expected to state definitions and prove theorems and they solve problems involving proofs and geometric constructions. Of course, there are Russian students who struggle with writing proofs and deriving formulae. But they are used to the concept of intrinsic reasoning and they know that they are expected to articulate it. In the U.S., ask a class of either seventh graders or freshman Calculus students why a particular fact or formula is true and the answer invariably will be “Because our teacher told us so ” or “Because it says so in the book”.