Mathematical Habits of Mind Webinar Series Practices 5 and 6 Presented by Allison Miller and Jill Savage

Questions from 2nd Webinar Practices 3 & 4 Find the webinars and the PowerPoint slides on the WV TREE under Professional Learning Math Webinar Presentation Science Integration Webinar Presentation

Mathematical Practices 5 and 6 Use appropriate tools strategically. Attend to precision.

Habit of Mind #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.

Kindergarten & 1st Grade

2nd Grade & 3rd Grade

4th Grade & 5th Grade

“… a tool is anything that aids in accomplishing a task.” John SanGiovanni http://www.heinemann.com/blog/smp5/

Habit of Mind Statement paper & pencil concrete models ruler protractor calculator spreadsheet computer algebra system; statistical package; dynamic geometry software

Purchased manipulative models

Student- or Teacher-Made Models

Virtual Manipulatives

2 Parts to this Habit of Mind #1 #2 I can use the right tool for math. I can explain why I chose to use it.

Elementary Grades Tools should be introduced one at a time. Students should be taught how to use a tool. Students should be given opportunities to use the new tool again and again. Use of a particular tool might be required for a time.

Elementary Grades Students need opportunities to decide which tool to use. Students need to provide reasoning for using a particular tool, or for choosing one tool over another. Students need to begin thinking about the potential of a tool. Students also need to think about the limitations of a tool.

What tools might be strategically used for this problem? Karen brought 10 cookies to school. She gave 2 to each of her best friends and had 2 left for herself. How many friends got cookies?

What tools might be strategically used for this problem? On Thursday, 345 bottles of milk were served in the cafeteria. On Friday, 427 bottles of milk were served. How many more bottles were served on Friday than on Thursday?

What tools might be strategically used for this problem? Draw a triangle in which one of the angles is double one of the others.

How would choosing a tool be different for finding: 13 – 8 = For a 1st Grader? For a 3rd Grader?

How would choosing a tool be different for finding: Family Hardware has a number of bicycles and tricycles for sale. Jerome counted a total of 60 wheels. How many bicycles and how many tricycles were for sale? For a 2nd Grader? For a 4th Grader?

You’re working on a problem and need to figure out 747 + 286 You’re working on a problem and need to figure out 747 + 286. Would it be reasonable to use the following tools? Why or why not? Teddy bear counters? 2-color counters? Place value blocks? 100 chart? Open number line? Paper/pencil? Calculator? Protractor? Ruler? Mental Math?

Key Questions Students Should Ask: Do I need a tool? What tool is the best to use? How does it work? Do the results make sense? John SanGiovanni http://www.heinemann.com/blog/smp5/

Students Need To: Learn how to use tools; Know what tools are available; Be given many opportunities to use math tools; Explain why they chose a particular tool; Recognize the power and limits of a tool; Decide for themselves which tool might serve them best.

Attend to Precision Student Teacher Communicate precisely using clear definitions.   State the meaning of symbols and provide accurate labels. Calculate accurately and efficiently. Label accurately when measuring and graphing.   Teacher Have students clarify their definitions by emphasizing the importance of precise communication. Encourage accuracy and efficiency in computation.   Number Talks is a great tool to use to have students attend to precision.

Attend to Precision When I do Number Talks with my students, the first thing that I tell them is that when explaining how to work the abstract problems, I want their answers to be: Easy Quick Precise  

Attend to Precision My first through third grade students usually ask me "What does precise mean?"  I explain to students that it means the correct answer.  The answer has to be exactly right!  It must be precise.   One example that I give to my students is that when we measure, we have to be precise!    If you are installing carpet, and you measure incorrectly, your may order too much or too little carpet.  

We need to teach our students that we need to be precise when: Giving directions Measurement Launching rockets Keeping time

Here students are finding the perimeter of different shapes Here students are finding the  perimeter of different shapes. Some of the sides of the shape are not given, so the students need to determine the length of the unknown sides.   Students need to be precise when counting the blocks to determine the length of each side.   Sometimes students will not count a corner cube twice.  If the length of the top side is 8 and the left side is 10, they must be precise when counting each side.    

This is an example of the commutative propery for addition. Vocabulary This is an example of the commutative propery for addition.   This picture show strategies for student to use when adding doubles and doubles plus one.  

How can we teach our students to be precise? Mathematically proficient students try to communicate precisely to others.  They try to use clear definitions in discussions with others and in their own reasoning.  They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.   36 = 6 X 6                 26 + 4 = ___ X 10 They calculate accurately and efficiently,

How can we teach our students to be precise? Teachers who are developing students’ capacity to "attend to precision" focus on clarity and accuracy of process and outcome in problem solving. 

There are several different ways students could answer this problem. 37 + 25 = There are several different ways students could answer this problem. Trading      1      37 +   25      62 Partial Sums 37 + 25 = 30 + 20 = 50 7 + 5 = 12 50 + 12 = 62 Compensation 37 + 25 = 37 + (3 + 22) = (37 + 3) + 22 = 40 + 22 = 62 (Associative Property) EASY, QUICK, PRECISE  Here is an example of clarity and accuracy of process and outcome in problem solving. 

Attend to Precision  Students need to communicate exactly what they mean when explaining the how they get their answers.   It's amazing how students will start using the correct terms such as  "commutative property" when explaining strategies of how to add or multiply.   If teachers use the proper vocabulary, their students will also, and this will help students be more precise in their answers and when explaining to others.  

Resources Think Math website: http://thinkmath.edc.org/resource/mp5 Blog by John SanGiovanni: http://www.heinemann.com/blog/smp5/ Debbie Waggoner website: http://www.debbiewaggoner.com/math-practice-standards.html

Next Webinar Thursday, February 16, 2017 Click this link to attend:  Join Skype Meeting