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Math Instruction

8 Mathematical Practices

One of the most exciting elements of the Common Core State Standards in Mathematics is the identification and development of the 8 Mathematical Practices for Kindergarten through 12th grade. These practices refute the myths surrounding math instruction and demystify what it means to think like a mathematician. No longer can students (and parents) say that they don't have the "math gene" and that they have always struggled to do math. What researchers found is that successful mathematicians have developed certain habits of mind that lend themselves to being a productive mathematician. Through practice and specific learning opportunities, all students can develop these math practices and increase their mathematical efficacy.  Here you will see a summary of the practices. If you want to know how you can help your child develop these habits of mind, click here to access some questions you might ask to help develop their mathematical thinking. For those of you who would like to listen to an explanation of these math practices with some examples, click AUDIO to access that resource. 

1. Make Sense of Problems and Perservere in Solving Them

  • Make meaning of the problem and look for a starting point. 
  • Plan a solution pathway instead of just looking for a solution
  • Analyze what is given, relationships between elements of the problem and the goal
  • Relate current situation to concepts or skills previously learned and connect mathematical ideas to one another
  • Monitor and evaluate progress and change course if necessary
  • Continually ask if this solution or strategy makes sense

2. Reason Abstractly and Quantitatively 

  • Makes of quantities and their relationships
  • Represent symbolically (create equations, expressions)
  • Create a logical representation of the problem
  • Think about the meanings of quantities, not just how to compute them

3. Construct Viable Arguments and Critique the Reasoning of Others

  • Understand and use definitions when constructing mathematical arguments (arguing for your answer)
  • Justify conclusions with mathematical ideas
  • Listen to the arguments of others and ask useful questions to determine if their argument makes sense
  • Ask clarifying questions or suggest ideas to improve/revise the argument
  • Compare two arguments and determine correct or flawed logic
  • Communicate and defend mathematical reasoning using objects, drawings, diagrams, actions, verbal and written communication 

4. Model with Math

  • Solve math problems arising in everyday life
  • Apply assumptions and approximations to simplify complicated tasks
  • Use tools such as diagrams, two-way tables, graphs, or concrete manipulatives to simplyfy tasks
  • Analyze relationships mathematically to draw conclusions
  • Reflect on whether the results make sense, possibly improving/revising the model

5. Use Appropriate Tools Strategically

  • Decide which tools will be the most helpful (i.e. ruler, calculator, protractor)
  • Use technological tools to explore and deepen understanding of concepts
  • Use estimation and other mathematical knowledge to detect possible errors

6. Attend to Precision

  • Communicate precisely with others and try to use clear mathematical language
  • Understand the meanings of symbols and use them consistently and appropriately
  • Calculate efficiently and accurately

7. Look For and Make Use of Structure

  • Look closely to see if there is a pattern or structure to help solve the problem
  • Step back for an overview and shift perspective
  • See complicated things as being composed of single objects or several smaller objects

8. Look for and Express Regularity in Repeated Reasoning

  • See repeated calculations and loof for generalizations or shortcuts
  • See the overall process of the problem and still attend to the details
  • Understand the broader application of patterns and see the structure in similar situations
  • Continually evaluate the reasonableness of their immediate results

Problem of the Month Process

Problem of the Month Process

Family Math Night

October 15th was a great success, with 54 families coming into the MPR to learn about Common Core Mathematics Instruction and how teachers at QR work to build the 8 Mathematical Practices into our classrooms. Mrs. Alamillo, Ms. Sun, Mrs. Cooke, Mrs. Tapia and Mrs. Tricaso guided our families through a condensed Problem of the Month, allowing students and their families to have rich mathematical discussions, as they worked to discover and justify their answers. After all of that group problem solving, we all had the opportunity to see and evaluate the work of other teams solving the same problems. It was an exciting night with lots of laughter and math learning.  For those unable to attend, or those who want to look back at the presentation, you may click on Power Point.  

Handouts from the night can also be downloaded and printed. 

CCSSM Critical Areas of Focus.pdf

Summary of Math Practices and Questioning Strategies.pdf

What are some good math websites for me and my child?

You Cubed- Stanford University has put together a site rich in tasks and resources to push our students' thinking in deeply contextual ways. The emphasis of this site is on developing reasoning and finding the space to discover mathematics. 

Math Playground- This site is a fantastic interactive site that supports students' critical thinking and concept development as it walks them through a variety of problems to solve. All of the work begins with a problem...a word problem, requiring the students to think in context. Also, this site is great practice for the CAASPP testing in that the work requires the students to manipulate objects on the screen, to read carefully, and to check their work. 

Illustrative Mathematics- This site explains the common core content standards for each grade level as well as the 8 math practices. There are some videos and some task examples for each. This is a good teacher and parent resource and some older students might also find it interesting. 

National Council of Teachers of Mathematics- This site is primarily a site used by educator/members but there are some great student explorations that are available to all. 

The Math Forum- This site has weekly problems to solve, puzzles and games, connections to other math projects with people around the world. There is even a question and answer format for kids. 

Eduplace-   This site is created by Houghton Mifflin publishers (we don't use their materials for math instruction in the SRVUSD). It has some interactive components for students to practice problems solving and mathematics in a fun format. 

Khan Academy- This site has resources for students and parents to learn more about various math concepts, strategies and procedures. It works like a tutorial. 

National Library of Virtual Manipulatives- This site offers a 30 day free trial for families or educators to explore a variety of math tools for a variety of grade levels to develop concept understanding and skill acquisition. These manipulatives allow students to discover which tools are best for which types of problems or tasks. 

Problem of the Month

The SRVUSD works closely with the Silicon Valley Mathematics Initiative (SVMI).  The Silicon Valley Mathematics Initiative is a comprehensive effort to improve math instruction and student learning. The Initiative is based on high performance expectations, ongoing professional development, examining student work, and improved math instruction.  Our partners at SVMI have developed some graduated math tasks called Problems of the Month (POMs) that our students will be experiencing once per trimester minimum.

Students are given a math task with levels A-E. A level problems are the most accessible and they become more complex and involve higher order thinking and math skills as the students move up the levels. After working independently on the tasks for a while, the teacher asks students at the same level to collaborate and share their thinking. This sharing is key to the process in that our students learn to construct viable arguments and defend their thinking. They also learn to listen to the reasoning of others and evaluate its merit. After this collaboration, the team creates a poster explaining their newly refined thinking and these posters are shared with the rest of the class and sometimes even the rest of the school. Other groups view and examine the various posters with guiding questions in mind.  Do I agree with this group's conclusions? Did they tackle the problem in a similar or a different way than we did? How so? Have I learned anything new that would make our poster stronger?

After this walk about and reflection, students return to their poster to revise it to reflect any new learning they discovered in the gallery walk. Then individually, students will reflect on their learning and write about their aha's.  This POM process is a wonderful opportunity for students to push their mathematical thinking and reasoning and to practice the 8 Mathematical Practices underlying the Common Core Standards for Mathematics. To learn more about POMs please see the POM Power Point from Family Math Night. 

At the bottom of this page, see a graphic to further explain this POM process. 

So how do I know what my children are learning in each grade?

Parents can download the complete Common Core Standards for Mathematics (CCSSM) by clicking here, but this can be a cumbersome resource. Attached you will find a document pulled from the CCSSM where we highlight the Critical Areas of Focus and the Required Fluencies for each grade, Kinder through 5th. The Critical Areas of Focus are the most important skills for students to master at each grade level. They are the priority standards that should be given the most time and attention. The Required Fluencies are the skills that students should have mastery over- allowing them to be quick and accurate. These required fluencies are areas where parents can really support their children with games, flashcards, and lots of extra practice. To find out more, watch this fascinating TED talk by Jo Boaler- Why Students in the US need Common Core Math

For the Joy in Math...

Below are a few links to videos where you and your kids can have fun with math. In the Abbot and Costello clips, ask them what's wrong with the "logic" being demonstrated.  For the musical clips, have fun and sing along. Maybe, you and your kids can create your own math videos/songs to share with the rest of us. 

Abbot and Costello 7x13

Mr. Duey and Fractions


Making Math Cool with Alex Kajitani

For the fun of it, check out this Ted Talk from Alex Kajitani. I had the pleasure of meeting Mr. Kajitani and just loved his focus on making math memorable and fun. He focuses his work on engagement and on making our math instruction relevant to the students we are teaching. 


Making Math Cool


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