Mathematics, Paideia, & Seminar

Mathematics requires higher order conceptual and analytical skills, therefore, it is often difficult to completely forego direct instruction or essential skills practice. For this reason, I developed three strategies to assist students in acquiring rigorous mathematical content at the expense of an instructor as an explicit pedagogue, in favor of the instructor as a nurturer, or participant in the process of paideia. In order the three strategies are:

1) Teacher-Presenter-Scribe
2) Classic Mathematics
3) Practice-Create-Reflect

In order, let me address the methodology of each one. First of all, Teacher-Presenter-Scribe is a two or possibly three day lesson in the “we do” segment of a lesson plan, i.e., within what is called “structured practice.” In short, the students have already received some form of introductory direct instruction and are now ready to practice with the aid of the teacher. Thus, I assign the students to cooperative groupings, which I call PODs, and then assign each of these PODs a problem. The problems vary in difficulty and my choice in assigning problems varies with the blend of the group. Furthermore, I reserve the right to create homogenous, heterogenous, and random groups as needed for student learning. The students partake in the design of this strategy with me and lay out the ground rules. Once the ground rules are decided upon, the students begin the formal stage of cooperative practice on the assigned problem. As I circulate, the students begin to formulate their answers. After the remaining part of one period and possibly the beginning of another, the students begin to “own” the assigned problem and start to debate how they will present it to the other students. I continue to circulate and assist in designing this presentation. Once the students finish, they are graded upon how effectively they introduce the material to their peers. In short, I assign or allow students to pick groups, I circulate and help them problem solve, and then the “presenter” uses the SMART board and other technology I have access to, in order to present the problem to his or her peers. It requires so much work and encourages so much higher order understanding, that I weight this assignment as a test, or a performance assessment.

Classic Mathematics is assigned weekly. I included a reading on Infinity as an example. The idea is that students make connections in their own lives with what they have read from these intriguing articles. The hope is that teachers including myself might measure a quantifiable increase in scores for the given year in question due to an increased level of motivation, and an appreciation of the history and context of mathematical findings. One might be struggling to get a sense of what these readings I call Classic Mathematics would look like so here are two examples … A selection from Yang Hui’s binomial triangle might be a reading appropriate for an Algebra II class, while Sir Isaac Barrow’s Geometric proof of the fundamental theorem of calculus might be applicable for a fourth year course. (In intervention courses, something simpler like selections from an illustrated – but accurate and rigorous – history of mathematics might be more accessible.) My argument is that by reading classical authors and classical ideas students will learn more effectively and acquire the skill of rigorous analysis. Lastly, it should be noted that classics is here defined as mathematical treatises that have stood the test of time, readings that mathematicians and scientists generally regard as foundational for the time period in which they were published. There is no judgment being made here between East and West, white or non-white, etc., rather, the judgment is between modern textbooks, i.e., secondary sources, and classical publications, i.e., primary sources. And the judgment is not that one must be scrapped in favor of the other, but that each text and subsequently each curriculum, must augment each other conceptually. Reading classical calculus without a textbook would ill prepare the next generation of engineers, for example, but teaching students calculus algorithmically and with no appreciation for proof, discovery, and historical process would create a generation of human computers instead of educated professionals. Every effort must be made to integrate these primary sources into instruction – making excuses and depriving students of the “why” and “where it came from” in mathematics is arguably why students are so broadly upset about mathematics.

My list primary instructional strategy in mathematics is Practice-Create-Reflect. This is an independent problem solving activity. For teachers who are new or struggle with discipline, or both, this strategy might be appealing. The general idea is that after introducing a concept or lesson, and after allowing for structured practice, students should be able to synthesize or create something new with the knowledge they have just acquired. Let’s say I delivered a lesson on the distributive property and allowed students to practice problems from the source text. What would come next? Assessment? Perhaps, there is no doubt that traditional summative assessment can be used effectively in mathematics with the proper development and interpretation. But teachers should consider this activity as either a penultimate exercise prior to the assessment, or as a type of performance assessment in lieu of the summative assessment. The students, in this case dealing with the distributive property, would first locate a problem that they wanted to practice. They find the problem and they solve it. Then, they have to create their own. Here is where the understanding steps in … Students are often surprised that they can use decimals or fractions to distribute, or that they can switch signs to negatives. In the case of factoring, students will manipulate the problem and create a prime trinomial and be at odds as to what to do next. In a sense, they are discovering the “union” of algebra in the distributive property, that to equate one side to the other with decimals or fractions will require a similar manipulation on the other side of the equation. Or, that they need other methods, e.g., the quadratic function. Lastly, they reflect in written form on their findings … For example, one student told me they thought they were wrong at first because they were using decimals (or fractions) when manipulating the two-step equation they chose, but then later realized it was because they had not picked natural numbers and multiples.

jonathan Written by:

Santa Fe area Math and CS teacher; class archives, instructional techniques, and musings on educational leadership.

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