Introductory Reasoning and Problem Solving in Mathematics

Author

Lynne Ryan

Associate Professor, Mathematics

Blue Ridge Community College

Citation

Ryan, L. (2014). Introductory Reasoning and Problem Solving in Mathematics. Blue Ridge Community College.

Description

This assignment covers material on reasoning and problem solving as presented to students in an introductory level, general education mathematics course for non-STEM majors. Most of their math education up to this point has been focused on producing the correct answer, usually by performing a computation. To set a different tone and expectation, this assignment introduces the idea that students will be expected to analyze and articulate their own thought processes, rather than just providing “the answer.”

Background and Context

This assignment covers material on reasoning and problem solving as presented to students in an introductory level, General Education mathematics course for non-STEM majors. “Mathematics for the Liberal Arts” is a freshman level, general education mathematics course intended to fulfill transfer requirements for students wishing to major in fields that are not math intensive (e.g. English, Humanities, Communication). Many of the students come into the course with weak math backgrounds, and attitudes ranging from math disinterest to full blown math phobia; students view the course as something they have to get through to fulfill their Gen Ed requirement and move on. When I’m meeting with students for the first time and describing the content of the course and the overall intent, I call it “math buffet” – it’s a survey of topics in sets, logic, numeration, and geometry presented at a very introductory level. Each chapter presents a new topic, and there’s not really a progression of material to a higher level of mastery, but the course does have a unifying theme: problem solving and analysis. For each little corner of math that the course explores, students are called upon to explain the process and how it relates to the problem being solved. That’s the aspect of the course which suggested it might be a good fit with the framework of the DQP, with the caveats concerning the scope and scale of the assignment noted above.

Mathematics in particular is difficult for students to break down and articulate precisely what’s going on, as the focus of most of their math education up to this point has been to produce the correct answer, usually by performing a computation. So the first step is that, given any topic or problem, they should be able to not only solve the problem or recognize and identify examples, but be able to explain why their solution is correct, or how this example fits with the topic under discussion. “Create” at this level certainly isn’t about “producing new scholarship”…but can they at least take what they’ve learned and produce a new example or question on their own, and explain why and how it fits into the topic? Mathematics courses of this type (and nearly every institution has some variant of a math course for humanities) can be an entry point to start teaching the basic analytic and explanatory skills that they’ll need to draw on as they progress. The first assignment sets the tone for the course, and introduces the idea that they will be expected to analyze and articulate their own thought processes, rather than just providing “the answer.”

The main challenge is the scale or scope of the DQP proficiencies – they describe learning outcomes that are appropriate for an entire program of study, representing the culmination of two years (at the Associates level) of integrating intellectual and practical skills. At the course level, most of the references to “DQP assignments” lean in the direction of capstone projects or research papers, where students are given the opportunity to demonstrate that they can “(1) analyze, (2) synthesize, (3) evaluate, (4) create (practice the discipline by producing new scholarship)” across an entire field of study, usually in a course that is falling near the end of their program.

This particular assignment is the first assignment in an introductory level mathematics course for non-math majors (see more on that below). A large scale assignment isn’t appropriate for the course content or the level of student preparedness. However, this also creates an opportunity to examine the ideas driving the DQP and consider how they would operate on a small scale, with respect to a single short assignment, or even a single question within an assignment. From my observations, we expect our graduating students to have absorbed (over the past two years of coursework) how to synthesize and integrate their knowledge without really pinpointing “exactly at what point and in what courses did we first introduce and encourage demonstration of these analytical skills, and give them a framework for how to proceed?”

In terms of the DQP, I view this introductory assignment as falling under the heading of “Broad, Integrative Knowledge”… if you replace every occurrence of “in the field” or “in the discipline” with “in the unit” or “in the topic.” I acknowledge that makes things not so “broad” any more, but that’s really the point – students can’t jump straight into “broad” without spending some time practicing integration on a smaller, narrow scale. Within the assignment, I would like to frame things so that students (paraphrasing the DQP)

  • Illustrate core concepts of the topic while executing analytical, practical, or creative tasks
  • Select and apply recognized methods to solve given problems in the topic
  • Assemble evidence relevant to problems in the topic, describe its significance, and use in analysis

And, since it’s a math course, there’s also a touch of “Intellectual Skills” going on – they’re going to need to need some Quantitative Fluency. In particular they should

  • Present accurate calculations and symbolic operations and explain their use relative to the topic and problem being solved

Reflections

For more context, it’s helpful to know what the original version of this assignment looked like. It covered the same topics (inductive and deductive reasoning, and problem solving with an emphasis on Polya’s four step process). It comes at the end of the first chapter, and at this point students have worked through the lecture material on the topic (the course is delivered online; lectures and examples are recorded video), and have also completed several sets of online practice problems. The online practice problems (e.g. given an example of an argument, identify it as inductive or deductive, predict the next number in this sequence, and so on) provide the formative steps, and the written chapter assignments provide additional problems that can’t be adequately addressed by an online homework system.

The original directions were brief:

Problem One:

Construct examples of various argument types, as requested below. Use the format

Premise A

Premise B

[Use many premises as you need. You might only have one.]

Conclusion

a) An argument which uses inductive reasoning, and has an obviously false conclusion.

b) An argument which uses inductive reasoning, and has an ambiguous (maybe it’s true, maybe it isn’t, we don’t know yet) conclusion.

c) An argument which uses deductive reasoning and has a true conclusion.

d) An argument which uses deductive reasoning and has a false conclusion.

Problem Two:

Here is a listing of some textbook problems from section 1.3 (on pages 25 – 28): #16, #26, #32, #40, #52, #62, #64

Pick ANY TWO of the listed problems to work. Don’t just give me the answer, but explain your reasoning; to receive credit you must do a writeup of each problem similar to what we were looking at in class, using Polya’s problem solving strategy. Your solution must show each of Polya’s four steps explicitly. No credit will be given for just an answer. Only partial credit will be given for something that’s explained, but doesn’t use Polya’s four steps in the explanation.

The assumption was that the students had worked through the examples, and completed their online homework. They have seen several examples of inductive/deductive arguments and problem solving using Polya’s four steps, and so will model their examples after those.

The reality was that the assignments tended to be either close to 100% correct…or completely incorrect. Examples of the various types or arguments would not resemble the structures or models developed in the lectures and homework. Some students quite obviously were not working through the course material.

As a result, I’d end up writing lengthy comments explaining why their examples were incorrect. It occurred to me that this was backwards – they were spending almost no time on the material, and that perhaps, instead of me writing paragraphs about why they were incorrect, they should write paragraphs about why they were correct! The assignment directions were completely rewritten to require students to justify why their example was a good example, and the explanation had to draw on points from the lecture material. A rubric for grading the Polya problems was included with the assignment, so students would see exactly how I was going to rate their solution, making it quite clear that “no Poyla’s four steps” = “almost no credit at all.”

After incorporating the revised assignment into the Spring 2014 online MTH 151 course, I was able to compare results to the previous Fall section. Since the questions had been restructured somewhat, I grouped the assignment questions that assessed inductive/deductive, and those which assessed Polya’s four step method, and computed scores as percentages of available points.

Fall 2014  
Inductive/deductive average score 74.3% 75.8%
Polya’s method average score 66.2% 74.6%
Total average score 71.2% 75.3%

The improvement in the Polya problems was striking – having a rubric in front of them saying “this is how I’m going to grade this and if you don’t use Polya, you’ll barely get credit” seemed to finally drive the point home. While not all problems were solved correctly, in Spring 2014, only one student completely overlooked the “Polya” part, in contrast to five students in Fall 2013.

The inductive/deductive parts showed only a slight improvement…but having the students do the explaining up front allowed me to provide much better feedback as to where their errors were.


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