1 Oct 2009, 4:03pm
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Can a 4th Grader Learn Trigonometry and Calculus?

I know.  It sounds crazy.  I thought so, too, when I first read Don Cohen’s brochure about teaching calculus to 7-year-olds.  As it turns out, kids are perfect candidates for learning tough subjects, especially trigonometry and calculus because of they’re endless curiosity and wiggly-ness.  They like to figure things out for themselves and they LOVE learning what the “big kids” are learning.

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.

Ian's Unit Circle

Ian's First Unit Circle

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.

Ian's Corrected Unit Circle

Ian's Corrected Unit Circle

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

Ian Calculate the Values of the Cosine Function

Ian Calculate the Values of the Cosine Function


Ian Calculates the Sine Ratio

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 Finds More Relationships For Sine and Cosine

Ian Work Finding Relationships For Sine and Cosine

Ian FInd the Relationship Between the Sine and Cosine

Ian Find the Relationship Between the Sine and Cosine

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.

Plotting on the Cartesian Coordinate System

Plotting on the Cartesian Coordinate System

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 Color Codes the Unit Circle

Ian Color Codes the Unit Circle

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.

An Identity for the Cosine

An Identity for the 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 graph of the Sine Function

Ian's Graph of the Sine Function

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

Changing the Frequency of the Sine Function.

Changing the Frequency of the Sine Function.

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.

Calculating the Tangent Function

Calculating the Tangent Function

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!)


Recently Juan Casteneda wrote a blog post about students with ADD/ADHD.  Juan and I are both tutors and have had many of the same experiences with our students.  What strikes us is the frequency that the ADD/ADHD label is handed out when a kid isn’t doing well, particularly in math, geometry, algebra, etc.

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.

“Don’t Tell Her THAT!”

Two of our high school girls had this conversation recently:

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. more »

Teaching Calculus to A “D” Math Student

The circumstances that lead me to teaching the tenacious Michelle are completely unexpected.  If Don Cohen the Mathman had not invited me to join Facebook, I still would not be a member.  But I did join and I find that it’s a good tool for encouraging my students even after they no longer need me for math.  So Facebook isn’t so bad and maybe I SHOULD check out this Twitter thing because, ya know, your business won’t go anywhere with out a blog of some kind.  So I joined that, too.  It took me a while to figure out the whole thing.  As it turns out the Twitterverse (all the Twitter people) is very helpful and I was following and being followed in no time.

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

more »

Why would anyone want to teach calculus to a 7-year-old!?!?!

The professor yelled when he asked this question.  I’m used to people being boggled at the thought of young people learning calculus.  I’m used to the blank stares, buggy eyes and puzzled looks, but this was a first.

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! more »

mathheadinc: the blog

Welcome! It’s finally here! This is the first post of MathHead Inc, the more casual side of MathHead Tutoring.  We get a kick bragging about our students and sharing all things related to math.  We’ll try to make it fun and interesting, two words not usually associated with math, algebra, geometry and least of all calculus.  You may not love math after reading this blog but hopefully you won’t cringe everytime someone mentions the word.  Maybe you’ll start to see the world differently, as a mathematical place full of beautiful patterns.  There should be just enough information that we lead you to the thrill of discovering things on your own.


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!