Category Archives: Games

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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Games as a tool for practicing math facts

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When it comes to the debate between basic skills practice vs. conceptual understanding, I stand firmly in the middle – both have to be in place for math progress to occur. A student who is great at the rote practice of algorithms, but who lacks understanding of the underlying concepts, will often look like a whiz through elementary school but may stall out when they get to the more abstract ideas in algebra. On the other hand, students who have a clear understanding of concepts, but who are slow with calculation, will be bogged down significantly in their problem solving, which can lead to boredom and frustration. This is my fancy way of saying that learning your times tables does not make a math education by itself, but is still very important.

Practicing calculation gets a bad rap, however, because it usually involves boring worksheets. It gets even worse when a timed element is added. I hate timed tests for elementary students! Either the student is good at them and doesn’t need that kind of practice, or they’re not good at them and their self-confidence crumbles with the ticking of the clock. Instead of worksheets, I recommend games for practicing basic skills. Even the most contrived game is going to be more fun than flashcards or a worksheet.

Recommendations vary a lot with the age of your child and what they need to work on, but for a student who is struggling with self-confidence or is math-phobic I like starting with commercial games that have some skills practice but don’t look too educational. In that category I suggest:

Zeus on the Loose – An incredible game for practicing rounding skills and mental addition to 100. We made our own variation that reverses the meaning of the cards to practice subtraction as well. My top choice in this category.

Lost Cities:The Card Game – A game that I actually play with other adults for fun, but also that just happens to involve a lot of mental addition, along with a bit of subtraction and multiplication.

Star Realms – A fast paced card game that involves keeping track of several different kinds of points (combat, trade, and life) and uses mental addition and subtraction on every turn. The numbers start small but build quickly. We love the space theme and cool art work.

If you need a little more focus, and your student doesn’t mind practice that’s more obviously about math I recommend the following:

Sum Swamp – Fun for the littlest elementary kids. Basic addition and subtraction for Pre-K to 2nd grade.

Double Shutter – Fun and hands-on way to practice breaking apart numbers from 1 to 12. Especially good for Pre-K to 1st grade.

Math Dice – This is a cheap and portable game that involves rolling dice and using addition, subtraction, multiplication, or division to reach a target number. There are a few variations including a junior version with larger dice, Math Dice Chase, and Math Dice Powers.

Prime Climb – A little bit of an investment but a beautiful game with phenomenal game play. A roll and move board game that involves practice of addition, subtraction, multiplication, and division, with an added introduction to prime numbers and factorization. My top pick in this category!

Finally, outside of commercial games there are a ton of facts practice games you can play with just a deck of ordinary playing cards. My favorite is “Multiplication War” – a simple adaptation of a childhood classic that involves each player turning over two cards and multiplying them together to compare scores. This handout describes several variations and many more math games that you can play with just a deck of cards.

Although computer programs and apps would seem a natural for practicing math facts, most I’ve seen are little better than dressed up worksheets or flash cards. Games that involve physical components like cards and dice are a much better way to practice mental agility with numbers because they are competitive and engaging. Cognitively, I believe more learning takes place when there is a human connection and something tangible to manipulate.

Games also fit well with my learning philosophy of starting small and working frequently.  For busy families, its a way of spending quality time together in a constructive way, and if you have multiple children parents don’t even have to be supervising.  Siblings can play by themselves or you can enlist grandparents, cousins, and other relatives.  Uncle Joe probably doesn’t want to supervise homework when he’s visiting, but he might just be willing to play a few rounds of Star Realms with the kids.

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