CC Standards for Mathematical Practice: Analyzing Key Information ...

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The Common Core for Math is a two-dimensional construct. Its first dimension is the content students are to learn; its second dimension is how students are to learn it.?The Common Core?s authors have organized it in this two-dimensional fashion because they understand that students learn mathematics by practicing it ? by doing it repeatedly in different ways and with many variations in initial conditions, strategies, levels of reasoning, and available information. The Common Core?s authors have organized its mathematics elements around the positive idea that mathematics has a culture of accomplishment that favors a set of habits.

The first general habit on the list ? the first Standard for Mathematical Practice ? is: ?Make sense of problems and persevere in solving them.? This is actually a heading that covers thirteen different specific habits, which I identified?here. In this post, I?ll address the third of these: ?Students analyze [given facts], constraints, relationships, and goals.?

This is the first explicit reference to a higher-level cognitive domain in the Common Core Standard for Mathematical Practice. It opens up a huge can of worms: how do we analyze facts, constraints, relationships, and goals?

Analysis has a several meanings in science and mathematics. My favorite words can reveal their meanings in their etymologies: analysis is one of my favorite words.

The root word of ?analysis? is ?lysis?, which in Greek means to loosen. The normal use of the word is to describe a process or the result of a process where the various parts of a single thing are ?loosened? from each other: we look at the parts of a thing to learn more about it. This is a key distinction.

If you?ve read enough of my posts, you are probably aware of my fondness for the fantasy fiction of J. R. R. Tolkien. One of my favorite lines in his?The Lord of the Rings?comes from the wizard character Gandalf, during a futile argument with another wizard, Saruman, in which he says ?He who breaks a thing to find out what it is has left the path of wisdom.? It is important in applying this word?s genuine meaning to the analysis of physical things that we maintain the integrity of the system or thing that we study. We have to distinguish between breaking or destroying something to learn about it and disassembling it in such a way that we can restore its integrity. This also spurs a good ethical debate about our methods of securing samples for study in life science.

When we apply this action concept to an abstract discipline like mathematics, however, we can be more direct in our plan of attack, because there is no doubt that we can and will reassemble the parts of our problems in order to assure that our answers are sensible and correct. In analyzing math problems, we need to identify the various pieces of information that a problem provides and organize them in a clear and useful way.

Useful ways of organizing information depend on the nature of the problem. Here are some ways of organizing pieces of information in ways that might help us conduct effective analysis:

1.?Ranking according to importance.
2. Ranking according to relative magnitudes.
3. Constructing a timeline.
4. Constructing a two-dimensional map.
5. Constructing a three-dimensional map.
6. Constructing a flowchart.
7. Ranking according to centrality in a concept map or Venn Diagram.
8. Establishing a table based on a verbal model or algebraic formula and assigning data to rows and columns based on the relations we perceive.

There are certainly more ways to do this: we should teach several if not all of these to students and encourage them to develop their own methods of visualizing the relationships among various pieces of information in a problem. Remember that no two students learn the same thing in exactly the same way, and that, in a classroom where students collaborate and reflect together on their thought processes ? practicing a shared and external brand of metacognition ? every student benefits from pooling ideas during the learning process.

Note, too, that when problems that we must analyze come from a textbook or worksheet, students? reading skills are absolutely crucial to this step; if we present the problem orally, then listening skills are essential. If a student suffers from poor reading comprehension or listening skills for any reason, then that student is at a decisive disadvantage. We should all be quite familiar by now with multiple intelligences theory: the intelligence that supports student success in dealing with written math problems is linguistic intelligence, because the act of producing an analysis of the information in a written or spoken word problem is linguistic in every way that matters. Reading comprehension, fluency of every type, vocabulary, and a very sound appreciation of the impact of grammar, punctuation, and delivery on the meaning of communication are necessary skills for this kind of problem.

If our assessment relies too heavily on media and situations that only invoke linguistic intelligence, such as equivalent audio and text, then we?re not providing authentic assessment: we should try to do better in this department if we really want to see the Common Core deliver on its ambitions. Scientists, engineers, and researchers don?t face problems that come from the ?Problems? section of a textbook assignment. No one reads a new problem to them periodically so that they can work on it. Someone determines that there is a problem and commission data collection solution from these kinds of professionals.

Rarely if ever will a scientist, researcher, or engineer begin the problem-solving process with sufficient information to solve anything: occasionally it will seem to them that they begin with very little information, if any at all. These professionals gather the data they need for their work using sophisticated instruments, their senses, and their experiences. It is this ability to take an ill-defined situation or problem and secure the necessary information to make the product meaningful that is most crucial to research and innovation. This is one of the crucial keys to success for innovators trying to make new things and make old things better.

It?s silly to call word problems ?real-world? problems, because we never encounter the problems we must solve in adult professional life in that form. We?ll gather evidence from interviews, experiments, observations, research, conversations with colleagues, and our reflections on our own experiences ? including the expertise we developed in school. Where?s the textbook that puts students through all that to do a problem? A few seconds? reflection should also guide you to the conclusion that the tests our students take to gain admission to colleges and universities are completely inauthentic: the ACT and SAT math sections are full of exercises and novelty problems, making them challenging to finish in the allotted time, but in no way are those ?real-world? problems. I won?t even deign to mention the various state NCLB tests.

To get students to analyze information from problems in ways that will support deep and durable learning of both math content and various processes of analysis, we must at some point make the tasks they do more authentic. This was one of the primary motivations for the development of the Common Core.?If we can make our tasks authentic and engaging, we will see students analyzing information in many ways; if we can engage them in collaboration with peers or with us that includes debating the comparative advantages of different analytic methods, then we can foster metacognition.

Note that, in the context of a word problem that students read, we can nearly reduce the analysis process to a note-taking process. This isn?t going to help our students become good analyzers. If we can get students to do the authentic version of this, we once again have provided the answer to ?Why do we have to study math?? Recall that we have already developed one answer, the cultivation of the virtuous habit of perseverance. This post provides a second answer: ?to learn how to create and share analyses of information.?

I am, as my more frequent readers have surely noticed, in the habit of railing against industrial and bureaucratic notions when we foolishly apply them to education. We can find one of the phenomena that irritate teachers most about the industrial-bureaucratic ideas that have invaded education in recent years?in a?clich?: ?paralysis by analysis.? I?m not sure we can teach our students very many strategies for constructive problem-solving action that are more important than this: when we reach a circular process or a dead end in analyzing word-problem information, then we must restart the entire process and change aspects of our approach that we believe led the previous attempt off course.

Analyzing facts, constraints, relationships, and goals is a complex behavior for us to teach. Mathematics has its own culture which favors those who develop certain habits, and analyzing has a subculture which also favors certain habits. Chief among these is the ability to match the information to a standard model or to a new kind of model that we create for the given problem. If we can engage students in authentic problems that combine analytic practice with sophisticated methods of collecting information, then we will help them make tremendous progress in developing metacognitive habits and learning to work in the higher cognitive domains.

Lege, explora, cogita. Quaere verum.

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Source: http://mikepoliquin.com/2012/08/08/cc-standards-for-mathematical-practice-analyzing-key-information/

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