Beneficial Practices

Beneficial Practice: Graphic Organizers

The benefit of graphic organizers is that they present information in two dimensions. They show the relationships between the elements appearing in the organizer. Since higher level mathematics involves learning concepts, patterns, and processes, the goal of higher-level mathematics is to recognize and learn the patterns that connect the numbers. Graphic organizers lend themselves to addressing this goal. Ives (2003) offers these guidelines for using graphic organizers in algebra instruction:

  • The content of the organizer must not be verbal (words, phrases, sentences). Content should be mathematical analogues—numbers and other symbols, expressions, and equations.
  • The graphic display, the spatial arrangement of the mathematical elements, must support the information to be learned.
  • Graphic organizers should be an integral part of instruction, not a substitute for instruction. The relationships shown in the graphic organizer should be explicitly taught and connected to the graphic organizer.

Types of Graphic Organizers

  1. Hierarchichal Diagrams
    • Main branch for the overall concept
    • Sub-branches of supporting information
    • Main concepts and sub-branches are linked by arrows, lines, colors, numbers, or phrases to show the connections of the information
  2. Hierarchical diagram showing examples of monomials (one term), binomials (two terms), and trinomials (three terms). The example includes non-examples.

  3. Sequence Charts
    • Shows a sequence of events or procedures
    • Arrows are used that flow in one direction
    • Numbers are used for each step to reinforce the flow of steps

Example of a sequence chart showing a problem-solving strategy

Students would question themselves as they work the strategy:

  1. Search the word problem
    1. Read the problem carefully
    2. Ask yourself questions: “What do I know? What do I need to find?”
    3. Write down the facts
  2. Translate the words into an equation in picture form.
  3. Answer the problem
  4. Review the solution
    1. Reread the problem
    2. Ask yourself questions: “Does the answer make sense? Why?”
    3. Check the answer

Effective uses of graphic organizers

  1. Consistent, coherent, and creative usage
  2. Use of both teacher-directed and student-directed approaches
  3. Address individual needs through curricular adaptations
    • Provide partially completed graphic organizers
    • Highlight information in text
    • Provide cues at the bottom of a blank graphic organizer
    • Provide group activities

Beneficial Practice: Strategy Instruction

Mathematical problem solving is a complex task that requires students to plan, organize, prioritize, and evaluate their work. For students who have difficulty organizing, planning, and prioritizing, mathematical problem solving is frustrating without templates and scaffolds to help them learn. After a while, students lose their confidence and motivation to succeed in math. Students with learning disabilities respond well to strategy instruction (Montague, 1992). A strategy is an instructional sequence of needed actions along with guidelines and rules in order to make effective decisions when solving a problem (Maccini & Gagnon 2005). Effective strategies include:

  • Memory devices to help students remember the strategy (example: first letter mnemonic)
  • Steps that use familiar words stated simply, using action verbs (example: Read the problem)
  • Steps that are sequenced accurately and lead to a positive outcome (example: Read the problem before solving the problem)
  • Steps that prompt students to use cognitive abilities
  • Metacognitive prompts for monitoring problem solving performance

Here are some examples.

Verbal Strategies

PEMDAS Acronym: Memory strategy for remembering the sequence of steps for simplifying algebraic equations

Parentheses Exponents Multiplication Division Addition Subtraction

KNOW Acronym: Memory strategy for remembering important problem-solving steps

Key words Numbers Operation Work it out

Visual Strategies

All students, especially those with learning disabilities, benefit from strategies that help them remember the steps in algebraic procedures. The visual representation of the strategy can help students with their conceptual understanding, as well as their attention and memory. For example, the FOIL and Face strategies provide support for factoring polynomials. FOIL is an acronym for multiplying the first terms, outer terms, inner terms, and last terms. This strategy can be pared with the Face strategy, which provides visual cues to help students pay attention to the sign of the middle terms. By drawing arcs from one variable/number in one expression to the correct variable/expression in the second expression, a face takes shape. (See the diagram below.) Visual representations of strategies can help students with their attention, memory, and conceptual understanding. Visual representation of multiplying two binomials, showing that the first terms are multiplied, then the outer terms, then the inner terms, then the last terms. Next the results are added together.

Problem-Solving Strategies

A growing body of research has evolved that supports the use of visual representation to help students better understand the meaning of word problems that require mathematical operations (Task Group on Instructional Practices 2008). Rather than focus on strategies that look for key words in order to determine what operation to use to solve the problem, students use a visual representation to analyze the problem and determine how to handle the relevant information (Xin 2005).

Example: Graphical Representation of a Word Problem

Maria bought 70 stamps at the post office in 40¢ and 25¢ denominations. If she paid $25.00 for the stamps, how many of each denomination did she buy?

Stamp Problem Graphic

In this graphic:
x represents the number of 40¢ stamps.
(70 – x) represents the number of 25¢ stamps.
The value for x can now be solved. Students may need a reminder of basic math related to combining like terms and the distributive property.

More on active reading strategy for math word problems

Instructional Steps for Teaching a Problem-solving Strategy
  1. Provide an advance organizer of the strategy that:
    • Relates previously learned information to the new lesson
    • States the new skill to be presented
    • Provides a rationale for learning new information
  2. Provide teacher modeling of the strategy steps
  3. Provide guided practice
  4. Provide independent student practice
  5. Provide feedback and correction
  6. Provide regular review to promote generalization of the strategy

Note-taking Strategies

A math strategy notebook can be a powerful tool to support all students in their attempt to organize the information they learn so that they can remember and refer to the most important concepts presented in the algebra class. A three-column note strategy works well for a strategy notebook, for example:

Concept/Main Idea Explanation Example
Slope of a line Slope is the ratio of the vertical change to the horizontal change in a line Slope = m m = change in ychange in x
Slope intercept form y = mx + b m = slope b = y intercept y = -3x-2 m = -31 b = (0, -2)
y intercept Where a given line crosses the y axis y = (0, b)

Another helpful note-taking strategy is for students to create their own template for solving equations of a particular type. Students with weaknesses in working memory, attention, and executive function may not know where to start when they have to graph an algebraic equation. Templates give students a plan and a sequence of steps to follow.

Template: Solving Slope
Step 1 Slope Intercept Formula y = mx + b
Step 2 m is slope m = _________
Step 3 b is y intercept b = __________
Step 4 Point of 4 intercepts (0,b) = (0, __ )
Step 5 Graph

For general class notes, we recommend a two-column note strategy.

References

Bower, M. (2009). Graphical representations of distance=rate x time, stamp problem, greatest common factor, problem solving template. Unpublished algebra problems.

Ives, B., Hoy, C. (2003). Graphic organizers applied to higher-level secondary mathematics. Learning Disabilities Research & Practice, 18(1), 36-51.

Maccini, P., Gagnon, J. (2005). Math graphic organizers for students with disabilities. Retrieved January 2, 2008 from www.k8accesscenter.org.

Maccini, P., Hughes, C. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15(1), 10-21.

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem-solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.

Roditi, B. (2007). The strategic math classroom. In L. Meltzer, (Ed.) Executive function in education: from theory to practice (pp. 237 – 260). New York: Guilford Press.

Roditi, B. (2006). Math strategy instruction. In L. Meltzer, (Ed.) Strategies for success: classroom teaching techniques for students with learning differences (pp. 95-128). Austin, TX: Pro-Ed.

U.S. Department of Education (2008). Report of the task group on instructional practices of the mathematics advisory panel. Retrieved February 19, 2008 from http://www.ed.gov/about/bdscomm/list/mathpanel/reports.html.

Xin, Y.P., Jitendra, A.K., & Deatline-Buchman, A. (2005). Effects of mathematical word problem-solving instruction on middle school students with learning problems. The Journal of Special Education, 39(3), 181–192.