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Monday, July 11, 2011

Principle Four

Model with mathematics


Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical  results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 


"Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Learning mathematics with understanding is essential." NCTM


Powerful message... as I read through this principle, everything in it points or refers to inquiry based learning and truly applying knowledge to solve a problem. It isn't about worksheets or daily practice, it speaks to real world application, relevance and embedded learning. Each site that I visited to gain more information or a clearer perspective, spoke to having students work through difficult, but concrete problems where they have to use the math they are learning to solve the problem, at all grade levels. This inquiry process involves Accountable Talk, group work and differentiation. Students will need lots of practice working in groups with manipulatives, solving open ended problems or conducting investigations to build understanding. Students who are able to "play" with the concepts will be more willing to apply what they have learned to solve a problem. 


Students must be able to use the knowledge  flexibly, appropriately applying what is learned in one setting to another. This blend of factual knowledge, conceptual understanding (guided principle 2) and the ability to use or apply the knowledge proficiently, enhances all three elements and makes the learning more powerful and lasting.




The URL below is to an article, "Teaching for Understanding: Guiding Principles",by Kathy Richardson who has listed 12 steps to keep in mind when implementing a mathematics program that gives high priority to the development of understanding. Definitely worth a read and perhaps even of printing for later review.


http://www.aps.k12.co.us/instruct/resources/math/sec_notebook/docs/teach4understanding.pdf







2 comments:

  1. Once again I think this principle lends itself to DI and could also be very motivating for students. For real life mathematical problems we can draw from Mittineague events, things in the community, or their own personal experiences. It'll certainly take more time planning, but doing less problems and delving deeper should offset some of the time. The outcome sounds like what we would all want as math teachers: Students who can apply their math knowledge to real life situations. Maybe we could develop fake accounts for the kids or something. Uh oh my wheels are starting to turn and I smell smoke!!!

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  2. I hope that your excitement is contagious! I, too feel that the possibilities are endless. I love the fact that these guiding principles are leading us to real life problem solving situations where we can dig deep for connections and meaning! Think about your lesson on measurement where you bring all the real life tools in and let the students try to find the best use for them.... Now add in some WONDER questions or some open ended inquiry types of exploration... Can you imagine the type of connections that can be made? The possibilities are endless and really exciting!

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