Combining Cognition and Metacognition During the Problem Solving Process

By Dawn Woods, RME Elementary Mathematics Coordinator

Mathematical problem solving extends beyond the application of mathematics skills and concepts to include the semantics and syntax of language and the situations that the language represents within social-cultural contexts. Sometimes when students consider word problems, they rely on coping strategies such as using key words or apply general strategies such as “draw a picture”, which can limit the student’s problem solving abilities (Clements & Sarama, 2009). However research is showing that when students are engaged in metacognition, or thinking about their thinking, that their problem solving competency increases through the awareness of their reasoning (Cambell & White, 1997; Goos, Galbraith & Renshaw, 2002; Caswell & Nisbet, 2005).

Intervention Central, an online RtI resource, outlines a research-based strategy designed to engage struggling students in the problem solving process. Based on Montague’s work, students apply a “Say-Ask-Check” routine to stimulate metacognition as they work through the cognitive steps of the problem solving process (1992). During each step of the problem solving process, students are taught to “say” or self-instruct by stating the purpose of the step; “ask” or self-question what he or she plans to do to complete the step; and “checks” by self-monitoring the successful completion of the step. This “Say-Ask-Check” routine with close teacher support during instruction can increase the likelihood of student success.

Following is an example of what the “Say-Ask-Check” routine could look like when applied to George Polya’s four-step mathematical problem solving techniques (1945; Wright, 2011).

Problem Solving Steps	"Say-Ask-Check" Routine
Understanding the Problem	Say (Self-Instruction):  “I will read the problem until I can restate the problem in my own words.” Ask (Self-Question):  “Do I understand the problem?” Check (Self-Monitor):  “I understand the problem.”
Devise a Plan	Say (Self-Instruction):   “I will create a plan to solve the problem.” Ask (Self-Question):  “What is my first step? What is the next step, etc.?” Check (Self-Monitor):  “My plan has the right steps to solve the problem.”
Carrying Out the Plan	Say (Self-Instruction):  “I will solve the problem” Ask (Self-Question):  “Is my answer reasonable?” Check (Self-Monitor):  “I carried out my plan to solve the problem.”
Looking Back	Say (Self-Instruction):  “I will check my work.” Ask (Self-Question):  “Did I check each step in my calculation?” Check (Self-Monitor):  “The problem appears to be correct.”

Combining cognition and metacognition through using the problem solving process and the Say-Ask-Check routine increases a students’ awareness in his/her reasoning thereby increasing the likelihood of his/her academic success.

References

Campbell, P., & White, D. (1997). Project IMPACT: Influencing ad supporting teacher change in predominately minority schools. In E. Fennema & B.Nelson (Eds.), Mathematics teachers in transition (pp 309-355). Mahway, NJ: Erlbaum.

Caswell, R., Nisbet, S. (2005). Enhancing mathematical understanding through self-assessment and self-regulation of learning: he value of meta-awareness. Building Connections: Research, Theory and Practice. Retrieved from http://www98.griffith.edu.au/dspace/handle/10072/2482 .

Clements, D. & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routlege.

Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Education Studies in Mathematics, 49 (2), 193-223.

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.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press

Wright, J. (2011). Math problem solving: Combining cognitive and metacognitive strategies. Retrieved from http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies.

ESTAR & MSTAR Assessments - Professional Development

By Savannah Hill, RME Professional Development Coordinator

Today, I want to spend some time talking about some of the professional development opportunities available with the Texas Algebra Ready Initiative. We have spent time before talking about the assessments that are available, but briefly, currently available is the ESTAR and MSTAR Universal Screeners (grades 2-4 and 5-8), coming in January will be the MSTAR Diagnostic Assessments (grades 5-8), followed by the ESTAR Diagnostic Assessments (grades 2-4) next year.

But in order to correctly implement those assessments and interpret the reports given, there is a learning process. Many teachers may not know that professional development is available. It is essential that teachers understand why and how to use the Universal Screener and the MSTAR Diagnostic Assessment and how it can support a Response to Intervention approach. Here is some of the available courses that teachers should take before giving an ESTAR or MSTAR assessment.

ESTAR and MSTAR Universal Screeners: The ESTAR and MSTAR Universal Screener is a formative assessment system administered to students in grades 5-8 to help teachers determine if students are on-track or at-risk for meeting curricular expectations in algebra and algebra-readiness. Currently, a course is available to prepare teachers to administer the ESTAR and MSTAR Universal Screener - Overview of the Universal Screeners. Training on the use of the Universal Screener is available through Project Share. An updated version (v4.0) will be released soon.

MSTAR Diagnostic Assessments (grades 5-8): This assessment, designed to follow the MSTAR Universal Screener, is administered to those students identified as at-risk on the Universal Screener. The Diagnostic Assessment will help identify WHY students are struggling with algebra-related core content, and provide information that can be used to plan supplemental instruction. Two courses will be available: MSTAR Learning Progressions and Overview of the MSTAR Diagnostic Assessments. Information on how to access these courses, which will provide suggestions on how to prepare for administration of the MSTAR Diagnostic Assessments and guidance on how to interpret results following administration, will be made available through various list-servs and Project Share Groups over the coming weeks.

All courses are online and can done individually. PLCs could also use time to review the material from the courses and review for remediation. For more information, contact your local Educational Service Center or visit www.projectsharetexas.org.

Technology as a Tool for Teaching Content

By Sharri Zachary, RME Mathematics Coordinator

Technology can assume many roles in education. It is often utilized as a resource, delivery system, or means of production (Yuan-Hsuan,Waxman, Jiun-Yu, Michko, & Lin, 2013). Previous research studies found that computer programs were particularly useful in instruction when they are purposeful in supporting the needs of all students, are factual, and provide students with new learning experiences. In addition, the research also revealed higher gains in academic performance when students were allowed to use computers in small groups rather than individually.

In a recent study by the authors for grades K-12, the effects of teaching and learning with technology on student cognitive outcomes and affective outcomes were revisited to inform current instructional practice. Outlined are some key things teachers can do to integrate technology in their instruction, such that there is improvement in student outcomes:

Cognitive

• Allow students to collaborate in pairs or small groups with technological devices 
• Develop instructional material that makes sense contextually 
• Incorporate project-based learning that allows students to bridge skills and subject matter

Affective

• Include challenging activities in your instructional materials 
• Ask higher-order questions 
• Work together with your students to produce a result via technological device 
• Emphasize collaboration in your teaching and their learning utilizing technology

Yuan-Hsuan, L., Waxman, H., Jiun-Yu Wu, Michko, G. & Lin, G. (2013). Revisit the effect of teaching and learning with technology. Journal of Educational Technology & Society, 16(1), 133-n/a.

Connecting the Area Model to the Standard Algorithm

By Cassandra Hatfield, RME Assessment Coordinator

Using the area model for multiplication and using the standard algorithm for multiplication are often put in two separate and unrelated categories. Often times textbooks spend very little time developing the conceptual understanding and focus on the procedure of the standard algorithm.

However, “as much time as necessary should be devoted to the conceptual development of the algorithm with the recording or written part coming later.” (Van De Walle, Karp, Bay-Williams, 2013). Students are more successful when they can relate their prior knowledge with a new concept. Designing lessons that connect the area model and partial products can then lead to the understanding of the standard algorithm. This powerful transition allows students to visually see the why behind the standard algorithm.

The model below uses color to amplify the connection between the area model, the partial products strategy and the standard algorithm with 2-digit multipliers. Notice that the area model was drawn proportionally, not as a “window pane.” The importance of drawing area models proportionally was discussed in one of my previous posts, It's Not a Window Pane... It's an Area Model.

It is important to consider the value of the digits rather than the digits themselves when using partial products or the standard algorithm. For example, when multiplying 20 x 20, use the base 10 language 2 tens times 2 tens is 4 hundreds or 20 times 20 is 400. Try to avoid “two times two.”

Students can use the partial products strategy just as effectively as the standard algorithm. In fact, it is of utmost importance to give students the opportunity to explore, explain, and demonstrating their understanding of the value of the digits over the digits themselves.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.