Ian is one of those kids. He came to us in 4th grade because his then tutor was taking a full time job. I will be forever grateful that she did. She had already been teaching him trigonometry **because he likes it.** I started teaching him infinite series and the unit circle. Over the next year and a half, Ian calculated the values for the sine, cosine and tangent functions, discovered some trigonometric identities, found the relationship of the sine and cosine and where they are positive and negative in the quadrants of the trig unit circle, and more. Some of his work is below.

This is Ian’s first attempt at a trig unit circle. He started with 0° angle on the left and increased clockwise instead of 0° angle on the right and increasing counterclockwise.

This is Ian’s corrected trig circle. The 0° angle is also the 360° angle.

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These are Ian’s calculations for the sine and cosine functions. On his trig circle, he measured the horizontal distance from the center for the special angles 0° through 90° and divided each by the measure of the radius. He measured the vertical distance for the sine. Then Ian used a calculator to find the values of the cosine and sine, then took the different between those and his calculations to find the error. The difference was very small. Ian’s measurements were excellent.

Ian started to compare the values of the sine and cosine and found that, as the cosine goes down, the sine goes up. It took several sessions and lots of experimentation to figure out that the sin(n)=cos(90-n). Ian was excited and enjoyed it. His comment at the end was, “why didn’t I see that sooner.!” Funny kid.

Most students who take trig are juniors and seniors and never truly understand this identity.

Ian learned how to plot on the Cartesian coordinate system. After being told that he always had to do the horizontal motion first, he wrote the instructions for plotting the point using coordinate pairs.

Ian noticed that the sine and cosine was positive or negative in certain quadrants. He noticed that the sine is positive in the upper quadrants, I and II, and negative in the lower quadrants, III and IV. The cosine is positive in the right quadrants, I and IV, and negative in the left quadrants, II and III.

Ian did this work at home. You may notice that the top two quadrants weren’t correct but he figured out his mistake. He tried using two colors but then decided that the four-color circle was best. It’s a great visualization of the positive and negative values of the the sine and cosine.

Ian figured out that if he added a multiple of 360 to the angle 60°, that he got the same result for the cosine. Ian also found that it didn’t matter if the multiple was a positive or negative multiple of 360. Good work!

Ian’s excellent graph of the sine function. When he finished he wanted to know “how to make more waves.” See below.

These are Ian’s notes for the results of using a graphing calculator to find different was to make more or fewer waves, i.e. changing the frequency. Ian found that multiplying the angle by 2 double the waves and dividing by 2 halved that waves.

One of the last things Ian did with us was calculate the tangent function. He divided the horizontal measurement from center by the vertical measure of the special angles, as before.

Ian did work on some infinite series but his love is trigonometry and why I focused on it in this post. If you think Ian is gifted or a genius, guess again. I’m not saying than Ian is average because he isn’t. Ian **LOVES **math and is willing to work on it even when he can barely keep his eyes open. Ian **IS** advanced and had an advanced start that was fed by two parents who have a fearless love of learning. Ian was interested so they just saw what he could do.

The same work that you see here, we can teach to a kid who is flunking math. We do it all the time. In fact anyone who can count can learn what you’ve seen Ian do.

*(UPDATE: *Since then, Ian has skipped a grade and is now taking Algebra I in 7th grade, a class normally taken in 9th grade. So, technically, he’s three years ahead in math. YAY!)

In over fifteen years of tutoring, I can count on one hand our students who had verifiable learning disabilities. In the case of supposed ADD/ADHD, we recommend that parents rely on the expertise of a certain licensed psychologist. He uses **all digital real-time Neuropathways EEG’s** (electroencephalograms) to diagnose brain abnormalities and then attempts to correct any abnormalities by means of neurofeedback instead of medication. The parents of one of our elementary school students had the testing done after being told that he had ADD/ADHD. (This 8 year old could not sit still to save his life, but when I let him doodle, it was easier for him to sit quietly and to focus on infinite series.) The testing uncovered petit mal seizures! And more often than not, our students have more dyslexic tendencies than ADD/ADHD. You can see why a proper diagnosis is extremely important when the usual mode of care is to medicate the child.

I’m always skeptical when I hear that a student has been told he has ADD/ADHD. In most cases, that student has no problem focusing on a video game or favorite hobby for hours on end. After asking the parent many questions, ADD/ADHD usually doesn’t accurately describe the students situation.

There are a number of other good habits that often help children who do not focus well. Few habits help more than hard work and diligent study. It overcomes a multitude of evils, namely, slack study habits and, yes, moderate health issues. In fact, the determining factor for the success of our students is how hard they study and practice. We can teach them all the cool tricks in the world, but if they don’t study, they can ‘fuggedabout’ getting better grades.

You may have noticed that I haven’t mentioned intelligence. That’s because we haven’t met a kid who wasn’t smart enough NOT to work hard. Every kid has the ability to work hard. It’s all about how much she wants better grades. Even our students with disabilities are as smart and often smarter than the average student **because **they have learned to work harder. It also doesn’t matter if a student doesn’t like the subject at hand. It matters if she likes getting A’s and B’s and how hard she is willing to work for it or how much she wants her cell phone or computer privileges back…

In the end, the declaration that a student has ADD/ADHD and subsequent medication of that child is an attempt to quiet the learner who needs to be taught another way and another attitude towards studying.

]]>MOLLIE: I [stink] at factoring.

LEIGH: Don’t tell her THAT! She’ll just make you work on it

MORE.

Sometimes my students make me laugh so hard my stomach hurts. I think it’s hilarious that Leigh said this as if I weren’t in the room, let alone that I was standing right next to her. It’s also funny that she would suggest that another student take the easy road because Leigh has raised her grade 10 percentage points since she started one month ago on March 16. Her progress was evident only **after** she started practicing harder.

After fifteen years of tutoring, if Joe and I know nothing else, it’s that convincing our students to practice diligently gets the best grades. Even students with the worst math skills or those with disabilities improve their grades with careful practice (Geoff Colvin calls it “deliberate practice” in his book *Talent is Overrated)*. For example, Lee Alderman became valedictorian despite his mild autism or Brandi Binder who graduated high school with honors after having half her brain removed. (Read about Lee here & here and Brandi here & here.)

Most of our students are kids with reasonably good health but some have decided or were given the impression that math is not their subject, so they give up too soon. They give up before the hard work begins, the very thing that will get them where they need to be.

You may think of yourself as a typical student but we want you to know that you have stunning ability to improve with directed training, “deliberate practice”. So, put forth your best effort. Work hard so you will keep improving. Not much else beats the satisfaction of getting a good grade knowing that you worked hard to get it.

]]>Michelle declares that she has always been a “D” math student.

So Kris Colvin follows me and I’m wondering why a design person is interested in following a math tutor but wasn’t brave enough to ask. Some time passes and Kris says that if anyone needs writing done, to talk to Michelle. I do need some writing done for my site, so I contact Michelle. We hit it off very nicely as I explained what I do and what I need (in case you didn’t know, I’m a math and calculus tutor and run a kid’s calculus program). Michelle declares that she has always been a “D” math student all the way through college. ( Here’s that odd behavior of fearlessly admitting that one is terrible at math but the same person would **NEVER** admit the same deficiency in reading! By the way, Michelle reading skills are tops.) OK, so she’s very bad at math. I find it extremely hard to believe. Anyone talking to Michelle for a short time would come to the conclusion that she is one sharp cookie. She’s very good at what she does now but would much rather have gone into socially if it hadn’t been for those stinky statistics.

After discussing the terms of our contract, I make a deal with Michelle: I will teach calculus to her on the condition that she blog about it. She had to share her feelings and emotions about learning a scary subject like calculus. She quickly agreed. I hoped she wouldn’t regret it later and back out. This would be a fantastic opportunity to show that any reasonably intelligent person can learn calculus if taught using Don Cohen’s methods.

Out first lesson was on March 5. One month later, I think it’s fair to say that Michelle is having as much fun as I am. Each post so far has made me laugh out load. I hope you will follow along with her on her “journey from frightened to fearless“.

Hopefully you will be inspired to learn or at least explore the math/trigonometry/calculus you were told or thought you couldn’t. If you need help getting started, contact us anytime: EMAIL, phone: 816.560.8098.

CORRECTION: Kris Colvin led me to Adelle Charles who led me to Michelle.

]]>I was at a math technology conference in Fall 2008, attended by professors, teachers, school administrators and tech coordinators. Everyone was interested in what new software and hardware could be useful in their classrooms. I was in a small slice of math geek paradise!

We were sitting down to lunch. I already knew Dr. Richard Delaware (University of Missouri at Kansas City) and Carolyn Anderson (Park University, Parkville, Missouri). Dr. Delaware was one of my husband Joe’s professors while he pursued his Masters in Statistics. Carolyn Anderson is an Associate Professor and has referred students to us for years. Both are excited about our kid’s calculus program and think kids ought to learn hard subjects.

We introduced ourselves to those we didn’t know and vice versa. After I said who I was and what I did, “I’m the owner of Mathhead Tutoring. I’m a math and calculus tutor and I teach calculus to kids 7-years-old and up”, the unnamed professor bellowed, “Why would anyone want to teach calculus to a 7-year-old!?!?! I want Johnny to be able to do ‘2+2 is 4. 4+4 is 8’!”. (Cue record scratch). Profs. Delaware, Anderson and I turned to look at him. Everything stopped for exactly 2.3 seconds. We couldn’t understand why he would be angry about someone else teaching calculus to kids.

After I saw the looks on Profs. Delaware and Anderson’s faces I got so tickled. I tried not to laugh. I imagined how my young students would react if he said that to them. They would probably shrug their shoulders and go back to what they were doing.

I teach calculus to kids **because I can**, because **they are smart enough** to learn it, because it gives them confidence and alleviates their fears about math. They love learning what the “big kids” are learning. The calculus gets them excited and they say to me, “Miss Lori, teach me THIS!!” In the process they get to do lots of fractions and decimals. They get to practice arithmetic without realizing it’s arithmetic and always come back for more. What’s bad about that? Nothing. The kids and I are having a blast. I know he’ll never have as much fun at his job as I do! I won’t stop as long as I can see the looks of delight on their faces when they discover something new.
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Here’s something to think about to get you started. Have you ever thought about flowers being mathematical? Have you noticed the clockwise and counter-clockwise spirals at the center of many of them? Try counting the spirals in each direction. Keep a notebook. What patterns do you see? Count several of the same kind of flowers, like sunflowers and daisies. Feel free to use the photo in this post until you grow your own. The numbers you see will be related to this series of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… Send us your discoveries.

Now, just have fun!

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