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Just Mumbling around Career Education

Episode 64: December 29, 2021

  • Episode 64: What's Beyond Learning, Ver.2 (December 29, 2021)

    Teruyuki Fujita (University of Tsukuba)


    Last month (November 2021), the Center for Curriculum Redesign (CCR) announced the "Modern Mathematics" global standards, covering pre-school education through the ninth grade, which corresponds to the third grade of junior high school in Japan. As many of you may already know, CCR had a major influence on the three pillars of qualities and abilities to be fostered in Japan's new national curriculum standards: the Courses of Study.

    For more details, I will leave it to the following website created by CCR:
    https://curriculumredesign.org/modern-Mathematics/

    Here, I would like to focus on the video introducing the global standards. I was struck by the urge to show this video to myself as a high school student when I firmly believed that there was no point in doing Mathematics. The following is a tentative Japanese translation* of the narration of the first half of the video, which I would like to share with you. (The video itself is short, less than 4 minutes, and almost all of the script is effectively laid out and presented on the screen as part of the video. Please take a look at the video at the URL above.)
    *In this English version, the original text is simply quoted below.

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    Think about your own educational experience with Math.
    Did you understand why you were learning it?
    What do you remember from it now?
    Were you able to apply it during this pandemic?

    Some of today's greatest life challenges- global warming, inequities, pandemics- can be addressed based on our collective understanding of Maths like Exponential, Probabilities, Complex Systems, and Game Theory.

    But a lot of applicable Math doesn't have a home in our educational systems, and we are all paying the price as a result. Sophisticated math has its place, but we need to learn math that is relevant and do we need to make room for the math that matters to everyone, not just college students.

    If we truly understood Exponentials are deceiving then Explosive how would our response to the pandemic be different? Wouldn't we have enacted a shut down early on, preventing soaring infection rates, a devastating death toll, and save trillions of dollars? If we learned Probability better, would our citizens be making better decisions? If we truly understood Algorithms and Complex Systems, would we have had a more successful vaccine rollout? If we learned Game Theory to help us predict human behavior, would we have communicated differently about the importance of wearing masks? Would we be better equipped to address vaccine hesitancy?

    The pandemic serves as a striking example of where modern Math could change the trajectory. There are many more ways our understanding of it will serve humanity.

    ✽

    Not only Mathematics, but the system of learning that we generally call "school subjects" become more advanced and abstract as the grade level of the subject goes up. Naturally, the level of understanding and proficiency among children becomes varied, and the gap between "those capable of learning" and "those who are not" becomes wider. Therefore, in subjects that require a so-called "build-up" approach to learning, curriculums for "capable children" and "less capable children" are designed, and in many cases, this contributes to the widening achievement gap.

    Curriculum reform in the United States in the 1970s was a perfect example of this. --If Algebra 1, the standard Math subject for the ninth graders (generally the first year students in high schools), is too difficult, students can first take Consumer Math, which is more relevant to their daily lives. Then, there will be "Algebra 1A" and "Algebra 1B", half the level of Algebra 1, so students don't have to worry about taking Algebra 1 in tenth grade. Just start with Algebra 1A for one year. Then, take Algebra 1B for the next year. You are welcome to take your time. --Such a policy is indeed far better than being forced to sit on the "sidelines" of a class you don't understand and spend your time doing nothing.

    However, the students deemed to be "capable" will advance to Calculus during their high school years, and some of them will even reach out to "Linear Algebra" offered in cooperation with universities, and in the process, they may have the opportunity to be exposed to Exponential, Probabilities, Complex Systems, and Game Theory. On the other hand, those who can't even touch a glimpse of such things are forced to approach the harmful effects of learning mathematical knowledge that may be lost after graduation to obtain a high school diploma, despairing, "Why learn Algebra? These subjects have nothing to do with my life." If this is the reality, is it truly "for the good of the child"? Is experiencing the excitement and joy that comes with the discovery that "Math has the power to change the course of our society and the trajectory of our future!" a luxury that those "not capable" students cannot afford?

    In the 1980s, the term "EQuality" was coined as part of educational reforms aimed at achieving excellence in education by simultaneously achieving quality and equality. In the 1990s, the No Child Left Behind (NCLB), a federal law with the principle of "leaving no one behind," was enacted and various measures to improve academic performance were developed under the law. As a result, the gap in academic performance between racial and ethnic groups showed a decreasing trend until the early 2010s, and overall academic performance (average score on the National Assessment of Educational Progress) improved. However, the most recent statistics available today, for 2020, confirm that the gap is widening again, and the overall improvement in academic achievement has also been braked. (Average scale scores for age 13 long-term trend Mathematics, in U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 1978, 1982, 1986, 1990, 1992, 1994, 1996, 1999, 2004, 2008, 2012, and 2020 Long-Term Trend Mathematics Assessments.）

    As a result of the national effort to improve academic performance in the U.S. since the 1980s, "capable" students are encouraged to learn as fast as possible even before they enter high school, and they are taught as if they are going to get a kick in the butt to get credit for Calculus while in high school. This also has led to a generalized policy of not giving "easy" options such as Consumer Math to "less capable" students. The "capable" students are expected to run up the stairs of elective subjects in the field of Mathematics, with Calculus at the top, without batting an eye. As a result, there are fewer opportunities to discover that Mathematics has the power to change the trajectory of our future. For the "less capable" students, it is becoming increasingly difficult to see the connection between Mathematics and their present and future lives.

    In Japan, an analysis of the results of the National Assessment of Academic Ability revealed that "students in schools with low school aid rates and municipalities with high-income levels have relatively higher academic performance." The correlation between the socioeconomic status of schools and municipalities and the academic performance of students has been identified in Japan as in most of the countries on the globe. Furthermore, various measures are being taken across the country, such as the allocation of more teachers and financial support to low-performing schools and local governments, and the active implementation of Lesson Studies and after-school learning support at each school. However, as many of the readers of this series probably already know, the situation of "polarization of academic achievement" has not been rectified due to complicated factors such as the temporary closure of schools due to the COVID-19. In addition, although Japanese junior high and high school students have extremely high academic achievement as indicated by TIMSS and PISA, surveys of student attitudes show that Japanese students are at the "the bottommost" in the world in terms of the percentage of students who find meaning in their schoolwork in relation to their future and in terms of their overall motivation and interest in learning. There has been no change in the situation where Japanese students are at the lowest end of the world.

    ✽

    Of course, no matter what measures are taken, it will probably be impossible to completely eliminate the achievement gap between "capable" and "less capable" students.

    However, we should never neglect our efforts to minimize the gap. If we fail to do so, the negative aspects of school education, which has been criticized as a reproduction mechanism of social disparity, will remain unchanged. To continue such efforts, we need to address the issues that have been internalized in all the efforts that have been made so far, including "attentive teaching," "individualized teaching," "optimal support for learning," not to mention the traditional way of teaching such as "slapping buttocks to raise scores". In short, the problems that it is difficult to provide an opportunity to rapidly improve the academic achievement of "less capable" students to the extent similar to a leap forward, and that it is also difficult to reduce the gap with "capable" students. A new approach is necessary to overcome these evils of long standing.

    The "Modern Mathematics" Global Standards proposed by CCR may be an opportunity to overcome this problem. At least, I personally have strong hopes for such possibilities.

    ✽

    To understand the details of CCR's "Modern Mathematics" Global Standards, you have no choice but to directly download the standards themselves, available free of charge upon request via the website mentioned at the beginning of this article. However, I would like to share with you some extracts of the essence of the standards from the report Mathematics for the Modern World: Standards for a Mathematically Literate Society, compiled in April 2021 prior to the release of the standards.

    1．For Whom has the Mathematics Curriculum Been Created?（p.6）
    *The subheadings here are added by the author for this article, and are not a translation ("citation" in this English version) of the report headings. The same applies hereafter.
    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?　
    --Calculus is the quintessence of high school success; it represents prestige for parents.

    2．The Aims of the "Modern Mathematics" Global Standards（pp.7-8）
    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 the responsibility of one's school to instill). So what are the minimal necessary skills for all citizens when it comes to math?

    Quantitatively literate citizens need to know more than formulas and equations. They need a predisposition to look at the world through mathematical eyes, to see the benefits (and risks) of thinking quantitatively about commonplace issues, and to approach complex problems with confidence in the value of careful reasoning. Quantitative literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently.

    The OECD defined Mathematical Literacy as: "...an individual's capacity to:
    · identify and understand the role that Mathematics plays in the world,
    · to make well-founded judgements and to use and
    · engage with Mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen."）

    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 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.

    3．The Distinction Between Consuming and Producing Mathematics（pp.15-16）
    This distinction, between consuming and producing Mathematics, is key to understanding how students will be expected to learn what appear to be more advanced topics. It is true that if one learns math to such a degree that they can produce it, they will have learned enough to be able to consume (i.e. interpret) math when they come across it. This is rooted in a long tradition of constructivist pedagogy.

    However, what we are seeing is that the large portion of students who never do master math, are also unable to interpret it; in other words, they are not math literate. It seems that when we "shoot for the moon" (all students producing math), we do not "land among the stars" (at least most students properly interpreting math). Instead, a small portion of students land on the moon, and the majority are alienated from Mathematics altogether. What if, instead of trying to make all students produce math to whatever degree they are able, we refocused instead on teaching all students to interpret math (i.e. making everyone mathematically literate). Like increasing the use of technology, this shift has the potential to extend how advanced a mathematical topic can be covered, and puts math literacy in its rightful place as the goal for all students.

    4．The Imperative of Addressing the Diverse Needs of Students（pp.14-15）
    The math standards must be refocused to be aimed at the majority of students, who will not go on to study STEM, but will need to understand how math applies to their lives. However, this creates a tension because those students who will go into a STEM career need to go into greater depth, which of course should be encouraged.

    One way we have captured this in our structure is by creating what we’ve called "Extension" standards (those that begin with [EXT] in their Name). This is to help support schools who would like to create more differentiation in their classrooms by providing students who have finished their work with more challenging material which is still on the same topic as the rest of the class. On the other end, we also provide certain Learning Objectives (those that begin with [LO] in the Name) when we believe there is a particularly important way to break down a dense standard into smaller standards or even activities.

    Furthermore, our standards reach a point after ninth grade where they reach a "trifurcation", i.e. get divided into three tracks. There is no value judgment placed on any of the tracks; rather the purpose is to give each student the math that would be most helpful to them. For that reason, we recommend classes aimed at three groups:
    · Those going to study STEM in university
    · Those going into a vocational career
    · Those going to university for Humanities or Arts

    Depending on their goals, students would be best served by classes that treat math in very different ways. For the students going on to STEM, classes would typically resemble what we have traditionally seen in math classrooms: going into great technical depth and preparing students for exams. For those going into a vocational career, there are certain other maths that are very important, and would need to be studied carefully, in the natural, applied, contexts of the vocations. Finally, for those going into Humanities and Arts, it is still important to cover math, but the goal is to prepare students to be critical consumers of math and math claims, rather than its producers.

    ✽

    I do realize that I have translated/quoted from the report at length, but this time I really wanted to share these with you.

    This report describes the Mathematics education in schools up to now as a staircase-like structure in which the content is arranged to lead to Calculus at the top, and students are asked to climb the stairs while acquiring technical/procedural details. The report also observes that it would be unacceptable for all students to be included only to identify those who are likely to specialize in STEM fields. The report also criticized the idea that forcing everyone to study such a structured curriculum to acquire mathematical knowledge and skills is too high a goal, like trying to reach the moon, and setting such unrealistic goals has alienated the majority of students from Mathematics.

    Based on these premises, the report states that for the majority of students, excluding those who intend to specialize in STEM at university, the skills they need are the literacy to utilize Mathematics to interpret the world, that is, the skills necessary to see the world through a mathematical lens and determine their path forward. In other words, the ability to use mathematical knowledge in a meaningful way to logically grasp various issues that arise in our daily lives and to make appropriate decisions. The challenges we face in our daily lives today, such as global warming, inequality, and pandemics, can be addressed based on a comprehensive understanding of Mathematics, including Exponential, Probabilities, Complex Systems, and Game Theory. Therefore, all students need to be literate enough to understand and utilize (interpret) these advanced mathematical topics in their daily lives.

    At the same time, it is a particular feature of this report that it entirely maintains the position that the utilization (interpretation) of these topics does not have to be accomplished solely by the individual. We can use calculators and software, look up things on the Internet, and ask questions to experts when we utilize those advanced Mathematics in our daily lives. It would be rather unrealistic to expect anyone to handle advanced Mathematics without any help. However, if you do not have the skills to identify the proper search phrase, know what you are looking for, and recognize whom to ask, you will not be able to take advantage of these tools and resources.

    Namely, except for those who specialize in STEM studies, the skills to independently and precisely handle complex computational problems under time pressure are not necessary at all. On the other hand, if you remain alienated from Mathematics without ever being exposed to advanced Math such as Topology, Exponential, Probabilities, Complex Systems, and Game Theory, you will not even come up with the idea of utilizing Mathematics to deal with the challenges you face in your daily life. And even if by chance you do come up with such an idea, you may not even know how to use the various tools and resources that can help you.

    The CCR's "Modern Mathematics" Global Standards will ensure that students acquire the arithmetic skills to process everyday events mathematically while cultivating mathematical literacy that all of us living today should share by the ninth grade (the third year of junior high school in Japan). After that, the courses will be developed based on "trifurcation", the concept of providing Mathematics education that suits each student's career prospects.

    ✽

    Here, let us reconfirm that the General Provisions of the new Courses of Study, the National Curriculum Guidelines, which have already been implemented in elementary and junior high schools and will be adopted in senior high schools in the coming school year (from April 2022), commonly state that:

    Each school should enhance career education in accordance with the characteristics of the subjects, with the primary focus on special activities, so that the students may anticipate the ties between what they learn and their own future and acquire the competencies that are the necessary foundation for social and occupational independence.

    It is an indispensable task for career education that each student should be able to "anticipate the ties between what they learn and their future." We need to devise ways to enable students to see the ties between what they learn and their future when dealing with the content designated for each grade level in every subject.

    However, as the video by CCR introduced at the beginning of this article points out, we also need to keep in mind that some of the biggest challenges in our daily lives today can be addressed with a highly integrated understanding of advanced Mathematics.

    If students can envision a pathway in which their current learning can be connected to their future through the learning that lies ahead, their sense of exertion and exhaustion may be significantly reduced. Of course, the computational skills of junior high school students cannot handle exponential functions, but we can help them recognize that there is indeed a source of power to comprehend society beyond what they are learning now and that it is within their reach as long as they use a calculator or other tools properly. This realization, I believe, can be expected to enhance the motivation to learn.

    This is the key to connecting career education with the "Modern Mathematics" Global Standards proposed by CCR.

    Nearly three and a half years ago now, in Episode 24 (August 6, 2017), I made the following remarks myself.

    The topics that the students are working on at schools are indeed connected to what they will learn at university, that they constitute a fundamental part of the knowledge that underlies our society, and that they contain aspects that are linked to issues that need to be further explored. It is a fact that the entirety of such systematic knowledge is not visible to the students (i.e., no attempt has been made to show it to them), which is why they say, "This kind of learning is useless and meaningless. It's boring, and it's like biting sand." Therefore, they may misunderstand the learning at school as a mere token for entrance examinations.

    There may be only a few young people who have the potential to contribute to changing the shape of the current knowledge system. Even at so-called "schools for academically talented students", such students are probably an exception. However, every child needs to realize that there is a continuous connection between his or her current studies and the greater system of knowledge, and that at the tip of that system, there are still unanswered challenges waiting for someone. How many people can continue up a steep mountain path without being informed of the full extent of the mountain ahead or the view from the top? I suspect that the number is extremely limited.

    At schools, not only in Mathematics but also in other subjects, students have been forced to "keep climbing up a steep mountain path without being informed of the full extent of the mountain ahead or the view from the top". Of course, there is a difference between the "literacy to look at the world through mathematical eyes" presented by CCR and the "whole mountain and its peak" that I had assumed in Episode 24. However, it should be my own "homework" to analyze the difference between the two, and I think it is time to end this article.

    It's already beyond redundant, isn't it?

    (Translated and uploaded on Janueary 2, 2022)

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