Hi. My name is Scott Laidlaw. I started teaching middle school mathematics six years ago after a three year tenure teaching at Appalachian State University. I've been a middle-school math teacher since, most of those years at a public school in New Mexico teaching 5th to 8th grade of all abilities.

I started using math games in 2003 when I moved to teaching in New Mexico where I faced the prospect of leading a classroom where the average student had less than 28% percent chance of scoring in the proficient category on the state test (not unlike many teachers throughout the country).

I had the idea to build an epic math game at 35,000 feet. At the time, our school taught thematically and we were studying the Renaissance. On a plane on the way to a conference, I was reading about the spice trade to bulk up on my personal knowledge of life in the 1600's, when I opened up "SkyMall" magazine to find an enormous map of the world, 10 feet high and 14 feet wide! I thought, "Hey cool, let's buy that map and have kids "sail around the world" to learn basic functions and ratios while they trade spices." I talked with my administrator about the idea, and 6 weeks later the map showed up. We covered it with 3000 sticky notes (a literal number) to hide the known world, made some wooden ships and rules, and the kids came in for an after-school kick-off of the math game called Piracy! It was amazing, wonderful, and absolutely fraught with problems that led me to pull out my hair.

Humbled by the ridiculous map, the next year, I worked with a colleague named Todd Wynward who really ramped-up the meaning of "math game" to become a meaningful means of teaching math content, and together we created a game called "Kingmaker," a story-book approach to gaming, where students started by learning that they'd been exiled from their Kingdom with 12 other peasants and a pig. As "rulers" of a small pitiful fiefdom, the students taxed their population and calculated disease as they grew a little Kingdom. Played individually, we learned that games could be an excellent medium to teach math with depth. With Kingmaker as the support piece of our math curriculum that year, our math state test scores soared.

Despite their eventual success, we made a lot of mistakes in those first two games, including no less than three mortifying experiences in the early years where the math games "broke," and with hundreds of hours poured into the curriculum, I felt like giving up. But I had supportive administrator, creative in his own right, who kept saying, "let's revise this and make this better." Over the years, it seemed like Todd revised the basic design of my math games about 100 times. "Through feedback," he would say, "we become excellent."

It was the following year when I launched a math game at school that finally worked with a hands down effective approach. I had been in Peru for the prior summer, studying ancient cultures and kept running across a major story-line of ancient rites of passage where adolescents readied themselves for adulthood through a personal wilderness journey. With the help of our school staff, parents and volunteers, major story line work by Todd, and gems of ideas from colleague Stephanie Owens, I created a math game called, "Ko: Rites of Passage," the story of a child living in wilderness 10,000 years ago. The aesthetics were beautiful: five deer hides hand-sewn as the platform for a hand-painted map, stone animal totems from Peru, an actual antique box, and a shaman robe. But the major upshot was that I found a basic math game design that worked solidly in the classroom. I linked "Ko" to national standards of learning outlined by the National Council of Teachers of Mathematics, and built a game that appealed to girls as much as boys. Through other aspects of our math program such as tutoring, math artwork, math writing and teaching students to transfer their knowledge to the state test, we easily doubled the average state test score in mathematics, even with a group of 5th grade students who entered the year below the 28% proficiency rate.

After that year, it was much more than test scores that kept me going. "Can we play the game?" became a mantra of days, weeks and months to come. The desire of the students kept me wanting to learn more, and get better at math game designs. At break, kids wanted to play the game. At lunch they wanted to play the game. Last spring break, I even got a call from a student who asked to be "cleared" to move to the next level. Really.

With this energy, in the years to come, two more math games were created, "Destiny" and "Empire." The former was about living in an old west gold town, and the latter about a child ruler in Ancient Mesopotamia. A software application created for Empire by Loren Johnson of Venado Partners allowed each student to maintain a personal "rulership". Empire eventually received national press and showed me just how powerful a software application could be in engaging students. And, Empire was "hard core" math, which required following an 8 step algebraic equation to complete each turn, checked by computer to get through. When I tested it the first time, it took me 45 minutes to complete one turn! But the kids were engaged and worked through problems I wondered if ever could be completed.

After a group of schools from around the country called and asked to purchase Empire, I realized it was time to help other teachers. Imagine Education was launched, and we chose Ko (now called Ko's Journey) as our first major product.

Ko's Journey is a on-line math game that will be released on January 12, 2010. As a web-based interactive story-based design, I can only say that the team building it has done so with a programming aesthetic not-too distant from the work of volunteers on those leather hides years ago, undergoing multiple revisions, with the ultimate goal of creating a simple to use, engaging and effective learning environment for middle student to learn the core threads of middle-school math.

-Scott Laidlaw, Ed.D.
Director, Imagine Education

Every teacher wants their students to enjoy math class, and often they turn to games to increase student engagement. Unfortunately, many teachers see short-lived interest or have a hard time finding games that teach more than just multiplication tables. But games can be powerful educational environments for learning math if structured correctly. Over the past six years, I created and used math games in a variety of forms, from quick, five-minute, single rule games to semester-long, adventure theme games. They became central components to my classes, and during that time, my students scored well on state tests even though they generally entered our school below average.

"Do we get to play the game?" became a mantra I've heard literally thousands of times, and it's that kind of engagement, coupled with clear teaching strategies, that works to create a wonderful classroom environment.

The nice thing is, it's relatively easy. I've compiled here a list of my best suggestions to make your own game in the classroom. The truth is, anyone can design their own math games, and if you use the following template, you'll have a math game up and running in a couple of days. Fortunately, most problems are easy to avoid, and you will quickly be on your way to a new, even enchanting, approach to teaching math in your classroom.

Building a Game:

1. Pick a theme or story. It really doesn't matter what it is. Just pick something in which you are interested and go with it. From rockets to pirates to cooking to camping to fishing to the economy, you have a story to tell. And, don't worry about teaching math yet, or how it will happen. It will come.

Example Theme: We were studying the Renaissance so I decided to make a spice trading game with companies, countries and pirates.

2. Make a simple goal for the game. Again, it doesn't matter what it is, but create a simple goal for students. In competitive games, fear among students and math is highly prevalent, so if you set the goal for winning against another team, allow for some re-start mechanism.

Example Goal: Acquire spices, bring them to home port, trade them and make money to purchase supplies, and even weaponry. For the re-set, if your ship was sunk, you could borrow money from the bank to upgrade (with interest of course!).

3. Create a simple turn-based structure. Not all of my games have a turn-based structure, but most do. It's a simple way to think about games. A single "turn" in the Renaissance game Piracy was one move of their ship. Think about the math you'd like involved and keep it simple!

Example Turn: Student A purchased a "schooner" and therefore can move .7 times the wind speed. The wind speed is determined by the roll of a die. Student A rolls a 3, and therefore can move 2.1 knots/hour. This is multiplied by 24 hours so the student can travel 48 + 2.4 or 50.4 knots on that day. This gets represented on a scaled map that they make themselves. 

4. Make a platform and test it! Have your students draw a map and scale it themselves after a quick lesson. Have them practice your new game by doing something simple and non-competitive, like sailing around a buoy. Don't add more to your game just yet. Just test and revise the turn-based structure a couple of times. Don't worry about keeping track of turns either. Just let the kids move their ships. If the game has angles, give them a challenge like moving on a small board without crashing into each other.

The kids won't be "wowed" at this point, but they'll have a little fun. Don't expect miracles, yet...  

A Couple of Guidelines

1. The math must make sense as part of the story. All games have a story line, even if the story is just shooting asteroids to save yourself, and math games are notoriously designed in a way that places math as the sore thumb the story line. In "math baseball" for example, you solve a problem to get a hit, but the mathematics concept is unrelated in context to the hit (i.e.- 4 x 4 = 16 is meaningless to a second base hit). The net effect is that students not only lose interest in a shorter time, but also that their conceptual learning will be low. In Monopoly, banking is a good example of a game in which the math makes sense to the story-line.  Another example I learned at a conference, in which we used chart paper with a drawn in map of a golf course. We first had to guess the angle of our shot, then used a protractor to line up the shot on the tee, and finally we multiplied the roll of a die by inverse fraction of a "club" we chose to determine distance. It was fun, and the math concepts at least made some basic sense. A 7 iron, for example, (equal to 1/7) shot less far than a two iron (1/2). 

Part of the point is that the math doesn't have to be perfect. For example using a bow and arrow, I often use the draw weight of a bow (how much force is on the string) multiplied by the angle of the shot up to 45 degrees as the distance it shoots. Modern physics tells us that the equation is much, much more complex. But, for a middle-school student, they understand that a 30 degree shot from a 30 pound bow will go further than a 20 degree shot from a 20 pound bow, and it makes sense. You've got them thinking.

2. Stay away from games where the math is totally arbitrary. It is the flip side of guideline 1, but too often the game becomes something like the following: solve this problem, get this prize. Not only does it miss the conceptual teaching you were hoping for, but your students will tire of this approach and you've actually done some harm-- the inherent message is, math isn't worth doing for its own sake, so we must then give an arbitrary reward. 

3. Choose a story-line, any story line, and then create a simple, repetitive, coherent way for math to be embedded. If all these guidelines sound the same, they are. As one of my mentors has said, "Learning is not just learning new things, but learning the same thing in a deeper, more meaningful way." (Dr. Art Combs) If you are designing from scratch, you can come up with just about any idea and build a math game around it. The richness around a subject will come as you study it more. 

Everyone has seen the classic word problem: Dave has 9 marbles, Sue has 6 marbles, and Greg has 4 marbles. Of the total number of marbles, what percentage of the marbles does Dave have?

For many of us, we immediately launch into solving the problem. We add the total number of marbles and start scribbling down the calculations. But for a large group of students, they literally just don't get the point. In their minds many are thinking, "Why in the world do I have to do this?" We can try to coax them with candy, gum, even good grades, but without authentic purpose, as teachers, we won't get the results we desire. So what next? Enter story-time. Really. Take a word problem, add characters and a rich plot, and end with a math problem to negotiate the challenge in the story and you've engaged the kids' imagination and their hearts.

Before it's dismissed, it's worth knowing that the learning architecture for story is ages old- for hundreds, thousands, even millions of years, it's been a fundamental pedagogy.

In modern day, however, it's almost taboo to have true story embedded as part of math class. I think this is due in part because math has been previously defined as lacking context. In it's daily use of course, we never see math without a rich context. Taxes are a rich, story-based context with characters and plot and sometimes even great sadness. Checkbooks also. Bank accounts, loan applications, bills, re-models of the floor, etc., all are surrounded by story, plot and characters. 

In the end though, we simply can't mimic the math we encounter in daily life expect to keep the attention of students. Taxes aren't authentic for a 13 year old. But if their minds are immersed into a rich story about the spice trade and pirates in the 1600's, and taxing "their population" means they are able to "buy" a new ship, taxes become relevant and engagement soars!

This is my list, compiled from 6 years of classroom teaching. It changes periodically, but I welcome comments. I work with this list repetitively throughout the year. Contrast these concepts with a more specific skill, such as learning to use a stem and leaf plot. If the stem and leaf plot can be authentically embedded into a project, we might work with the concept extensively. More often than not, I teach about 12 to 15 concepts in-depth, and the wide array of specific concepts one day a week in a catch-up tutoring class, or during "test-training" months. Students might see some exact concept only 20 to 40 minutes during the course of a year. Yes, students do miss something by not having a longer period to work with these ideas, but because the concepts we are learning at deep, meaningful level are authentically embedded in context, student engagement remains high and the long-term outcome is positive. Here they are, my top 10 suggestion areas of study that lead to middle-school math success.

Students should have ample and repetitive practice:

1. Using the Number Line. I feel the number line is the building block that forms the foundation of our mathematical thinking. As a base 10 system (decimal), students ought to know that our math comes from 10 fingers, 10 toes and is very old evolutionarily. It's more important, in my opinion, to spend time building a deep understanding of the number line rather than moving to more advanced topics if you notice students without these fundamental understandings.

2. Converting Fractions to Decimals to Percents. Of all the computations I hold as a standard, the ablility for a student to convert fractions such as 3/5 to decimals (.60) to percents (60%) is the core concept I define as giving students the most flexibility within the broader array of math.

3. Estimating. Estimating may be the most important tool for students to learn to approach mathematics in a way that makes sense. Of all the tools in estimation, I recommend that the bulk of the practice of estimation be in a mental format. Too often we are focused on students computing in paper and pencil format. I refer to mental estimation as the light saber of tools in mathematics, referring to its incredible importance and power.

4. Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic rules. Sure, it's a standard that could mean a lot, but I don't worry about choosing one particular method of approaching this concept. It's much, much more important to spend a significant amount using a tool in an authentic manner and gain mastery than it is to abandon deeper for the sake of "going on." Sure, there is a  balance here, but collecting real data and playing with it graphically will lead to a greater sense of purpose.

5. Visualizing and solving problems involving surface area and volume. This is such a specific type of content, but like the number line, I feel it forms the cornerstone of geometry. I can't prove it with any numberical data, but I believe surface area and volume create a type of thinking crucial to how students learn mathematically.

6. Converting from one unit to another. When I teach this concept, I usually have students develop their own units of measure and create a challenge for other students.

7. Formulating questions, designing studies, and collecting data.

8. Using and applying scales, ratios and proportions to quantitative relationships.

9. Enhanced practicing of mental math and solving problems in various methods.

10. Multiplication and Division Tables!

 Sources:  Angela Flicker and the National Council of Teachers of Mathematics and my own years of teaching.

When E.D. Hirsch writes about Depth vs. Breadth in education, he suggest it's a false polarity, and that we ought to "strike a reasonable balance between deep generalizable concepts and broad based facts." The problem is, it's not reason that is guiding math content in middle-school classroom-- it is a textbook, whether on-line or paper form-- and the top names generating textbooks are necessarily known for their comprehensive coverage of topics. In a purvue of recent math texts, the top publisher normally had between 11 and 14 chapters, each covering 10 topics per chapter in up to eight contexts for each sub-topic. Just from the birds eye view of looking at the scope of the content, I was left feeling overwhelmed and scared, and this is coming from someone who performed well on the both SAT and GRE math sections.

If the argument of depth versus breadth is moot from a purely philosophical perspective on what should be, the emotional impact in actual classrooms is nevertheless tragic. When surveyed, over 80% of students had something negative to say when they were asked how they felt about math. And teachers of public schools are often torn between the cognitive preparation of their students and covering content printed in a text. With impunity and the best of motives, teachers often feel compelled to go on. 

We shouldn't spin it too far. I'm not suggesting that only a few concepts ought to be taught over the course of a year. Still, the teachers who have the most success in the classroom do not solely follow a textbook and tend toward a more limited scope of content to teach. In a 2001 study out of University of Virginia, high school physics students of teachers who did not rely on a text as their primary teaching source ended up with better grades in college, intimating that teachers who don't try to "cover the world" tend to have more success.

Personally, I don't think Hirsch is incorrect. We ought to strike a balance between depth and breadth as each impact the other. I think the real problem is that we nowhere near a balance. I'd argue that when a top selling text proudly advertises as it's number one asset a total of 671 pages for a single course in middle-school math, we're a little off kilter.