An understanding can never be “covered” if it is to be understood. Wiggins and McTighe (2005, p. 229)
There has been a big buzz out there that math has changed it the last few years, but math has never changed. The way math should be taught and understood is what received a new make over. In the pass, the focus has been on getting the right answer. Now, the focus is on "how did you get that answer." As a society, we want our children to look deeply into the "process" of how you do something instead of emphasizing the end result which is just getting the answer.
Teachers are now having to shift their understanding of math and reflect on how they are facilitating math.
"What is understanding? Understanding is being able to think and act flexibly with a topic or concept. It goes beyond knowing; it is more than a collection of information, facts, or data. It is more than being able to follow steps in a procedure. One hallmark of mathematical understanding is a student’s ability to justify why a given mathematical claim or answer is true or why a mathematical rule makes sense (Council of Chief State School Officers, 2010)."
The days of showing students the tricks to just get to the answer quicker, are gone. Teachers never had to truly understand the math themselves. All they had to do was show the procedure, and students would mimic the memorized procedure to get the answer. There are critics out there that say, "So what is wrong with that?" Well there are many problems with that line of thinking! For example, it's like showing a plummer the procedures of how to stop a leak. At first it worked, the plummer followed step by step as he was told and stopped the leak, but then by doing that, it caused another leak at a different spot. When the plummer used the same procedure that he was taught to fix the next leak, guess what? It didn't work!! Why??? Well, the problem is if the plummer was taught about how pipes work and truly had an understanding of the pipe system, he would of been able to flexibly think and generate a new or different way to fix the second leak. My example is a true depiction of what happens in the math classroom today. Students are falling behind because their so called "tricks" and procedures are not working. Don't get me wrong procedures shouldn't be left out. Fluency is very important, but the most essential piece of the puzzle is 'the understanding' of the concept. How can you connect what you learn if you have no understanding?
"Procedural proficiency—a main focus of mathematics instruction in the past—remains important today, but conceptual understanding is an equally important goal (National Council of Teachers of Mathematics, 2000; National Research Council, 2001; CCSSO, 2010)."
Teachers are having to alter how they instruct mathematics. Instead of teachers being the main source of information
( teacher-centered), spewing out facts and students absorbing information, the tables have turned. The new math classroom is now student-centered as the teacher is facilitating the learning. Math is about getting out of your comfort zone and taking risks. It's a place where mistakes are wanted! The foundation is based on true authentic understanding and built upon being able to represent student thinking whether it be by explaining verbally , written, with pictures, or numbers the ' the how' of their process. " Understanding must be a primary goal for all of the mathematics you teach."( Van_de_Walle)
The NCTM explains why this new emphasizes is so important.
"The National Council of Teachers of Mathematics (NCTM, 2000) identifies the process standards of problem solving, reasoning and proof, representation, communication, and connections as ways to think about how children should engage in learning the content as they develop both procedural fluency and conceptual understanding. Children engaged in the process of problem solving build mathematical knowledge and understanding by grappling with and solving genuine problems, as opposed to completing routine exercises. They use reasoning and proof to make sense of mathematical tasks and concepts and to develop, justify, and evaluate mathematical arguments and solutions. Children create and use representations (e.g., diagrams, graphs, symbols, and manipulatives) to reason through problems. They also engage in communication as they explain their ideas and reasoning verbally, in writing, and through representations. Children develop and use connections between mathematical ideas as they learn new mathematical concepts and procedures. They also build connections between mathematics and other disciplines by applying mathematics to real‐world situations. By engaging in these processes, children learn mathematics by doing mathematics. Consequently, the process standards should not be taught separately from but in conjunction with mathematics as ways of learning mathematics."
I really couldn't say it better myself. My next blog post will be an example of how this new way of teaching math looks in the classroom! Stay tuned. Watch this great video of what a math classroom should look like and sound like.