Students must be able to correctly interpret math graphics in order to correctly answer many applied math problems. Struggling learners in math often misread or misinterpret math graphics. For example, students may:
overlook important details of the math graphic.
treat irrelevant data on the math graphic as 'relevant'.
fail to pay close attention to the question before turning to the math graphic to find the answer.
not engage their prior knowledge both to extend the information on the math graphic and to act as a possible 'reality check' on the data that it presents.
expect the answer to be displayed in plain sight on the math graphic, when in fact the graphic may require that readers first to interpret the data, then to plug the data into an equation to solve the problem.
Teachers need an instructional strategy to encourage students to be more savvy interpreters of graphics in applied math problems. One idea is to have them apply a reading comprehension strategy, Question-Answer Relationships (QARs) as a tool for analyzing math graphics. The four QAR question types (Raphael, 1982, 1986) are as follows:
RIGHT THERE questions are fact-based and can be found in a single sentence, often accompanied by 'clue' words that also appear in the question.
THINK AND SEARCH questions can be answered by information in the text--but require the scanning of text and the making of connections between disparate pieces of factual information found in different sections of the reading.
AUTHOR AND YOU questions require that students take information or opinions that appear in the text and combine them with the reader's own experiences or opinions to formulate an answer.
ON MY OWN questions are based on the students' own experiences and do not require knowledge of the text to answer.
Teachers use a 4-step instructional sequence to teach students to use Question-Answer Relationships (QARs) to better interpret math graphics:
1. Distinguishing Among Different Kinds of Graphics
Students are first taught to differentiate between five common types of math graphics: table (grid with information contained in cells), chart (boxes with possible connecting lines or arrows), picture (figure with labels), line graph, bar graph.
Students note significant differences between the various types of graphics, while the teacher records those observations on a wall chart. Next students are shown examples of graphics and directed to identify the general graphic type (table, chart, picture, line graph, bar graph) that each sample represents.
As homework, students are assigned to go on a 'graphics hunt', locating graphics in magazines and newspapers, labeling them, and bringing them to class to review.
2. Interpreting Information in Graphics
Over several instructional sessions, students learn to interpret information contained in various types of math graphics. For these activities, students are paired off, with stronger students matched with less strong ones.
The teacher sets aside a separate session to introduce each of the graphics categories. The presentation sequence is ordered so that students begin with examples of the most concrete graphics and move toward the more abstract. The graphics sequence in order of increasing difficulty is: Pictures > tables > bar graphs > charts > line graphs.
At each session, student pairs examine examples of graphics from the category being explored that day and discuss questions such as: "What information does this graphic present? What are strengths of this type of graphic for presenting data? What are possible weaknesses?" Student pairs record their findings and share them with the large group at the end of the session.
3. Linking the Use of Question-Answer Relations (QARs) to Graphics
In advance of this lesson, the teacher prepares a series of data questions and correct answers. Each question and answer is paired with a math graphic that contains information essential for finding the answer.
At the start of the lesson, students are each given a set of 4 index cards with titles and descriptions of each of the 4 QAR questions: RIGHT THERE, THINK AND SEARCH, AUTHOR AND YOU, ON MY OWN. (TMESAVING TIP: Students can create their own copies of these QAR review cards as an in-class activity.)
Working first in small groups and then individually, students read each teacher-prepared question, study the matching graphic, and 'verify' the provided answer as correct. They then identify the type of question being posed in that applied problem, using their QAR index cards as a reference.
4. Using Question-Answer Relationships (QARs) Independently to Interpret Math Graphics
Students are now ready to use the QAR strategy independently to interpret graphics. They are given a laminated card as a reference with 6 steps to follow whenever they attempt to solve an applied problem that includes a math graphic:
Read the question,
Review the graphic,
Reread the question,
Choose the appropriate QAR,
Answer the question, and
Locate the answer derived from the graphic in the answer choices offered.
Students are strongly encouraged NOT to read the answer choices offered on a multiple-choice item until they have first derived their own answer-to prevent those choices from short-circuiting their inquiry.
Solving an advanced math problem independently requires the coordination of a number of complex skills. The student must have the capacity to reliably implement the specific steps of a particular problem-solving process, or cognitive strategy. At least as important, though, is that the student must also possess the necessary metacognitive skills to analyze the problem, select an appropriate strategy to solve that problem from an array of possible alternatives, and monitor the problem-solving process to ensure that it is carried out correctly.
The following strategies combine both cognitive and metacognitive elements (Montague, 1992; Montague & Dietz, 2009). First, the student is taught a 7-step process for attacking a math word problem (cognitive strategy). Second, the instructor trains the student to use a three-part self-coaching routine for each of the seven problem-solving steps (metacognitive strategy).
In the cognitive part of this multi-strategy intervention, the student learns an explicit series of steps to analyze and solve a math problem. Those steps include:
Reading the problem. The student reads the problem carefully, noting and attempting to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).
Paraphrasing the problem. The student restates the problem in his or her own words.
‘Drawing’ the problem. The student creates a drawing of the problem, creating a visual representation of the word problem.
Creating a plan to solve the problem. The student decides on the best way to solve the problem and develops a plan to do so.
Predicting/Estimating the answer. The student estimates or predicts what the answer to the problem will be. The student may compute a quick approximation of the answer, using rounding or other shortcuts.
Computing the answer. The student follows the plan developed earlier to compute the answer to the problem.
Checking the answer. The student methodically checks the calculations for each step of the problem. The student also compares the actual answer to the estimated answer calculated in a previous step to ensure that there is general agreement between the two values.
The metacognitive component of the intervention is a three-part routine that follows a sequence of ‘Say’, ‘Ask, ‘Check’. For each of the 7 problem-solving steps reviewed above:
The student first self-instructs by stating, or ‘saying’, the purpose of the step (‘Say’).
The student next self-questions by ‘asking’ what he or she intends to do to complete the step (‘Ask’).
The student concludes the step by self-monitoring, or ‘checking’, the successful completion of the step (‘Check’).
While the Say-Ask-Check sequence is repeated across all 7 problem-solving steps, the actual content of the student self-coaching comments changes across the steps.
Table 1 (as well as the attachment at the bottom of the page) shows how each of the steps in the word problem cognitive strategy is matched to the three-part Say-Ask-Check sequence:
Students will benefit from close teacher support when learning to combine the 7-step cognitive strategy to attack math word problems with the iterative 3-step metacognitive Say-Ask-Check sequence. Teachers can increase the likelihood that the student will successfully acquire these skills by using research-supported instructional practices (Burns, VanDerHeyden, & Boice, 2008), including:
Verifying that the student has the necessary foundation skills to solve math word problems
Using explicit instruction techniques to teach the cognitive and metacognitive strategies
Ensuring that all instructional tasks allow the student to experience an adequate rate of success
Providing regular opportunities for the student to be engaged in active accurate academic responding
Offering frequent performance feedback to motivate the student and shape his or her learning.
DESCRIPTION: The teacher analyzes a particular student's pattern of errors commonly made when solving a math algorithm (on either computation or word problems) and develops a brief error self-correction checklist unique to that student. The student then uses this checklist to self-monitor—and when necessary correct—his or her performance on math worksheets before turning them in (Dunlap & Dunlap, 1989; Uberti et al., 2004).
MATERIALS:
Customized student math error self-correction checklist (described below). (View attachments at the end of this article to view both sample and blank Math Self-Correction Checklists.)
Worksheets or assignments containing math problems matched to the error self-correction checklist
INTERVENTION STEPS: The intervention with customized math error self-correction checklists includes these steps (adapted from Dunlap & Dunlap, 1989; Uberti et al., 2004):
Develop the Checklist. The teacher draws on multiple sources of data available in the classroom to create a list of errors that the student commonly makes on a specific type of math computation or word problem. Good sources of information for analyzing a student's unique pattern of math-related errors include review of completed worksheets and other work products, interviewing the student, asking the student to solve a math problem using a 'think aloud' approach to walk through the steps of an algorithm, and observing the student completing math problems in a cooperative learning activity with other children.
Based on this error analysis, the teacher creates a short (4-to-5 item) student self-correction checklist that includes the most common errors made by that student. Items on the checklist are written in the first person and when possible are stated as 'replacement' or goal behaviors. This checklist might include steps in an algorithm that challenge the student (e.g., "I underlined all numbers at the top of the subtraction problem that were smaller than their matching numbers at the bottom of the problem") as well as goals tied to any other errors that impede math performance (e.g., "I wrote all numbers carefully so that I could read them easily and not mistake them for other numbers").
NOTE: To reduce copying costs, the teacher can laminate the self-correction checklist and provide the student with an erasable marker to allow for multiple re-use of the form. (View attachments at the end of this article to view both sample and blank Math Self-Correction Checklists.)
Introduce the Checklist. The teacher shows the student the self-correction checklist customized for that student. The teacher states that the student is to use the checklist to check his or her work before turning it in so that the student can identify and correct the most common errors.
Prompt the Student to Use the Checklist to Evaluate Each Problem. The student is directed to briefly review all items on the checklist before starting any worksheet or assignment containing the math problems that it targets.
When working on the math worksheet or assignment, the student uses the checklist after every problem to check his or her work—for example, marking each checklist item with a plus sign ( '+') if correctly followed or a minus sign ( '-') if not correctly followed. If any checklist item receives a minus rating, the student is directed to leave the original solution to the problem untouched, to solve the problem again, and again to use the checklist to check the work. Upon finishing the assignment, the student turns it in, along with the completed self-correction checklists.
Provide Performance Feedback, Praise, and Encouragement. Soon after the student submits any math worksheets associated with the intervention, the teacher should provide that student with timely feedback about errors, praise for correct responses, and encouragement to continue to apply best effort.
[OPTIONAL] Provide Reinforcement for Checklist Use. If the student appears to need additional incentives to increase motivation for the intervention, the teacher can assign the student points for intervention compliance: (1) the student earns one point on any assignment for each correct answer, and (2) the student earns an additional point for each problem on which the student committed none of the errors listed on the self-correction checklist. The student is allowed to collect points and to redeem them for privileges or other rewards in a manner to be determined by the teacher.
Fade the Intervention. The error self-correction checklist can be discontinued when the student is found to perform on the targeted math skill(s) at a level that the teacher defines as successful (e.g., 90 percent success or greater).
Common Core Standards: Varied
Setting: Whole Class, Small Group, Individual
Focus Area: Acquisition, Fluency and Generalization
Overview: The purpose of interleaving worked problems and problems to solve is to provide scaffolding through models or examples for students as they proceed through a set of math problems
Common Core Standards: Varied
Setting: Whole Class
Focus Area: Acquisition, Fluency and Generalization
Overview: Math PALS is a whole-class peer-tutoring program that can be used with students across grades K-6. With PALS, students work step-by-step on grade-level mathematics skills. The PALS model allows for students to practice mathematics skills with immediate feedback and to engage in discussion about mathematics.
Pirate Math
Common Core Standards: Varied
Setting: Whole Class and Individual
Focus Area: Acquisition, Fluency and Generalization
Overview:
Common Core Standards: Multiple
Setting: Small Group or Individual
Focus Area: Acquisition and Generalization
Overview: As indicated by Montague (2003), “The purpose of Solve it! Is to teach students to be good problem solvers.” Solve It! is a scripted curriculum designed to teach mathematical problem solving by engaging students in a series of steps that allow them to actively participate in metacognitive processing and demonstrate higher-order problem solving skills.
Common Core Standards: Multiple
Setting: Whole Class, Small Group or Individual
Focus Area: Multiple
Overview: A schema is a way to organize or pattern information within a structured framework of known and unknown information. Within word-problem work, the learner identifies the type of problem (i.e., schema), which lends itself to solving the problem using a given organizational pattern. The main focus of Schema-Based Instruction or Schema-Broadening Instruction (SBI) is to teach word-problem solving using identification of a problem schema, representation using diagrams or equations to represent the schema, and solving the word problem. Scaffolding of student learning is provided throughout.
Common Core Standards: Multiple
Setting: Whole Class, Small Group and Individual
Focus Area: Acquisition, Fluency and Generalization
Overview: The purpose of a word-problem mnemonic is to provide students with a framework for solving word problems. The mnemonic reminds students to work step-by-step through a word problem. Some word-problem mnemonics can be used for problem solving beyond basic word problems.