Monthly Archives: July 2016

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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