A Year in the Life: Ambient Math Wins the Race to the Top!
For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.” And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.”
Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. In reviewing the Standard for Mathematical Practice #1, I find that these standards are too dense and complex to include two of them in one blog post. Today will be #1, tomorrow #6, and the other three pairs will follow. The standard will be broken up into parts, a sentence at a time (in blue), followed by their ambient Kindergarten counterparts. I will of course be omitting references to higher mathematics since they are irrelevant here.
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
We could begin by taking the word “sense” quite literally, by applying it to the senses. If the math problem is approached in a sensory way, it’s operated upon by more than just the intellect, so the heart and hands have a fair say in the solution as well. Perhaps we should even question the use of the word “problem” in relation to math. In the Kindergarten it may be more appropriate to think of the problems as phenomena to be observed and marveled at first. The best, most creative minds and hearts throughout history learned through just this sort of acute observation. If the two parameters of learning through the senses and attentive observation are employed, it may be possible for Kindergartners to explain the meaning of the phenomena to themselves and look for how it may resolve itself, albeit with some help and guidance.
They analyze givens, constraints, relationships, and goals.
At this level, all of these qualities are inherent in the way the material is presented. The Kindergartner takes any given, constraint, relationship, or goal that is implied in any situation on faith and trust, so it’s incumbent upon the teacher to communicate those qualities in an age-appropriate way, keeping in mind that play is paramount now and reserving academics for later is usually better.
They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
Kindergartners do this every time they engage in play, beginning a course of action in response to a problem and not hesitating to change or regroup if necessary until a solution is found. If allowed to fully play this out, an unshakable foundation is built, later translating this process to academics. It is best if play and creativity stay alive and well throughout, informing all academic endeavors with their bright color and art.
They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.
This again is a quality endemic to all serious play, and as stated above, if allowed free reign until age 7, will build the sturdiest foundation for later learning. Regarding math at this age, most concepts will be made accessible by their concreteness and sensory approach, so the child(ren) will enter on math’s ground floor and progress to its higher levels gradually and with much support.
They monitor and evaluate their progress and change course if necessary.
May I say here that the “teaching to the test” employed in most schools and at all levels at this time is not a good venue for this sort of free trial and error exploration. If the primary (and in some cases only) goal is to pass the test, real learning cannot happen in a healthy way. If the Kindergarten ambient math materials are made available in the free play area after they are demonstrated, this sort of play learning and exploration may very well take place.
Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.
All of the ambient materials suggested here meet this description, and problems may be conceptualized and solved in the context of free play, as described above.
Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
This will happen at this age within careful guidelines provided by the teacher. The same concept is continually presented in different ways and it becomes obvious that the same result can be obtained through different means. It becomes clear that the methods used make good sense if a secure scaffolding is maintained throughout.
They can understand the approaches of others to solving complex problem s and identify correspondences between different approaches.
As mentioned above, all concepts are consistently presented in a multifaceted and diverse way. Another solid foundation is built here: the recognition of the fact that others’ thoughts, approaches, and methods are equally valid. This develops into a healthy awareness of the correspondences between different approaches. Good citizenship too can be extrapolated from this realization, which brings another important point to light. That good teaching and learning by their very nature always have a strong moral fiber.
Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal. Tune in tomorrow for the Standard for Mathematical Practice #6.