Tutoring Stories
From IAE-Pedia
Contents |
Introduction
This page is part of the Information Age Education Math Tutoring Project. General background information about math tutoring is available in the free book:
- Moursund, David and Albrecht, Robert (9/2/2011). Becoming a better math tutor. Eugene. OR: Information Age Education. Both a PDF file and a Microsoft Word file are available. Click here to view the TOC, Preface, the first two chapters, and the two Appendices.
Scenarios from the Math Tutor Book
George, A dyslexic Student
In his early childhood, George was raised by a combination of his parents and two grandparents who lived near his home. George was both physically and mentally above average. He prospered under the loving care—think of this as lots of individual tutoring—provided by his parents and grandparents. He enjoyed being read to and this was a routine part of his preschool days.
George was enrolled in a local neighborhood school and enjoyed school. However, his parents learned that George had a learning problem when they received his end of second grade report card. The teacher indicated that George had made no progress in reading during that entire year and was having considerable difficulty with math word problems.
His parents were surprised by the fact that George actually passed second grade, and that the teacher had not made a major intervention sometime during the school year.
A grandparent had heard about dyslexia, and so the parents and grandparents did some reading in this area. Dyslexia is a type of brain wiring that makes it difficult to learn to read. And sometimes makes it difficult to learn arithmetic. It was obvious that George was dyslexic.
Under strong pressure from George’s parents, the school tested George, and it turned out that he had severe dyslexia. With the help of an IEP (Individual Education Program) that included a substantial amount of tutoring by reading specialists for more than a year, George learned to read and more than caught up with his classmates.
This is a success story. Dyslexia is a well-known learning disability that makes it difficult to learn to read and that also can make it difficult to learn to do arithmetic. Extensive individual tutoring leads to a rewiring of the tutee’s brain. This rewiring allows the reading-related structures in the tutee’s brain to function much more like they do in a student that does not have dyslexia.
Kim, A Fourth Grader Tutored in Her Home
Kim was a fourth-grade student who did not like math. Alas, early in the school year, her math grade was a D. Kim did better in other subjects. Kim's mother Jodi was sure that Kim could do much better with a little help, so she hired a tutor (Bob) who would come to their home once a week, help Kim do her math homework, and hopefully help Kim to like math better, or at least dislike it less. Jodi knew that Kim did well in subjects she liked.
Jodi and the tutor talked. "Aha" thought the tutor, who loved math games. "This is a splendid opportunity to use games to make math fun for Kim." The tutor suggested to Jodi that each tutoring gig spend some time playing games as well as doing the homework. Jodi readily agreed.
Tutoring began. Each tutoring session, Kim and the tutor spent 30 to 40 minutes doing homework and then played math games. Kim loved the math games. After a few tutoring sessions, she became more at ease doing the homework because she knew that she would soon play a game. Better yet, she began trying to do more homework before the tutor arrived in order to have more time to play games.
Kim became very good at playing games, including games at a higher math maturity level than usual for a fourth grader. It became clear to the tutor that Kim was very smart in math.
Kim and the tutor played many games. Her favorite game was Number Race 0 to 12, a game in which you try to move racers from 0 to 12 on five tracks. (See Chapter 5 for a detailed description of this game.) To move your racer, you roll three 6-faced dice (3D6) and use the numbers on the dice to create numerical expressions to move the racers on their tracks.
As the weeks rolled by, Kim became better and better at creating numerical expressions. After a few weeks, she became as good as the tutor in rolling 3D6 and using addition, subtraction, multiplication, and parentheses to create numbers to move her five racers on their five tracks.
Spring rolled around and Science Fair project beckoned. Kim and her mother asked the tutor to suggest science fair topics. He did. Among the topics was one of his favorites, making homemade batteries from fruit, vegetables, and metal electrodes. Kim liked this idea and chose it as her science fair project.
Kim, with great support from her mother, made batteries using apples, bananas, lemons, oranges, potatoes, and other electrolytes. She experimented with pairs of electrodes selected from iron, aluminum, carbon, zinc, and copper. Jodi bought a good quality multimeter (about $40) for Kim to use in order to measure the voltages produced by various combinations of fruit, vegetables, and metals. Kim found that copper and zinc electrodes produced the highest voltage using several fruits and vegetables as electrolytes.
This story has a very happy ending. Kim’s Science Fair project was outstanding! And, Kim became a very good math student! In retrospect, we can conjecture that Kim’s previous home and school environments had not appropriately fostered and engaged Kim’s abilities in math and science. The combination of two tutors (mother and paid tutor) helped Kim to develop her interests and talents in both math and science.
The active engagement of Kim’s mother was a very important part of this success story. Jodi was an excellent role model of a woman quite interested in and engaged in learning and doing science. This story also illustrates the power of a team engaged in the tutor/tutee process. The active engagement of all three members of this tutor/tutee team was outstanding.
This story also illustrates another important point. The tutor had a very broad range of knowledge, skills, and approaches to getting a tutee engaged. The real breakthrough came via games and the Science Fair project rather than through the original “contract” on homework tutoring.
With the help of the paid tutor and her mother, the tutee became a very good math and science student.
Two 8-th Grade Girls in Algebra
One of my first tutoring gigs was tutoring two 8th-grade girls in algebra. The three of us met twice a week for the entire school year in the home of one of the girls.
For the first few weeks, we spent our hour doing the assigned homework. The tutees did not do the assignment prior to my visit, but waited until I arrived. Then we slogged through the assignment together.
One day we finished early, so I asked, "Want to play a game?" They said, "OK."
We played Pig for the rest of the hour, and I stayed on for a while afterwards because they were having so much fun.
Before I left, I said, "Hey, if you do your homework before I arrive, we can go over it, and then play games. I have lots of games."
From that day on, they did their homework before I arrived and we went over it. Because we were not pressed for time, we could delve more deeply into what they were learning and/or could be learning in doing the homework assignment problems. We always finished with ample time to play a fun math game.
This example illustrates using a potential reward to shape behavior. Behaviorist learning theory is discussed later in this chapter. The example also illustrates that the tutees were quite capable of doing their homework without the aid of a tutor. The tutoring environment provided a type of structured social and educational learning situation that made the homework more fun.
A Mentoring Example
When I was the math/science mentor in Richard Zimmer’s Mars Society (fall semester) and Mars Habitat (spring semester) courses at Sonoma State University, I posed the task of designing learning environments for the children in the three communities on Mars that our students were designing.
At first, each group designed “schools” like the schools they were used to. I encouraged them to jump out of that box and indulge their most outrageous fantasies about learning environments that they would like to grow in from age 0 and up, on Mars or on Earth.
Beautiful! Loverly! They responded with learning environments that used the entire community as the learning environment and included learning much stuff from the computer-based artificial intelligence network, low-road stuff, and also high-road stuff using AI simulations and AI-human dialogs. And, tra la, tra la, apprenticeships. Every person in the community who had knowledge and skills became a possible mentor for anyone who wanted to acquire that knowledge or those skills at any age.
On Mars, no schools. Instead, everyone helps everyone to learn. Serendipity!
Asperger's Syndrome Student
In this scenario, the tutor is working with a student identified by the initials SD. The scenario is written in the present tense.
SD is diagnosed Asperger's Syndrome. He spends much time in the high school’s Special Services Center. I tutor him 1st period Tuesday and Thursday, and frequently see him when I am tutoring other students. He rarely smiles, and he frequently seems tired and/or distracted.
One Thursday, SD’s Geometry homework was light and we finished it in 25 minutes. I was about to ask, "What shall we talk about the rest of the period?" SD preempted me and asked, "What shall we talk about the rest of the period?" I said, "You choose." He laughed, smiled, and said, "Dungeons & Dragons." SD's body language went from neutral to gleeful anticipation.
For the rest of the period, SD was animated, smiling, laughing, and clearly having a good time. During the next 30 minutes, SD was highly interactive, engaging in a dialog that we shared equally.
I shared some of my personal experience with D&D and other Role-Playing Games, and described how I had used them with elementary school students. We discussed the six characteristics of a D&D character: Strength (STR), Constitution (CON), Dexterity (DEX), Intelligence (INT), Wisdom (WIS), and Charisma (CHA). SD knows about rolling 3D6 to create a beginning character so that each characteristic value is 3 to 18 inclusive.
Game Masters are likely to let you roll 4D6 and choose the “best” 3D6 in order to avoid a very low score for a characteristic. We calculated the probability of rolling 3D6 and getting three ones and the probability of rolling 4D6 and getting four ones. We agreed that, rolling 3D6, an average characteristic value is (3 + 18)/2 = 10.5. But, no roll of three dice can produce 10.5. So, we talked about the idea that the average of a set of numbers can be different than any number in the set.
We continued our investigation of the math involved in using dice to create D&D characters until the bell rang.
It is exciting and rewarding to participate in such a tutoring session. The tutee chose a topic and the tutor was able to bring both personal experience and math into the topic. Bob (the tutor) has a huge repertoire of games to draw on, as gaming has been one of his life passions.
Square Root Black Holes
In a recent tutoring session, we finished the required work before the period was over. So, I showed my tutee the Square Root Black Hole.
The process: Start with any positive real number. Take the square root of that number. Then take the square root of the answer. Keep on truckin'—take the square root of each answer.
This process is easy to carry out on any calculator that has a square root key. We were using a TI-84. I showed my tutee how to use [2nd] ANS on the TI-84, and then he began the task, starting with 100. Below I use SQRT to indicate the TI-84's square root function.
SQRT(100) [Enter] = 10
SQRT(ANS) [Enter] = 3.16227766
SQRT(ANS) [Enter] = 1.77827941
SQRT(ANS) [Enter] = 1.333521432
Then my tutee and I discovered that he could press [Enter] without SQRT and get the next answer.
[Enter] = 1.154781985
[Enter] = 1.074607828
[Enter] = 1.036632928
[Enter] = 1.018151722
[Enter] = 1.0090035045
[Enter] = 1.004507364
[Enter] = 1.002251148
[Enter] = 1.001124941
[Enter] = 1.000562313
I asked my tutee to make conjectures about what is happening. His conjectures:
• The numbers are decreasing.
• The numbers are all greater than 1.
• The numbers are getting closer to 1.
• The decimal to the right of the decimal point goes down by a factor of about two each time.
Abracadabra! Alakazam! I had never noticed that last conjecture. Score math maturity points for my tutee.
The tutoring period ended before we had time to explore using the same process for positive numbers less than 1. In addition, we did not have time to discuss rounding errors in calculator arithmetic. All in all, this is a quite interesting topic! My tutee agreed that it was amazing, fun, et cetera, et cetera.
High School Student with Asperger's Syndrome
DQ was a high school student when I tutored him for two years. He has Asperger syndrome. Quoting from the Wikipedia:
- The lack of demonstrated empathy is possibly the most dysfunctional aspect of Asperger syndrome. Individuals with AS experience difficulties in basic elements of social interaction, which may include a failure to develop friendships or to seek shared enjoyments or achievements with others (for example, showing others objects of interest), a lack of social or emotional reciprocity, and impaired nonverbal behaviors in areas such as eye contact, facial expression, posture, and gesture.
Your authors know people in the computer field who have AS and have been very successful in their careers. We emphasize with their social problems and admire their intellectual achievements.
The relationship I developed with DQ was a combination of being a math tutor and mentor. When I first met DQ, I soon learned that he is a Star Trek fan and identifies with Spock and Data. He likes logic and frequently talks about logic. One day I asked him, “You like logic and you are a Trekkie, so is it OK if I call you SpockData?” Big smile and an enthusiastic “Yes!”
SD (formerly DQ) has well defined aspirations for his future. He wants to go to college, get an MBA, quickly make a lot of money, and then spend the rest of his life going to college and learning about everything. A major goal in tutoring SD was helping him to make progress toward achieving his goals. He has strong intrinsic motivation and our tutoring built on this intrinsic motivation.
Since these goals include a college education, some tutoring time is spent helping him to understand what is required to get into a college and what it takes to be successful in college. Many high school students have little insight into the nature of the academic challenges and requirements of higher education. SD is doing well in moving toward his future.
Tutoring to Raise Score on State Test
In the school where Bob tutors, some students receive special tutoring designed to help them score higher on the state test. In these tutoring sessions, I with a group of one or two tutees. In the situation described below, I worked with a total of 17 tutees, with no more than two at one time. Most were high school sophomores and juniors. Sample questions from previous tests were used in these tutoring sessions.
I make use of the following scale to help describe student responses to various math questions.
- Abracadabra! Alakazam! Solved with no help or almost no help. Oh happy day!
- Solved with here a nudge, there a hint.
- Solved with much ado and heaps of help.
- Alas, alack, and oh heck. Led through every step of the solution and still boggled.
Here are three sample test questions. Student average scores (using the above scale) are included. When the average score is less than 3.0, I try not to cry.
Example 1. Average score = 1.8 (17 students)
No student noticed that (1/2) raised to the 4th power is less than 1 and therefore can be ignored as a possible answer.
I had to show almost everyone how to use a TI-84 calculator to calculate (1/2) raised to the 4th power.
Some students asked where the fraction key was. How sad.
Example 2. Average score = 0.6 (17 students)
Most did not know how to calculate percent.
Students who knew how to calculate percent calculated 60% of 800 and selected A. 480.
I had to point out "60% greater than …" and explain what it meant.
03. Average score = 0.8 (17 students)
Most selected A. two times as great. No one drew diagrams. I drew diagrams of a 4 by 2 rectangle and an 8 by 4 rectangle with dashed lines showing four 4 by 2 rectangles inside the 8 by 4 rectangle. Some still looked puzzled.
Over time, students forget most of the math that they cover in the courses they take. Teachers and curriculum developers know this, so curriculum includes a great deal of review. Nowadays, it is common for math classroom teachers to also spend considerable time teaching to upcoming high stakes tests. Thus, the tutoring scenario could just as well have occurred in a regular math classroom.
Teaching or tutoring using sample test questions can have several different purposes. For example:
- Acclimate students to the types of questions and the testing environment that they will encounter. This helps them to get used to the time pressure, the tools they will have available (pencil, paper, graph paper, calculator, formulas) and how they will indicate their answers (such as a mark sense form or on a computer), and how they can go back and make corrections.
- Help students relearn content and problem-solving strategies that they have covered in the past.
- Help students gain increased skill in transferring their math knowledge and skills into a good performance in a high stakes testing situation.
- Help students learn to play the game of taking multiple choice math tests. This includes learning how to deal with questions that may be poorly stated.
Math educators agree that it is important for students to learn to check their answers. One way to check an answer is via estimation. Another way is determine if the result “makes sense.”
Lets look at the first example from the test questions given earlier.Listen to me as I (one of your authors) imagine “playing the game” of answering the question.
- The statement of the question suggests that there is only one correct answer. I glance at the four answers and notice that the first one is quite small relative to the others. So, the correct answer is B, C, or D. I carry around in my head that the square root of 5 is approximately 2.236. I quickly compare this with 7/3, which I know is 2.333…That eliminates answer B, and a tiny bit more thinking allows me so see it also eliminates answer D. In a modest number of seconds I conclude the answer is C, using only my brain as a calculator and information retrieval device.
- If I did not remember an approximate value for the square root of 5, I could have used my calculator. A calculator provides an alternative to memory, mental calculation, and paper and pencil calculation.
Your authors are well aware that this type of thinking is way beyond most of the students that receive tutoring. For the most part, it is a type of math maturity that is not taught in their classes. The quickest way for them to find the answer is to use a calculator. Alas, some students did not know how to use their calculator to calculate (1/2) raised to the 4th power..
Now, listen to me as I (one of your authors) analyze the second test question.
- At first glance, my brain is slightly befuddled. More students prefer cola than prefer milk. Maybe the first possible answer will help un-befuddle my mind. Yes, 480 is too small, because it is not larger than 800. So, the answer must be B, C, or D. I need to think more carefully about “how many more” students prefer cola. Yes, it is 60% more (than the 800 who prefer milk). Hmm. What is 60% larger than 800. Aha, that is easy enough. I can easily do 60% of 800 in my head, and add the result to 800 in my head. The answer is C.
- Alternatively, I could just use an approximation that 60% is a little more than a half. So, the answer needs to be a little more than 800 plus half of 800. That eliminates answers B (too small) and D (way too large), and I am done.
In both of these examples, I am playing a type of test taking game. The techniques I use are a type of math maturity that a fifth or sixth grade student might well have developed given appropriate instruction and practice. And therein lies a major problem in our math education system. Students are not given appropriate instruction and practice of this type of analysis and thinking because it would require too much class time.
Finally, consider the 3rd sample test question.
- The statement of the question suggests that the answer is independent of any particular rectangle. So, if I draw a rectangle and then one that is twice as long and twice as wide, all I need to de is to compare the areas. This is particularly easy if I have graph paper or a ruler. I do not need to know a formula for the area of a rectangle.
- Drawing a relatively precise diagram can be useful in solving a variety of multiple choice test questions. Contrast this question with the following questions that require “constructed” solutions.
A. Prove that if the length and width of a rectangle are doubled, the resulting rectangle has four times the area of the initial rectangle.
B. Prove that if the length and width of a rectangle are doubled, the resulting rectangle has twice the perimeter of the initial rectangle.
A Teacher's Lament
The following story from Sonya Richardson isn't exactly about tutoring. However, it reflects a situation in which tutors are often called in to help. It also reflects frustrations with our math education system. Quoting an 11/10/2011 email message from Sonya Richardson to David Moursund:
- Many of these students have had high school algebra (both 1 and 2) and geometry. They just didn't learn it. Chronic problems are not knowing the multiplication tables, making it difficult to find factors of numbers for various operations. The other real biggie is fractions. Students just don't understand them and how to work with them. Changing from decimals to percents and the reverse are also problems. These students do not understand the number system.
- MTH 65 [A course offered in college but that does not carry college credit toward a degree] is about equivalent to high school algebra. The same kinds of problems persist into MTH 95. It is about equivalent to high school Algebra 2.
- The most failed course in the Math Department is MTH 111. I am aware that students continue to go on with math to the next level even though they fail or get a D in MTH 65 and MTH 95 and then when it counts, they don't make it. It's a tough situation.
- I wish I could corral all these students into a basic skills class where we could model the basic concepts and use hands on learning to make sure the students learn them and get comfortable with them before going on.
- One of the changes I would love to see is allowing students to fail in the early elementary grades and on so that they have a reason to study math and get it together before they get to the kind of situation we see today. It would take some major attitude changes on the part of administrators and parents!
- I had to fight the administration a number of times to allow a student to repeat part of course so that they could become successful. The parents wanted it, the students wanted it, but the administration was not on board with us. When we did accomplish this, the students really did become successful. It was wonderful to see.
- I used to get students in Algebra 2 or geometry who had failed Algebra 1. Then I was supposed to make them successful! It's just so unrealistic.
- I had a student one year who had failed Algebra 1 in a previous school (we didn't know) and when he transferred after school started was automatically put into Geometry that he couldn't handle, and when his records finally came, he was put back into Algebra one. He proceeded to do nothing with it, not paying attention to doing assignments, etc. Then he had an accident, had some brain damage, was put on an IEP, and I was supposed to make him successful in Algebra 1. The parents and administration really came down on me for not getting him to succeed. Crazy system.
- Many of the students in MTH 65 work their hearts out, trying to accomplish what they need to. It just doesn't work for them with the background they have. I hate to see it.
Additional Scenarios
Readers: Please share your math tutoring experiences—both successes and failures. Send your "stories"' to [moursund@uoregon.edu Dave Moursund] or [starshipgaia1@msn.com Bob Albrecht.]
Author or Authors
This page was created by Bob Albrecht and Dave Moursund.
