The Institute For Mathematics and Computer Science (IMACS) has received a grant of $5,000 from the Multiplied Foundation Fund of the Community Foundation of Broward to provide full scholarships worth over $10,000 for 20 students to enroll in IMACS' 2016 Hi-Tech Summer Camp.
The Multiplied Foundation was founded by 14-year old IMACS student, Peyton Robertson, with the mission of supporting and expanding STEM (science, technology, engineering and math) education. Peyton has a deep appreciation for how early exposure to enriching STEM activities can motivate a young person. At 11 years old, he won the Discovery Education 3M Young Scientist Challenge. By 14, he was awarded three patents.
Peyton credits his academic accomplishments, in part, to the strong foundation in mathematical thinking that he developed while attending IMACS: "The early skills I developed at IMACS helped me to have a deeper understanding of the math and computer science classes that I have taken in school. My hope is that other students will benefit from the foundation that the IMACS program can provide."
"IMACS is honored to be working with the Multiplied Foundation to provide scholarships to 20 very deserving students," said IMACS President Terry Kaufman. "Bright and curious minds come from all backgrounds, and we all need to do more to identify and nurture these kids. We thank the Community Foundation of Broward for making this opportunity possible."
Camp scholarships were awarded to rising 4th through 9th graders who have a desire to build their math and logical reasoning abilities but who would otherwise not have the resources to attend. Recipients were selected from applicants at Piney Grove Boys Academy (PGBA) in Lauderdale Lakes and "I Have A Dream" Foundation in Miami.
James Wilson III, a rising 5th grader at PGBA, is excited to attend the camp. "Every day we get to do a cool project and learn something new. I can't wait for tomorrow," exclaimed James. "It's great to see my son, who is very athletic and into sports, also be so intrigued and interested in technology thanks to his time with the program," observed James's mother, Melissa Mata. "The exposure he's getting at IMACS is definitely priceless."
Frances Bolden, Educational Administrator at PGBA, is also impressed with IMACS: "I could tell from meeting the staff and touring the facilities that IMACS is about challenging students through the latest technology to expand their knowledge to a new level." She added, "Everyone gave us a warm welcome, and I left knowing that our students were in good hands."
IMACS Hi-Tech Summer Camp program consists of logic puzzles, computer programming, virtual robotics, electronics, and an element of competition. Working solo and in teams, kids learn how to think logically and creatively while having fun.
About the Multiplied Foundation
The Multiplied Foundation's mission is to support and expand STEM (science, technology, engineering, and math) education. The Multiplied Foundation was founded by Peyton Robertson and seeded with the $100,000 he won during the 2015 Pebble Beach Pro Am's Chip Off Challenge. Each year, the Multiplied Foundation distributes 5% of its 12 quarter rated average value to organizations supporting STEM education. For more information, visit multipliedfoundation.org.
The Institute For Mathematics and Computer Science is an independent teaching and educational research institute focused on helping students reach their highest potential in math, computer science and logical reasoning. For more information, visit imacs.org.
About the Community Foundation of Broward
Founded in 1984, Community Foundation of Broward helps families, individuals, and corporations create personalized charitable Funds that deliver game-changing philanthropic impact. We provide leadership on community solutions, and foster philanthropy that connects people who care with causes that matter. Our 450 charitable Funds represent $173 million in assets and have distributed $89 million to create positive change. For Good. For Ever. For more information about Community Foundation of Broward, visit cfbroward.org or call
954.761.9503. Connect at #cfbroward @cfbroward
Toolboxes are handy things to have around, especially when they are filled with useful tools. A maker might include a hack saw, hammer and soldering iron in her toolbox. A musician might include a metronome, pitch pipe and recording device in his toolbox. What would you find in the toolbox of a problem solver? Many things, to be sure, but among them is likely to be an array of logical reasoning skills.
Logic promotes clarity of thought in understanding ideas. As the complexity of our world increases, each of us will face a growing number of unfamiliar and challenging situations. The shift to a knowledge-based economy keeps progressing. The significant decisions we must make keep multiplying. People are needing to think critically more and more but with less and less time and information. In a world such as this, logical reasoning is a powerful skill that can help a person sort through the noise and think clearly. When faced with new and difficult questions, the experience of having solved hard problems with logical thinking can be the difference between a panicked "Where do I even begin?!" and a calm "Let's see where to begin."
For example, you might start by looking at the problem from different angles to see what's inside. Investigate further as you anticipate obstacles and how they might affect your analysis. Then build a mental map to help you navigate your way. What are the different paths you can take? As you develop a feel for the problem, divide it into more manageable pieces. Working your way through the pieces will likely require sideways thinking with creative approaches and capturing those tricky insights that flutter in and out of focus. Connect the dots as you build a way forward. When you've got all your pieces solved, glue them together for a complete solution!
People who learn to think logically are simply better equipped to analyze complex problems. From Brexit to antibiotic-resistant bacteria, unanticipated developments often lead to unprecedented situations that leave many smart, experienced people wondering what to do. Solving problems does not stop at the schoolhouse door. It is when you step into the "real world" that, if you are fortunate, the most interesting and difficult problems present you with opportunities throughout your life to make a meaningful difference. What will be in your child's toolbox when those opportunities arise?
There was a time when the ability to faithfully recall information was a strong determinant of success. Now we have highly-sophisticated search engines and revolutionary chatbots that speak answers to your spoken questions. What you know and think is no longer the definitive measure of being smart. What matters now and in the future is how you make use of the vast knowledge that is readily available. The modern measure of being smart is in how you think.
Leverage and Extend Existing Knowledge
When it comes to knowledge, success today is more about whether you can combine and extend what is already known in order to address novel situations. For example, competitive business environments shift more rapidly than in the past. What worked yesterday no longer works today. As a result, companies have less need for employees to retrieve information or repeat past processes. Instead they have a growing need for people who can reconfigure knowledge to solve problems in real time. Being an encyclopedia of knowledge might make you a Jeopardy! champion, but in the information economy, if that is all you have to offer, it makes you obsolete.
Of course, you need to have some existing knowledge before you can extend it. But talented, young learners can often master the "what" quickly and thus need earlier and more frequent opportunities to explore deeply the "how" and "why". Parents and teachers can support these children by minimizing repetitive work and shifting learning toward activities that ask students to use what they know rather than show what they know.
There is still a meaningful way that existing knowledge by itself can help to develop thinking skills — by understanding how the knowledge came to be. In most educational settings knowledge is delivered as fully formed and tidy, which bears no resemblance to the reality of the non-linear struggles and strides that define the paths to new discoveries. Research shows that understanding the challenges that famous scientists faced motivates students to put forth greater effort on hard problems than merely knowing about the scientists' accomplishments. Another study focusing on teachers' knowledge of math history suggests that teachers who understand that mathematics is an active and creative field can help their students see math as a place for inventive problem-solving where questioning and investigating are highly valued.
Solve Problems but also Learn to Spot Them
Many education-focused organizations emphasize problem-solving skills. Such skills can be thought of on three broad levels. The entry level is to solve problems that are given to you with conventional approaches that are taught to you. At this basic level, success is a matter of how well you reproduce the steps that you have memorized. But as with knowledge, being smart today is about more than this simple kind of problem-solving.
The next level is to solve problems that are given to you with approaches that you try to construct yourself. This level requires a more sophisticated way of thinking because you will need to build on what you already know to come up with approaches that are new to you. To develop these kinds of problem-solving skills, you need more than your existing knowledge; you need analytical tools to make use of that knowledge. For example, you need to be able to see inside the larger problem to its component parts, match up what you know to these smaller problems, and then glue the solution together. Developing problem-solving skills at this level is an extended process best started when children are young. But doing so is well worth the investment because without this foundation, children cannot move on to the next level.
At the most sophisticated level, problem-solving starts with being able to identify problems and ask questions that are less apparent to others. Problem-spotting pushes thinking to the next level because it requires a critical eye toward the conventional and a forward-thinking mind about what the future will bring. This is the kind of thinking that is highly valued in university life and in a fast-paced, global economy because it is the engine for innovative ideas that move humanity forwards. Parents and teachers can plant the seeds for this highest level of problem-solving by creating an intellectual environment in which young people feel safe in both challenging conventional thinking and in having their thinking challenged.
Bright, young learners want more than to merely know what is known and do what has already been done. Satisfying their intellectual curiosity must include opportunities to do more than demonstrate mastery of existing knowledge and procedures. Talented students want to be smart by today's standards. Parents and teachers can support them by providing educational opportunities that leverage their knowledge and stretch their problem-solving skills.
As parents we instinctively want to help our children when they struggle. Our impulse is often to just fix the problem ourselves. While this approach makes sense when children are very young, as they grow it becomes counterproductive. In rescuing our kids, we provide a quick fix but deny them opportunities to practice important skills that they will need to become independent adults.
This is true in general and certainly with respect to education. As children grow, we should encourage them to develop the thinking skills that they will need in college and in their chosen careers. In what may seem like counterintuitive advice, teaching kids to think sometimes involves parents doing less, not more. Here are five tips on how parents can help children learn to think for themselves.
1. Don't be so quick to offer help. If your child asks you for help with school work, first consider how much independent effort she has truly given before turning to you. In a world of online programs that provide hints at the click of a button, it's easy to fall into the trap of thinking that getting to the answer quickly is good and struggling is bad. But math education research shows that a healthy amount of struggle actually leads to deeper learning. By answering your child's questions too soon, you can adversely affect how well she learns. Instead, remind her that it is perfectly normal to take a good amount time to work through problems, allow sufficient time in her schedule to do so, and encourage her to be patient with herself. Once you sense that she has put in a reasonable amount of independent effort on a problem, it's okay to offer hints that give just enough information to get her thinking again without giving away too much.
2. Don't point out mistakes before children have had a chance to discover the mistakes themselves. When you point out mistakes to your child too soon, he is less likely check his own work because he knows you'll do it for him. You also curtail the learning and brain growth that happens when he has to determine on his own that he made an error and what the error is. Finally, you may be signaling that you disapprove of making mistakes, which can lead to other problems such as an unwillingness to try challenging activities. When it's time to bring mistakes to your child's attention, a better way is to let him know that he has made at least one error but not tell him what, where, or how many. This approach necessitates that he will have to look carefully over all of his work and encourages the good habit of doing so on his own as he gets older.
3. Do encourage children to correct their mistakes. Too often, bright students just want to forget about the mistakes they make on school work, especially if they have been led to believe that smart people don't make mistakes. But a mistake means that the problem was challenging for your child, so she should definitely look back at it and work to correct her understanding. Often times we learn better from mistakes than from successes, and research shows that our brains actually grow more from making and correcting mistakes. Help your child see the good in mistakes by using them as opportunities to learn more.
4. Do ask open-ended questions. Asking questions is a highly effective way of guiding your child to think his own way to the answer — What information do you have? What information are you missing that could be helpful? How can you break up the problem into more manageable pieces? When your child was young, he probably asked you a lot of "Why?" questions. As he gets older, turn the tables and ask him why in the same inquisitive, non-judgmental way. Why do you make this assumption? Why do you take that approach? Why does this lead to that? As you know from being on the answering end of "Why?" questions, formulating a thoughtful response takes effort. For a growing mind, that's a good thing. One caveat: Be sure to avoid doing the thinking for your child by giving away too much information in the form of a question such as, "Shouldn't you combine like terms first?"
5. Do provide challenges that require deep thinking. Just as your muscles won't grow from lifting light weights, your child's thinking skills won't develop from doing easy problems. What's more, curious, young minds are naturally engaged by interesting, hard problems as long as those minds are not yet overcome with a fear of making mistakes. Intellectual challenges also provide opportunities to develop good study habits, resiliency from surviving the inevitable mistakes, and genuine confidence that comes from knowing you can handle hard problems.
Ladies Learning Code and the Institute for Mathematics and Computer Science (IMACS) are partnering to award over $3,000 in scholarships to participants in the 2016 Girls Learning Code Camps and members of GLC’s Teen:Club. The two full and three half scholarships will allow recipients to enroll in IMACS’ introductory computer science course, University Computer Science I (UCS1).
240 girls ages nine to twelve are expected to attend Girls Learning Code Camps in Toronto this March and Summer. The camps are designed to help girls see technology in a whole new light — as a medium for self-expression, and as a means for changing the world. IMACS will award one full scholarship and one half scholarship among applicants who attend the camps. All campers are invited to apply.
Shuli Jones attended the very first Girls Learning Code camp as a 6th grader back in 2012 where she won a half scholarship to UCS1. She is now a sophomore in high school and recently scored a ’5′ on the AP Computer Science A exam as a freshman. Since starting IMACS’ computer science courses soon after the camp, Shuli has been one of IMACS’ top CS students.
IMACS will award the other full scholarship and two half scholarships to members of GLC’s Teen:Club. In Teen:Club, 13 to 17 year old girls focus on solving problems, turning ideas into reality and learning about what a future in technology could look like. All members are invited to apply.
"Now that we have been running youth programs for 4 years, we are really starting to see our participants move on to technical career paths, pursuing Computer Science in University and coming back to participate in our programs as mentors, inspiring the next generation of girls," says Laura Plant of Ladies Learning Code. "We’re excited for our girls to have an opportunity to study computer science at a high level with IMACS."
"IMACS believes in the potential of bright girls and young women to shape a better future for all people," adds IMACS president Terry Kaufman. "We are thrilled to be working with Ladies Learning Code again to encourage more girls to pursue computer science in college and in their careers."
The course, UCS1, is a challenging and engaging introductory programming course that gives students the tools they need to excel as a computer science major in college. UCS1 teaches the fundamental principles of computer science at the university level but was developed for talented middle and high school students.
Scholarship applicants must submit a 100-250 word essay to Ladies Learning Code on what they hope to get out of this program. They are also required to pass the online eIMACS Aptitude Test. The application deadline for these scholarships is August 31st 2016.
Applicants may register for the eIMACS Aptitude Test at www.eimacs.com/aptitude. On the online registration form, enter "Girls Learning Code 2016" in the box for "How did you hear about IMACS?"
About Ladies Learning Code
Ladies Learning Code is a Canada-wide not-for-profit organization that runs workshops for women and youth who want to learn beginner-friendly computer programming and other technical skills in a social and collaborative way. Today they have 22 chapters (and counting) across Canada, a thriving girls’ program called Girls Learning Code and co-ed kids program called Kids Learning Code. With sponsors and community partners that include TELUS, Microsoft, Facebook, Autodesk, Google and many more, Ladies Learning Code has become synonymous with technology education in Canada.
Founded in 1993 by a team of mathematicians, computer scientists and educators, IMACS offers talented middle and high school students a range of online courses in university-level mathematics and computer science that are recognized by many prestigious universities. IMACS also offers math and computer enrichment classes at its US centers in Florida, North Carolina, Missouri, Connecticut and California. Over 4,500 students throughout the US and the world attend local IMACS classes or study its online courses. For more information on IMACS visit www.imacs.org.
For Ladies Learning Code:
Laura Plant, Co-Executive Director
Natasha Chen, Communications Director
Logic is the systematic study of reasoned argument. Why study logic? For students planning to take university-level math courses, which typically involve reading and writing proofs, the benefits are evident. But even for students heading in a different direction, developing strong logical reasoning skills has benefits too.
Logic Supports Learning Mathematics
Few would dispute that arithmetic is a fundamental base upon which the successful learning of math is built. Consequently, the development of arithmetic skills receives enormous attention in elementary school. Logic is equally fundamental to learning all other areas of math but is rarely taught in elementary school. Were students to develop a strong foundation in logic at a young age, their capacity to understand middle and high school math would be greatly enhanced.
In arithmetic, we learn how to combine or break apart numbers in order to proceed from the information given to the answer sought. If we bake three dozen cookies, multiplication tells us how many cookies we have in all. If 17 cookies are left, subtraction tells us how many were eaten. Honing arithmetic skills in elementary school allows students to quickly and accurately call upon them later, thereby making it is easier to learn more advanced math in the future.
Logic is a similarly powerful and essential tool for learning more advanced math. Whereas arithmetic is used in constructing and deconstructing information that is numeric in nature, logic is essential in constructing and deconstructing assertions that are mathematical in nature.
Students typically encounter a weak version of proofs and logic for the first time in high school geometry. But by then, pre-algebra and elementary algebra have already been taught as mostly procedural rules and algorithms to accept at face value. As a result students do not develop a genuine understanding of the derivation of those rules and algorithms and why they work. By the time students reach high school, their natural curiosity about mathematical ideas has been reduced to a hard-to-break habit of memorizing and applying rules blindly.
A worthy goal of the Common Core math standards is to develop mathematical understanding in addition to procedural skills. True mathematical understanding comes with seeing how knowledge is pieced together step-by-step on the path to deriving an important result. This is similar to what we know about developing scientific understanding in young minds — hands-on projects make learning stick better than memorizing facts. Math’s version of "hands-on" is your brain working its way through the reasoning that mathematicians followed in order to provide us with neat and tidy rules.
With a strong foundation in logical reasoning, students would have the key analytical tool needed to work their way successfully through the reasoned arguments that lead to the rules. Some may even wish to achieve the deepest understanding by using logic to independently derive the mathematical truths they were once asked only to memorize.
Logic Supports Complex Problem Solving Beyond Mathematics
The amazing thing about logic is that it makes us better thinkers and problem solvers well beyond mathematics. Computer science could not exist without logic. If you want to understand the underlying principles and algorithms and be able to apply them in any programming environment, you must understand logic. Engineering disciplines rely on logic too, especially electrical engineering as any Minecrafter worth her redstone will tell you. The entire practice of law is founded upon reasoned argument, which is why the LSAT has sections on logical reasoning and analytical reasoning.
People who learn to think logically are better at analyzing complex problems. Mathematics, with its diversity of complex problems, is a natural thought playground for developing logical thinking skills. The first steps in solving any math problem are assessing the situation and coming up with a plan of attack. Can you view the larger problem as a combination of more manageable cases? Can you take what you know and derive what you need to know? Would additional information lead you to certain conclusions? Answering these types of questions requires logical thinking.
People who learn to think logically are also better at assembling solutions. Once you have assessed the problem, broken it down, and gathered the necessary information, how do you put it all together to arrive at a solution? When faced with a novel situation, can you devise an approach where there was not one before? As the pace of change in our world accelerates, every field increasingly needs people who can strongly grasp core concepts and combine them sensibly to solve any problem, especially unfamiliar ones. Learning to apply logical thinking to mathematical situations is excellent training for handling complex problems, no matter which college major or career you choose.
Logic has a way of promoting clarity of thought in understanding ideas that were previously unfocused or muddled. Whether you are planning on becoming a professional mathematician or just trying to get through math homework on your way to a different intellectually demanding career, there is no doubt that strong logical reasoning skills would ease your path.
This blog article is part of the Hoagies' Gifted Education Page March 2016 Blog Hop on Math Education Enrichment. Please click on the Blog Hop image above to learn more about Math Education Enrichment from other Hoagies' Blog Hop participants.
—Keith Devlin, Professor of Mathematics, Stanford University
Several of our Math Enrichment parents have asked why IMACS wasn’t mentioned in The Atlantic’s recent article entitled "The Math Revolution". Well, we’re the first to admit that we’re not sexy! To journalists, that is. Or rather what we do isn’t sexy, especially when compared to the panache of producing math competition winners. But we believe in what we do, which is more than just math. IMACS prepares children to be better thinkers for life. And after twenty-three years of seeing our approach succeed, we wouldn’t change for all the media coverage in the world.
For readers unfamiliar with IMACS, we specialize in mathematics and computer science for bright, talented and gifted K-12 students. The core of our program is our Mathematics Enrichment curriculum for elementary school students, which is taught at 29 locations throughout the US. IMACS Math Enrichment is unique in that it is intentionally designed to be completely different from school math. One might say it’s "mathematician" math because it teaches even our youngest students about beautiful and powerful mathematical structures studied by professional mathematicians and typically not introduced until college.
Why do we take this approach instead of starting with what kids have already seen and expanding on it? (It sure would be a lot easier on us to market curriculum that parents recognize and that journalists think they can write about!) This goes back to the overarching purpose of everything we do — making better thinkers for life. We’re not just about turning good math students into even better ones, although we do that. We’re not just about providing gifted students with an outlet for their creative mathematical thinking, although we do that a lot. We’re not just about teaching problem-solving skills, although we certainly do that. IMACS is about teaching children how to think in creative, logical and flexible ways that will benefit them throughout their lives, whether within or outside the mathematical arena. We equip them so that, long after they have left our classrooms and entered into the "real world", they will be able to react intelligently and constructively to problems and situations that no one alive today has even imagined.
How do you learn to face those kinds of novel situations? Not by working on problems that look familiar. As IMACS alumnus Zachary Kaufman put it, "Graduate school at Stanford was so much easier with the logical thinking skills I learned at IMACS. While classmates tried to memorize each type of problem, I was able to strongly grasp core concepts and use them to solve any problem, even if it was different from those I had seen." Zachary is a better thinker for life. Or alumna Rachel Cuozzo: "An MIT degree says to the world that you are capable of complex problem solving and abstract thinking. IMACS gave me a jumpstart on my MIT education, as it taught me to exercise the same parts of my mind that I use now on a daily basis." Rachel is a better thinker for life. This is what IMACS is about. Unfortunately, it doesn’t always make good copy in an article about math whizzes.
Furthermore, IMACS has never been only for kids who already excel at math or families seeking a challenging math education for their gifted children. Of course, the majority of our students are very mathematically talented, and they happily attend our Math Enrichment classes or take our online math courses. But IMACS also nurtures bright students with untapped abilities whose prior negative experiences turned them away from math. Usually, they have been misled into thinking that being good at math means calculating quickly, scoring high on standardized math tests, or enjoying math competitions. For many talented students, none of those apply. IMACS is able to take those kids — who often have natural talent for math expressed in other ways, such as a love for games, puzzles and patterns — and show them (1) what it means to be genuinely good at math and (2) how incredibly good at math they truly are. IMACS makes them, too, better thinkers for life.
So if you are a current IMACS parent wondering if you’re missing out on something, or if you are a prospective parent considering IMACS, we encourage you to confirm that the math enrichment providers you are considering actually live up to their claims. Some will tell you that they go beyond what schools do. Some will tell you that they don’t do "drill and kill". Verify for yourself. Ask to observe a class and to see samples of the curriculum. Does it look familiar? Is it largely similar to what schools already teach? That will tell you a lot about whether your child’s mind is going to be expanded as much and in ways that you’re hoping for and quite possibly paying for.
IMACS (The Institute for Mathematics and Computer Science) will open a new teaching center in Silicon Valley in Fall 2016. IMACS California will offer all five levels of Mathematics Enrichment classes as well as Computer Programming & Virtual Robotics classes for after-school, weekend and homeschool students. These classes are designed for gifted and talented elementary and middle school children who enjoy intellectual challenge and for those with untapped academic skills.
IMACS students are grouped by ability, not by age or grade. In order to ensure that each child is challenged appropriately and placed at the right level, each prospective student is required to attend a free one-hour Placement Class. This is also an important opportunity for the child and the parents to see what IMACS is about. Placement Classes in the Bay Area will start in July 2016.
If you are a Bay Area parent who would like to receive informational updates on location, class times, Placement Class registration and Open House dates, please send an email to firstname.lastname@example.org with the subject line "updates". Your email address will be kept confidential and used for this purpose only.
Last New Year’s Eve, Mary, Nancy and Peter each made a different resolution about desserts.
- One of them said "No ice cream."
- One of them said "All of my desserts will have fruit! (That makes it healthy, right?)"
- One of them said "No chocolate."
In January, the three friends went out to eat three times. Loving social media, they each declared their dessert choice online during the meal. During each meal …
- exactly one person kept his or her resolution and was honest about it,
- exactly one person broke his or her resolution and was honest about it, and
- exactly one person broke his or her resolution and lied about it.
Over the three meals, each person took each action once. Here were their declarations:
|At Meal #1||peach cobbler||apple pie with vanilla ice cream||chocolate ice cream|
|At Meal #2||chocolate covered strawberries||plum pudding||peach cobbler|
|At Meal #3||apple pie with vanilla ice cream||chocolate cake||cheesecake|
For each person, what resolution was made?
Try your best before looking at the solution to this year’s IMACS Holiday Logic Puzzle.
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!
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