Fostering independence



In my last post I discussed how its ok if elementary aged students are not yet working independently. Reading problems out loud, working side-by-side with your child, or providing other scaffolding to help them get through their work isn’t dooming them to use you as a crutch for the rest of their lives. It’s totally fine and developmentally normal for kids and young teens to still need some help getting work completed.

At the same time, moving towards independence in completing work and some degree of self-teaching is important. Incremental steps towards this goal can be taken gradually, without pressure, stress, or constant nagging. Here are some of my favorite techniques:

  1. Cut out the busy work!  While there’s a certain logic in assigning routine, easy, fill-in-the-blank style problems to complete independently, the boring and contrived nature of that work often lead to less focus and more goofing off than something that requires real thought. Cut out the busy work altogether, practice routine skills through games, and find bite-sized problems that are just the right level to engage independently for a short period of time.
  2. Do your own work. Just like all of us children can get lonely or feel isolated when completing school work on their own. Sitting right next to them at the kitchen table and completing your own work (even if its just catching up on emails to a friend), not only provides a role model for the student, but also company and connection.
  3. Make it the final thing. As a gradual transition towards more independence do the first problems together and then leave the last few to work on independently. This can give students the emotional support and confidence they need for the problems while gradually improving stamina to complete more on their own.
  4. Pomodoro technique. This is a really simple time management technique where work is broken down into focused 25 minute intervals with a 5 minute break. I’ve found that some students work much more effectively on their own when they know there is a limit to how long they will sit at the table.  For students just starting out or who have attention issues, starting with 10 or 15 minutes at first might be more appropriate. The time can gradually be adjusted upward. A visual timer helps some students stay motivated, but others will find a timer distracting or anxiety producing, so keeping it hidden and calling out 5 minute intervals might be more effective depending on the temperament of the child.
  5. One problem at a time. Some students find a whole page of problems intimidating, especially when asked to work through them alone. An incredibly successful technique for me has been to write problems on note cards and have the student work through them one at a time. It’s a little more work for the teacher, but working through a stack of problems one-by-one gives a sense of accomplishment after each card is completed. There’s also less chance for the students to get intimidated by a long page of work.

In the end, independence with school work is nothing to stress about. Small, gradual steps, with patience and a sense of humor can make this a natural process rather than something difficult or stressful.

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How much work should be independent?


The picture above is of my son doing a Venn diagram challenge problem in the first chapter of Art of Problem Solving’s Counting and Probability. It’s a tough book, and a tough problem, especially for a 10 year-old. He knew exactly how to do it, but had made a few careless arithmetic errors so I asked him to try again and be a little more careful.

He sighed: “Can you just read it out loud to me? I want to do it together.”

By “do it together” he didn’t mean give hints, tell him the answer, or hold his hand through the work. He just wanted to work through his thought processes out loud to someone else. So I read the problem out loud. Then I  listened and repeated some of his intermediate steps back to him. A lot of our math work is like this – me right by his side providing feedback or just a sympathetic ear. And that’s ok! I see a lot of parents get wrapped around the idea that their children should be working mostly independently at this age and they get disappointed when they can’t just hand their child a worksheet and have it done efficiently. It’s wonderful if your kid can go off and finish a few worksheet pages by themselves but there’s also nothing wrong if they can’t.

I would also argue that there’s such a thing as too much independent work. Human interaction almost always enhances learning, making math (or any subject) much more pleasant and also helping retention. The ability of a parent or teacher to provide instant feedback should also not be underestimated. Being able to correct mistakes immediately, provide guidance, or simply see where a child is at, are all easier when you’re sitting right next to them.

Working independently is an important skill that all students will need someday, but its important to remember that someday doesn’t have to be today! In my next post I’ll discuss some of the techniques you can use for getting students to work a bit more independently in a natural and gradual way.

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How much is enough?

Recently two books about math education crossed my path. The first is The Math Myth and Other STEM Delusions by Andrew Hacker and the other is A Mathematician’s Lament by Paul Lockhart. Although quite different in their perspective, each author questions the current math education being offered by traditional schools.

The Math Myth challenges the standard high school math sequence and Hacker questions the value of any math beyond Algebra I. He makes some interesting points about math being elevated as a subject simply because it is easily measured through standardized testing and how its used as a proxy for overall intelligence. He argues that few professions have any need for advanced algebra, trigonometry, or calculus and that requiring these subjects for college entrance acts as a defacto gate-keeping mechanism that primarily hurts poor and minority students who have less access to tutoring and math-rich environments. The author essentially asks: Why should someone be denied the opportunity to study for a career in graphic design or a job as a vet tech simply because they haven’t mastered the binomial theorem and other obscure math principles that they will never use in their personal or professional life? I found myself agreeing with some of Hacker’s points but I object to the idea that K-12 education is solely about practicality and that mathematics does not help one learn how to think.

A Mathematician’s Lament is also severely critical of the current state of math education, but Lockhart takes a different and more startling view – math is more similar to fine arts, he argues, and should be treated as we do drawing, painting, and music. These artistic disciplines definitely require technical skills and knowledge to do well, but unlike mathematics there’s also an emphasis on performance, creativity, and joy. He makes an analogy that how most students learn math now would be equivalent to teaching painting with a  “paint by numbers” program that only requires students to match colors to predetermined blocks on a canvas. In Lockhart’s world, math education would center around discovery, originality, abstraction, and beauty. He abhors the idea of making math more practical and relevant: “In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide — a relief from daily life, an anodyne to the practical workaday world.” He makes a strong argument for the kind of interesting puzzles and pondering that I see excites my students.

I see merit in both of these arguments, and one way to reconcile both viewpoints, that I strongly argue for, is a greater emphasis on discrete mathematics in our school system. Pushing every student into a math sequence that leads to calculus is inappropriate and unnecessary. Topics like combinatorics, number theory, logic, and graph theory are very approachable to students of varying ability, teach mathematical reasoning skills, and are just plain fun. So many of these topics can be framed as puzzles with elegant solutions which elevates mathematics into something more than tedious calculations. For example, my 10 year-old son and I are working through a book on group theory (the study of symmetry) and we just created some really cool diagrams of the group formed when certain rules are applied to switching art on 3 walls. There are no numbers involved, but its much closer to what real mathematicians do than adding up columns of sums.

From a practical standpoint discrete math is foundational to the study of computer science, and its no less “rigorous” (whatever that means) than calculus. I would also love to see a statistics course added to the high school sequence, especially one that emphasizes interpreting statistics, data visualization, and graphic design. Students going into any science related field, including social sciences like political science, psychology, and sociology, would benefit enormously from this kind of background and in today’s data driven world, making sense of data and newspaper graphics is part of being an informed citizen.

I don’t hold out hope that mathematics teaching in schools is going to change any time soon, but whenever I hear criticism of Common Core and a call for “back to basics” in math education, I inwardly cringe. I understand the frustration, but unfortunately there is no turning back. The math taught to our parents – routine memorizing of algorithms and focus on arithmetic computation has no place in a 21st century education. I welcome discussions, like those started by Hacker and Lockhart, that challenge the status quo, even if its not something I wholly agree with.

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Fun math videos for all ages

A while ago I made some recommendations for fun living math books that would inspire and educate students who have an interest in mathematics. Along those lines, here is a list of some of the videos we’ve enjoyed. Many can be found online at various websites or on Amazon or Netflix, but I’m leaving out links since availability changes quickly.

The Story of One  (2005) – This hour long BBC documentary about the history of numbers is my favorite! Presented in a humorous and highly entertaining style, it is perfect for kids ages 6 and up, but also educational and interesting enough for adults. The Amazon DVD is overpriced but the video is available at various places online and some libraries.

Between the Folds (2008) and The Origami Revolution (2017) – These fascinating documentaries are about origami as an art form with mathematical insights. I recommend for ages 10 and up, or younger if highly interested in origami.

Flatland: The Film (2007) – A very well done animated interpretation of the classic by Edwin Abbott. My son was riveted by the drama and concepts of this world. Available for purchase on Amazon and also online. Ages 7 and up.

The Secret Rules of Modern Living: Algorithms (2015) – Available on Netflix, this hour long documentary covers algorithms of various types, including traveling salesman, sorting, and other optimization problems. My son learned a great geometric representation of Euclid’s algorithm for finding greatest common factors simply from watching this show (his demonstration pictured above). Great for ages 8 and up.

Fermat’s Last Theorem (1996) – This documentary about Andrew Wiles, who solved one of the world’s most famous math problems, is absolutely gripping. The emotions and tensions involved and the peek into the lives of how academics actually work on problems makes for a great film for the budding mathematician. Recommended for ages 12 and up. Available at various places online.

The Great Math Mystery (2015) – A one hour NOVA documentary that discusses some of the mysteries of mathematics like finding Fibonacci numbers in botany, the connection between Pi and probability, and the idea that the mathematical underpinnings of reality might be proof that we are living in a simulation. This show is a survey of topics and thin on explanations but high on wow factor. Great for inspiring students to appreciate and find mathematical connections. Recommended for ages 10 and up. Available online and on DVD.

The Story of Maths (2008) – This four part BBC documentary about the history of mathematical development is a little more in-depth and serious than some of my previous recommendations, so younger students might find it dry. Its great for ages 12 and up who are interested in math and history. Currently available on Netflix.

The Code (2011) – This three part BBC series has a slightly sensationalistic style and explores how numbers and shapes play a role in the natural world. My son found the third episode on prediction the most interesting. Recommended for ages 8 and up. Currently available on Netflix.

Finally, no discussion about math movies would be complete without mentioning the 1959 classic Donald Duck in Mathmagic Land! Great for all ages and available online.

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What’s so terrible about Khan Academy?

I remember the moment well – it was almost bedtime, and my son, already in pajamas, asked me about the derivation of the quadratic formula. Yeah, yeah, that’s kind of thing just happens in our family. Since it had been a while (like 25 years) since I had studied this myself I looked on the internet for the best way to present the idea of completing the square.

I loathe Khan Academy, but decided it might be useful for this kind of refresher. Boy was I wrong! The opening of the video started off ok as he discussed the fact that quadratic equations that are perfect squares are super easy to solve.  Check! If we can make any quadratic look like its a perfect square we can solve that easily too. Great! I won’t bore you with the details, but needless to say it went downhill from there as the video immediately dived into a bunch of manipulations and never once explained things conceptually. You see, completing the square is actually really easy to understand if you actually draw a square. Hey look, we’re taking a rectangle and cutting off some bits to make it a square and adding another little bit to finish it off! You can draw it out and show the student visually exactly what’s happening. Here’s an example of a site that gets it right.

This is the crux of my problem with Khan Academy – a lot of procedural mathematics with little focus on understanding. It might be ok for test prep review when concepts have already been mastered, but I absolutely cringe when I hear of homeschoolers using it as their primary math program or parents of gifted kids using it to accelerate. These kids need to be challenged, inspired, and taught problem solving skills, not memorization of mindless algorithms. For a child that is behind or already turned off from math, seeing numbers fly around the screen with no idea of the underlying concepts could lead to even more confusion. Real math is not about manipulation, its about knowing how to think and set up a problem.

I do think there are some good videos out there for learning math, for instance AOPS has some that follow along with their booksEducation Unboxed has outstanding teacher training videos for using Cuisenaire rods, and The Great Courses has some cool lectures that we’ve really enjoyed. Additionally, online courses can be a great way for motivated older students to learn. However, when it comes to elementary and middle school math in particular, I really believe that human interaction, discussion, hands-on manipulatives, and instant feedback are essential components of learning how to think about numbers. For a more detailed and articulate critique of Khan Academy I highly recommend this article from the Washington Post: Khan Academy: the revolution that isn’t. The title says it all.

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Review: The National Museum of Mathematics

When it comes to museums I’m pretty picky. I wasn’t expecting much when I planned on visiting MoMath, the National Museum of Mathematics, during our trip to New York City two weekends ago. Their website is pretty minimalist and few people have even heard of this museum, but it turns out that MoMath on 26th street in Manhattan is quite a gem. My son was delighted at the idea of a whole museum devoted to mathematics, and the giant “Pi” handle on the double glass entrance doors just raised our expectations of what was inside. There were two floors of interactive exhibits and tons of friendly staff to help explain and interpret the math.

A few of our favorites:

The bicycle with square wheels

My son heard about this concept a while ago, so he was super excited to finally see one in person and try it out. There’s a great explanation of how this works here.

Enigma Cafe

I love science museums that have an area to sit down and explore small puzzles and games at your own pace and this one was especially well done – nestled in a cozy wood paneled area of the bottom floor. The puzzles themselves were highly engaging and tactile but with small screens to explain the goal, solution, and mathematics behind each.  If only it had been a real cafe and served coffee, I would have been in heaven.

Tessellation wall

Younger visitors to the museum gravitated to this station which had a huge wall with different types of magnetic tessellations. We loved the fact that the interactive panel nearby offered challenges appropriate for older students and we enjoyed the chance to build our own designs.

The cryptography machine

This was a natural fit for my son who has been studying cryptography since he was in second grade using CryptoClub.

Shape Ranger

This interactive light table used to explore the efficiency of shape packing was a favorite of mine.
Suggestions for future exhibits: I would love to see something focusing on cellular automata through the Game of Life or Abelian sandpiles (a concept my son and I are playing with in our homeschool). Also an exhibit to explain the Monty Hall problem would be a lot of fun. I feel both of these lend themselves easily to cool interactive explorations.

My only real critique of the museum, and of a lot of museums actually, is that they require you to stand and read a lot of background information on the nearby displays to really get the most out of the exhibits. For example on the bottom floor they had a wonderfully done plinko machine with marbles dropping in to demonstrate binomial probability distributions. This was great for my son, who has been exposed to this idea many times, to see as a live demonstration. However, for an elementary student who is just learning about this concept, a 15 minute read-through at a kiosk is asking a lot when surrounded by so much excitement going on around them. The Harmony of the Spheres exhibit proposing to connect music and math was similarly distracting – kids touching the balls to light up and make sounds without really learning much. From a cognitive science standpoint I would love to see more museums integrate the learning into the exhibits by requiring predictions and other user feedback rather than having explanatory text that is separate from the physical demonstration.

At the same time I do have to give credit to MoMath for putting a lot of thought into the information they present. Rather than overwhelm users with a lot at once they offer enough for each visitor to get started and then offer a choice of more information for those who are interested.  They also have plenty of staff who help to explain and interpret the exhibits.

MoMath is a great time, especially for a math loving child. It is centrally located across from Madison Square Park and reasonably priced at $15 for adults and $10 for students. The museum claims to target children in the 4th-8th grade range, which I thought was very accurate though there is plenty for adults as well. The gift shop is a special treat with lots of math related games, puzzles, and art work. My son wanted to return and we will definitely make it a priority on our next visit to New York City.

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How to add variety to your child’s math education


In my previous post I talked about how variety is a key component of math acceleration. But what does that look like in practice? For my last two years of elementary school I was lucky enough to attend an innovative full-time gifted program that did an incredible job of covering the basics while simultaneously doing creative project-based learning. Math for that program was split into two parts: 1) regular math class which was a standard program but entirely self-paced and 2) a math laboratory class which covered unusual topics like probability, different base number systems, graph theory, and cool projects like visualizing a million. I loved this idea so much that I decided to use it with my own student. On a weekly basis we spend approximately 2/3 of our time working in our main math book (currently Art of Problem Solving’s Introduction to Algebra) while another 1/3 of our time is spent on fun and interesting topics outside of the mainstream. A typical day might look like this:

Morning Main lesson: AOPS Algebra – 10 minute challenge problem from a previous chapter, 10 minutes of easier review exercises, 20 minute lesson on new material such as factoring quadratic equations.

Mid-morning break: 20 minutes working on a Scratch programming project that draws a line based on user inputed point and slope.

Most evenings:  30-60  minutes of discussion about Venn diagrams, logic, probability, triangle numbers, geometric drawing, playing with Zometool, or whatever strikes our fancy.  These discussions are led by me but not forced and usually start out with “Have you ever heard of…?”

I have a math-loving student and provide a rich educational environment at home so access to fun books, games, apps, toys, and videos are also part of his math education. All of this often adds up to 2 hours or more of daily mathematics, and it adds up effortlessly. Even for a student like mine who absolutely loves math, plodding through pages of standard exercises and working until mastery in every area would be an uphill battle. For students who are working ahead of grade level, keeping their chronological age in mind and being realistic about attention span is essential.  Changing topics, interleaving review with new material, and combining difficult problems with easier questions keeps attention and interest high which allows us to work for a longer period of time and cover more material. It also has the added benefit of helping the student make new connections. Despite what we’ve been told, mathematics is not a single lane track leading from arithmetic to calculus.

The idea of a math laboratory is also perfect for parents who want to supplement their child’s school work with something at home. After a long day of school and homework trying to pile on more math learning during evenings and weekends can be a tough sell to a worn out kid. However, if the topics are fun and unusual, it can be a natural and joyful activity. Most of my math laboratory activities are lesson plans that I put together myself, but a few of my favorite off-the-shelf math enrichment resources are listed here:

Mathematics: A Human Endeavor
Alien Math
Crypto Club
Leonardo’s Mirror and Other Puzzles
Time Travel Math
Patty Paper Geometry
Scratch Programming

I plan to review some of these books and resources in the future, but all are excellent. With Scratch programming I like to challenge my student to make math related programs that review a previous concept or explore something interesting like finding prime numbers or perfect numbers.

The main thing is don’t be afraid to mix things up, head off course, backtrack, and follow those rabbit trails of learning wherever they might lead.  You’ll be surprised at how much material you can cover when you don’t stick to a single minded plan or curriculum.

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Variety is the spice of life… and the key to math acceleration


Sometimes I get asked how my son came to be so accelerated in math. I’m sure some assume we must be doing a shallow just-the-basics curriculum or that I stand over him Tiger-mom style as he whips through pages of worksheets. Neither is the case, and the math study we do is both very deep and the most joyful part of our school day. There are a few secrets though:

  • Math is his thing. It happens to be my thing too, so this creates a synergy between teacher and student that is hard to force. Simply put, it makes magic. This is the most important ingredient to math acceleration. I would not be able to coerce the amount of learning we do: it comes from the intrinsic interest of the student.
  • We homeschool, and do so year-round. Nothing beats a one-on-one learning environment. With homeschooling we can take time to work through concepts that require more attention and skip the busywork, all with my student getting immediate feedback on mistakes. The lack of a long summer break means we can keep moving forward without a loss of learning or time wasted on extensive review, and although we do take plenty of vacation time, we also do math on most weekends and while traveling. I estimate the combination of summer and weekend work probably doubles the amount of math we get done in a year compared to a student in public school.
  • I make it fun. Hands on activities, games, and a unique approach to skill-building along with humor and a light hearted attitude by me, keeps us moving forward even when work gets tough.
  • Variety. The first three points might be obvious but the issue of “what to study” in the first place is not quite as clear and what I’m here to talk about because it applies to all learners, accelerated or not, math-loving or less so.

The biggest lesson I’ve learned in working with gifted children is that no matter how brilliant the student is, stamina and attention span are still strongly tied to chronological age. Without a sustained attention span the amount of material one can cover is going to be limited.  What does this mean in practical terms? It means even if you have a rare Kindergartner who can do long division, they probably can’t do a whole page of practice problems the same way a fourth grader could. In fact, they can probably only do one problem before they are bored or mentally exhausted. Even students who are not extremely advanced will get bored with too much repetition, and this can hinder math progress as well as leave students thinking math is uninteresting and uncreative when nothing could be further from the truth.

My radical solution is to give up on the idea of mastery, albeit temporarily. Some educators (though fortunately not all) are obsessed with the idea that students must memorize their multiplication tables, perform spectacularly on timed tests, and get 100% on assigned work before they can move onto more interesting stuff, but this is entirely backwards. The more advanced the student is, the more backwards it is. To let you in on a secret: the narrow track we’ve created for math education is almost entirely artificial. Math is such a broad and rich field that there’s simply no reason you have to stay on path. You can mix arithmetic practice with probability with some beginning algebra with number theory and cover more than one topic at once. I have especially high regard for discrete math which is all but ignored by traditional K-12 schooling. Of course all students should master multiplication tables and become automatic at traditional algorithms, but not to the exclusion of other learning – by exploring multiple threads of math at the same time its much easier to keep interest high, tedium low, work longer, and gradually build stamina for more difficult work.

Just the other day I came across this profound statement in Art of Problem Solving: The Basics, Volume I:

We strongly feel that a student should learn all subjects simultaneously. There are two reasons for this. First, it helps to convey the interconnectedness of it all; how geometry naturally leads to coordinates and how those coordinates make it easy to define conic sections and the complex plane; how counting leads to probability, the binomial theorem, and number theoretical ideas. Second, it all sinks in better. Overloading on a single subject can cause students to acquire a surface understanding which doesn’t connect to any deeper comprehension, and is thus rapidly lost.

In my next post I’ll give some specific examples of how to include more variety in your students math education.

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Take a break from the workbooks



Whether you have a student who lives and breathes math or a reluctant one who balks at the idea of worksheets, living math books, where math is integrated into a story, might be the perfect tool for extra math learning.  Unfortunately many story based math books are low quality in the sense that the math itself is either very basic or the storyline is contrived and uninteresting. The following resources have met my criteria for both covering math exceptionally well and also being something that a child might be motivated to pick up and read over-and-over again on their own.

Sir Cumference

This is a wonderful series of picture books by Cindy Neuschwander for the youngest of students. The stories follow the adventures of the Knight Sir Cumference of the Round Table, his wife, the Lady Di of Ameter, and their son Radius as they uncover geometric properties of various shapes and other math truths. Books in the series:

Sir Cumference and the First Round Table
Sir Cumference and the Dragon of Pi
Sir Cumference and the Great Knight of Angleland
Sir Cumference and the Sword in the Cone
Sir Cumference and the Isle of Immeter
Sir Cumference and All the Kings Tens
Sir Cumference and the Viking’s Map
Sir Cumference and the Off-the-Charts Dessert
Sir Cumference and the Roundabout Battle

Penrose the Mathematical Cat

The Adventures of Penrose the Mathematical Cat by Theoni Pappas is by far our favorite living math book and great for kids aged 6-12, especially ones that enjoy animals and unusual math topics. When he was younger my son went through a phase of liking books that explore the perspective of animals trying to understand the human world and this one definitely fits into that category while also having an incredible amount of interesting math layered into it. The story part of it is a bit thin, but Penrose the Cat is such a fun character that you don’t notice too much. Topics include things that normally fall well outside of a typical elementary math curriculum including chapters on binary numbers, Fibonacci sequence, tangrams, and infinity. The sequels –  The Further Adventures of Penrose and Puzzles from Penrose  are also worth getting if the first book is a hit with your student.

The Number Devil

The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger is about a boy visited by a math loving demon in his dreams. Covers both basic and advanced number concepts in a casual way with a highly integrated and whimsical story. The chapters on Fibonacci numbers and Pascals’ triangle are particularly well done. Perfect for ages 8 and up.

Murderous Maths

The Murderous Maths series by Kjartan Poskitt is our latest acquisition of living math books and I’m glad I splurged on the box set, because its been a huge hit! Each book covers a different topic and is a dense chapter book embedded with comics, diagrams, and boatloads of humor. The topics range widely from everyday math basics like geometry, measurement, and probability to quick mental math tricks to math trivia to far out there subjects like graph theory. The humor and presentation make it perfect for math lovers and not-so-mathy students alike, especially for those in the 9-12 year-old range. This series is published in the UK but easy to find on Amazon at a reasonable price and a few are also available in Kindle format.


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Math roundup this week!

I spend a lot of time trying to keep math interest high by introducing concepts in a fun way, exploring old material from new angles, and bringing hands-on learning to our “Math Lab” whenever possible. This isn’t always easy when delving into more advanced math topics – baking with fractions is one thing but information theory, combinatorics, and algebra are quite another. However, I firmly believe this is the only way to teach talented math students at the elementary level without burning them out or turning them into grinds. Fortunately I love math and computer science so the hours I spend researching and planning activities is never a chore. It does involve a lot of experimentation and dabbling though. Not every topic is brought to completion in a neat, well thought out package. A lot of math enrichment is about throwing cool ideas out there and seeing what sticks. My hope is that even if something is not a hit, I’m planting a seed for future investigations. This week I’m giving you a peek behind the curtain to see how random our weekly math explorations really are sometimes. Fortunately most of them were a hit, but that’s not always the case!



If you don’t know what a Flexagon is, you are really missing out, and even if you’ve heard of them you may not be aware of all of these awesome mathematical properties. First discovered by British graduate student Arthur Stone in 1929, Flexagons are geometric models that have hidden faces when flexed in certain ways. They are super simple to make out of paper and a fun project for home or the classroom. I was pretty dismissive of the math behind them until I started doing some research and then was blown away by their awesomeness. The famous physicist Richard Feyman even invented visual state diagrams, which are especially complex for hexa-hexaflexagons and my student has had a great time trying to make his own.

Non-Transitive Dice Paradoxes


We’ve been using two new resources this week to study probability in a novel way. The Amazing Mathematical Amusement Arcade by Brian Bolt is a fantastic book that distills famous math puzzles into a style that is more visually appealing for kids. I picked this up at the library last week, and just coincidentally a puzzle in this book called the Gambler’s Secret Strategy completely mirrored a more technical chapter I was reading on non-transitive dice paradoxes in Martin Gardner’s Colossal Book of Mathematics. The crux of the puzzle involves dice that have equal total values, but different numbers on each face, such that they allow a gambler to always have a better chance of winning in a head-to-head roll even when letting his opponent choose first.  We created paper dice and built conditional probability trees to prove that the gambler always has the advantage. Definitely a topic we will be revisiting.

Pentagonal Numbers


I put this sequence on the board one day and told my student to ponder it.

1, 5, 12, 22, 35, 51, 70, 92, 117…

The first day he told me that he didn’t know what the pattern meant but that he had figured what the next number in the sequence was (145). He then created a formula for figuring out the next number in the sequence given the two previous. Then, since he had trouble visualizing what these numbers meant, we drew pictures to illustrate why they are called pentagonal numbers. Now we’re trying to figure out what the connection is between this series and the series of triangular numbers. After that we will explore the set of integers that cannot be expressed by a sum of three pentagonal numbers. This one has been a huge hit. I love these long term math explorations!

Recent reading


I have a series planned on recreational math books for kids. Our latest acquisition in this category is a British series called Murderous Maths. Surprisingly dense and a whole lot of fun. Expect a detailed review soon.

Beast Academy 5c. The long awaited next book in this elementary math series from Art of Problem Solving. My student has moved beyond the math in this series but he still loves reading the comic book adventures of the little monsters as they solve math problems, and the workbook makes for great review.

The Thrilling Adventures of Lovelace and Babbage – a fun and fascinating romp about two of the founders of the computer age. Only slightly educational but a great graphic novel for any geek. I wasn’t expecting my son to get into, but he’s been dragging it around and re-reading like crazy.



We’ve also been taking our digital logic to the next level with a free circuit design tool called Logism. It’s very easy to use and allows for quick design and simulation of functional circuits. He had no problem completing simple projects of building a half-adder and full-adder, and is now working on a 7 segment display controller. So many possibilities here and an easy way to explore electronics without an expensive science kit.

That’s our math roundup for the week! We’ve been trying to enjoy our gorgeous fall weather here on the east coast by doing less school and spending more time outdoors, but we always have time to squeeze in lots of math so stay tuned.

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