# Mathematics Based On Assumptions And Critical Thinking

When one of my daughters was in high school, a student in her math class stood up in disgust and exclaimed “Why do we have to learn math for 12 years when we are never going to use any of it?” You might think that as a mathematics educator I would find this statement upsetting. Instead, the student’s question got me thinking about the fact that she saw no connection between the mathematics and her future, even though her curriculum was full of story problems that at the time I would have called “real-world problems.” Every mathematician has probably encountered an “I’m not fond of math” confession. Choose any subject and you can probably find someone who dislikes it or does not care to practice it. But when I have talked with strangers about my experience teaching English and shop and history and physical education, I rarely (if ever) have encountered a negative response. Because math can be a pathway to many careers, the problem seems important to address.

Mathematics in its purest forms has incredible power and beauty. New mathematics is key to innovations in most science, technology, engineering and mathematics-related (STEM) fields. Often at the time new mathematics is invented, we don’t yet know how it will relate to other ideas and have impact in the world. Mathematical modelers use ideas from mathematics (as well as computational algorithms and techniques from statistics and operations research) to tackle big, messy, real problems. The models often optimize a limited resource such as time, money, energy, distance, safety, or health. But rather than finding a perfect answer, the solutions are “good enough” for the real-life requirements. These problems can be motivating for mathematics students, who can relate to mathematics that solves problems that are important to them.

To solve modeling problems, mathematicians make assumptions, choose a mathematical approach, get a solution, assess the solution for usefulness and accuracy, and then rework and adjust the model as needed until it provides an accurate and predictive enough understanding of the situation. Communicating the model and its implications in a clear, compelling way can be as critical to a model’s success as the solution itself. Even very young students can engage in mathematical modeling. For example, you could ask students of any age how to decide which food to choose at the cafeteria and then mathematize that decision-making process by choosing what characteristics of the food are important and then rating the foods in the cafeteria by those standards. The National Council of Teachers of Mathematics (NCTM) is providing leadership in communicating to teachers, students, and parents what mathematical modeling looks like in K–12 levels. The 2015 Focus issue of NCTM’s *Mathematics Teaching in the Middle School* will be about mathematical modeling and the 2016 *Annual Perspectives in Mathematics Education* will also focus on the topic.

How might mathematical modeling improve the experience of mathematics for students such as the one in my daughter’s class?

School mathematics is often presented as a set of steps to be followed in a particular order. Students can follow procedures without understanding (and perhaps not caring to understand) how the steps are connected and why they work. Then, they may rightly see this doing-without-understanding as a useless skill—even though correctly applying the procedures can lead to success on many standardized tests, which are developed to have predictable, standard, easy-to-grade answers. In contrast, mathematical modeling problems are big, messy, real, and open-ended. In modeling, students need to make genuine choices about what is important, decide what mathematics to apply and determine whether their solution is useful. Modeling provides opportunities for students to develop and practice mathematics-related skills, then communicate their understanding and interpretation of the problem.

When people say they dislike math, I imagine them staring at a paper with lots of X-marks all over their work. Think about how we grade school mathematics: Textbook problems provide the correct methodology, and solution manuals contain the correct answers. Problems are designed to facilitate grading, which can deemphasize creativity, elegance, efficiency, and communication. Instead, mathematical modeling problems require answers that not only use valid mathematical arguments but also make sense in context. Good models provide compelling approximations to solutions that can’t be exactly nailed down because the problem is complex, open-ended and messy. You can check out the **Moody’s Mega Math Challenge** or **Mathematical Contest in Modeling** for examples of mathematical modeling problems. Of course, some models are better than others, and justifying the solution is a critical aspect of the process.

The way students generally learn mathematics in school does not resemble the way a mathematician does it or the way it is practiced in other fields, such as business, science, computing, and engineering. Problems that ask uninteresting questions about real things, such as apples, are not much of an improvement. Many people have asked me: Is there really any new math to be discovered? The information in textbooks rarely hints at interesting unanswered questions (such as the **Millennium problems**). In addition to the interesting abstract questions out there, mathematical modeling problems are generally practical. They relate to issues someone needs to understand or decisions someone needs to make, such as when a drug is safe and effective enough to make it available to the public. Modeling makes mathematics relevant to real problems from life.

When people tell you that they are bad at mathematics, they will often recount the moment they hit the wall and gave up. They recall a class, a teacher, or a test and perpetuate the idea that if you hit a wall in mathematics you are no good at it. This idea is reinforced by the fact that in school you have to learn particular ideas in a given amount of time or you fail. But here’s the truth: Every mathematician, even one called a genius, hits a wall at some point. Sometimes we get stuck on a problem for years. When we hit a wall, we have to practice harder and longer. We acquire more tools and information. We talk with our colleagues. And like an athlete who misses a shot or loses a game, we only find success if we try new strategies and do not give up. The open-ended nature of mathematical modeling problems can allow students to employ the mathematical tools that they prefer as well as practice skills they need to reinforce. The fact that the process itself involves iteration (evaluation and reworking of the model) clearly communicates that a straight path to success is unlikely.

I’d just as soon lose the term genius, but if we have to use it, here’s how I see it. A genius is someone whose brain is tickled and delighted by certain ideas. A genius is inclined to think about these ideas far more than most other people, and this perseverance enables them sometimes to think about the ideas in new ways. They are focused, creative, hard-working rule-breakers who put their work ahead of other pursuits. Their work is recognized as exceptional and groundbreaking. My problem with the term comes from the stereotype of a genius (who comes to mind for you?). Assumptions about the characteristics of a genius can have a negative impact on those who don’t fit the mold, in terms of recognition of their work, attribution of their success to cleverness (as well as hard work), and their own wrestling with **imposter syndrome**. Geniuses are seen as people who don’t need any help or collaboration to succeed. But the truth is that we mathematicians often work with other people and have social lives, like anyone else. Those of us who work alone are still reading and building on the ideas of others. Real mathematical modeling problems are so big and messy that in order to solve a problem on a reasonably short deadline, a team approach is almost always valuable. The problems are multi-faceted and open to multiple approaches, so students can contribute their strengths to their team, while learning from the approaches taken by others. Everyone can help make the solution and communication stronger.

Math teachers are working to help real mathematical modeling become a regular part of the K–16 mathematics curriculum. We are shifting our perspective on what qualifies as an engaging “real-world problem.” As modelers, students will tackle problems that matter to them and to society. They will decide what information is relevant, make reasonable approximations, use appropriate mathematical tools wisely, and communicate clear, compelling results. As modeling teams, students will persevere through challenges and surprise us all with the ways they can use mathematics to improve the world. I look forward to a future in which I introduce myself as a mathematician and instead of saying something negative (or something about how I am different from and smarter than them), people tell me about a cool problem they once tackled.

The purpose of this research was to determine the critical thinking ability of mathematics from junior high school students based on FI and FD cognitive style. Data of this research were taken from students grade VIII at SMPN 2 Ambarawa. The research method used a descriptive qualitative approach. Data was taken with a testing method; the critical thinking was measured with WGCTA which is modified with mathematical problems, the cognitive style was measured with GEFT. The student's test result was analysed, then four students were selected, the two of them are FI cognitive style, and the others are FD cognitive style, for qualitative analysis. The result showed that the ability of mathematics critical thinking students with FI cognitive style is better than FD cognitive style on the ability of inference, assumption, deduction, and interpretation. While on the aspect of argument evaluation, mathematics critical thinking ability of students with FD cognitive style is a little better than students with FI cognitive style.

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- Student's rigorous mathematical thinking based on cognitive style

H Fitriyani and U Khasanah 2017*Journal of Physics: Conference Series***943**012055IOPscience

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