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

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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

 

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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!

Flexagons

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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

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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

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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

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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.

Logic

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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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Playing with logic

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I still remember the first time I took a formal symbolic logic class in college. My first thought was: “Where have you been all of my life!” As a computer science major with tons of additional coursework in mathematics, I was no slouch when it came to logic and systematic thinking. However, it was a simple formal logic class offered through the philosophy department that revolutionized my math education.

Unbelievably, I had never been taught how to do formal proofs, and since high school they were something I had muscled my way through with a little bit of intuition and a whole lot of imitation. Had I learned the basics of formal logic from the beginning, I would have saved hours of pain over the years and greatly deepened my understanding of mathematics. So when my 9 year-old son started to show an interest in logic, I decided to run with it, hoping to build a foundation that might someday lead to the skills he will need higher level math courses.

Learning logic doesn’t have to be difficult as it naturally lends itself to games and puzzles. We’ve had a great time with the following activities:

Sudoku Puzzles. These are an excellent way to get started with logical problem solving. I find the typical 9 digit sudoku size too large for most children starting out, so I like to begin with 4 digits and work up from there. You can search online for sudoku puzzles of all sizes. A magnetic travel set based around 6 digits has been perfect for my student and has lasted for years. In addition to being able to set up dozens of configurations at differing challenge levels, moving the magnetic pieces is easier than constantly erasing and rewriting.

Chocolate Fix. If you’re willing to invest a few more bucks in developing logic skills, this Sudoku-like puzzle by ThinkFun has a lot of visual appeal. Players arrange trays of chocolate based on clues given in the booklet.

Mastermind.  This commonly-seen family game is incomparable when it comes to systematic thinking. One player creates a color code and the other person tries to guess the code based on information provided in response to each guess (white for correct color and red for correct color and position).  To make it easier at first I started my son out using only 3 of the 4 spaces and no repeated colors until he was able to play the full game.

If you’re looking for something that students can do more independently, the following workbooks have been a hit here:

Lollipop Logic. A series for the younger set (K-2). Introduces the fundamentals of logical thinking including picture sequences, patterns, analogies and deduction.

Grid Perplexors. We absolutely love this series of workbooks from Mindware that involve telling a story and using clues to fill in the missing information.  The out-of-print Venn Perplexors are also excellent, if you can find them.

Beast Academy 4B. I’ve talked about Art of Problem Solving many times and how their Beast Academy series uses a comic book format that makes new topics very approachable. Even if you don’t do the workbook or use the rest of the curriculum, the guidebook for 4B introduces logic in a really entertaining and engaging way. The puzzles in the workbook are outstanding as well.

If you end up with a logic-loving student like I have, there are a few choices that really build skill and provide a natural pathway to more formal logic studies:

Liars and Truth teller Puzzles. These are traditionally known as  “Knight and Knaves” puzzles and for whatever reason my son is obsessed with them. These problems involve an island where all of the inhabitants either always tell the truth (Knights) or always lie (Knave) and you have to figure which is which. A classic example is:

Two men are standing at a fork in the road. One is standing in front of the left road, and one is standing in front of the right road. One of them is a truth teller and the other a liar, but you don’t know which is which. One road leads to your destination, the other to death. By asking one yes/no question, can you determine the road to take?

I downloaded this Knight and Knave Puzzle Generator that gives us an infinite supply of new problems and solutions with varying amounts of statements. Right now we are solving them using truth tables, but since the generator also includes the formal symbolic logic representation of the statements, these puzzles will be the natural transition for my son to learn symbolic logic.

Digital Logic

If you have a  background in engineering or computer science you might have fun introducing your students to the basics of digital logic. Refreshing my memory of the basic digital logic circuit design class I took in college, I made paper AND, OR, NOT, etc. gates for my student to build simple logic circuits and used colored Go pieces to provide inputs and outputs. If you are also familiar with Boolean Algebra, this is a great intermediate step to formal symbolic logic. The MIT OpenCourse Introductory Systems Laboratory has a problem set that is not too difficult for beginners.

Whatever approach you choose, know that a little bit of logic can go a long way towards furthering mathematical understanding. This won’t be the last time we visit the subject on this blog so stay tuned.

 

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How to teach binary numbers to a Kindergartner (and above)

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I was fortunate enough to first be exposed to binary numbers, also known as base 2, in a fifth- grade gifted math program. But even in the upper grades or as an enrichment topic, few students get introduced to binary even though its the basis of all computing.

This is a shame, because binary numbers are quite easy to get started with, even for very young students of kindergarten age. If a child can understand the concept of doubling and some very simple addition then they can understand binary numbers.

Before delving into some methods for teaching binary numbers, first I’d like to talk about why. First off, it’s really fun! I disagree with the idea that something has to be useful to be worth studying. In fact, I think we do children a disservice by emphasizing the utility of math instead of the fun and beauty. No one tells a ten-year-old they should do art because they might be a famous painter someday; instead we focus on the idea of creativity and self-expression. However, math is often treated differently, with parents and teachers talking about the need to balance a check book, calculate area, or pass algebra to get into nursing or engineering school. There is not enough emphasis on the aesthetics and reward of puzzle-like problem solving. When it comes to binary in particular, I like to treat it as a secret code. Kids love the idea of secret codes, and it’s all the more fun because binary numbers look like base-10 numbers in disguise (it’s not a thousand and one, it’s nine!).

Also–and this is a topic for another lengthy post–I believe anytime you can stretch the brain with a new mathematical concept, even at a very introductory level, it helps solidify understanding of mathematics as a whole and can accelerate overall learning, regardless of its immediate practicality.

To get started, you need a set of Cuisenaire Rods (C-rods), some colored construction paper, and a few stones or tokens.

Here’s my method:

  1. Take the C-rods for 1, 2, 4, and 8, put them in order, and have the student talk about what pattern is represented. It’s important in teaching math concepts to let the student come up with their own solution, rather than just giving the answer, as they will remember it much better. But eventually you should guide the student to come to the conclusion that each rod is double the previous one. You can expand on this as much as you want to in order to meet your students understanding level. If they are familiar with multiplication you can talk about “times 2,” and if they understand exponent notation you can add that in as well, but for a kindergartner, doubling is enough.
  2. Let them have one of each color rod (1,2,4, and 8) and challenge them to make other numbers. If you only have one of each of these rods, can you make a 3? A 5? How high a number can you make only having one of each of these four colors? (The answer is 15)
  3. Bring out the Papy Minicomputer! This teaching tool was invented by Fredrique Papy as part of CSMP, a “new math” program of the 1960s. It’s basically just a four-color square grid in the same colors as the C-rods, very easy to make one out of construction paper. Tell your student that the colors on the minicomputer are the same as the C-rods: white=1, red=2, purple=4, brown=8. BTW, I love using the word “minicomputer” because it piques a child’s interest quite a bit, (“Wow! How is this piece of paper a computer?”)P1040712
  4. Give the child four stones or identical tokens and tell them that we’re going to practice making numbers, just like we did with the C-rods, except this will be faster because its on the minicomputer. Tell them the one rule to remember is that there can only be one stone on each color square. Put stones on the spaces to demonstrate – for instance one stone on white equals 1, but a stone on white and red equals 1+2 or 3. The minicomputer picture above represents the number 6 (4+2). Spend quite a bit of time practicing with this and bring out the C-rods in conjunction if they have trouble visualizing it.
  5. After having them practice reading numbers off the minicomputer for a while, tell them it’s their turn to make numbers with the stones and the minicomputer.  For example, you can say: “What does 7 look like on the minicomputer?” If they try to put two stones on one space gently remind them that the rule is only one stone per space and to try and look for another option.
  6. After they are comfortable building numbers, have them learn to read you the configuration of their minicomputer starting with the biggest place value first and working down to white. “Zero stones on brown, one stone on purple, zero stones on red, one stone on white.”
  7. Eventually, as they get more comfortable, this will get shortened to just the number of stones. For instance: “zero, one, one, zero.” Have them practice writing this down. You can even make a game of it where they have to build a number on their minicomputer in secret, tell you the “code” of ones and zeroes, and then you can use the minicomputer to recreate it to guess their number.
  8. With practice, many children will start to do the work in their head. If your child is as nerdy as mine, they’ll start playing with the concept in the real world: “Hey my shoes are in cubby one, one, zero, one!”

Natural questions about what happens if we want to represent a number greater than 15 will arise, and you can add in more digits for 16, 32, etc.  Also, once they are fluent in binary, introducing other base number systems is a much easier process (especially if you stick to only two digits to start). Our family developed a whole animal math game in which we calculated in base 4 “ostrich math”, base 6 “ladybug math”, and base 8 “spider math.” Unfortunately there are not man animals with an odd number of feet!

Obviously this method does not yield the same depth of understanding as for a student who really gets place value and exponents, but its a whole lot of fun and a great jump start on a concept that many won’t see until their first Intro to Computer Science course.  Plus, having your kindergartner understand 4-digit binary is just a really cool accomplishment!

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I’m not a fan of math apps, except for this one.

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When it comes to long car rides or waiting at the doctor’s office, books and audiobooks are my first choice for entertaining my son. However, when it becomes necessary to pull out the iPad, I’m grateful for the wide range of educational apps that are a better alternative to the mindless entertainment of Angry Birds. But the problem with educational apps is that most promise a lot more than they can deliver and are frequently little more than dressed up flash card programs. There’s a place for this kind of practice, but from a cognitive standpoint things that involve human interaction and tangible manipulation like board games, songs, and Cuisenaire Rods are going to lead to better retention even for something as basic as learning multiplication facts. For a child that really needs to work on math skills, computer apps should definitely not replace one-one-one tutoring and problem solving.

The DragonBox games, however, are a big exception to my general “no app” math learning policy. Unlike math flashcard programs, which focus on rote arithmetic practice, DragonBox introduces the fundamentals of algebra and does so with unique visual puzzles. Each level begins with a split screen and a scattering of boxes with different pictures on them. The object of the game is always the same – to get the main glowing box all alone on one side of the screen, the equivalent of isolating the variable in an equation. You can manipulate the boxes in different ways that are analogous to arithmetic operations, for instance you can remove a box from the screen by placing a another box on top of it that is identical except for reverse colors – the equivalent of adding an inverse. What’s tricky however, is that whatever you do to one side of the screen you have to do to the other, imitating the balance necessary in manipulating equations. Most of the learning in DragonBox takes place gradually through exploration and experimentation and without a lot of direct instruction, a real plus in my opinion as I’m a big fan of the discovery approach to mathematics.

The geometry version of the game, DragonBox Elements, is even better in my opinion. The graphics are first rate, and the game introduces and reinforces geometric concepts in an incredibly engaging way. As the tagline for the app states: “Every puzzle is a geometric proof.” In DragonBox Elements the player brings creatures to life by completing the proof-like puzzles, for example tracing equivalent lines and angles in a geometric construction. It’s a fun way of exploring the relationships between shapes, and also highly addictive!

Both of these games have received a lot of press, with some of it a little exaggerated like this Forbes article claiming that it takes only42 minutes to learn algebra through this app. Obviously there’s a lot more to learning algebra than just moving boxes around on a screen, but as long as your expectations are reasonable it’s a terrific math tool. It’s not a magic bullet that will catapult your student into advanced mathematics, but it does really shine when it comes to basic skills like learning to simplify terms as much as possible. DragonBox Elements is my favorite in the series, but I found my son needed quite a bit of reinforcement to really understand the geometric constructions because it lacks any degree measurements; it’s not a flaw of the game, just a limitation. Despite the press, the designers of DragonBox seem to recognize its inherent strengths and weaknesses and so offer pdf supplements to parents and educators that can help translate concepts in the game to the classroom. This is a great resource if you have time to use it.

The bottom line is that DragonBox is a fun and inexpensive way to introduce some basic algebra and geometry when you have a few minutes to kill on a tablet. The original DragonBox comes in two levels: basic for ages 5 and up, and advanced for ages 12 and up, while Elements is appropriate for all ages. They are priced at $4.99 through the Apple App store, Google Play, or Amazon.

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