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MATHEMATICS—MRS. JESSICA MANGINI

 

As we continue exploring the Mathematical Practice Standards, it is important to keep in mind that these standards work hand in hand with the content standards. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

Source: corsetandards.org

 

To refresh, here are the 8 Standards for Mathematical Practice:

  1. Make sense of problems and persevere in solving them
  2. Reason abstractly and quantitatively
  3. Construct viable arguments and critique the reasoning of others
  4. Model with mathematics
  5. Use appropriate tools strategically
  6. Attend to precision
  7. Look for and make use of structure
  8. Look for and express regularity in repeated reasoning

This month I will highlight Standard 7- Look for and make use of structure.

 The standard states:
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…. 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…. —CCSS

WHAT DOES IT REALLY MEAN?

When students “look for and make use of structure” they are finding patterns and repeated reasoning that can help solve more complex problems. For young students this might be recognizing fact families, inverses, or the distributive property. As students get older, they can break apart problems and numbers into familiar relationships.

 

Teachers help students developing a capacity to “look for and make use of structure” by helping learners identify and evaluate efficient strategies for solution. Teachers in the lower elementary grades might help students identify why using “counting on” is preferable to counting each addend by one, or why multiplication or division can be preferable to repeated addition or subtraction. Teachers in the upper elementary grades might help students discern patterns in a function table to “guess my rule.”  Students are taught to identify multiple strategies and then select the best one. They repeatedly break apart numbers and problems into different parts and use what they know is true to solve a new problem. Teachers ask students to prove solutions without relying on the algorithm. For example, when students are changing mixed numbers into improper fractions, they have to prove to that they have the right answer without using the “steps.”

 

Teachers ask:

  • Why does this happen?
  • How is ____ related to ____?
  • Why is this important to the problem?
  • What do you know about ____ that you can apply to this situation?
  • How can you use what you know to explain why this works?
  • What patterns do you see?

Source: Insidemathematics.org

What are students doing?

In order to identify the structure of a mathematical problem, students often need to engage in some form of visual learning or visualization. In early algebra this may mean that students are making a pattern or displaying data in a table to determine where a pattern exists. Structures are used to build place value and number sense. Students use various models in order to understand numbers and how they can be composed and decomposed.

With data analysis and statistics students use the structure of the graph or other data presentation to make sense of the problem and find a solution. It is therefore essential that students learn multiple ways to represent and analyze data, numbers, and shapes in order to determine and understand the structure of any problem.

 

As we move into the spring months, please continue to work with your children on their Basic Fact Fluency.  Consistent routines help ensure your child understands the importance of knowing those facts!  Practice in the car on the way to soccer, baseball, lacrosse, etc. is a great way to get fit this in.

 

Tiger Times April 2016