Posted by Jim Jewell at 5/25/2013
i. Math is Foundational
ii. Math Test Scores are decreasing
a. Faulty foundation leads to shaky understandings
Student thinking consists of many things. Formulas, relevance, tedium, and enjoyment are part of their attitudes and thinking about mathematics. One problem that leads to very serious learning difficulties in mathematics is the set of misconceptions students may have from previous inadequate teaching, informal thinking, or poor remembrance. We begin with a definition. Pines (1985) offers the following definition: Certain conceptual relations that are acquired may be inappropriate within a certain context. We term such relations as "misconceptions." A misconception does not exist independently, but is contingent upon a certain existing conceptual framework. Misconceptions can change or disappear with the framework changes. http://mtc.tamu.edu/9-12/index_9-12.htm?9-12M2L1.htm
iii. What’s the solution?
i. The 5 Strands
Learning about numbers, the operations, algebra, geometry, measurement and probability lay the foundation for math. While number sense is paramount to the foundation the other strands are integrally woven into the picture and yet still mastered in this order. Each strand depends upon the other for math to be fun and yet it’s not just about this sequence. Yes, for mastery, sequence is important, but the journey provides the cement that holds everything together.
ii. Process thinking
Process thinking is next in making math fun. James, as the traveler must take the steps by himself or nothing is learned and the foundation begins to crumble. Knowing the hows and whys are as important as anything in math or better yet in life. When he knows the hows and whys, he can justify and explain the importance of anything. Being able to struggle and learn this for himself builds upon a strong math foundation and makes math fun.
Solutions are the natural outcome of a solid foundation and taking the procedural steps to solve any problem. Why is finding the solution so important? The solution is the Victory. Without victories how can he measure himself against life’s challenges? As the very nature of math is foundational, he gains power from the victory and continues to solidify his foundation further, thus ensuring additional learning and reinforces the sense that math is fun!
Once he learns the solutions and can justify these solutions thus reconfirming that he is indeed standing on a solid foundation is he ready to learn that there is another way. My mom showed me some shortcuts to Geometry in college, while I was learning how to be a better teacher. I can remember how this frustrated the professor. But it wasn’t just a shortcut I knew, but the foundations and process that got me there also. I could explain the hows and whys to win over not only my classmates, but the dean of education as well when it came to explaining why I was right and my professor wrong and just prideful. The thing I do remember from this was that while my mom was a math teacher she waited until I was in college to share these shortcuts that could have made Geometry in middle school a whole lot easier. In hindsight, she knew my foundation wasn’t ready yet for those shortcuts. My Mom was a wise lady; I miss her dearly, but see her fruits in my own instruction everyday.
iii. Who’s helping whom?
As parents, teachers and fellow shapers of young minds we must constantly attack this challenge from a professional standpoint. Yes we want bask in the victory that comes from opening that report card or grade book that is filled with As and Bs, but mostly As. Even teachers bend the rules as the stress to perform on standardized tests becomes too difficult to ignore.
a. The Student
The student in our situation is a young man in the third grade who has seen his diagnostic test scores in math gradually drop from the mid 90s to the mid 60s in a matter of a few years. He likes school and he still likes math, but primarily because his scores have been padded to hide a weak math teacher’s teaching skills and a reluctant parent from intervening.
b. Weak foundation
He has a weak foundation and is beginning to experience frustration at the mere appearance of numbers in challenges he faces everyday. He stumbles through the operations, occasionally guessing here and there with relative success, but when multiple step problems arise he gets bogged down in the details because he is not sure of himself anymore.
i. Contributing factors
As stated earlier, the shapers have not taken the learner or the traveler to heart, but held them up as a reflection of themselves or their teaching skill. Parents do this to deflect any questions about their own character or upbringing. Teachers, who forget their job, do this mistakenly thinking that by doing so they will win favor with the administration. In short, the parents and teachers have placed their pride before their obligation to teach the student.
ii. No victory
When we place ourselves before the learner, we steal the victories from them and leave them feeling dependent and hollow.
c. The Teachers
We often think that to learn math, a math teacher must be present and this just simply isn’t true. Everything teaches, life teaches everyday with each new challenged placed before us and when we get in the way we are no longer teachers, but additional obstacles in the way. We have good intentions initially, but we forget we are trying to create independent free thinkers who can stand on their own and not dependent children who constantly cling to parents or teachers like crutches.
Yes, pride plays a large roll in the picture, but impatience plays a nearly equal role. Primarily parents who are already burdened with the challenges that life is currently bringing down upon them play out this role in their children’s lives. It takes time, precious time to walk through the process and learn the necessities for themselves, but it has to be done or the scores will continue to tumble and math will no longer be fun.
This brings to mind an encounter where a wife asked her husband to make the Kraft Mac & Cheese for their son. The husband, thinking here’s an opportunity to teach a math skill and provide an opportunity for his son’s victory he suggested to his son that he learn to make the Mac & Cheese himself. The husband offered to teach him how to do things the same way he learned. During the process of learning the wife grew impatient and began to make suggestions on how to solve the problem quicker, not realizing she was robbing her son of an opportunity learn for himself. He had to add 2/3 cup of water to the pasta. He asked for the measuring cups and the Husband directed him to the dry measuring cup set. The son sorted through the cups and knew that a cup was too much and that ½ cup was next size down. He also knew that 2/3 was between ½ and 1 cup, but he wasn’t sure on how to solve this problem. Unable to endure this any longer the wife barged in saying, “this is silly, there’s an easier way” and grabbed the liquid measuring cup off the shelf and told her son exactly what to do. I have no idea what motivated the wife to do this; surely she would have chosen not to intervene had she known the consequences of her actions. Sometimes, I wish math was easier to learn, but it’s just not in the cards. Anything worthwhile takes time and until we learn that we are doomed mediocrity.
iv. An Allegory
A boy attends school and finds he likes math. It’s as easy as 1,2,3 and he begins to build his foundation to this structure called the Math Tower. The numbers 1-100 are on the bottom, place value comes next and he begins to realize that math can give him a better view of life’s challenges as he stands upon the first floor of his new Math Tower. The boy begins to learn the basic operations, and is encouraged by a passerby to not worry about how many studs he uses along the walls and spaces the studs at every 36 inches. He finds this saves him time and money so he can quickly move on to the next floor. His tower is beginning to loom over most everything around him and as the Spring thunderstorms come he finds he is no longer so content with the security his tower once provided him. As the city grew around him the boy realized both his parents and teachers also had towers of their own and some of them stretched into the clouds. Later renters begin to move into the boy’s tower they complained of drafty rooms and some rooms were noisy. One parent suggests to bring in an inspector to ensure the foundation is up to specks and the other says lets just give it a new coat of paint, stating no one will notice anything then. The painting wins out as it is much cheaper and less time intensive. One day a parent reaches from the penthouse floor of their tower and pulls their son’s tower up to nearly the floor of the penthouse. The boy thinks to himself, why did I work so hard when this was so much easier. The boy says to himself, “I tower over many of my peers and I am successful. One day the boy decides to take his tower to another neighborhood and moves away from his parents and teachers. Life in the new neighborhood is okay at first, but as heavy winds come they cause the tower to topple upon the buildings below. Many residents are angry, but none more than the boy who vows to never build a tower again.