The Eighth Principle

  Look for and express regularity in repeated reasoning.

     This principle reads:

 Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal.  As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 

The central idea here is that mathematics is open to drawing general results (or at least good conjectures) from trying examples, looking for regularity, and describing the pattern both in what you have done and in the results. 

This principle relates so closely to the last where students are looking for patterns and structure to base their reasoning and explain their mathematical thinking. The investigative or inquiry approach to math leads students to discover patterns and structure in mathematical concepts. 

One example from Thinkmath is that "children can recognize that adding 9 can be simplified by treating it as adding 10 and subtracting 1and that this can be a discovery rather than a taught strategy. In one activity—there are obviously many other ways of doing this—children start, e.g., with 28 and respond as the teacher repeat only the words “ten more” (38), “ten more” (48), “ten more” (58), and so on. They may even be counting, initially, to verify that they are actually adding 10, but they soon hear the pattern in their responses (because no other explanatory or instruction words are interfering) and express that discovery from their repeated reasoning by saying the 68, 78, 88 almost without even the request for “ten more.” When, at some point, the teacher changes and asks for “9 more,” even young students often see it as “almost ten more” and make the correction spontaneously. Describing the discovery then becomes a case of “expressing regularity” that was found through “repeated reasoning.” Young students then find it very exciting to add 99 the same way, first by repeating the experience of getting used to a simple computation, adding 100, and then by coming up with their own adjustment to add 99." (Thinkmath)

"Through investigation and discovery, students can use repeated addition to begin their exploration of linear growth and repeated multiplication to extend to geometric growth. Drawing on their observations of geometric properties, students make and test conjectures, develop generalizations, and write formulas."  (Discovering Math) The more children can relate repeated patterns the more connections they will build and the more meaning they will bring to their math.

Principle 7 Look for and make use of structure

7            Look for and make use of structure.

This principle reads: 

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

As I was browsing online I came across this description/explanation of the seventh mathematical principle on Thinkmath and felt that I needed to include it exactly as written... it so speaks to the importance of children understanding and connecting the natural patterns/structure found in math and the realization that without that understanding their math ability is based on rote memory and computation.

The article reads:

"Children naturally seek and make use of structure. It is one of the reasons why young children may say “foots” or “policemans,” which they have never heard from adults, instead of feet or policemen, which they do hear. They induce a structure for plurals from the vast quantity of words they learn and make use of that structure even where it does not apply.

Mathematics has far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent. If, for example, students’ first encounter with the addition of same-denominator fractions drew on their well-established spoken structure for adding the counts of things—two sheep plus three sheep makes five sheep, two hundred plus three hundred makes five hundred, and two wugs plus three wugs makes five wugs, no matter what a wug might be—then they would already be sure that two eighths plus three eighths makes five eighths. Instead, they often first encounter the addition of fractions in writing, as 2/8 + 3/8, and they therefore invoke a different pattern they’ve learned—add everything in sight—resulting in the incorrect and nonsensical 5/16. Kindergarteners who have no real idea how big “hundred” or “thousand” are (though they’ve heard the words) are completely comfortable, amused, and proud to add such big numbers as “two thousand plus two thousand” when the numbers are spoken, even though children a year older might have had no idea how to do “2000 + 3000” presented on paper.

This CCSS standard refers to students recognizing that “7 × 8 equals the well-remembered 7 × 5 + 7 × 3.” Array pictures help (see MP standard 5, “Use appropriate tools strategically”), but so does students’ linguistic knowledge, if the connection is made. The written symbols 5×7 + 3×7 = 8×7 are very compact, but the meaning they condense into just eleven characters is something that students understood well even before they learned multiplication. Before they have any idea what a collection of sevens is, they know that five of them plus three of them equals eight of them. It’s just five wugs plus three wugs again.

“Standard arithmetic” can be taught with or without attention to pattern. The CCSS acknowledges that students do need to know arithmetic facts, but random-order fact drills rely on memory alone, where patterned practice can develop a sense for structure as well. Learning to add 8 to anything—not just to single digit numbers—by thinking of it as adding 10 and subtracting 2 can develop just as fast recall of the facts as random-order practice, but it also allows students quickly to generalize and add 18 or 28 to anything mentally. The structure is a general one, not just a set of memorized facts, so students can use it to add 19 or 39, or 21 or 41, to anything, too. With a bit of adjustment, they can use the same thinking to subtract mentally. This is, of course, exactly the way we hope students will mentally perform 350 – 99.

Structure allows sensible definition of odd and even: pairs with or without something left over.

In elementary school, attention to structure also includes the ability to defer evaluation for certain kinds of tasks. For example, when presented with 7 + 5 7 + 4 and asked to fill in <, =, or > to compare the two expressions, second graders are often drawn—and may even be explicitly told—to perform the calculations first. But this is a situation in which we want the students’ attention on the structure, ♣ + 5 ♣ + 4 or even  + 5  + 4, rather than on the arithmetic. Without any reference to symbols “standing for” numbers, which might well be distracting or even confusing to second graders, they readily see that  + 5  + 4 if the same number is under each hand. This same skill of deferring evaluation—putting off calculation until one sees the overall structure—helps students notice that they don’t have to find common denominators for 1¾ – ⅓ + 3 + ¼ – ⅔ but can simply rearrange the terms to make such a trivial computation that they can do it in their heads." Thinkmath (http://thinkmath.edc.org/index.php/Look_for_and_make_use_of_structure)

Remember that...."Children naturally seek and make use of structure." (Thinkmath)  What a beautiful, meaningful and natural way to learn math. It is our responsibility to give them the opportunities to find and make those connections. 

WOW! What a conference!

Hi Everyone!  We are just about finished the week.  We have lots to practice and lots to do to earn our PDP's.  We will all be coming back here on December 14th for the final day.  It is exciting to know how much new tools we are able to bring back to our new school year.  We have high hopes our technology will support what we are trying to do. 

READABILITY WEBSITE

WE JUST HEARD ABOUT THIS WEBSITE TO HELP STRUGGLING/ALL READERS.  CHECK IT OUT!http://www.readability.com/  GREAT STUFF!

Bagels and Laptops

A new idea for sharing... a school had limited funding for tech equipment but they had funding for bagels.  So the principal started an early morning group called Bagels and Laptops.  Teachers would bring their laptops, grab a bagel and meet for 15 minutes before school.  They would share what they are doing online with each other each week.  With their laptops there, they could enter the ideas and sites into the computer right then and there.  Sounds like it could be fun!

New Literacies Institute

Mrs. Lyons, Miss Metcalf and Mrs. Ramos  are attending the New Literacies Institute in Boston for the week.  We are enjoying learning how to use new technology.  We are with the West Springfield Team of 6.  Our teacher leader is Kathy Hillman.  We are using an IPAD 2 that was given to our district and we all were given flip cams(the newest version) to bring home with us.  It is very exciting!  We will have lots to share in the fall.  Maybe sooner...

Principle 6: Attend to Precision

Attend to precision

The principle states:

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

When reading through this principle, it brought me to think about three things, determination, discipline and self control. These are difficult but necessary tenets of not only math, but all education. If children are going to be successful learners we need to help them develop the work ethic necessary to be precise, to be willing to slow down, revise their work, think about their reasoning and to communicate it clearly. Determination, discipline and self control are characteristics or traits that need to be developed.  These are traits that are often connected to academic rigor. 

One of the age old questions is how..... How can I get my students to be more precise, to slow down and really concentrate on what they are doing? How can I get my students to delve more deeply with their thinking? Think of those questions in connection with the two definitions of academic rigor....the combination of inquiry and curiosity, student engagement, confidence, meaningfulness, critical thinking, problem solving and hard work. The blend of determination and efficacy towards learning. True mathematical engagement should lead to understanding; that is the goal of all mathematics. Finding ways to engage your students through investigation and inquiry is a good start. 

Students who are invested in a search for understanding are often rewarded with not only increased knowledge, but with a deeper development of that "inner tool set" (self-control, discipline, determination) the tools that will help them to become life-long learners.


The two definitions of rigor come from the article Kim shared with us in May and are as follows: 


The first defines rigor as "the goal of helping students develop the capacity to understand content that is complex, ambiguous, provocative, and personally or emotionally challenging." 

The second defines rigorous as "demanding strict attention to rules and procedures; allowing no deviation from a standard"

The article states that is the combination of the two that truly create academic rigor.

The next Principle

Math Principle 5:  Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.


This principle is pretty straightforward....if you expect that students will be able to use the tools necessary for any math operation, they need time and practice using them. All tools, no matter how simple they may appear, need practice. Students can not be expected to use any of the measurement tools necessary to problem solve unless they have first hand knowledge of how everything works. Even a tool as simple as a ruler needs explicit  instruction and follow up practice.


Once again as I read into these practices, I see how Inquiry Based learning helps our students to apply and use the skills they need to be proficient mathematicians. I think next year is looking like an exciting year for math!!

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

The Third Guiding Principle for Mathematical Practice

The third standard reads: Construct viable arguments and critique the reasoning  of others. 


The explanation is that mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and -- if there is a flaw in an argument -- explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grade levels can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 


To develop the reasoning that this standard asks children to communicate, the mathematical tasks we give need depth. If students are used to working on simple, single step problems with one finite answer, it is hard to get them to explain, so you tend to get one word answers like, I added.  Problems that require thought, discovery and multiple attempts almost make it easier for children to talk about. In order for students to be able to communicate a process they need to be able to give a clear articulation of a sequence of steps or the chronology of a problem or strategy. The more "action" they have in solving, the more articulation is possible. 


"The way children learn language, including mathematical and academic language, is by producing it as well as by hearing it used. When students are given a suitably challenging task and allowed to work on it together, their natural drive to communicate helps develop the academic language they will need in order to construct viable arguments and critique the reasoning of others.” (Thinkmath)


This process becomes more defined when you put it into the context of collaboration and group work.If given an open ended or discovery question, materials and resources to aid them in their exploration and a group or learning partner to work with, children begin to discover and "talk" about what they learn. These "learning conversations" are the backbone to reasoning and communicating their thoughts both verbally and in writing. 


Acknowledging the work that another group has done and critiquing their thoughts is difficult for most young children; constructing an argument that challenges their work, and proving or justifying the challenge is very difficult at any level. Math students must not only be proficient, but able to deconstruct and reconstruct the problem at hand and then explain their reasoning behind it. The ramifications of this standard are huge, but, if the inquiry and collaborative pieces are in place and if Accountable Talk is embedded in student practice then "prove it" becomes the norm.


Can you put your students into this situation? If given the opportunity, the tools and the time can you see this becoming a normal part of your math discoveries? 

Standards for Mathematical Practice 2

The second standard of practice is Reason abstractly and quantitatively. 


The explanation reads: Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complimentary abilities to bear on problems involving quantitative relationships: the ability to decontextualize -to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents - and, the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 


The last phrase explains the what standard two is really all about. Students need strategies of how to look at problems different ways, to be able to isolate and solve a part of a problem, and to be able to look at and see connections and relationships within a problem and how to manipulate that information to help them solve the problem. 


Again, across grade levels and abilities this can take on many shapes.. for instance, a first grader who understands that 4 + 3 = 7 and can demonstrate this by showing OOOO +  OOO = OOOOOOO is able to represent the equation symbolically. If they can then relate the symbols to the number and show how 4+3 and 3+4 are interchangeable then they are using quantitative reasoning about number and structure. 


Second graders who are learning how to write numerical expressions may be given the challenge of writing numerical expressions that describe the number of tiles in this figure 

in different ways. Given experience with similar problems so that they know what is being asked of them, students might write 1+2+3+4+3+2+1 (the heights of the stairsteps from left to right) or 1+3+5+7 (the width of the layers from top to bottom) or 10+6 (the number of each color) or various other expressions that capture what they see. These are all decontextualizations—representations that preserve some of the original structure of the display, but just in number and not in shape or other features of the picture. Not any expression that totals 16 makes sense—for example, it would seem hard to justify 2+14—but a child who writes, for example, 8+8 and explains it as “a sandwich”—the number of blocks in the middle two layers plus the number of blocks in the top and bottom—has taken an abstract idea and added contextual meaning to it.

More generally, Mathematical Practice #2 asks students to be able to translate a problem situation into a number sentence (with or without blanks) and, after they solve the arithmetic part (any way), to be able to recognize the connection between all the elements of the sentence and the original problem. It involves making sure that the units, (objects!) in a problem make sense. So, for example, if fourth graders  are asked to solve a problem that asks how many busses are needed for 99 students if each bus seats 44, they might decontextualize a problem  and write 99÷44. But after calculating 2r11 or 2¼ or 2.25, the student must recontextualize: the context requires a whole number answer, and not, in this case, just the nearest whole number. Successful recontextualization also means that the student knows that the answer is 3 busses, not 3 children or just 3.

Our goal throughout all of these NCTM standards is to promote deeper and richer mathematical understandings.  Think about this standard as you become more familiar with the Common Core and how you might  design lessons, investigations or activities that will enable your students to have a richer experience and a deeper understanding of number.  

Not to sound like a broken record, but I feel that students will need inquiry experiences working  with manipulatives, investigating in groups, drawing or modeling their problems and spending time discussing, questioning and proving their thoughts. Accountable Talk.... Inquiry Circles....