# Category Archives: Uncategorized

## Playing with logic

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)

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?”)
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.

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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## Stretching math through the summer (part 4)

This post is about middle and high school summer math, and it’s going to be short, because let’s face it, teens don’t have a ton of time for complicated programs.  It’s difficult for busy families to find the time to supplement their child’s math education, even if the student is highly motivated and loves math.  This is especially true for middle and high school students who often have busy social lives, jobs, activities, and their own opinions about where their education should be headed.

I believe in the power of doing small amounts of regular practice: 15 minutes of problem-solving, done consistently, is far more valuable than a marathon session once a month. However, my problem-a-day approach for elementary students – cutting out word problems and putting them on the fridge – may not seem quite so cute to a teenager.  Also, unless you have a particularly strong math background yourself, it can be tough to know where to direct your older students and to correct their work.  Fortunately there is a great tool out their for this age group called Alcumus.

Art of Problem solving is my favorite math eduction organization, and Alcumus is their free online learning tool, with over 13,000 problems.  Students select a subject at their level (such as pre-Algebra), pick a topic within that subject (such as decimals), then start solving problems in order to gain experience points, work on quests, and earn badges all with unique art work and goals.  For instance, the “Third Times a Charm” quest gets completed when 3 new topics are mastered, and the “Life of Pi” badge gets earned when correctly answering a question in terms of pi.

Statistics on student performance are displayed in a similarly fun manner, as Dungeons and Dragon-style character attributes.  Experience points are earned for playing regularly, power points are earned for solving problems consistently above level, and resilience points for not giving up after wrong answers.  The look and feel of Alcumus is ideal for teen students: fun and motivating, but without being cute or overly gimmicky.

Alcumus is adaptive and gives extremely detailed and complete solutions to problems. The detailed solutions help students understand their answers, where they went wrong, and alternative ways to think about the problem. When students get answers right, the challenge level increases, while incorrect ones dial back the difficulty to give practice at lower levels.  This keeps the frustration level low while still stretching student learning.

To get started with Alcumus go to the Art of Problem Solving website here, register for an account, and jump right in!

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## Stretching math through the summer (part 3)

So far I’ve focused on summer math for elementary students who are ahead and for those who are right on track. What about students who are behind and need remedial work to catch up? Summer is an ideal time to work on math concepts away from the pressure of school. Creating an environment where math is fun and relaxing is important for all students but especially for those that have low confidence in math.

If you have a student who is behind in math, its important to look at where the problem really lies. If they have a good grasp of basic concepts but just need a little more fluency with basic math facts, then games are an ideal way to practice without stress. You can use workbooks for more advanced topics like long division or fractions, but keep it short and simple: only a few problems per day. Dense pages of similar problems can lead to fatigue, and tired students make mistakes.

In many cases though, the problem isn’t just an issue of practice, but a true lack of understanding. Unfortunately, when a student doesn’t understand basic concepts there are no short cuts; they need one-on-one attention with a focus on really building number sense rather than memorizing mindless procedures. Hands-on activities should be the first resort for remedial students, and so enters my favorite math manipulative… the C-rod.

Cuisenaire rods (or C-rods for short) are one of the most amazing math tools of all time. Developed in the 1920s by a Belgian school teacher and popularized in the 1950s, the set of colored rods representing numbers 1-10 (each rod a different length and color) can be used to teach almost any basic arithmetic concept.  I like to start little ones out with simple games like identifying rods by number, building a staircase of numbers by laying the rods in order, and hiding one of the rods to guess which one is missing.  Basic arithmetic begins with the simple idea of adding one, then by identifying pairs that add up to 10, a foundational concept necessary for the kind of regrouping needed for addition and subtraction of larger numbers.

Why C-rods and not coins, counting bears, or other discrete objects?  C-rods move children towards thinking in terms of whole numbers, rather than seeing each number as simply a collection of ones.  It’s a subtle distinction, but an important one for developing the ability to break numbers apart and put them back together again.  It seems counterintuitive, but counting is inherently inaccurate, and the faster you can move a child away from counting, the better. Cuisenaire rods are great for basic concepts, but also really shine when it comes to more advanced topics like multiplication, division, and fractions.

It’s hard to understand how they work without a real demonstration, and so rather than try to describe how to use them in words, I like point people to Education Unboxed, a website wholly devoted to teaching with cuisenaire rods.   Rosie, the site’s creator, uses her own three daughters in these charming and engaging videos to demonstrate how to teach everything from addition to fractions to decimals to basic polynomial factoring.  I recommend watching the videos in advance and teaching your child directly, but if you are really short on time, you and your child can watch the videos together to try and work through the concepts.

What about Khan Academy videos?  This is a topic for another post, but in general Khan Academy is incredibly overrated.  The videos on their website mostly show procedures and tricks, and do not teach for understanding. While this can be somewhat useful when reviewing material that was once mastered a long time ago, it doesn’t really help a student who never understood the concepts in the first place.

If you want to get started with cuisenaire rods I recommend a basic set of 155 rods for about \$20 on Amazon.  The Education Unboxed videos are available for free.

Next up:  Summer Math for the advanced high school student!

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## Stretching math through the summer (part 2)

Whether its warm days spent at the lake, roasting s’mores on a family campout, or long afternoons spent reading in the backyard, summer is a great time for relaxing and decompressing from the rigors of the school year. A break from routine and a chance to get outdoors is good for everyone’s mental health, and children are no exception. For many families, however, summer also causes stress. Aside from issues of childcare and keeping kids occupied during the summer, a three month break from academics can be way too long for some students. All skills are subject to a “use it or lose it” deterioration, but math skills are the most perishable.  A small amount of regular math practice over the summer can mean big gains in the fall, in terms of less review and less frustration.

However, there’s no need to ruin the family vacation with pages of boring worksheets. Parents should be partners in student learning, not taskmasters or adversaries. Math practice at home, particularly in the summer, should be fun and engaging. Last week I covered Art of Problem Solving’s Beast Academy, my favorite program for high-flying elementary students. This week, I’ll talk about two more excellent books: Primary Challenge Math and Real-World Algebra, both by Edward Zaccaro. These books are ideal for students who have a good grasp of basic concepts, and want a little more challenge, but aren’t necessarily ahead.  I recommend Zaccaro’s books for summer math because they are substantive with a focus on problem solving, but they don’t feel like workbooks. With cartoon illustrations and plenty of white space, the pages are inviting rather than intimidating.

Primary Challenge Math is a great starting point for elementary students in grades 1-4. Each chapter covers a particular type of problem with cute headings like “How much does it cost?” (money) or “Oh no! I have to change the recipe” (fractions) and includes 4 different levels of problems from novice to “Einstein.”  This format makes it very adaptable for different ages and abilities. In first grade my son worked to Einstein level in a few chapters, but for topics we hadn’t covered yet in our regular math program like decimals and percent, he only attempted the novice level problems, saving the rest for later.

Real World Algebra is very similar to Primary Challenge Math but is geared towards students in grades 4 and up. It gives younger students a lightweight taste of algebra and older students the chance to build confidence before middle school math. What impresses me the most about Real World Algebra is the focus on using algebra as a language to structure word problems. While there is some focus on procedural steps to problem solving, most of the book is simply about setting up problems correctly by translating the words into equations. As you know, word problems are my favorite because they really build and test the student’s mathematical understanding. Along those lines this book makes algebra seem like an easy way to solve word problems rather than an intimidating conceptual leap.

You can purchase Primary Challenge Math, Real World Algebra, and other books by Edward Zaccaro, from Amazon or other retailers.

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## Stretching math through the summer (part 1)

Summer is still a few months away but I know many parents are already looking for ways to build or maintain their child’s math learning over the summer. There are several great options that I’ll be discussing in the coming weeks for every age and ability level, but for math-talented students in grades 3-5 my first recommendation is usually Beast Academy from the Art of Problem Solving (AoPS).  Although Beast Academy is a full year-long comprehensive math curriculum, it comes in an incredibly fun and cute package, which makes it ideal for summer.

Every grade level has 4 parts (A-D) each consisting of a comic book, which serves as the textbook, and also a corresponding workbook.  The Beast Academy characters in the comic books are fun and talk the student through new concepts, while the workbook problems themselves are very puzzle and game-like. True confession – my son has been through the entire available Beast Academy program and still likes to use the comic books for bathroom reading. They are that engaging!

My only word of warning is that Beast Academy is both challenging and unconventional. Art of Problem Solving made a name for itself with advanced high school and math contest prep materials, and Beast Academy carries their tradition of using a discovery based approach to mathematics and deep problem solving to the elementary level. It is ideal for students who love math and love to figure things out for themselves, but I would not recommend the workbook for a student who already struggles with math, as the problems found in the book can sometimes be intimidating.

When using as a summer supplement expect to get through only one book, unless you’re using it a grade or two behind as review, something that I actually recommend as most fifth graders who have been taught conventionally could still get a lot out of the lower grade Beast Academy books. The depth of this series is incredible – in addition to standard fare like multiplication, fractions, and decimals, Beast Academy covers some topics that are seldom seen in elementary school such as logic and basic combinatorics.   At the same time I also wouldn’t hesitate to try these materials with a very advanced math student who is a grade or two younger (my son started 3A in first grade). The fun presentation and packaging make it ideal for a student who is intellectually ahead but lacking the maturity to jump into more serious math texts.

AoPS has full sample pages of both the guidebooks and workbooks for each level on their website.  Materials can be purchased directly from AoPS or through some discount homeschool catalogs like Rainbow Resource.

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