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.Â Most of their math education up to this point has been focused on producing the correct answer, usually by performing a computation.Â 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 â€œthe answer.â€
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. â€œMathematics for the Liberal Artsâ€ 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 Iâ€™m meeting with students for the first time and describing the content of the course and the overall intent, I call it â€œmath buffetâ€ â€“ itâ€™s a survey of topics in sets, logic, numeration, and geometry presented at a very introductory level. Each chapter presents a new topic, and thereâ€™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. Thatâ€™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 whatâ€™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. â€œCreateâ€ at this level certainly isnâ€™t about â€œproducing new scholarshipâ€â€¦but can they at least take what theyâ€™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 theyâ€™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 â€œthe answer.â€
The main challenge is the scale or scope of the DQP proficiencies â€“ 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 â€œDQP assignmentsâ€ lean in the direction of capstone projects or research papers, where students are given the opportunity to demonstrate that they can â€œ(1) analyze, (2) synthesize, (3) evaluate, (4) create (practice the discipline by producing new scholarship)â€ across an entire field of study, usually in aÂ 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 isnâ€™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 â€œexactly 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?â€
In terms of the DQP, I view this introductory assignment as falling under the heading of â€œBroad, Integrative Knowledgeâ€â€¦ if you replace every occurrence of â€œin the fieldâ€ or â€œin the disciplineâ€ with â€œin the unitâ€ or â€œin the topic.â€Â I acknowledge that makes things not so â€œbroadâ€ any more, but thatâ€™s really the point â€“ students canâ€™t jump straight into â€œbroadâ€ 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 itâ€™s a math course, thereâ€™s also a touch of â€œIntellectual Skillsâ€ going on â€“ theyâ€™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
For more context, itâ€™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 Polyaâ€™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 canâ€™t be adequately addressed by an online homework system.
The original directions were brief:
Construct examples of various argument types, as requested below. Use the format
[Use many premises as you need. You might only have one.]
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 itâ€™s true, maybe it isnâ€™t, we donâ€™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.
Here is a listing of some textbook problems from section 1.3 (on pages 25 â€“ 28): #16, #26, #32, #40, #52, #62, #64
Pick ANY TWO of the listed problems to work. Donâ€™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 Polyaâ€™s problem solving strategy. Your solution must show each of Polyaâ€™s four steps explicitly. No credit will be given for just an answer. Only partial credit will be given for something thatâ€™s explained, but doesnâ€™t use Polyaâ€™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 Polyaâ€™s four steps, and so will model their examples after those.
The reality was that the assignments tended to be either close to 100% correctâ€¦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, Iâ€™d end up writing lengthy comments explaining why their examples were incorrect. It occurred to me that this was backwards â€“ 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 â€œno Poylaâ€™s four stepsâ€ = â€œalmost no credit at all.â€
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 Polyaâ€™s four step method, and computed scores as percentages of available points.
|Inductive/deductive average score||74.3%||75.8%|
|Polyaâ€™s method average score||66.2%||74.6%|
|Total average score||71.2%||75.3%|
The improvement in the Polya problems was striking â€“ having a rubric in front of them saying â€œthis is how Iâ€™m going to grade this and if you donâ€™t use Polya, youâ€™ll barely get creditâ€ seemed to finally drive the point home. While not all problems were solved correctly, in Spring 2014, only one student completely overlooked the â€œPolyaâ€ part, in contrast to five students in Fall 2013.
The inductive/deductive parts showed only a slight improvementâ€¦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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