Teaching

 Teaching

An effective mathematics program is based on a carefully designed set of content standards that are clear and specific, focused, and articulated over time as a coherent sequence.

The sequence of topics and performances should be based on what is known about how students’ mathematical knowledge, skill, and understanding develop over time. What and how students are taught should reflect not only the topics within mathematics but also the key ideas that determine how knowledge is organized and generated within mathematics. (See Standard for Mathematical Practice 7: Look for and make use of structure.) Students should be asked to apply their learning and to show their mathematical thinking and understanding. This requires teachers who have a deep knowledge of mathematics as a discipline.

Mathematical problem solving is the hallmark of an effective mathematics program. Skill in mathematical problem solving requires practice with a variety of mathematical problems as well as a firm grasp of mathematical techniques and their underlying principles. 

Armed with this deeper knowledge, the student can then use mathematics in a flexible way to attack various problems and devise different ways of solving any particular problem. (See Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning.) 

Mathematical problem solving calls for reflective thinking, persistence, learning from the ideas of others, and going back over one's own work with a critical eye. 

Students should be able to construct viable arguments and critique the reasoning of others. They should analyze situations and justify their conclusions, communicate their conclusions to others, and respond to the arguments of others. (See Standard for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.) 

Students at all grades should be able to listen or read the arguments of others, decide whether they make sense, and ask questions to clarify or improve the arguments.

Mathematical problem solving provides students with experiences to develop other mathematical practices. Success in solving mathematical problems helps to create an abiding interest in mathematics. 

This principle says it all.... It isn't about facts, or completing worksheets or even success at tests; it is about  thinking,and analyzing, reasoning and problem solving. The ability to apply, utilize and even synthesize what we know about math. It is truly about looking at math as a life long skill, one that we use everyday in a myriad of ways. This is the expectation at all grade levels. 

This is imposing, exciting and a little overwhelming, but undoubtedly the way math should be taught.

These next pieces deal with pedagogy, the practice of math.... It is the intent of the CCSS to blend the guiding principles of content that we just explored with these principles of practice. The first is learning....

 Learning

Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of mathematics, and develop depth of understanding.

Students need to understand mathematics deeply and use it effectively. 

The Standards for Mathematical Practice describe ways in which students increasingly engage with the subject matter as they grow in mathematical maturity and expertise through the elementary, middle, and high school years.

To achieve mathematical understanding, students should have a balance of mathematical procedures and conceptual understanding. 

Students should be actively engaged in doing meaningful mathematics, discussing mathematical ideas, and applying mathematics in interesting, thought-provoking situations. Student understanding is further developed through ongoing reflection about cognitively demanding and worthwhile tasks.

Tasks should be designed to challenge students in multiple ways. Short- and long-term investigations that connect procedures and skills with conceptual understanding are integral components of an effective mathematics program. Activities should build upon curiosity and prior knowledge, and enable students to solve progressively deeper, broader, and more sophisticated problems. (See Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them.) Mathematical tasks reflecting sound and significant mathematics should generate active classroom talk, promote the development of conjectures, and lead to an understanding of the necessity for mathematical reasoning. (See Standard for Mathematical Practice 2: Reason abstractly and quantitatively.)

Wow...... Inquiry Based Learning and Accountable Talk! I feel like this principle is speaking directly to us and telling us that we are on the right path. We have a lot of work to do, but it feels really good that we chose the journey that we did. As you read through it, I would love to hear if you feel the same.

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!