Packing Mathematical Strategy Suitcases for the Next Grade Level

By Cathy Buyrn, M.Ed.

  sc   As you start helping your students pack up their book bags and prepare to move on to the next grade, take some time to reflect on their mathematical skill journey over the past year.   Where did they start?  What did you discover about them when they arrived in your classroom?  What skills did they have?  What did you have to fill in? Where are they now?  Are they good problem solvers?  Can they reason mathematically?  Can they use communication skills to articulate their understanding of mathematical concepts?  Are they able to make connections between mathematical procedures and ideas?  Do they use mathematical representations to interpret and solve problems? 

     Think about helping them pack their suitcases with versatile essentials that will support their transition to the next grade.  While you hope that they take every content skill detailed by the curriculum standards, consider deeper process skills that will serve them well regardless of the grade or content on their mathematical journey.  What will they be able to pull out of their strategy suitcases and use independently to move themselves forward?  What will help them become mathematical thinkers prepared to attack new concepts? 

Math Strategy Packing List

Problem Solving     pb

     Many problem-solving strategies involve acronyms that provide students with a list of steps to follow when working through problems.  While these strategies can be effective for many students, consider teaching students to develop their own list of individualized steps when learning how to work through specific types of problems.  Students can generate lists “from scratch” or alter the steps of existing lists.  Transitioning students from dependence on memorizing predetermined steps to being able to develop their own list helps develop metacognition (Uberti, Mastropieri, & Scruggs, 2004).

Acronym Strategy Example

Student-Generated Steps

RPV-HECC

Read for understanding
Paraphrase--in your own words
Visualize--draw a picture or a diagram
Hypothesize--make a plan
Estimate--predict the answer
Compute--do the arithmetic
Check--make sure everything is right
                                                         (Montague, 2005)

1.  Read it two times.
2.  What do I need to figure out?
3.  Try to say it in a different way.
4.  Organize the information.
5.  Think about what to do with the information.
6.  Try it out.
7.  Test the answer to see if it fits.


Reasoning

r     Develop student reasoning by introducing open-ended math questions.  Open-ended math questions do not have one path to a single solution but force students to consider the information presented in a variety of ways.  Teachers can explore students’ ability to reason mathematically by observing their responses to open-ended math problems.   As students develop reasoning skills, they will be better able to attack traditional types of math problems.  Open-ended math problems are available from a variety of online sources, such as those listed below, or teachers can turn traditional math problems into open-ended items by removing information.

Open-Ended Math Problem Resources

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http://books.heinemann.com/math/about_site.cfm

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Writing and Scoring Open-Ended Question in Math

opm

http://www.fi.edu/school/math2/index.html


Communication

    c Encourage students to engage in mathematical discussions.  Let their questions and ideas drive the direction of lessons.  Support them as they explain their own thinking and paraphrase others’ explanations.  Build in opportunties for students to engage in math talk in pairs or groups instead of teacher-directed whole-group structures.  Rubrics, such as the one below, can be used to assess students’ ability to communicate and help them develop an awareness of high-quality math communication skills.

Math Communication Rubric
Adapted from http://www.exemplars.com/

EXPERT

  • A sense of audience and purpose is communicated.
  • Communication at the practitioner level is achieved, and communication of arguments is supported by mathematical properties used.
  • Precise math language and symbolic notation are used to consolidate math thinking and to communicate ideas.

PRACTITIONER

  • A sense of audience or purpose is communicated.
  • Communication of an approach is evident through a methodical, organized, coherent, sequenced, and labeled response.
  • Formal math language is used throughout the solution to share and clarify ideas.

APPRENTICE

  • Some awareness of audience or purpose is communicated, and may take place in the form of paraphrasing of the task.
  • Some communication of an approach is evident through verbal/written accounts and explanations, use of diagrams or objects, writing, and mathematical symbols.
  • Some formal math language is used, and examples are provided to communicate ideas.

NOVICE

  • No awareness of audience or purpose is communicated.
  • Little or no communication of an approach is evident.
  • Everyday, familiar language is used to communicate ideas.


Connections

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      Students need to make connections between math concepts and processes and should be able to see how they can be useful in meaningful contexts.  Consider helping pull it altogether for your students by starting with real-life situations and having them discover what types of concepts and processes they will need to solve problems.  For example, in the video below, see how Dan Meyer illustrates a learning environment where “the math serves the conversation.  The conversation doesn’t serve the math” (Meyer, 2010).

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Click here or on the image to link to the Ted Talk Video Clip

Representation

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     Students need to be able to create and interpret representations of mathematical problems and processes.  They encounter graphicrepresentations on math assessments and in their daily lives.  A powerful method for building their skills with  graphic representations is using Question-Answer Relationships (QARs) in math.  While the the four types of QAR questions were designed as a reading strategy, they can also be applied to math problems.

QAR Question Types
Adapted from http://www.readingquest.org/strat/qar.html

Right there

The answer can be found  easily in the passage or problem.

Think and search

The answer requires some searching but can be gained directly from the passage or problem.

Author and you

The answer requires the student to put his or her own thinking together with the information presented in the passage or problem.

On my own

The answer requires the student to use his or her own ideas and information.


     To apply this strategy, teachers must first provide explicit instruction on the different kinds of math graphics.  Once students are familiar with the different kinds of graphics, they must begin to explore the different types of information in those graphics.  After students can identify the different graphics and information, teachers can start to help them make connections with the four types of QAR questions (Wright, n.d.).  A detailed process for implementing this strategy with students may be found at the link below.

http://images.pcmac.org/Uploads/FranklinCountyGA/FranklinCountyGA/SubDepartments/Forms/
QARs%20to%20Interpret%20Math%20Graphics.pdf

References and Resources

Meyer, D. (2010, March). Math class needs a makeover [Video file].  Retrieved from http://www.ted.com/talks/lang/eng/dan_meyer_math_curriculum_makeover.html

Montague, M. (2005, February 28). Math problem solving for upper elementary students with disabilities. Retrieved from http://www.k8accesscenter.org/training_resources/MathPrblSlving_upperelem.asp

Uberti, H., Mastropieri, M. A., & Scruggs, T. E. (2004). Check it off: Individualizing a math algorithm for students with disabilities via self-monitoring checklists. Intervention in School & Clinic, 39(5), 269-275. Retrieved from EBSCOhost.

Wright, J. (n.d.).  Applied math problems: Using question-answer relationships (QARs) to interpret math graphics.   Retrieved from http://images.pcmac.org/Uploads/FranklinCountyGA/FranklinCountyGA/SubDepartments/Forms/
QARs%20to%20Interpret%20Math%20Graphics.pdf