MathRacing – I work with a wide age range of kids ages 6-16 and I have yet to encounter a student who doesn’t get excited when this game starts. The competitor in all of us comes out and the brain starts working. In this two-person racing game, the screen is split into two with a child working on either side. For every correct problem solved, the car moves further down the track. The goal is simple: cross the finish line first. Easy, medium, and hard settings allow students of all ability levels to play against each other at appropriate levels. Adjusting the speed of the cars can shorten or lengthen make the game. All in all, this is an excellent game for practicing whole number addition, subtraction, multiplication, and division.

Combine 4! – One of my favorite games that I have spent incalculable hours playing. (Look on the global leader board for the guy named “Harris” and see how much I’ve played!). This app, based on the math card game “24”, is a game for middle school students to learn the order of operations and to improve number sense. The goal is to create the number 24 with four cards – numbered 1 through 12 – while using addition, subtraction, multiplication, and division. Sounds easy? Not necessarily. The process is the hard part. With 1362 ways to make 24, this game will keep you busy for days. A great thinking game for small kids and big kids alike!

Elevated Math – This app is an excellent teaching app that teaches basic math through Algebra 1. Utilizing cartoons and humor, this app engages kids on various levels. Being an Algebra 1 teacher myself, I appreciate how this app breaks down Algebra into small, easy to understand parts. It pauses frequently to allow students to write notes and complete problems on a provided whiteboard feature. My favorite feature of the program is its ability to help teachers answer the age-old question, “When am I ever going to use this stuff?” Its “career video” section has interviews with professionals discussing how they use math in their everyday lives. A great way to show kids why they need to learn math! elevatedmath.com

Motion Math – In this app, the understanding of fractions and percents are presented in a kinesthetic way. The user must move their device left and right to correctly place a floating ball containing a fraction or percent on the right place of a zero to one number line. Fractions are represented both visually and numerically. Just when you think you have the hang of it, Motion Math introduces a new number line such as -2 to 2 or 0 to 2. This game is a great way to teach students about fractions and to feel comfortable working with them. If you don’t place the ball on the exact right spot, the app assists you on where it should go. When the game is mastered, you’ll have an exceptional understanding of fractions and percents.

Cross Fingers – In this tangram puzzle game, your brain will be put to the test. Maneuvering pieces into the right places using one, two, or sometimes three hands, the game quickly lives up to its name. This game will test your dexterity and your spatial awareness as you move different shaped pieces around a board filled with obstacles. While the game is easy through the first ten levels or so, the difficulty increases and provides a challenge for students of all ages and ability levels. With over 660 levels now available, this game is a winner for its simplistic, yet challenging nature.

- Jeff Harris teaches at Hawthorne School in Beverly Hills and runs the summer program “The Beverly Hills Math Factor” (www.bhmathfactor.com)

Jeff and three of his students

Pages from Da Vinci's notebook - Photography by Maia C

Last week a principal introduced me to a math teacher in my school district. The principal proudly stated that over the past three years this teacher’s students had averaged 98% advanced or proficiency in Algebra 1.

“Wow! How did you do that?” I exclaimed. “Is there something special in your teaching technique?” Obviously, this teacher knew her subject, but so do many teachers and without achieving these results.

“I care for my students,” she responded.

Okay. Yes, caring does have a lot to do with a teacher’s success. We wrote a blog about it in February called We MUST Engage Our Kids. Caring was one of four ways that we suggested. But 98% advanced? Can “caring” account for that kind of success? After I pressed for more information, this teacher finally revealed that somehow she is able to tell if her students understand her instruction or not. For instance, she can tell if a student in the back row has done his homework simply by his facial expressions and body language. Her students are always amazed that she knows them so well.

I realized that this math teacher had a natural skill for observation. Is this something we learn or is it a talent that some of us have?

I think it’s both. I’m convinced that becoming better at observation is something we can learn. Here’s why:

Years ago, when riding in the car with my six-year-old daughter, we played a game. We counted Volkswagen Beetles. Whenever we saw one, we would call out its color. The prize was a black VW, which would earn us a cookie when we arrived at our destination. (a black VW bug was my first car) At first, every week or so, we would spot a black VW, but then, gradually, they became more frequent to the point where I would spot one every day. Even now, I notice them quite frequently. From this experience I realized that we are capable of training ourselves to become more observant. But what is the best way?

I teach a class on creativity at a design college and have researched how creative people work. More often than not, I have found they keep journals. Andy Warhol kept one. So did Leonardo da Vinci. The Atlantic recently reported that 3500 notebooks of Thomas Edison’s were recently discovered.  “The notebooks are filled with fascinating observations and insights–many pertaining to unrelated projects, in a seeming free flow of associations.”

A few years ago, I met a historian who was reading Christopher Columbus’ journals. He told me that Columbus would write down every minute detail of what he saw and felt when he was at sea – the way the waves moved, the debris found in the water, the kind of birds and the direction they were flying, even how the air made his bones feel. It makes sense that Columbus, through his observations, had a hunch that if he sailed far enough west he wouldn’t fall off the earth, he would find land.

I require my students to keep a journal and write in it. I encourage teachers to do the same. As I tell my students, it doesn’t matter what you write as long as you do it consistently. I usually require at least two pages a day.

Use the pages to record things you see or have seen. It doesn’t matter what it is. Describe the ceiling in the coffee shop, the way your dog looks at you, the color and style of the shoes the person across the room is wearing, the clutter on your desk. Gradually, you’ll find yourself becoming more observant. Continue this when classes start in the fall, but turn your attention to your students: describe what they are wearing (tennis shoes, loafers, ironed or wrinkled shirts), how they sit at their desks (straight, slouching), their personality characteristics (outgoing, shy, teasing, cautious), how they speak up in class (confidently, timidly, arrogantly), and most important, their knowledge of the material and if and how they falter. I suspect you will start picking up subtle nuances in your students’ behavior and know intuitively if they are learning the material properly. If they don’t understand something, you’ll catch it right away and make the correction in your teaching.

I urge you to try it. Go to a stationary store and buy a nice notebook. Writing in it will take some discipline, and starting will be the first hurdle, but if you write every day, it will become a habit. Isn’t this what you require of your students – to study everyday?

If you do give this a try, please let us know how it goes. There is no science behind this idea that I know of, but I strongly believe that nurturing your skills in observation will help make you a better teacher.

Math is hard. Of course it can be fun and gratifying, but at times it can also be discouraging. Can a teacher teach students how to properly respond to failure?

Finding the right answer is sometimes not as important as learning to approach a problem correctly, having the willingness to study a mathematic question from different angles and resisting the urge to give up. This is best taught in middle school. In grades 6 – 8 if a student fails a test or gets a “D” or “F” in class, it has no effect on his or her college admission records. But this failure can become a valuable life lesson and, in fact, if approached properly a teacher can help students prepare themselves for high school, for college and even find good jobs. Here is what is needed:

First, become conscious that you, as a teacher, have a dual role – to teach math concepts AND to teach character development.

Next, share as many examples as you can of people who faced failure but didn’t give up. Johannes Kepler, the German mathematician, filled 10,000 folio pages with calculations before discovering that the planets circled the sun in an ellipse. Teachers told Thomas Edison that he was “too stupid to learn anything.” And he failed almost a thousand times before he designed a light bulb that worked. Here are another 49 famous people who failed at first before they succeeded.

Finally, build a culture in your class or home where it’s okay to make mistakes. We learn from mistakes. And if your students fail, find out why. It could very well be that they need to study more, or study more effectively. Empower them to dig deeper into the problem by showing them the opportunities they have. Assure them that they have the capability to work a little more, a little harder.

If you can teach both math and character development in your class, you’ll go from being a good teacher to a great one. You won’t just help the students learn math skills, but you will teach them to handle the uncertainties in life. Leonardo Da Vinci called it sfumato (literally, “up in smoke”). He thought it was important that people become comfortable with life’s ambiguities: joy and sorrow, beauty and ugliness, success and failure. Life will feel more balanced. Character becomes stronger.

Studies have shown that students who succeed in college are not necessarily the ones who do well academically in high school; instead they have  exceptional character strengths, like optimism and persistence and social intelligence. They are the ones who are able to recover from a bad test and quickly decide to do better next time; to bounce back from a fight with their friends; to resist spending time with their friends and study instead.

By helping your students to develop strong characters, you might help them avoid some of the bad habits that people develop because of an inability to handle doubts and fears of the uncertainty of life. These habits include obsessive fantasies, talking excessively, smoking, drinking, taking drugs.

I bet you didn’t think being a math teacher was so important!

Two months ago we posted an article entitled: We MUST Engage Our Kids. Here we listed what we considered the necessary ingredients for a teacher to conduct a successful math class. These were passion, real-world problems, humor, and caring. The other day I was chatting with Vijay, a math tutor working in Romania, and he sent me a short article he had written about how he had started tutoring. I found it fascinating. Two of the ingredients really stood out (though I’m sure he uses all four). Can you guess which two? Here is his article.

It all started about two and a half years ago when I was told to leave Oracle where I was working in Bucharest as an educational consultant. I felt dejected, confounded, and lost until my dear wife suggested that I look at my Curriculum Vitae and pointed out that I had achieved the highest grades in my high school math classes. I had always taken a special interest in math, encouraged by a teacher who used to coach me extra hours on the weekends in her home. With her guidance and my continuous practice I achieved first in the math public exam in Mumbai.

“Why don’t you work as a math tutor,” my wife suggested, “a freelance math instructor.”

With my wife’s help I created a brief about myself and distributed fliers in the various international schools here in Romania as well as to all my expat contacts. A few days later, I received a letter from a parent looking for a math tutor for her child.  She requested a face-to-face interview and after talking with the parents I formed a class for their child, meeting once a week. This began my career as a math tutor.

Today, I have 28 students, all coached by me, with ages ranging from those in 2nd grade to 12th.  Boy, am I ever inspired to give the best to them. Just watching them progress inspires me. I had a student who jumped from Grade D to Grade A in one academic year.

They work in tandem with me and do all the homework and tests I give them. I count on internal motivation and encourage them a lot. They get plenty of positive feedback from me and I give them choices in their homework.

I work seven days a week, totaling around 60 hours a week. As the great Chinese philosopher said, “If you love what you are doing, you will never work the rest of your life.” I love math because it allows me to exercise my brain continuously and also gives me a lot of self-confidence. It’s nice to know that I’m responsible for shaping so many students’ careers.

Vijay Krishnan  Bucharest, Romania http://www.bucharesttutor.com @bucharesttutor

We know many teachers do not like parents teaching their kids tricks, but once students have demonstrated conceptual understanding, learning tricks makes math so much more fun.

In our lesson M1.2 Divisibility Rules we provide some “tricks” for helping students learn if a number is divisible by another number, which sure helps them when identifying factors. In this clip we illustrate the rules for determining if a number is divisible by 3, 6, and 9.

(A part of the lesson on rules for division with the autopauses deleted.)

Tell us in the comment section if you know any other arithmetic tricks. We would love to know what they are and we’re sure other teachers would too.

MATH TEST

2 + 2 x 2 + 2 x 2 – 2 x 2 =

a.     6

b.     16

c.     40

And I was shocked to see that so many people got the wrong answer.

Knowing the Order of Operations is important for a student’s success in math – so important that we included two lessons in the Elevated Math iPad app dealing exclusively with this subject.

If you are absolutely sure of the right answer, don’t bother watching the following videos. BUT if not… or if you want a peek at how Elevated Math teaches the Order of Operations, take a look at the following two videos.

The first one shows a section of the lesson in Numbers and Operations (M1.1) where we introduce the word PEMDAS to help students remember the mathematical order. Although understanding a concept is far more important than memorizing rules, we felt that students and teachers would appreciate having something to remember.

(a section of the Elevated Math iPad app lesson, “Order of Operations” with the autopauses removed)

Next, in A1.5 of the “Getting Ready for Algebra” module, we explain the Order of Operations again, but this time the stress is placed not on the rules but on the application. Here is how we explain the use of grouping symbols.

(a section of the Elevated Math iPad app lesson, “Applying the Order of Operations” with the autopauses removed)

Now, in case you’re still stumped by the question above and don’t want to spend the $1.99 for one of these lessons, we’ve kindly given you the answer:

2 + 2 x 2 + 2 x 2 – 2 x 2 =

2 + 4 + 4 – 4 =

6 + 4 – 4 =

10 – 4 =

6

Photography by Ta-Duc

When running for school board, one question was asked a number of times, especially in the debates and usually asked by students, “How will we provide more teacher/student interaction?” I assumed that students wanted a more personal experience in their learning until an article last week made me realize that a lot more was behind this question. The BBC News wrote, “Secondary school pupils are so scared of looking stupid in maths lessons they will not tell their teachers if they do not understand, suggests research.” The article continued, “The reasons pupils gave for not asking for help more often were that they were worried about looking foolish, were embarrassed or did not want to draw attention to themselves.” In other words, they lack confidence, which could be overcome if teachers had the time to spend more one-on-one with their students.

Are there other ways to build a student’s confidence in math?

Well, hiring a tutor helps. A good tutor. And teachers are great tutors. But tutors are expensive.

As we discussed in a previous blog, games can make math fun, which in turn can increase skills and build confidence.

A mother explained in Four Things a Parent Could Do To Help a Struggling Student that she thought Khan Academy would be fun for her son and he might find the points system and accumulation of badges motivating. She was right. “Before he knew it, an hour had passed and he was still interested in the exercises. More importantly, he was not getting frustrated if he got something wrong like he does in class or during homework. He wanted those points and, by golly, he was going to get them.”

The Elevated Math iPad app is not a game, but it’s fun. With ten million dollars invested in the lessons, we had to place a nominal price on them. But if parents can afford the cost of a small cup of coffee, they can afford a lesson for their child that not only builds math skills but also solidifies understanding of critical math concepts.  More skills?  Understanding? How can this not increase confidence?

But though we love what Elevated Math (and Khan Academy) accomplishes, nothing replaces a teacher who can recognize students’ strengths and nurture them through their weaknesses. So how do we free up teachers so they can spend more individual time with their students?

I was talking to a mother the other day who is looking for a school for her daughter. One private girls’ school she visited explained that all their math classes were flipped.

“Because this produces high achievement,” I said knowingly.

“No,” the mother answered.  The reason floored me. The math teachers found they were not completing their lessons during class time because the girls were always raising their hands and asking questions.  The flipped classroom became a viable solution for this school.

In a typical co-ed math classroom boys usually are more assertive when it comes to asking and answering questions; girls are more likely to refrain from  asking questions for a variety of reasons.  The girls’ school previously described  prides itself in allowing girls to express themselves in class.  As a result, the girls in this all-girls school seemed to have too much confidence if that could be possible.

Flipped classrooms do help build confidence. If students don’t get the instructional video with the first viewing, they can review the video again. Or they can ask a friend to explain it. Or show the video to their parents for help. If none of these steps explain the concept, students can confidently strut into class knowing that the problem lies not in their lack of understanding but in their teacher’s explanation.

Our educational system is in crisis mode. Last week Los Angeles Unified announced that 11,000 preliminary pink slips are to be issued, and teacher job satisfaction is plummeting.  It sounds as if not only the students need confidence, but also the teachers. In the college class I teach the greatest satisfaction comes when I work one-on-one with a student. More teacher/student interaction could be a solution that helps everyone.

Math students who begin their journey into absolute value usually evaluate expressions with absolute value as “always positive.” That is until they encounter the absolute value of zero, and then their answers become “always positive or zero.”

The formal definition of absolute value is |x| = x if x ≥ 0 or –x if x < 0. The negative x confuses students, and they never quite understand that it is the absolute value that is always positive or zero. Unless this misunderstanding is corrected, the situation becomes more problematic when solving inequalities that involve absolute value, which can lead to unhappy teachers and muddled students who usually conclude, “we don’t like math.”

In our Elevated Math lessons we make it clear that absolute value is distance, and distance is always positive or zero.  We begin in lesson M3.1 with instruction on negative numbers followed by problems, and then we introduce the concept of opposite numbers before explaining absolute value:

Here is how we do it:

Lesson M3.1 continues with an explanation of integers, how they relate on a number line, and how to order integers.

In our algebra lessons absolute value is defined again in lesson A6.1 followed by the solving of basic absolute value equations.  Here is a portion of that lesson:

The lesson continues with problems and more instruction that includes solving absolute value equations with one and no solutions.

A subsequent lesson, A6.2, gives instruction and solves problems for more advanced absolute value equations, which include two-step linear equations.

When the student reaches lessons A6.3 “Inequalities with ‘Absolute Value is Less Than’” and A6.4 “Inequalities with ‘Absolute Value is Greater Than,’” they now have a firm grasp of the absolute value concept.

In the last lesson of this module, A6.5 “Problems with Absolute Value Equations,” modeling scenarios are modeled using an absolute value equation or inequality.  Using all of these Elevated Math lessons can lead to happy teachers and successful, confident students who conclude, “we like math very much!”

Illustration by Damián Navas

A short article written in a 2006 issue of NCTM’s mathematics journal, Teaching in the Middle School, caught my eye. It was entitled “Some Students Do Not Like Mathematics”. The reasons stated were the same as we have heard for years: “We don’t engage our students,” “Parents are not involved,” “Students don’t know how to expand their thinking when they solve a problem.”

I object to hearing a problem discussed without including at least one concrete solution, and this got me thinking: What solution(s) would I offer if I had written this article.

Of course, my first advice would be to buy an iPad and download the Elevated Math lessons. Most students enjoy math when they watch the videos and work the problems.

But that’s an obvious answer. What else could I say?

I read the article again, and zeroed in on the statement “students who dislike math have problems with persistence and motivation.” It seems easy to give up on such students. I mean, how the heck do you teach persistence? And how do you motivate an unmotivated kid? But maybe these are not the questions to ask.

When searching for a creative solution it’s helpful to ask a different question. For instance, if you need to make dinner for 20 people, you can dwell for days on how best to cook your favorite chicken dish.  But you might also ask, “What dish is easier to make?” Or even better, “Can each person bring a dish so we have a variety of foods and I only have to make one small entree?!”

The same goes with answering the question, “How do I help students like mathematics?” Maybe the questions to ask are “What do students like?” and “How can we tie math into that?”

Of course! Games! Kids like to play games, and they like to compete. Games teach persistence, and healthy competition motivates, right?  Maybe we should not be so quick to discourage our kids from playing video and computer games.

Two middle school math teachers came to my place last summer so I could help them load the Elevated Math lessons onto their iPads. They had a math camp and wanted to try out the lessons. As they waited for the upload to complete, they played a game called Five-O. Their students, they said, loved the game and liked how math was fun.

Last week I read an article in the Las Vegas Sun. The article was about a school district that had seen vast improvement in math scores by using a computer game called ST Math by Mind Research. Quoting from the article, “Teachers were reluctant at first about surrendering valuable class time to a computer game… But that changed once administrators and teachers saw how effective the game is at keeping students focused on math.”

This idea is far from new. Many articles have been written proclaiming the effectiveness of math games in learning and applying concepts and skills.  But with the launch of the iPad less than two years ago and the subsequent development of math apps with game features, teachers and parents now have a wealth of resources available in the iTunes Store to help students like math, to help them become more persistent through competition, and to motivate and engage. Asking the right questions can indeed lead to effective solutions.

Here are some math apps we have tested and like:

Math Showdown Multiplication   [StartingStrong.com]

Multiples x   [MathTappers Apps…   designers are math educators and researchers, Tim Pelton and Leslee Francis]

Splash Math 3  [StudyPad, Inc.]

EverydayMathematics Baseball Multiplication with 1 – 12 Facts  [McGraw-Hill]

Math Wars  [Life Skills Games]

Aliens Kids Math HD – for iPad   [Brain Counts]

Math Bingo  [ABCya.com]

Monkey Math School Sunshine  [THUP Games]

Rocket Math   [Dan Russell-Pinson]

Math—Rounding Numbers   [King’s Apps]

Motion Math Zoom   [Motion Math]

Math Board    [palaware]

Multiplying Acorns HD   [Operatio]

4^th Grade Math;  Splash Math Worksheets App    [StudyPad, Inc.]

Math Ball  [Wuxi Haohai Information Technology Co., Ltd.]

Timed Math   [Elmore]

The cartoon introduces the Elevated Math algebra lesson entitled, “Solving Systems of Linear Equations by Substitution.” Graphing and elimination are both methods for solving, but substitution is another one. All three methods produce the same solution, but depending upon the problem, one method might be easier. Find this lesson in the iPad app, Elevated Math.