Isabella Yeung completed the EMF Math program during grades 6-8 through her public school district. She wrote the following guest blog post to share her memories of the program and advice for current and future EMF students.
EMF truly is an amazing program. (No, I’m not saying this just because …; I mean it with all my heart.) Looking back at the last three years, I am grateful for the opportunity to have participated in EMF. Unquestionably, I have grown from when I was a 6th grader just starting EMF. This rigorous program has expanded my knowledge about the world of mathematics, stimulated my brain cells, boosted my confidence level, and let me grow as a student.
In fact, because of EMF and the support of my teachers and EMF professors, I scored 100% on the University of Florida Dual Enrollment math placement exam on my first try. Thanks to my score, I will be taking UF’s Calculus 1 and Analytical Geometry course as a 9th grader.
EMF has played a crucial role in my academic life and in my social life as well; I have created numerous everlasting memories with my best friend who was also in EMF at the same school. (As EMF students, we sat in classrooms with non-EMF students doing the usual accelerated math.) My friend and I gave each other encouragement when EMF exercises were difficult, and we amused each other with our imaginations during the regular classroom lessons.
There is definitely a noticeable difference between EMF students and those who don’t take EMF. My math teachers certainly noticed it. Whenever teachers would give my best friend and me a math test along with the rest of their class, we would score the highest in the class even though we didn’t pay much attention to the regular lessons and barely studied for these tests.
In 7th grade when my friend and I were in the Algebra 2 Honors classroom, we would finish our EMF work and then quietly make up crazy stories and comics to keep from getting bored by the regular lesson. We both got the top scores of our class on the end-of-course exam because of EMF. He got a near-perfect score, while I got a perfect score. The experience in 8th grade was similar. When the teacher would warn us about a Geometry quiz happening the next day, my friend and I would usually just skim through the regular textbook chapter for 10 minutes that night and obtain the highest marks in the class, either a 100% or just one question off.
Thus, if you stay on schedule with EMF, the content definitely prepares you for school math finals or state math exams. But mostly, it’s the way of critical thinking and logic skills that EMF instilled in me that has ensured my success on the tests and exams.
Let me be frank; this course is not for the faint of heart. Sure, you might find it challenging, frustrating, and downright annoying at times, but this is all part of the growing and learning process. This is a program that forces you to look at concepts from multiple perspectives and drives you to think outside the box. It seriously takes your processing and logic skills to the next level! Because of this, my analytical skills have definitely been refined to a sharp point.
The three most important traits you need to possess in order to succeed in EMF are resolve, perseverance, and dedication. If you have these three things, you will undoubtedly make it through the program and earn the title of "EMF Graduate." No matter how puzzling and infuriating a problem might seem, you must not give up until you have solved it. The satisfaction you get when you have accurately unraveled a seemingly impossible exercise is unparalleled. EMF requires that you work diligently and prepares you mentally for most, if not all, obstacles you will face in high school and beyond. If you do end up procrastinating on EMF, you will have to work very hard to catch up. (I know this from experience, having spent some of my school holidays to get back on track.)
As a word of advice to future and current EMF students: Know that there are no shortcuts in EMF. Work hard. In the end, the rewards you reap will surely be worth it. I have not and absolutely will not ever regret partaking in EMF. It has paved a plethora of new roads, ones that I will readily embark upon. I encourage you all to march on and face EMF head-on. Best of luck to all!
The Institute for Mathematics and Computer Science (IMACS) is thrilled to announce that the Elements of Mathematics: Foundations (EMF) online program for talented middle school students is now complete with the release of the 18th and final EMF course, Precalculus Coda.
Precalculus Coda brings together topics from earlier EMF courses and expands on them with an extensive study of vectors in the real number plane and space, linear transformations (including orthogonal mappings), and matrices and their application in solving systems of equations.
Upon successful completion of the EMF program, students will have covered all national and state math standards for pre-algebra through precalculus but from an entirely modern mathematical approach. In addition, they will have enjoyed thorough introductions to college-level topics such as Abstract Algebra, Logic, Set Theory, Number Theory and Topology, and be well-prepared to excel in a college-level Calculus course.
Why Modern Mathematics
Mathematically talented students typically master school math with little effort. As a result, they tend to become bored with and disengaged from math if limited to traditional curriclum. They may also fail to develop the mental resiliency and study skills that come with having to work hard at math problems designed for their ability level.
The EMF curriculum is designed specifically to engage young, gifted thinkers in ways typically reserved for university math majors. It is useful to distinguish the two ways in which EMF is college-level: (1) the sophisticated use of logic and reasoning via proofs, and (2) the introduction of modern areas of mathematics such as Topology.
EMF incorporates all the concepts from a traditional curriculum but uses a logic and proof-based approach not usually entrusted to students until college. Talented middle school students who are intellectually mature and motivated can benefit from this more sophisticated approach; it fosters the development of creative thinking skills that are not easily replaced by computer processing power as algebraic manipulation skills already have. In other words, EMF prepares students with skills for the future, not skills of the past.
EMF's incorporation of exciting topics in modern mathematics—topics ignored by traditional school math—makes it far more interesting and intellectually engaging. These topics represent areas of active research as opposed to traditional school math subjects that were understood completely by the late 17th century. The idea of cutting off a child’s education in literature, history or science at the 17th century is ludicrous. The same applies to mathematics.
For children in many parts of the US and around the world, EMF now fulfills the promise of opening up access to a comprehensive, unified, proof-based approach to modern mathematics. IMACS certainly believes in the value of talented and dedicated teachers, but we also believe that lack of access to such teachers has unfairly limited for too many bright students the right to be challenged at their ability level on the way to achieving their full potential in mathematics. We aim to change that with EMF.
Course 1 < $20
In celebration of EMF's completion, tuition for the first course has been dropped to $19.95 (regularly $59.95) for a limited time. Through August 31, 2018, new families save 67% on the first EMF course or 25% on a bundle of all 18 courses. Visit EMFmath.com to learn more and enroll.
EMF in Schools
Did you know that numerous districts, schools and after-school programs license EMF? In fact, the first cohort of EMF students in Broward County Public Schools, Florida completed the program last year. As a result, many of them will be enrolling in Calculus as 9th graders this fall. If you would like to see EMF offered in your school, share the EMFmath.com/schools website with the relevant decision-makers in your district.
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.
The Institute for Mathematics and Computer Science (IMACS) believes in making sure kids exercise their minds this summer, in addition to their bodies, with fun yet educational summer camp experiences. Spending summers having fun and being outside is definitely important, but a small dose of educational summer camp is essential for all students too. Here are three reasons why:
1. Retain Knowledge: Summer vacation can often lead to forgetfulness and overall loss of learning. According to a study by the RAND Corporation, students lose, on average, one month of learning during the summer, all students lose some learning in math, and summer learning loss is cumulative over time. When kids aren't working certain parts of their brains during summer, they end up spending the first few weeks of the academic year refreshing what was lost. Educational summer camp activities keep those areas of the brain active so that children are ready to engage in new learning when school begins again.
2. Develop New Interests: Educational summer camps are a great way to explore new academic areas that kids don't have time for during the year because of school commitments and extracurricular activities. IMACS Hi-Tech Summer Camp, for example, often sparks the interest of kids who never considered math, electronics or virtual robotics as something they would enjoy. This is especially true for girls who often don't get enough exposure to these kinds of activities. At IMACS, we have had a number of students realize they want to study engineering after being immersed in hands-on projects and the logical and creative ways engineers think at our Hi-Tech Summer Camp.
3. Make New Friends: If your child enjoys fun, academic challenges in mathematics, computer programming and gaming already, they will not only have the opportunity to advance their skills, they will meet other kids their age with similar interests. Educational summer camp attendees are smart, fun, and enjoy a little friendly competition. It's a truly unique opportunity for your child to connect with other kids who appreciate their unique way of thinking. It's also a time for kids to be inspired by instructors who are genuinely passionate about their field.
Educational summer camp should be a part of every child's summer activities. It helps stop summer "brain drain", exposes kids to new ideas and pursuits, and leads to great friendships and memories. When choosing an educational summer camp, be sure to ask if the camp is staffed by high school and college kids or by highly-qualified and experienced instructors. After all, these kinds of camps should involve real thinking in addition to real fun!
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
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!
October 13, 2015 is Ada Lovelace Day, a day to honor the achievements of women in science, technology, engineering and math. IMACS asks that you join us in celebrating this day by encouraging a girl to pursue her interests in the STEM subjects.
She may be your daughter, sister, student or friend. She may be enthusiastically expressive about her love for STEM, or she may be the quiet type who will share deep thoughts if you ask. Or she may be especially in need of your encouragement because she’s not yet received that message or, worse yet, has been actively discouraged from pursuing her passion for STEM.
What can you do to encourage a girl in STEM today or any day? If you have the knowledge and time to share, become a mentor to her. If you don’t have the time to commit to mentoring, help her find appropriate enrichment activities such as local events at the science museum or after-school programming classes that will keep her engaged, especially when social pressure can push her off track. Even something as simple as sharing stories about talented women in STEM who can serve as role models can make a difference.
IMACS is honored to have been a meaningful part of the education of numerous high-achieving girls who have gone on to amazing college and professional careers. They now serve as inspirational role models for our younger students. Who knows? The girl you encourage today may one day be an IMACS alumna studying STEM at a top university!
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)
The following letter is extracted from the new book, Letters to a Young Math Teacher, by Gerald Rising and Ray Patenaude, which is available from Amazon.com and other sources. Gerry Rising is Distinguished Teaching Professor Emeritus at State University of New York at Buffalo where he co-founded the university’s Gifted Math Program for highly-qualified regional students in grades seven through twelve. Gerry Rising was for many years associated with IMACS activities and is a strong supporter of our work.
Letter Seventeen: A Bag of Tricks
Paul Rosenbloom enjoyed an international reputation as a senior mathematician when I joined his Minnemath Project at the University of Minnesota as his assistant director. His Elements of Mathematical Logic remains today a basic resource recognized worldwide to be of historical as well as academic significance.
In our very first conversation he told me that he considered teaching outside the classroom an important aspect of a mathematician’s life and he urged me to develop what he called “a bag of tricks” from which to draw math-related lessons for people of all ages. I would later see him draw from his own collection in a number of settings. He could captivate anyone from nursery school students to scientific colleagues and even senior political figures.
Within a month of that conversation I found myself seated on an airplane flying from Minneapolis to Denver. My seat partners were a bored nine-year-old and her mother. When the mother learned that I was a math teacher, she asked me if I would be willing to “show Marjorie some math.”
Fortunately, I had been reading David Silverman’s book, Your Move, and I had some pennies with me. I played a series of Nim games with Marjorie, finding her an interested opponent and a remarkably quick study.
Nim, for readers unfamiliar with the word, is German for “take” and in this context it applies to games with players taking one or more counters like coins from a collection following a given set of rules.
We started with One Pile Nim. I set out a line of pennies on Marjorie’s tray and explained the rules. We would take turns, at each turn removing one, two or three coins. The winner would be the player who could take the last penny or pennies.* When she quickly determined the strategy for playing, I suggested she challenge her mother. Her eyes lit up as she outplayed her mom.
Now she wanted more. I suggested the misère form of the game: same rules for play but this time the loser forced to take the last coin. Now I had her mother’s full attention as well, but again Marjorie, once she found the winning strategy, could outplay mom.
We went on to play the very simple Two Pile Nim game in which each player takes any number of coins from one of two piles with the winner taking the last coin, but then the less transparent game, Woolworth, which is isomorphic to Two Pile Nim. Woolworth is named for the five-and-ten-cent stores of the 1950s that have been reincarnated as today’s Dollar Stores. On a sheet of paper I drew the following diagram:
I placed nickels and dimes as shown in the figure. One player controls the coins on the left, the other the coins on the right. Players alternate moving either of their coins any number of squares to left or right but without joining or passing the opponent’s coins. The object of the game is to squeeze your opponent so that no moves remain.
By the time I showed Marjorie the connection between Two Pile Nim and Woolworth we were well on our descent into the Denver airport. I was rewarded for our hour together when Marjorie turned to her mother to tell her that she wanted to become a mathematician.
Where do you find lessons like these that require no background and yet are suitable to given age groups? I found mine from two sources: my reading of journals and books about mathematics and my observation of lessons taught by others. Appendix 2 lists a few of the many available book sources.
Now, because it is a good story, I will tell you about my use of a lesson copied from the remarkable teacher, Robert Wirtz.^
Late one school year when I was a math supervisor I was asked by a primary school teacher to demonstrate Stern blocks for her students. Stern blocks are inch cross-section blocks that come in lengths from one to ten inches. Cuisenaire rods are similar but with centimeter dimensions.
I took several sets of Stern blocks to this teacher’s classroom and showed the six-year-olds some of the relationships among them.
The students were excited by the opportunity to mix play with learning, and I was convinced that they were gaining from the lesson. I noticed, however, that their teacher was not happy with the children’s mix of play with learning. Several times she interrupted their activities to tell individuals to be quiet.
It was clear to me that the teacher was convinced that I wasn’t disciplining the children in the way she wished, so I suggested that I return the next day to teach another lesson. Despite her reservations about my conduct of this class, she jumped at the chance to have me return. It was clear that she was happy to pass responsibility for these irrepressible kids off onto anyone she could find.
I had seen Bob teach a lesson to a similar group and the next day followed his model. Before the class I had the teacher gather the students in front of the chalkboard. Saying nothing to them I drew a square on the board and within it marked two star shapes. What I had drawn looked like this:
I pointed to that little tail at the bottom of the figure and turned to the class, holding out the chalk.
Several hands waved and I offered the chalk to one of the wavers. The boy came up and carefully wrote a “2” below the diagram.
That got us started. I drew similar diagrams with different shapes and the children quickly caught on, competing to write the numbers below the figures. Then I began to complicate the requests by erasing their numbers and connecting two figures like this:
No problem. A volunteer wrote “5” below the boxes.
I continued in this fashion, soon replacing the boxes with numbers but using the same connecting segments. All of the children were eagerly participating, but not one word was said by me or by any of them.
After a half hour of this activity, I finally spoke up, calling the children’s attention to the fact that none of us had spoken until then. And I turned to their teacher to congratulate them on what we had accomplished.
Unfortunately, my lesson proved nothing to this young woman. Her response: “Yes, after you left yesterday I really scolded them for their behavior and you saw the result this morning.”
Even with the best of efforts, you cannot always win.
I have talked about these mathematical extras mostly in terms of their use outside of your classroom, but they can play a role in your classes as well. Here is an example described by Swedish math educator Barbro Grevholm:
One of the teachers worked with a smaller group of pupils that had difficulties with mathematics. It was Friday and the pupils were not concentrating. The teacher announced that if they worked well he would play a game with them for the final ten minutes. This motivated the pupils to pay attention to the lesson. When the teacher finally said that it was time for the game, the students were all alert and extremely concentrated. In the game the teacher threw a single die nine times and after each throw read the result aloud. The pupils drew on sheets of paper three by three grids and chose to put the die calls successively in any one of the nine squares. The winner was the one who, by adding up the resulting three three-digit numbers, produced to a sum closest to one thousand. The game was played several times and all students took part eagerly. There was almost complete silence during the game and everyone made the additions quickly. For some of the pupils obviously more calculations were made during those ten minutes than during the earlier part of the lesson.
Please take the examples I have used only as samples of my own and others’ collections of activities. You may or may not add them to your collection. What is important is that you accumulate such activities that work for you. They will serve you well.
Where Do You Find Such Activities
Sources of such activities are all around you. You just have to be alert to them and adapt them creatively.
Many such activities come from your reading. Mathematics and mathematics education journals describe topics and lessons that you can both enjoy and share with others. Of course, you should credit your source when you replicate such a lesson, but every writer I know would be delighted to know that his or her lesson was being duplicated.
Some non-standard texts are full of such lessons. Authors like Harold Jacobs and Sherman Stein are two who have gathered and present such useful topics. And some authors simply collect such presentations. Among these are Ian Stewart, Ross Honsberger and Howard Eves.
In a class by himself is Martin Gardner whose collections of such topics are unsurpassed. His range is amazing, covering everything from hexaflexagons to fractal music, almost all of them serving a perfect basis for well-planned demonstrations.
Even your daily newspaper can provide activities. Although many people are drawn to the Sudoku puzzles, I much prefer three other popular forms: Kakuro, KenKen and Numbrix. These puzzels are adaptable to students of all ages (I solve one or two each day) and can challenge them all. The simplest appearing is Marilyn vos Savant’s Numbrix. While this puzzle task is appropriate for primary grade students (all it involves is listing the integers from 1 to 81 in order in a 9×9 grid), individual Numbrix puzzles range widely in difficulty.
In addition to their intellectual challenge, there are two things about the Kakuro and KenKen puzzles that I find attractive: (1) they involve basic calculation facts that reinforce the solver’s skills and (2) they appear as free apps on mobile phones and tablets. This availability makes them perfect puzzles to introduce to fellow travelers.
* The strategies for playing these Nim games are included in Appendix 4. I encourage you to play the games before you look at those strategies.
^ Bob Wirtz and his wife adopted a number of children with severe learning problems and they developed math materials to teach them. With Mark Botel, then president of the International Reading Association, he gathered those materials into books for elementary school students. Sadly, they are difficult to find today for they include some very attractive activities.
Abstract reasoning ability entered the national conversation this year as the Common Core State Standards in mathematics were broadly implemented in the United States. In particular, one of the eight Standards for Mathematical Practice is to “reason abstractly and quantitatively.” The so-called STEM subjects — science, technology, engineering and math — are well-known for emphasizing this skill. Given that STEM-related fields are where most high-skilled job growth is predicted, today’s students would do well to develop their ability to think abstractly.
So what is abstract reasoning, and why is it so important? Let’s break it down: To reason is to use logic in piecing together information, usually with the goal of forming an inference or conclusion. Abstract simply means that this process is a thought-based exercise of the mind as opposed to being based in concrete experience. For example, if you know that ice melts at temperatures above 32°F, you can reason abstractly that an ice cube placed on the counter of your room temperature kitchen will melt. You don’t have to take an actual ice cube out of your freezer and observe it for an hour to arrive at this conclusion.
Of the subjects that you could study in order to develop strong abstract reasoning skills, computer science is a natural and practical choice, as well as being a highly creative and exciting area in which to learn and work. The programming aspect of computer science is well-known and is one area where abstract thinking matters a great deal. Programming, after all, is the creation of a set of instructions that a computer can follow to perform a specific task. Such tasks typically involve the manipulation of digital information, decidedly not the kind of stuff you can grab hold of to see how it reacts in the tangible world.
Learning to program well involves developing the ability to think logically and abstractly so that you can anticipate how the computer will react to the instructions you give. Great programmers are actually capable of writing simple code without having to check it with a computer because they have the ability to analyze processes in their minds. If you cannot think abstractly, you may still be able to get your code to “work” with trial-and-error tinkering, but that approach lacks the robustness needed to solve meaningful problems that tend to be more complex.
The rich experience of learning computer science, however, is so much more than coding. When you study computer science, you engage in computational thinking, in which logic, abstraction and creativity come together to help solve intellectually interesting problems. As Professor Jeannette Wing of Carnegie Mellon University argues in her seminal article* on the topic, computational thinking is a skill set from which everyone would benefit no matter their career path.
Why so? Because when you study computer science, your mind learns to grapple with high-level questions such as: How can existing information be used to deduce further information that will help solve the problem? How should a complex system be designed in order to maximize simplicity and usability? How can a complex problem be broken down into smaller pieces that are easier to solve? Can a common approach be devised to efficiently handle similar problems?
If these questions seem like they would be applicable in a wide variety of fields, STEM and non-STEM, it’s because they are. In essence, when you study computer science you learn the valuable skill of thinking abstractly like a computer scientist even if you don’t plan on becoming one.
*Wing, Jeannette M. “Computational Thinking.” Communications of the ACM 49:3 (March 2006) 33-35.
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