The following is an excerpt from Geometry: Incidence and Transformations, the first of three Geometry courses in the self-paced Elements of Mathematics: Foundations (EMF) series. EMF covers Pre-Algebra through Precalculus, plus several university-level topics, with the depth and sophisitication needed to challenge and mathematically talented children. Limited Time Offer: The first EMF course is available at 25% off regular tuition for students who enroll on or before October 31, 2017. Learn more at www.EMFmath.com.
In this course and the next our main focus is on Euclidean geometry. As you have undoubtedly noticed, in the preceding courses we have quite frequently talked about, considered, and even worked with a wide variety of geometrical concepts. But we have been relying on common knowledge and your intuitive understanding of such things; we have only rarely hinted at the formal underpinnings of this very important and pervasive area of mathematics.
The history of the study of geometry is very long and honorable, springing out of humanity’s attempts to describe mathematically the forms, shapes, and patterns seen in the real world.
Roughly speaking, our intended plan of study in this course is as follows: Each of us has a store of experiences with real objects, their forms, and their shapes. These experiences will be refined as we draw pictures of objects, observe specially constructed models of objects, and think about experiments that could be performed with these objects. As we sharpen our experiences with physical objects, we will describe corresponding geometric objects (such as points, lines, and planes), and decide what properties these geometric objects should have if they are to be mathematical replicas of the real objects. Of course, geometric objects, like all other mathematical objects, are abstractions, so the only role that can be played by our drawings and experiments with physical points, lines, and planes is that of serving as a guide to the properties that should be ascribed to their abstract geometric counterparts.
The properties we ascribe to the geometric objects will be called agreements. That is, we will agree to accept a certain property about geometric points and lines, say, because our experience strongly suggests that physical points and lines have that property. Of course, agreements in geometry, as in real life, have consequences. This then is the activity of geometry: to find out what the consequences of the basic agreements are.
In other words, we shall agree that the geometric objects we wish to study have certain basic properties. We shall then deduce that geometric objects with these properties must also have certain other properties. We will discover these new properties, not by looking at the objects (remember, geometric objects are abstractions!) but by thinking about them. We will express our thoughts in arguments of the following general form: Since we have accepted our basic agreements, then we must also accept that such and such is the case. But then we are constrained to accept that thus and so, etc. Finally we examine these new properties in order to decide whether they too are compatible with our experience of physical objects. In this way, Euclidean geometry becomes the mathematics of the shapes and forms of the world around us.
A typical pre-algebra curriculum is designed to promote computational or procedural fluency. Unfortunately, that is rarely enough to engage gifted minds. Pre-Algebra Plus, an online course within the Elements of Mathematics: Foundations program, offers gifted students a better option.
Five Reasons Gifted Students Prefer Pre-Algebra Plus
- Pre-Algebra Plus was created specifically for the way gifted minds work. Course material invites students to play with numbers, patterns, and puzzles. Unnecessary repetition is traded away for deep exploration of complex ideas. This approach appeals to both traditionally "mathy" kids and those with untapped mathematical talent.
- Instead of telling students what the rules are, Pre-Algebra Plus takes students on the journey of discovering the rules through observation, conjecture, and reasoning. As one parent put it, Pre-Algebra Plus teaches math from the inside out.
- Pre-Algebra Plus introduces basic logic and proof techniques. Logic and creativity complement each other naturally in the study of genuine mathematics. Creative thinking helps students leverage what they know and observe into new (to them) ideas about mathematical concepts. Logical thinking helps order and cement those ideas into cohesive arguments and conclusions.
- Pre-Algebra Plus is hard! The course explores several university-level topics not normally studied until college. For a mathematically talented child, being challenged is a joy. After all, what's the point of a Formula One racecar if you only get to drive it on a straightaway, even at accelerated speeds?
- Pre-Algebra Plus is individually-paced and available online 24/7, allowing each student to work on the course when convenient and go at the pace that works best for his or her learning.
Is your mathematically talented child ready for pre-algebra? Does he or she need a flexible, online course designed for the gifted thinkers? Pre-Algebra Plus is that course. Visit the Pre-Algebra Plus website to learn more about course content and enrollment.
Last week, The Atlantic published an excellent article questioning the trend toward requiring convoluted explanations of mathematical thinking in Common Core-aligned math classes. The authors rightly pointed out that verbal explanations are hardly the only way of determining whether a student understands a concept and that many of the brightest mathematical minds are verbally challenged.
The theory that if you cannot explain, you do not understand does logically lead to the conclusion that if you understand, you can explain. (IMACS Mathematical Logic students will recognize this as an example of contrapositive inference.) That theory, however, is completely false, yet it continues to drive the misguided practices of Common Core-aligned pedagogy to the detriment of another generation of students who, we worry, will be irreparably damaged in their understanding and appreciation of math.
Visual explanations are often a natural way to demonstrate mathematical understanding when designed thoughtfully and taught well, but even they can be gamed. There is another way, however, that is an effective measure of true understanding — demonstrating how well you can apply your knowledge to a novel situation. This approach does not call specifically for a verbal, visual or symbolic explanation, but it does require that teachers have the mathematical depth to recognize understanding when presented in a variety of explanatory modes, not just how a scoring rubric of model answers dictates.
Consider for a moment the art and science of cooking. Some people can only follow a recipe, and many have compared this to when students can only "plug 'n chug" math formulas and algorithms. Some people read cooking magazines or watch the Food Network and then impress with the right vocabulary at parties, the same way verbally skilled students will learn the right Common Core-friendly phrases to use in answering certain types of problems. Then there are the chefs, the ones who understand why certain ingredients and/or cooking methods work well together and what the fancy foodie talk actually means. What can they do with their genuine understanding?
If you've never watched the television show Chopped, take a moment. Competing against the clock and other chefs, each contestant must use everything in a basket of mystery ingredients to prepare an appetizer, entrée or dessert. When the ingredients are unveiled, it is not unusual for them to include such oddities as grasshoppers, gummy bears and leftover pizza. Talk about a novel situation!
It's hard to imagine a single traditional recipe that calls for such ingredients, or what erudite words you could utter about the culinary characteristics of a grasshopper. But leave it to the chefs on Chopped to shred the pizza crust, melt the gummy bears and use them with other ingredients to make breaded and glazed gourmet grasshoppers with a pepperoni pâté!
Mathematics is a lot like cooking. When you have a genuine understanding of mathematical concepts, you know what to do when faced with a problem that is unlike any you’ve seen before but that requires putting your knowledge together in a new way. Whether you can impress the Pulitzer Prize people as well is beside the point. Bon appétit!
The Institute for Mathematics and Computer Science (IMACS) recently released its first online algebra course, Algebra: Groups, Rings and Fields. This is the tenth course in the Elements of Mathematics: Foundations (EMF) program for talented secondary school students. Our latest self-paced offering has generated a fair amount of inquiries from parents seeking options for their mathematically advanced child. The answers to some of those questions can be found in the FAQ at elementsofmathematics.com. IMACS responds to others, which have been consolidated by topic, in this week’s blog post.
Q: My child aced pre-algebra and is ready for algebra. Why can’t I just enroll her in the first EMF algebra course?
When people use the term "algebra," they’re usually talking about high school algebra or what mathematicians call "elementary algebra." By contrast, EMF teaches the kind of algebra that a mathematics major at university learns called "abstract algebra." A student who has mastered pre-algebra is, no doubt, ready for high school algebra. However, this same student, no matter how talented, is simply not ready to jump directly into abstract algebra.
Why? Just as success in high school algebra is built on the foundation laid by elementary school math up through pre-algebra, success in abstract algebra requires a strong foundation in various mathematical structures and reasoning techniques that are rarely taught outside of a university setting. This important foundation is built up through the first nine courses of the EMF program. In fact, one might think of these early courses as constituting "pre-abstract algebra." As such, they are an integral part of the EMF program and essential to success in the later courses.
In case you’re wondering, students who complete the EMF algebra courses will have learned all of elementary algebra and be able to solve any high school algebra problem with ease. But they will also have learned a great deal more and be well-prepared to study the high-level mathematics that is at the heart of important disciplines such as particle physics and cryptography.
Q: What if my child already took high school algebra and geometry? Is there anything left for him to learn in EMF?
As IMACS principal founder and EMF co-author, Burt Kaufman, once wrote, "It is surely a sad state of affairs that in the traditional high school curricula, the student encounters very few, if any, mathematical ideas that postdate the seventeenth century. … It would be ludicrous if an English curriculum for the high school never contemplated confronting the student with a piece of literature written after Shakespeare."* That’s one key reason why the EMF curriculum was created—to teach modern mathematics to talented, young students who are capable of benefiting from advanced material that goes beyond the outdated math curricula used in schools.
Naturally, more experienced students will find some of the EMF material familiar, but EMF approaches these topics from a far more sophisticated standpoint. Between the new mathematical structures and techniques for reasoned argument that they will be learning, there is much for these students to gain in EMF if they are motivated to learn real mathematics as opposed to just school math.
Q: We tried other math programs for advanced kids, but they just seemed to be about going faster or preparing for competitions. That’s not working for our son who’s more of a deep thinker. I’ve heard that EMF takes a different approach. Can you explain?
First, a pair of quotes:
—Jim Simons, Mathematician and Founder of Renaissance Technologies
—Jo Boaler, Professor of Mathematics Education, Stanford University
The competition-inspired approach to math has its merits. But it’s hardly the only approach worthy of mathematically talented kids. The gifted population is filled with individuals who have exceptional talent and prefer to take their time. EMF is an ideal option for these students because the program is self-paced and encourages patience in coming to a deeper understanding of complex and beautiful ideas. At the same time, we’ve had numerous EMF students who also enjoy and excel at competition math. In fact, in situations where speed is of the essence, the non-standard mathematics to which EMF students are exposed gives them a distinct advantage over others who are seeing these ideas for the first time during a competition.
Q: You say that EMF is "mathematician math," and that it’s taught the way a math major at university would be taught. That’s nice, but what are the benefits for a talented child who has no interest in being a math major, let alone a mathematician?
Before your child writes off being a math major completely, especially if that choice is based on experiences with school math, consider the following:
—Keith Devlin, Professor of Mathematics, Stanford University
Whether your child decides to pursue a math major or not, there are several important skills that the EMF approach to mathematics teaches. As with all IMACS programs, EMF fosters the development of logical reasoning skills in young students. While some people believe that logic is cold and inhibits creativity (think: Star Trek’s Spock!), our experience teaching the EMF curriculum over the past 30+ years suggests otherwise. To the contrary, we have found that when IMACS students are equipped with the logic skills to construct their own well-reasoned arguments and critique those of others, the clarity of thought that this produces unleashes creative and innovative ideas that were previously unfocused or muddled.
Which brings us to how EMF encourages creative thinking. As Professor Devlin reminds us, mathematics is a creative discipline. The fact that what passes as "math" in schools is devoid of creativity should not be taken as evidence that true mathematics is indifferent to creative thinking. School math tends to take a "tell-then-drill" approach where the teacher states a rule and then students apply the rule to umpteen haphazard problem sets. By contrast, EMF uses carefully constructed exercises and interactive technology to guide students to their own "discovery" of mathematical results. To be successful in EMF, a student simply must think creatively in order to cross the bridge from keen mathematical observations to the "A-ha!" moments of intuitive understanding. And when these moments happen, the joy and pride of having climbed the intellectual mountain are profound.
Another skill that EMF promotes is abstract thinking. Imagine a world in which most jobs involve people interacting with tangible objects in the present. Perhaps you envisioned the Industrial Revolution, a time when mechanical inventions led to an explosion in manufacturing. Today, we find ourselves amidst a Knowledge Revolution wherein technological inventions mean that well-paying jobs require abstract thinking about intangible ideas such as code. This is obviously true in tech, but because tech touches every industry now, it’s also true for wide-ranging fields from medicine to music to law to film. It’s the question on many people’s minds: Are you going to design and program the robot, or will you be replaced by the robot? Whatever career your child pursues, he or she will almost certainly need to think abstractly. Abstract thinking is fundamental to the study of genuine mathematics, which is what EMF teaches.
Is EMF right for your child? IMACS created a 30-minute, online Aptitude Test to help prospective parents and students answer this question. Register to take the FREE test at elementsofmathematics.com. Extensive sample content from actual EMF courses is also available at elementsofmathematics.com.
* Kaufman, Burt, Jack Fitzgerald, and Jim Harpel. MEGSSS in Action. St. Louis: CEMREL, Inc., 1981.
A recent study published in the Philosophy of Mathematics Education Journal confirms that teachers’ images of mathematics and their mathematics history knowledge are interlinked. According to the study’s lead author, Danielle Goodwin of the Institute for Mathematics and Computer Science (IMACS), "By and large, the teachers with low history scores in this study were the teachers who exhibited narrow, negative views of mathematics."
Key findings from the study include:
- Respondents with low history scores
- were more likely to indicate that they believed mathematics overall was like "cooking a meal" or "a tool for use in everyday life."
- were more likely to believe that mathematics is a disjointed collection of facts, rules and skills than respondents with high history scores.
- appeared to be more likely to agree with the statement that "the process of doing mathematics is predictable" than those with higher history scores.
- Respondents with high history scores
- exhibited more favorable views of mathematics.
- were more likely to indicate that they believed mathematics overall is like "doing a dance" or "an art, a creative activity, the product of the imagination."
- disagreed more often with the statement "everything important about mathematics is already known" than did their low-scoring counterparts.
Attitudes Influence Decisions that Affect Students
Why does this matter? Because educators’ views of mathematics affect student learning experiences in a variety of ways, from daily classroom instruction to curriculum selection and development to far-reaching proposals for national math education reform.
Teachers’ images of math are typically based on their own limited experiences as young students, and so teacher education programs should incorporate mathematics history into their curriculum as a way of reshaping attitudes, the study suggests. Doing so would help future teachers develop an appreciation for and understanding of math as a subject that is alive and fundamentally creative. Fostering this viewpoint could help teachers help their students understand that mathematics is a natural place for inventive problem-solving where questioning and investigating are highly valued.
"Teachers who have rule-oriented images of mathematics can weaken student learning by representing mathematics in misleading ways," says Goodwin. Instead of conveying as healthy the struggle of intellectual discovery that naturally takes place in mathematics when new ideas are explored, "struggle" in US K-12 math classrooms has come to mean being "bad at math." This unfortunate association has left generations of Americans hating math and believing in the myth that they are not "math people."
Current teachers and pre-service teachers who want to improve their ability to teach math don’t have to wait for curriculum changes at schools of education. There are wonderful and accessible resources that provide a willing and curious mind with a deeper understanding of mathematics in the context of its rich history.
Recommended Reading and Viewing
If you’re still looking for a holiday gift for your child’s math teacher, perhaps one of the recommended books below would be appreciated. For the visually-inclined,
the videos and movies that follow provide many hours of awe-inspiring and sometimes humorous enlightenment.
- Journey through Genius: The Great Theorems of Mathematics by William Dunham
- The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth by Paul Hoffman
- e: The Story of a Number by Eli Maor
- Women in Mathematics by Lynn M. Osen
- The Joy of Pi by David Blatner
- Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
- Videos and Movies:
- Mathematics: Making the Invisible Visible, a five-lecture survey course by Stanford mathematics professor Keith Devlin
- A Mathematical Mystery Tour, BBC documentary looking at some of the greatest problems in the history of mathematics, some of which have since been solved
- Fermat’s Last Theorem, BBC documentary about mathematician Andrew Wiles’ proof of Fermat’s Last Theorem
- The Story of 1, BBC documentary about the history of numbers
- A Beautiful Mind starring Russell Crowe as mathematician John Nash (PG-13)
- The Imitation Game starring Benedict Cumberbatch as mathematician and computer scientist Alan Turing (PG-13)
UPDATE, July 28, 2014: IMACS has completed the update of our AP® Computer Science: Java Programming course to include eight fully-elaborated labs that far exceed the minimum requirements of the College Board. IMACS’ Be Prepared for the AP® Computer Science Exam online course has been updated as well. Students who are enrolled directly through eIMACS in our AP® Computer Science: Java Programming online course receive free access to the Be Prepared course.
Following a recent review of the AP® Computer Science A course and exam, the College Board has decided to replace its case study requirement with a requirement to complete a minimum of 20 hours of hands-on lab experiences. This change, which will take effect for the 2014-2015 school year, is being implemented to more effectively support student learning of core concepts in computer science. IMACS continues to follow closely all communications from the College Board, as well as discussions within the APCS community, on the forthcoming changes and will act accordingly.
From the beginning, IMACS’s philosophy has been to emphasize computational thinking and mastery of foundational ideas in computer science. This approach is reflected in how our Curriculum Development Group has meticulously designed our CS courses and, more importantly, in the success our CS graduates find in college, graduate school and at top tech companies. As such, IMACS fully expects that our AP® Computer Science: Java Programming course will continue to exceed, as it always has, all of the College Board’s requirements and remain College Board-approved.
GridWorld Case Study
Since the 2007-2008 school year, AP® Computer Science A has used the GridWorld Case Study to reinforce lessons on object-oriented programming.* GridWorld provides Java code designed to simulate the behavior of objects (Rock, Flower, Bug and Critter) in a grid. Ground rules such as Rocks cannot move, Critters eat Flowers and Bugs move forward and turn 45 degrees if blocked are part of the initial set-up. Given these starting parameters, students then write additional code that extends these various classes of objects. A student’s understanding of computer science concepts in the context of the GridWorld code is then tested on the AP exam with one free-response question and a handful of multiple choice questions.
College Curriculum Study
In 2011, the College Board undertook a College Curriculum Study in which institutions of higher education were surveyed about the AP® Computer Science A course case study.^ Of the 117 institutions that responded, 91% said they were not likely to change their credit/placement policy for AP® CS A if questions on the case study were not included in the exam. About two-thirds of respondents rated the inclusion of a case study as not important or only somewhat important.
“Although case studies have important benefits, their size and complexity have constrained the AP® CS program in adapting to new course content and pedagogy.”
— AP® CS A Exploration of a Change from GridWorld to Labs
Clearly, GridWorld is now past its prime. As the College Board noted on its website, the case study requirement in AP® Computer Science A needed updating “to stay aligned with the most recent practices in the continually changing field of computer science.”
Labs, Labs, and More Labs
This March, the College Board plans to release details of three sample AP® Computer Science A labs as examples of how the new lab experience requirement may be implemented. One expectation is that their shorter length will make the labs easier to integrate into the course curriculum throughout the school year. Teachers and curriculum developers will have the flexibility to include sample labs or other comparable labs at points they feel are most relevant and pedagogically effective. It is also expected that the sample labs will be more connected to real-world situations, perhaps increasing student interest in taking the course and studying computer science.
Most importantly, labs are expected to support student learning of fundamental ideas in computer science. Whereas the case study questions on the current exam are tied heavily to the context of the GridWorld code, the 2015 AP® Computer Science A Exam will test a student’s understanding of core concepts that are reinforced by hands-on lab experience, not knowledge specific to any particular lab. As an educational institution that has always emphasized foundational concepts in CS over code manipulation skills in the programming language du jour, IMACS is pleased to see the College Board take this important step.
Learn how you can give your child an unfair advantage in computer science. To find an IMACS teaching center near you, visit www.imacs.org. Talented middle and high school students can take university-level computer science online through our eIMACS distance-learning division.
*For readers who may be unfamiliar with object-oriented programming, it’s an approach in which the programmer creates “objects” with specified attributes and behaviors as modular, reusable code.
The new school year is now a month old. By this time, most children who attend a public K-12 school in the US will have experienced the new Common Core State Standards in Mathematics (CCSSM). On the one hand, IMACS is pleased to see that key elements of the teaching philosophy we have lived by for more than 20 years are reflected in the Common Core Standards for Mathematical Practice. For a variety of reasons, however, we maintain a healthy amount of skepticism about whether the implementation of the CCSSM will lead to meaningful, positive change in mathematics education, particularly for our most talented youth.
Common Core Was Not Designed for Gifted Kids
First, the CCSSM was not designed with exceptional kids in mind. The official Common Core Web site states plainly that:
“The Standards set grade-specific standards but do not define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations.”
The Web site further acknowledges that Common Core, like its predecessors, cannot adequately address the unique needs of individual learners:
“No set of grade-specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement levels of students in any given classroom.”
As to what educators should do about serving the diverse needs of a student body, Common Core guidance leaves them with unresolved internal conflict, offering both:
(i) “Learning opportunities will continue to vary across schools and school systems, and educators should make every effort to meet the needs of individual students based on their current understanding.”
(ii) “The Standards should be read as allowing for the widest possible range of students to participate fully from the outset, along with appropriate accommodations to ensure maximum participaton (sic) of students with special education needs.”
[Note that Common Core does not include gifted children as an example of “students with special education needs.”]
But children in the right-hand tail of the distribution do have special education needs. Whether due to a failure to understand this fact, budgetary pressure, or some other constraint, some school districts seem to be latching on to (ii) above, using the arrival of Common Core as a reason to reduce or eliminate services or accommodations for gifted students. Should this become a national trend in education policy, our country will surely suffer as the majority of gifted children who rely on public education are left without appropriate alternatives.
What About Creative Problem Solvers?
Notwithstanding the potential for improving the thinking skills of typical students, the CCSSM are simply not built to inspire or nurture the creative problem solver. The unfortunate embracing of computerized testing as a cheap means of measuring “learning” — consequently resulting in a culture of teaching to the test — has made the K-12 classroom a place to dread for many unique thinkers. The plan to continue use of computerized testing under the new standards suggests that the non-standard thinker may still be out of place in the Common Core classroom.
IMACS recently asked Gerald R. Rising, SUNY Distinguished Teaching Professor Emeritus at the University at Buffalo, how he thought the CCSSM would affect mathematics education for bright children, to which he replied, “Any imposed curriculum can have a depressing effect on special programs for gifted students.” He also shared the following anecdote about the limits of standardized testing:
“On one of the tests appeared the trivial-sounding question that went something like this: ‘A workman seeks to pass a 20-foot long board through an opening with rectangular 6-foot by 8-foot cross-section. What is the maximum width of the board that is possible?’ The answer choices were: 8 feet, 9 feet, 10 feet and 11 feet. Several of our students answered 9 feet, because the board would necessarily have some thickness that would prevent a 10-foot wide board from passing through the opening. They lost full credit for thinking that was perfectly reasonable but that did not fit the professional test constructor’s overly simplistic model.”
IMACS has been delivering courses and administering tests online to bright and creative children for over 15 years, so we know a thing or two about designing effective computerized assessments of high-level thinking skills. Let’s just say that it takes tremendous creativity, foresight, and a deep understanding of how to leverage the power of technology. If high-stakes testing is here to stay, as it appears to be, we sincerely hope that the consortia working on Common Core-aligned assessments will find ways to reward (or at least not penalize) creative problem solvers.
Inadequate Investment in Training
Common Core marks a major change in teaching philosophy for math education in the US. The intent is to move away from just teaching procedural skills by giving equal weight to conceptual understanding. Teaching math with an emphasis on thinking and understanding, however, is not something one becomes proficient in after a few hours of training, which is all that many districts have provided to their teachers.
Such a radical shift in mindset can be especially challenging for some who have taught math with a completely different focus for many, many years. This is not to say that teachers are incapable of learning to teach a new way — quite the contrary. But, as with any field undergoing fundamental change, extensive training and professional development are necessary if districts and schools want a successful implementation of Common Core. So far, the evidence suggests that they cannot or will not be making that investment.
“Common Core” Textbooks In Name Only
Many of the textbooks currently on the market that say they are aligned to the Common Core standards were developed before the creation of the CCSSM. Note the example below of pages from old and new versions of a math text currently being used in California. The pages on the left were from the edition published in 2009, the year before states began adopting the CCSSM. The pages on the right are from the current edition that proclaims “Common Core” on the cover. (Click on an image to enlarge.)
Furthermore, such textbooks often only align to the specific content skills listed in the CCSSM rather than subscribing to the overall philosophy of the CCSSM. Many that claim to be aligned to the CCSSM do not include problems or tasks that involve the higher-level thinking skills that are supposed to be measured by the new Common Core standardized tests being developed.
Awareness and Advocacy Are More Important Now Than Ever
What does all this mean if you are the parent of a talented child? Probably more work for you. Just over a month ago, nearly two-thirds of respondents to a poll on education said they had never even heard of Common Core! So, if you’re thinking that someone else will speak up first, don’t count on it. Advocacy for a gifted student has never been easy given the lack of awareness and amount of misinformation about their unique educational needs. With the potential for Common Core to bring more harm than good to the education of exceptionally bright kids, it is more important now than ever to be heard.
This year IMACS celebrates 20 years of educating talented, young students in mathematics and computer science. In all this time, we have never wavered in our philosophy that providing children with a deep and strong foundation in logical reasoning would enable them to take on virtually any intellectual pursuit with ease and confidence.
In mathematics, we continue to receive regular confirmation of our approach. Recent IMACS graduates often write to tell us of how advanced they are compared to their college math classmates, even at elite universities. Non-IMACS students who were so deftly skilled at applying formulas and algorithms in high school suddenly found themselves in college turning to our graduates for help in proving why these formulas and algorithms worked. It seems this phenomenon is steadily growing in computer science.
As strong advocates of K-12 computer science education, we are heartened by the broad realization that teaching children about this amazing and empowering field is of great importance. At the same time, IMACS urges parents, educators, and policy makers to understand the difference between coding and computational thinking, as well as the consequences of promoting one path over the other. As CS education decisions are made, we must not repeat the ruinous mistakes of math education policy lest we end up with computationally illiterate generation after generation as well.
Learning to Code Isn’t Enough
In a recent article titled “Learning to Code Isn’t Enough,” computer scientist Shuchi Grover offered the most articulate and convincing argument we’ve read on the shortcomings of the “learn to code” craze. In particular, Ms. Grover notes that the cognitive benefits gained through the process of good programming often fail to develop in online coding academies:
“Decades of research with children suggests that young learners who may be programming don’t necessarily learn problem solving well, and many, in fact, struggle with algorithmic concepts especially if they are left to tinker in programming environments, or if the learning is not scaffolded and designed using the right problems and pedagogies.”
“While the fun features afforded by these programming environments make for great engagement, they often draw away focus to the artifacts, many of which employ relatively thin use of computational thinking.”
The IMACS Approach
At IMACS, we have taken a considerably different approach to teaching computer science than the trendy, new organizations. Most importantly, we focus on universal thinking and problem-solving skills. That’s really what any programming exercise comes down to: thinking clearly about how to solve a particular problem. As Ms. Grover points out:
“If the goal is to develop robust thinking skills while kids are being creative, collaborative, participatory and all that other good stuff, the focus of the learning needs to go beyond the tool, the syntax of a programming language and even the work products to the deeper thinking skills.”
In our introductory computer science classes, IMACS deliberately uses programming languages that have trivial amounts of easily-mastered syntax. As a result, our students are able to concentrate their mental energy on learning the core concepts in computer science instead of on memorizing rules of syntax. Rather than focusing narrowly on ideas that only apply to a specific environment, IMACS classes develop computational thinking skills that can be applied to any programming situation.
Learning to Think with Logo
Children may begin taking IMACS Computer Enrichment classes as early as 3rd grade. Computer Enrichment uses Logo, an easy-to-learn language with a strong graphical component, to introduce students to programming ideas. Using a language with graphical components allows even our youngest students to understand and master advanced programming and problem-solving techniques.
IMACS Computer Enrichment places a heavy emphasis on computational thinking — thinking about logic, thinking about processes, thinking about good design. (All this takes place in a fun-filled class that incorporates interesting puzzles and problems.) A working program is not the main goal; rather, it is understanding how and why a program works or doesn’t. With a firm foundation rooted in computational thinking, IMACS students as young as 11 or 12 are well-prepared to move up to our university-level classes in computer science.
University-Level Computer Science
The IMACS curriculum continues with our Modern Computer Science track comprised of three university-level classes. The first course, UCS1, teaches the fundamental principles of computer science using Scheme. Scheme’s expressive yet simple syntax allows students to focus on learning universal concepts applicable in any programming language, even future languages not yet invented.
The second course, UCS2, begins in Scheme, but by the end students are programming in Haskell and Python. One reason that we introduced these additional languages into UCS2 was to show our students just how easy it is for them to learn new languages given their solid foundation.
The third course is our College Board-approved Advanced Placement Computer Science course in Java. This summer IMACS will be updating our APCS course with a new section on how to write Android phone apps. Although app development is not part of the AP Computer Science curriculum, the new component will allow IMACS students to gain experience in developing real applications.
The IMACS Advantage
While it sounds impressive to say that students who complete the entire IMACS computer science curriculum will graduate with significant experience in five diverse programming languages, what matters is that they leave us with something even more highly-prized: the ability to succeed in virtually any coding environment. Incidentally, whether or not IMACS graduates go on to study or pursue careers in computer-related fields, they gain an unfair advantage over their peers throughout their lifetimes thanks to their unmatched ability to dissect problems and articulate solutions. IMACS CS alumni, we look forward to receiving your emails.
The Institute for Mathematics and Computer Science (IMACS) is pleased to announce Elements of Mathematics: Foundations, a new series of online courses designed for bright secondary school students. EMF is a self-contained program that allows the talented student to complete all of middle and high school mathematics up to Calculus before leaving middle school. The curriculum is the result of more than a decade of research and development by an international team of mathematicians and educators and has been in use with gifted and talented students for over 20 years.
Acceleration vs. The EMF Approach
For mathematically talented schoolchildren, subject acceleration is an oft-advised tool for addressing their need to learn more challenging material. Through subject acceleration, a student works on math curriculum that is normally taught at a higher grade level. While acceleration does help bright students avoid repetition of material in which they are already proficient, by definition it cannot help them avoid the tedium that is the standard US mathematics curriculum.
EMF is not an accelerated version of the standard US mathematics curriculum. Instead it provides a deep and intuitive understanding of foundational concepts. This allows the suitably talented child to progress quickly through material for which others would require significant drill and practice. The curriculum then proceeds to cover concepts in a mathematically consistent way, going well beyond the typical gifted math class offered in schools or online. Topics from the standard curriculum – and much, much more – are taught in an intellectually engaging way.
Six Ways In Which EMF Is Unique
• The EMF curriculum was designed from scratch specifically for gifted and talented children to leverage their advanced capacity for learning and to engage their unique ways of thinking.
• EMF provides a deep, intuitive, and lasting understanding of mathematics as a cohesive body of knowledge that opens the door to scientific discovery and technological advancement.
• EMF focuses on the powerful and elegant ideas of mathematics, the kind that gifted and talented children find deeply satisfying and inspiring.
• The EMF curriculum exposes students to subject areas not found in the standard curriculum such as operational systems, set theory, number theory, abstract algebra, and probability and statistics.
• EMF maintains a level of mathematical rigor found typically at the university level while making advanced concepts accessible and fun for a younger audience.
• EMF gives students a true sense of what it takes to excel in college math courses, which is not the same as the skills needed to do well in standard math classes or at math competitions. EMF students do not have to “unlearn” certain habits before they can move forward with more rigorous math courses.
Is EMF Right For Your Child?
EMF courses are self-study and require a certain level of intellectual maturity. Talented students who have completed all of elementary school math but have not yet completed algebra and geometry would gain the most from EMF. However, students who already have some experience of algebra and/or geometry may still find benefit because EMF introduces concepts that are not covered in standard high school mathematics classes.
Parents who register their child at www.elementsofmathematics.com will be offered the option of having their child take a free online aptitude test to help determine their child’s level of readiness.
In our previous IMACS blog post, we began our response to Professor Andrew Hacker’s op-ed piece entitled “Is Algebra Necessary?” by taking a critical look at his reasoning in favor of eliminating the requirement for high school algebra. We argued instead that the approach to teaching algebra, and more broadly all of mathematics, should be changed significantly in the US to benefit all students, from those who are struggling to ones who are at maximum achievement under the current limited system. In this week’s post, IMACS discusses key elements that we believe should be part of any effective curriculum in mathematics.
We were pleased to see that Prof. Hacker quoted mathematics professor Peter Braunfeld of the University of Illinois as saying, “Our civilization would collapse without mathematics.” Prof. Braunfeld is not new to the mathematics education debate, having co-authored an article* on the subject with IMACS principal founder, Burt Kaufman, and IMACS curriculum contributor, Professor Vincent Haag, nearly 40 years ago. (Prof. Braunfeld was also a contributor to the IMACS curriculum.) Their article outlined five principles that have and continue to guide IMACS in our curriculum development.
Everything Old is New Again
A 40-year-old article!? How can that be relevant now? Sadly, the circumstances lamented then by the co-authors remain a plague on our US math curriculum to this day. Have you heard anything like the following excerpts lately?
On the mindless drudgery that passes for school mathematics: “A student has simply been shortchanged if after nine to 12 years’ study of mathematics, he leaves school with the notion that mathematics consists of a large collection of routine and boring algorithms that enable him to get ‘correct answers’ to certain, usually contrived, questions.”
It’s no wonder that students find math dull and tedious. The trivialized curriculum forced upon them has been stripped of all the wonder and beauty of mathematics.
On technology as the cure: “Some educators appear to believe that the basic problem lies not in the meager and often irrelevant content of school mathematics but in the inadequacy of the delivery systems. … [I]t is surely putting the cart before the horse to concentrate on improving delivery systems without at the same time making a concerted effort to improve and reorganize the mathematics that these systems are to deliver.”
In just the past year, the articles we’ve read suggesting that video tutorials, massive open online courses, and the iPad are going to “revolutionize” education are too many to count. The drive-thru window may have changed how Mickey D’s was served, but the stuff in the paper bag remained of questionable nutritional value for a long time. (Yet even the Golden Arches eventually overhauled its menu!)
Describing the approach then referred to as “behavioral goals”: “As we understand it … we must first very carefully set down our aims—just exactly what we expect the children to know at each stage in their progress. … Once this is done, materials can be produced that explicitly address themselves to the stated aims. Periodic tests and checks should be administered to determine whether the children have met the prestated behavioral goals, i.e., they can indeed ‘do’ the things that the materials purport to teach.”
Can we say teach to the test? The idea of “industrialized education” came about long before No Child Left Behind. What’s unfortunate is that such an ill-conceived notion wasn’t what was left behind.
Five Guiding Principles of Mathematics Education
So what are the five guiding principles that the co-authors proposed? The excerpts that follow summarize how they believed mathematics should and can be taught to children and how IMACS teaches today:
“1. Mathematics is an important intellectual discipline—not merely a collection of algorithms for performing calculations. One of the primary aims of a good mathematics curriculum should be to exhibit mathematics as a method of inquiry that enables us to answer interesting and important questions. We will never achieve this aim if we set our sights so low that we teach only the trivial—we must not, for example, become obsessed with teaching only algorithms.”
“2. The subject-matter of mathematics is ideas, not notation. … [T]he unfortunate fact is that more often than not mathematics is presented to children as if it were the study of certain kinds of printed marks on paper. A good example is the standard treatment of polynomials in high school algebra: students are told that polynomials are ‘expressions of a certain form’ and are then simply given a number of rules on the ‘proper’ way to ‘manipulate’ such expressions. We submit that if mathematics is presented as a subdiscipline of typography, it cannot play a significant role in the intellectual life of children.”
“3. Mathematics is an organized body of knowledge. … A mathematics curriculum has not done well by a student if it leaves him with the impression that mathematics consists of a myriad of unrelated bits and pieces. … If we are to present mathematics to the student as a coherent whole, we shall first have to become clear on what is fundamental and central to the discipline and what is peripheral. The fundamental ideas should be introduced to the student as early as possible so that they can then serve to unify the entire curriculum.”
“4. Mathematics gives us understanding and power over the ‘real’ world. … [T]he power of mathematics to give us solutions to ‘real’ problems is certainly not well exhibited by the stilted and artificial ‘applications’ we actually see in most curricula. … What we must provide, rather, is a wide variety of situations and problems with genuine life and spirit in them—problems that engage the student’s attention and arouse his curiosity. Surely a problem is ‘practical’ for a child if, and only if, it is one to which he would really like to know the answer.”
“5. Mathematics is a form of artistic expression. … A mathematics program that takes the poetry out of mathematics is a bad program for the simple reason that mathematics, like poetry, music, painting or dancing, deals in aesthetic values. … Nothing can replace the importance of a child’s pleasure in seeing an elegant piece of mathematics or, even better, in creating a piece of mathematics for himself. Learning mathematics and doing mathematics may at times be hard work, but it must never become mere drudgery.”
Exasperated students continue to ask when they are ever going to need high school math in the real world, and who could blame them? Yet you never hear such broad and fervent protest about high school science even though relatively few kids will put that knowledge to work. Why less complaining? Because science curricula, for the most part, still incorporate grand ideas that elicit awe in young minds. Why are parents appalled by cuts in the arts at school? Not because most think their kids will pursue careers in creative fields, but because they understand that the aesthetic experience lifts the human spirit.
Mathematics is brimming with this kind of elemental beauty too. Yet decade after decade, schools use curricula that deprive math students of the good stuff. It’s like feeding the cardboard box instead of the cereal to a kid and saying “See, it says ‘cereal’ right there on the label. How can you say it’s tasteless?” Children have been telling us for too long that we need to change the menu. It’s time we listen.
Editor’s Note: IMACS will soon be rolling out a series of interactive online math courses designed with the five guiding principles discussed above. These courses will allow talented students to complete all of middle and high school mathematics with the exception of calculus before leaving middle school. Check back at www.eimacs.com or like us on Facebook for exciting details to come!
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