Of Stomach Aches, Janitors Closets, and Ooblek: My Math Journey
We started our school year two weeks ago. Things were going great until one day I sold out in math with my second grader. Up until now, her math curriculum has been entirely hands-on with little paper and pencil. I decided to give a standard workbook a try. So, I flipped through the pages and saw "Skip Counting." Because heaven forbid you should use big, scary mathematical words, even if they are the most accurate. Anyway, on one page, students were meant to practice skip counting by 2s, 5s and 10s.
It went like this. For the first set, you had about six faces in a row and underneath each face was a blank. Most of the blanks already had traceable numbers under the faces: 2, 4, 6, 8.... The directions: Count the eyes. I cringed and groaned aloud. I tried to give in and do the page with my daughter even though I knew it wasn't right. She and I are so much alike and I could tell that she was struggling with the page. She is actually very mathematical/logical in her thinking and this was the first math lesson we had done where she was totally confused.
There were several problems with the exercise. One, it wasn't a good representation or practice of counting. When you count things, either physically or symbolically, with numerals in the case of a large number of items, there is a one-to-one correspondence between each number and its corresponding item. It drives me crazy to see parents ask their toddler to count for adults and then that child proceeds to simply recite numbers in their correct order.
A child on the floor, pulling blocks toward himself as he names each block, "One, two, three..." and then tells you the total number of blocks has counted. But put that kid in front of dinner guests and people will be aghast the poor child has to use blocks to count!
The second problem with the exercise relates to the first. The writers made the problem more abstract, not concrete for the child. The first face is correct. Under the face, the total number of eyes would be two. It doesn't work that way for the successive faces and blanks beneath, though. On the second face, the child is to trace the number four, but there is only one face above that blank. The child is supposed to keep a mental tally and the pictures above the blanks don't represent the numbers underneath.
There's an easy solution to make this a better paper-and-pencil exercise.
Instead of presenting the faces in a horizontal format, they could be represented vertically. So, on line one: one face with a blank beside it, line two: two faces with a blank beside it, line three: three faces, etc... Now, the students are actually counting and the blank is actually giving the total represented by the graphic. The second line would have two faces and therefore two sets of eyes represented, so the number four is an accurate total of the eyes represented by its corresponding graphic.
After this little fiasco, memories of my own math journey have kept coming to mind. During my first three years of college, as a young, bright-faced eager elementary education major, if you had asked me what I wanted to do in the future, I would have responded, "I want to teach English and Speech to middle schoolers." I love words, both written and spoken. I experience sheer joy when reading a good novel, short story, or play and then going back and savoring it, plumbing its depths for symbolism and layers of meaning. I dog-ear pages so I can go back and experience again those perfectly-worded sentences and phrases--the ones that make you wish you had said that or that if the author never wrote anything else, that was something of which to be most proud. I love the different nuances of oral language and speeches. I was going to teach my students to write well, to express themselves in their own voices. I was going to expose them to great literature and lead them to experience the power of the written word. I was going to teach them to speak and to recite. I was going to be An English Teacher.
Then came my senior year. The typical senior year for a four-year completion as an elementary education major meant Methods of Teaching in the fall and student teaching in the spring. Now, that was finally the meat of the program! For three years, I had heard upperclassmen speak of this mythical Methods in the most exhausted of manners. It was as if student teaching would be the proverbial cake walk in comparison to ...Methods....From what I had gleaned over the years, I still did not understand the organization of the Methods semester and no real idea of what to expect. It was to be the best time of my college years and it would have a profound effect upon me as a teacher and a person.
The Methods courses were organized as a block of classes taught by four teachers. The language arts instructor was not a permanent part of the core group of professors that semester. The core group of three were amazing. Together, they developed and implemented a Methods program that could compete on equal footing with any in the nation. I couldn't wait for Language Arts and Social Studies. Ah, the Humanities! Science I was interested in because my dad was always complaining that I didn't have enough science in my degree plan. Then there was math. Ugh. I would just get through it as best as I could. Why ugh?
The workbook page I gave E transported me back to second grade and memories of my first problem with math. It was skip counting and I just couldn't get it. I'm sure it was because of the similar workbook representations. The other kids were busy memorizing and reciting the even numbers to 100 and I couldn't get past the page not making sense. Although at the time, I would not have been able to express that. By December of that second grade year, I was starting to "feel bad" every morning in hopes of staying home.
It was a gamble because in our house, if you stayed home sick, you had to go to the doctor. I didn't miss school much. I used to think it would be fun to have stupid parents.
Mama was wise at reading situations and she knew several things about this one. I always seemed to feel better by eleven o'clock and I asked to go to school for the rest of the day. We changed classes in second grade, between three teachers. Math with Mrs. Knighten would have been over by around eleven.
Mrs. Knighten was several firsts in one for me. Mama's first inclination was to think I didn't like her because she was the first teacher who didn't baby me or view me as a teacher's pet. That was partly true. My kindergarten teacher used to rock us in her big rocker and I was definitely in her top list. My first grade teacher was beautiful and just seemed like a perfect teacher. I was treated as a favorite there, also. Mrs. Knighten treated everyone the same; imagine that.
Until that time, she was the oldest teacher I had and she was my first African American teacher. I feel guilty because I benefited from the times. Many of the best teachers I had in K-12 were African American women. Most of them were older and when they were choosing careers in the deep south, opportunities were pretty limited. These were intelligent women who could have been anything. They dressed and acted in a completely professional manner. From their appearance and performance, they could have been lawyers appearing in court or doctors presenting a new paper, instead of teachers in a south Louisiana classroom. I don't mean to say that teaching is any less of a profession than these two examples.
I don't think her race had anything to do with how I felt about her. Racism wasn't acceptable in our home. The problem was math. It was the first time I had trouble with something in school and I was terrified of Mrs. Knighten. Why? The handshake. That was back in the days of paddles on hooks. Mrs. Knighten didn't need a paddle. She would squeeze a misbehaving boy's hand so hard, they would literally fall to their knees and beg for mercy. By the end of the year, she let up a little and the boys would ask to have their hands squeezed and compete to see who could stand it the longest. It was scary stuff in the beginning of the year, though.
So, Mama met with Mrs. Knighten and told her she thought I was afraid of her. She did it in a very matter-of-fact way and told her that it was the first time I wasn't coddled in a classroom. She also let her know that she and my dad thought that was fabulous and just what I needed. I wondered why I couldn't have the irate parents who rushed to my defense for every little thing. See previous comment about stupid parents.
Mrs. Knighten put her arm around me, told met that we were going to have a great rest of the year and...did absolutely nothing different that what she usually did, except one thing. She was wise, like Mama, and she had looked for the cause of my distress. She finally saw it in my eyes one day and she stopped and helped me. After that, I knew I could ask her for help. At the end of the year, I cross-stitched her a "World's Greatest Teacher" piece and Mama framed it as my end-of-the-year gift. Once, after she retired, she subbed for our class. She told the whole class about my gift and said it was hanging in her home.
The next few years I had average math teachers and I was able to make As and Bs by relying on memorization. Even though I was identified as Gifted and Talented, by sixth grade the cracks in my math foundation were showing and it was all starting to fall apart. I overheard Mama say my teacher thought I had a "math problem." That was all I needed. Instant crutch. From that point, I quit giving my best and simply excused myself with, "Oh, I can't do that because I have a Math Problem."
By high school, I had given up. I was a year ahead in English honors courses, with straight As. I won a trip to Washington, D.C. in an essay contest. When I decided to try out for the flag corp my sophomore year, though, I didn't qualify because of my Algebra grade. Teachers even tried to get the rules bent for me. Why not? I had a Math Problem.
My parents were surprisingly easy-going about my math grades, though. If I slipped at mid-term in other course, it was a big deal, but for some reason, it didn't matter to them in math. Turns out, my dad's secretary was a math major, taking graduate courses at night. I'll just say she had observed first-hand the problem with my high school math instruction. My parents didn't tell me what they knew until years later. They didn't want me to use it as an excuse or open my mouth to tell other students.
"Life's not fair," was popular in our home!
Then, we moved during the summer before my senior year. I decided to take Trig/Advanced Math at my new school. Oh, joy. I was finished with English and now I got to take math in a new school surrounded by strangers. On the first day, Mrs. Smith walked in. Her stature was such that on the floor, running the entire length of the chalkboard, was a platform that was probably 1 1/2 feet high. She stood on the platform to write on the board. She was feisty and she could teach math. She loved it and she seemed to do even better with explanations when people asked questions. Nothing could trip her up.
She picked up on me right after the first weekly quiz. "You need help. Come see me after school. If I'm not in the room, just go knock on the janitor's closet across the hall. I go in there to smoke." So, I started going to her room after school most days.
She also gave me a teacher's edition of the textbook. One, she trusted me, and two, if her instincts were wrong, it would become obvious that I cheated if my daily homework score suddenly went up. Oh, the homework grades. Every Monday- Thursday, with a quiz on Friday. The first six weeks, I made a D. Mama scheduled a conference and Mrs. Smith told her I was the hardest worker in the class. She said my homework grades brought my grade down. I made all mistakes during the week and showed huge improvement on the quizzes. Later, my grades rose and for the first time, since first grade, I didn't hate math. Mrs. Smith was so good she could spot my gaps.
"It's as if you never had Algebra II," she said.
I thought back to that class, filled with days of playing cards, reading, and talking in groups. Literally. I remember the coach once said, "I've got to make up some grades." He then asked us our school activities to determine what grade we deserved. I was pretty active in clubs,, so that was my best grade in math.
Since I had spare time in my schedule, Mrs. Smith arranged for me to audit an Algebra II class. I loved it. At some point, I began to realize that the same reasoning skills I applied to a piece of literature as I analyzed it could be applied to mathematics. Eureka. Finally. One look at my scratch work would make a math major tear out their hair, but it led to the correct answer and I understood how I got there. Even better, I could usually manage to get there again!
Armed with a more positive experience with math, I was ready to meet Dr. Mary Reeves on the first day of the dreaded Math Methods course. I was not prepared for what would happen next. I was first stuck by her because she looked so much like my sister. She was about the same age and even had some similar mannerisms. Plus, my sister possessed the elusive "math and science gene" that I then honestly believed to exist.
Then, I was impressed by her teaching style. It was a rare balance of casual authority. She was able to command respect while still treating us as if were colleagues. That's a tough one; you can't fake that. She was also witty and sarcastic. And there wasn't a darn thing about her that was cutesy.
If there was one thing I loathed about being an elementary education major who wanted to teach middle schoolers, it was Cutesy. We had to take a Children's Lit/Library course and one focused on music in the classroom. The big project in the library class was completion of a bulletin board in the Education building. The board had to depict a children's book. Mine was easy to spot amongst the brightly colored boards featuring beloved characters, woodland creatures, and three-dimensional features:
|When you pack and move, you find all kinds of things--like a picture of that bulletin board!|
Over the course of the Methods semester, we would learn the "whys" behind mathematics. No longer would it be acceptable to teach kids simple algorithms and expect them to memorize and repeat. I remember the marvel to me that was place value. My foundations with place value as a child included memorizing the place value names in the correct order. When it came time to use place value in subtraction that involved "borrowing," I learned in elementary school:
1. Look at the number on the bottom in each place value column. If it's larger than the top one, you have to borrow.
2. Cross out the top number and put a one in front of it. But, depending on the column, you may just cross it out and make it one less.
3. Keep moving left and do this until you come to a number on top that is more than the number on the bottom.
4. Subtract in each column. Pray that you got lucky.
I had absolutely no clue what the place value names really meant or why on earth you put a "1" in front of the top numbers.
Enter: place value base 10 rod kits. The basis of our Math Methods class was not a textbook; it was Dr. Reeves and a math manipulative kit that was full of teaching items that helped make the abstract items of math more concrete for students. Place value rocked my world. I was so excited. My friend Amber had to listen to my excitement after each class. Poor Amber. That's a good friend. She's brilliant; it must have been so much fun for her to listen to me talk about finally grasping an elementary math concept!
I came to abhor the term, "borrowing," because it wasn't as complete a term for what was really taking place. I learned how to teach the concept of regrouping. You could take (borrow) a set of tens and regroup it into single ones. That's why you wrote the "1." Eureka.
Then came more topics: fractions, decimals, multiplication. We had to build classroom kits in our methods classes. When we left the semester, we had activities ready-to-go for our first teaching jobs. I used some of those throughout my nine years of teaching. My first few years, more experienced teachers would borrow from my kits when their observations by the principal were scheduled.
To perfectly dovetail with the revelation I came to in Mrs. Smith's class about the connection between analyzing literature and math, Dr. Reeves made us keep a math journal. I was able to sort out my thoughts about math concepts and think them out more clearly so my brain could really process them. That journal is on one of our bookshelves and I still look at it from time to time.
During that semester, I was also introduced to someone who would become my other math hero: Miss Toliver.
She respected students, taught for understanding of the concept, and made learning fun while still requiring students to improve accuracy with practice, practice, practice. Kids walked in to find trays of Oobleck or toothpicks on their desks. Sometimes, she wore a costume to pique their curiosity. That's what I wanted to be like as a teacher! I even wondered what life might be like as a math teacher instead of An English Teacher.
|Worksheets are fine for practice!|
I would find out after graduation. I worked that summer at the university middle laboratory school. The principal, Mr. Berry, was one of my instructors during my sophomore year and I did methods teaching in his school. I fell in love with the middle school model. Many schools are called middle schools, but few actually follow the model which includes teaming, advisory, exploratory courses, etc...
Mr. Berry offered me a graduate assistantship to teach at the lab school while I completed a master's degree, but I really wanted to head back to south Louisiana to teach. I lived at home during college and it was time to live on my own. When he realized I was determined, he made a phone call to the president-elect of the state middle school association. He had one of the few middle schools in south Louisiana. I stood outside the office while the call was made.
"I've got this little gal here who's a great teacher and you want her at your school," he began, in true deep south fashion. I had three different teaching positions in my career and every one of them was the result of similar phone calls.
It seemed that the principal had only one opening. It was a new position for an exploratory course called Exploratory Mathematics. Experienced teachers wouldn't touch it with the proverbial ten-foot-pole. It was a semester-long class required by all seventh graders (except those who were both Gifted and in Band; those classes took up their exploratory spots). They would take it in addition to their regular math class and he wanted the following:
1) At no point should they feel like they were in a math class.
2) Studens would have no textbooks or workbooks.
3) Standardized test scores in math should improve.
Well, what bright-eyed, eager, Miss Toliver groupie, first -year- teacher could turn down an offer like that? I went for the interview and was told the job was mine, based solely on Mr. Berry's recommendation.
So, I became a math teacher. My dad laughed when I told him.
It was actually something that he taught me years before, though, that made me think I might be able to do this. He introduced me to Notre Dame football because he was an admirer of Coach Lou Holtz. He told me once how the best athletes don't always make the best coaches. Lou Holtz was a skinny, uncoordinated kid whose position on the football team was what my dad and others would jokingly refer to as "end and guard." He sat on the end of the bench and guarded the water bucket.
|Love Coach Holtz!|
Daddy explained that sometimes, people like that made the best coaches since they had to work so much harder to learn everything about the game compared to a person with more natural ability. That's how math was for me ever since those days in Mrs. Smith's Trigonometry classes. I had to come at math from the back door and learn it in a much longer, detailed fashion with lots of practice. It gave me different approaches and strategies for explaining it to others. My own thought processes were sometimes much more clear to me than the processes of the person who could "just do it," but didn't know how to explain what came more organically, with fewer steps.
I spent the next three years teaching seventh grade Exploratory Mathematics. I dressed in an M & M costume when we explored probability in M & M packages. The kids would walk in and find things like whole pineapples or toothpicks and marshmallows on their desks. They were excited to find out what they were doing that class period. I would draw elaborate puzzles on the board each day for their Math Mind Warm-Ups.
|My first classroom.|
We also kept journals and I taught them to write about mathematics and their thought processes for solving problems. I did monthly Family Math nights where I taught families math games using playing cards, beans, toothpicks, and other household items. Kids enjoyed math and I loved teaching them. The test scores did go up, although I don't know how much I actually had to do with that.
When I told the principal we were moving, he made a decision on the spot. No more exploratory mathematics because he said he couldn't find another person who could do it the way I had done. Then, he told me I could take any of the math manipulatives and books I wanted because, "You and I both know they will just sit on a shelf and not be used after you leave."
It was a tremendous compliment that made me very sad. The next year, instead of Exploratory Mathematics, there was a class based on a computer program, with a teacher monitoring the students on computers. Someone missed out on the thrill of being A Math Teacher.
I would go on to teach jr. high honors Math and English for two years before I made the switch to elementary school. I wanted a kinder, gentler environment and I wanted to get "first shot" at teaching basic foundations of math and writing. It's easier to do that than it is to un-do things later.
I stayed in touch with Dr. Reeves for a while after graduation. She and her husband drove over three hours to attend our wedding. Their gift, a painting, still has a special place in our home. Recently, we found each other again through facebook. We may be on opposite sides of the political spectrum, but we find ourselves having common ground quite often and she's still a special person to me.
Now, I'm a homeschooling mom who gets to decide how it's all taught from the beginning. Sitting at my kitchen with those who are truly my "own" students, I'm not exactly alone. Whenever I teach, there's a little of Mrs. Knighten, Mrs. Smith, Dr. Reeves, and Miss Toliver in what I do. I will always be grateful to those ladies for their guidance on my math journey.