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Mathematics In The Modern World PDF
The modern world for which schools must prepare students bears little resemblance to the world for which our education systems were designed. The first challenge is the lack of a cohesive vision for systemic improvement ; without a cohesive plan, each effort, even if successful, gets lost in the noise 1 of all of the other efforts.
The Center for Curriculum Redesign endeavors to answer the question, “What should students learn for the 21st Century?” by defining the Knowledge, Skills, Character, and Meta-Learning abilities that will prepare students for their personal, civic, and professional lives.
And yet, even with a set of goals in mind, it is extremely challenging to overcome the inertia in the field of education due to the confluence of many forces perpetuating the present system.3 Society has undergone massive changes in recent times, which have transformed the way math is used and thus must be taught.
Much of the procedural aspects of mathematics are no longer performed by humans. The phrase “you won’t have a calculator with you every day” has become patently false, as we now all carry with us not only calculators but supercomputers and libraries and teachers in our pockets and in the cloud.
Yet curriculum remains firmly rooted in its tradition and history, extending all the way back to ancient times. Further, math is a discipline that builds on itself in a fundamental way; it is not possible to simply skip learning multiplication, for example, and continue to Algebra.
If students do not learn what they are meant to at a given time, they fall behind, which causes them to continue to fall behind. On the other hand, many occupations that did not used to require math are beginning to incorporate quantitative methods into their expectations.
Figure 1 below shows the relative importance of math 4 based on O*NET and Bureau of Labor Statistics data about importance and number of people employed in jobs that require different traditional knowledge areas.
Most important are speaking and reading comprehension, understandably. But following not far behind are Math and writing, which are almost equal. After that, there is a sharp drop off to the rest of the subjects.
notice is that the majority of occupations requiring math, require a medium level of math, roughly corresponding to “Analyze data to determine areas with the highest output”.
Mathematics is a wide and varied field, and in preparing students for the workforce, it is important to not get swept up in traditional notions of mathematics education, but rather, design for the world of the future.
The current moment adds one more layer to the challenge at hand; Covid-19 has forced schools to figure out how to continue education with many students not physically present at school.
In many cases, this necessarily means less instructional time. So, how can we decide what to prioritize given that we simply cannot cover everything? Even in normal times, there has been a vast chasm between the intended curriculum and the implemented curriculum, due to the insufficient time available for all of the assigned (and ever-expanding!) content.
The situation has only been exacerbated by COVID-19. Some jurisdictions have taken steps to offer guidance on which topics to trim and which to focus on. In this paper, CCR has taken this one step further and fundamentally re-imagined math standards as a whole.
Although this document focuses exclusively on standards, CCR recognizes that a change in standards alone is not enough to result in meaningful change. While we offer some starting points, it will be crucial to build upon this work with aligned curriculum, courseware, and assessment.
Goals for All For whom is the mathematics curriculum designed? If one were to look solely at how mathematics is treated in schools, they might conclude that math is really only needed for those who continue on to a STEM (Science, Technology, Engineering, Mathematics) career, and the only reason it is taught to everyone is in order to identify those that may be able to utilize it professionally.
The curriculum, in terms of its structure, prepares students for Calculus, which students are pressured to take. Why?
Worries about inequities are focused on the students who “fail to” make it in a STEM track, and those students, rather than receiving a math education that would prepare them for how they might actually use math, receive remedial math instruction, still attempting to push them as far as possible along the Calculus track (and further solidifying their distaste for anything described as “math”).
Besides the inherent judgment of non-STEM careers to be inferior to STEM careers, there are a number of other problems with this conceptualization.
CCR believes that there is value to mathematics education for all, and in fact, it is crucial that *all* students internalize the important takeaways of math, without necessarily mastering all the procedures.
By identifying the ultimate goals of each item in a set of math standards, we can begin to trace which content is only necessary for those going into STEM and begin to make room for the mathematics required for all citizens of the 21st century.
So what should be the takeaways of math education by those who do not go on to a STEM career (the vast majority of students)? The word “literacy” has been adopted from discussions about the reading and writing skills that all citizens need to participate in society, to refer to a set of minimal skills within any given particular modality that are absolutely necessary for all citizens to possess (and thus 6 the responsibility of one’s school to instill). So what are the minimal necessary skills for all citizens when it comes to math?
These are both great descriptions of what we hope students will take away from Mathematics education, yet math curricula continue to get bogged down in technical/procedural details of Mathematics. This is not unique to mathematics — all disciplines fall in this trap; for instance, in the traditional sense of the word literacy, some language curricula have fallen into a similar trap of focusing on drilling grammar rules.
We believe that rather than drilling grammar rules, a more useful approach would be to instill in students an understanding of linguistics so they can recognize and understand grammars in many different languages and settings if they wish. Similarly, we believe the goal of mathematics education is to teach students the skills they need to navigate the world through a mathematical lens.
Many mathematicians and some math educators will stress the importance of learning the technical details in order to gain a deeper understanding with the same certainty that parents stress eating vegetables before having dessert. But is procedural learning truly a necessary prerequisite to attain deeper mathematical learning?
A consensus is growing that the answer is no. Programs and curricula that aim to teach advanced topics of math, such as calculus and topology, to small children are cropping up, showing that the long-held wisdom about the inaccessibility of higher-level math may be based on flawed reasoning.
Namely, it is true that for those going into a STEM track, it is best to 9 learn the prerequisites to their full depth before moving forward. But for the majority of students, who aren’t on the STEM track, it is absolutely possible and necessary that they are exposed to a broader swath of the field of math,
so that they gain an understanding of its many uses and value, without necessarily being able to reproduce it all themselves with no support and under time pressure. In life, after all, one can always use a calculator or software, look things up online, or ask an expert; the trick is knowing what to type, what to search, and whom to ask.
The traditional, pragmatic approach to mathematics education is not only counterproductive to students’ conception of the field of mathematics, but it also does not align with the true use of mathematics in industry.
In professions that use mathematics as part of their work but are not centered around it, such as nursing, studies have shown that professionals use “more efficient, informal strategies that are typical of the situation” and many times those who do not score well on traditional tests of school mathematics are extremely accurate in their specific implementations.
10 Even for jobs truly focused around mathematics, the on-the-job math is not a natural extension of school math; “In school, the professor formulates the problem and you solve it—you hope. In industry, you formulate the problem and the software solves it—you hope”.
11 Conrad Wolfram describes four categories in Math education: Recognizing where mathematics is applicable, Translating practical problems into mathematical problems, Solving mathematical problems, and Interpreting and evaluating the outcomes.
Of these, “Solving” is the only one that gets “systematically addressed in mathematics education, and that this is exactly the step that is increasingly carried out by computers!”12 Such a heavy focus on math in industry also neglects the importance of mathematical literacy for everyday life and for a democratic society.
In 2020, the world witnessed firsthand how the understanding or lack thereof by policy makers of exponentials (“deceiving then explosive”) and complex systems (“small changes, big impact”) led to very different approaches to handling the Covid outbreak and the vaccine distribution.
Because of how different “school math” can be from real world math, there is a long-standing challenge to teach math such that it transfers into the real world. We believe a large part of the problem is that “school math” does not pay enough attention to the translation steps between the real world and math, and back.
Part of the issue is that this gets into the very perception of what is a math problem, which teachers see effortlessly due to their embedded expertise (aka “curse of knowledge” ), and thus rarely give students the chance to practice this crucial 13 step.
By focusing on the formal difficulty of the mathematical content, experts miss the important goal of preparing adults who can apply basic math concepts to everyday situations.
While the math itself might be basic, the applications can be quite varied and far from straightforward! “Opportunity to Learn” (OTL) refers to the coverage of topics that a given student is exposed to; according to this construct, the greater the number of topics, the greater is their “opportunity”.
| Language | English |
| No. of Pages | 55 |
| PDF Size | 3.6 MB |
| Category | Mathematics |
| Source/Credits | curriculumredesign.org |
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