Today, I tutored a sixth grader named Mizan. I asked her if she brought any homework to do, but she did not have anything with her. No backpack, no folder, no papers, no pencils. For a moment, I was at a bit of a loss of what to do. I asked her what she was learning in school, but all she said was “triangles.” She would not elaborate further on what they were doing with triangles. I had no idea if they were learning about the different types of triangles and angles, finding area and perimeter, the Pythagorean Theorem, etc.
I decided to start with the basics. I had her draw four different types of triangles--equilateral, isosceles, scalene, right--and then describe the defining characteristics of each. She did fairly well with this.
“Have you ever found the area of a triangle before?” I asked.
“No, but I know how to find the area of a rectangle,” she replied.
She knew that the area of a rectangle could be found using the formula A=b*h. I proceeded to show her how a rectangle can be divided diagonally into two right triangles. Therefore, I showed her, the area of a triangle could be found using the formula A=1/2*b*h.
I had given her a problem in which the base of the triangle was 2 and the height was 5, making the area 5. She quickly memorized this equation and put it to use successfully. However, there was just one problem.
“How do I multiply 10 times 1/2?” she asked me.
As a student at the end of sixth grade, I had not even anticipated that Mizan would need help with multiplying fractions.
One thing that I have noticed happens frequently with students from underserved school districts is that they often are missing important pieces. It’s analogous to constructing a building but leaving many important structural components out of the foundation. Especially in calculus this year, I have learned more than ever how important a good foundation in math is, and the middle school years are some of the most critical.
Mizan and I sidetracked from our triangle lesson for a while to instead focus on fractions.
“I don’t want to do fractions. I hate fractions.”
I showed her what I consider to be the easiest way to multiply a whole number times a fraction:
10 x 1 ---> multiply the top two numbers = 10 = 5
1 2 ---> multiply the bottom two numbers 2
She understood this concept very quickly and correctly answered every problem on multiplying fractions. On occasion, she would forget to simplify the fractions after multiplying; once I would remind her, however, she had absolutely no trouble reducing fractions.
While we were at it, I decided to show her how to divide fractions. While she had technically learned this in school a few years ago, she could not remember how to do it (and had probably never mastered the topic when first taught).
I taught her that dividing fractions was just like multiplying by the reciprocal. I think the term “reciprocal” scared her a little at first. Once I showed her that you just had to flip the number in the numerator with the number in the denominator, she no longer struggled with dividing fractions.
Next, I made a bold move and decided to show her the Pythagorean Theorem. Again, the was initially really freaked out by the title of the concept. While she wasn’t particularly jumping for joy over learning some new and harder math material, she didn’t object to it, either.
I gave her a problem with a right triangle that had a base of three, a height of four, and thus, a hypotenuse of five (which is the most simple of the Pythagorean identities). Mizan understood how to square three, four, and five, but she had absolutely no idea how to square root twenty-five. She didn’t even know what a square root was. Again, I was slightly surprised, just because squaring and square rooting sort of go hand in hand, like addition and subtraction or multiplication and division. I briefly showed her how to square root a number and she subsequently got the right answer for the problem, even though it took some time.
After such a challenge, I decided to review something a bit simpler with Mizan. We reviewed how to find the area and perimeter of a rectangle, along with the basic properties of a rectangle. After just a few practice problems, she became very quick at solving anything that had to do with a rectangle.
It was now halfway through the tutoring session, so I decided to give Mizan a series of problems over all of the topics we had covered today, along with a “challenge problem.” At first, I was only going to give her five problems.
“Do you think five problems is enough?” I asked, anticipating that she would tell me that it was actually more than enough.
“I think you could actually give me six or seven,” she replied with a sly smile.
I felt victorious. At the beginning of the tutoring session, Mizan could only talk about how much she did not like math, especially fractions. Now, she was asking for more questions!
So I gave her seven problems, with the last one being a challenge problem. She got nearly every problem right. The few errors she did have were just with forgetting to simplify fractions and the like.
Mizan even came very close to solving the challenge problem. The problem I gave her was of a shape made up of a right triangle and a rectangle. She was able to see that the shape could be broken down into a triangle and a rectangle. She also found the area of both the triangle and the rectangle independently. After giving her a small hint about adding the two areas together, she did so correctly, successfully conquering the challenge problem.
At this point, Mizan seemed much more open with me, and I could tell how much better she felt about her progress in math.
A few minutes later, it was juice and donut time. Mizan and I talked for a while, actually more than most kids are willing to talk. She asked me about my school, about the classes I was taking, and how my spring break was. She was also very surprised to learn that I was just a high schooler and not a college student. Only students at the Saturday Tutoring Program think that I am older than I am (5’2” problems).
After juice and donuts, Mizan received the writing assignment for the day, which was relatively simple since it was the last session. Mizan had to set three academic-related goals for herself and sign a contract that she would follow through on these goals over the summer.
Mizan made the following goals:
1. Read 40 minutes a day
2. Go to the library at least once a week
3. Practice her math skills
After finishing the contract, I told her that we could play a game for the remainder of the session. She chose BrainQuest, which is a game where you ask trivia questions from various subjects. However, Mizan would only play if she asked me questions as well.
Even though the questions were designed for 6th graders, a few of them even stumped me! The nice thing about these questions was that they prompted further discussion regarding academic topics.
For example, one question read:
Which is the smallest unit of an element: an atom or a cell?
Mizan said a cell, which is obviously the incorrect answer. However, this prompted our conversation about the difference between living and non-living things, what an element was, etc.
We continued playing BrainQuest until the session was over. Mizan was very polite and helped me put all of the pencils, books, and supplies away. She thanked me for helping her and wished me a good spring break.
“I think I’m gonna save these notes and these problems you gave me,” she said as we started to leave. “I want to keep working on this math until I understand it fully.”
I was very happy with Mizan for all of the progress she made and for being such a cooperative student.