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....

Standards for Mathematical Practice

With the advent of the Common Core and the changes that have already been made to the teaching of mathematics, I thought it important to list the standards for practice here. The first of these are the NCTM process standards of problem solving, reasoning and PROOF, communication, representation and connections. Thinking about just those pieces, it would be hard to pick one out that would be "more valuable" than the others. Add to these: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency and productive disposition (habitual inclination to see mathematics as sensible, useful and worthwhile and belief in one's own efficacy and we are looking at a the teaching of  mathematics in a whole new way.

The Common Core forces us to dig deeper and to probe for clearer, stronger and richer understandings in the concepts we are teaching and we must build on those foundational understandings that allow our students to effectively use and apply what they know, emphasizing the process and the inquiry that leads to an even greater understanding.

The first of these Mathematical Standards is Make sense of problems and persevere in solving them...
Here is the explanation" Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Mathematically proficient students  check their answers and can explain correspondences between equations, verbal descriptions, tables or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Students continually ask themselves, "Does this make sense?" They can understand the different approaches of others to solving problems and can  identify similarities between those approaches.

Think about this in relation to your students  (all grades) and their math learning. This is just the first standard of practice.  What does this look like in your classroom. What changes may have to be made in order to keep to this standard? Can you see how this connects to our journey with Accountable Talk and Inquiry Circles?? Please share your thoughts!

I will continue to post every few days with another standard. There are eight in all.

I am including a link that will take you to an Arizona site. Arizona has also adopted the Common Core and they have created a document that breaks down the power standards included in the core. The only piece that is not included are the MA only standards. It is user friendly and definitely worth looking at!
To access, just plug in the URL and click on the word doc or PDF for your grade.

http://www.ade.az.gov/standards/math/2010MathStandards/