“New comprehension does not automatically occur, even after evaluation and reflection. Specific strategies for documentation, analysis, and discussion are needed.” (Shulman, 1987, p.19)
This quote is crucial to evaluating mathematical word problem solving skills. Students require direct instructions to support the development to skilfully extract information from word problems, identify missing attributes, select most approach analytical method and compute and make sounded conclusions. GRASP is an explicit metacognitive strategy that trains learners to effortlessly process word problems.
Using the GRASP Method to teach word problem solving skills
What is GRASP?
GRASP is a strategy to solve mathematical word problems. Below is the one of the many definitions of GRASP.
G – Give Information (i.e. operation signal words/ phrases)
Students are to extract information from a given word problem.
R – Required Information
Students are to identify parts that need to be solved.
A – Analyze (i.e. mathematical strategies)
Students are to select appropriate strategy (or strategies) to develop an answer.
S – Solve the question (i.e. computation)
Students are to use the select strategy (or strategies) to solve the question.
P – Paraphrase the answer
Students are to communicate their answer in a way that solves the problem.
Pedagogical Knowledge
Pedagogically, the GRASP method is a suitable framework to teach students how to solve word problems because this strategies shows thinking processes. The method helps make thinking visible by asking students to isolate and document components of the word problems. Linn (2000) believes that making thinking visible makes the metacogntive processes inspectable. In the case of GRASP, students can reflect upon their work and inspect the areas that led to an erroneous answer. For example, if it were a computation error, the analysis part would contain mistakes.
Teachers can clearly insepect how a student comprehends authentic word problems. Since students have to mindfully select appropriate analysis strategies based on given information, it mobilizes an integrated understanding of problem solving strategies. Moreover, authentic word problems make comprehension and problem solving accessible and relatable. Students are also more motivated to solve problems that were created by their peers.
Additionally, this strategy requires teaching practices such as scaffolding and modeling. Without this pedagogical strategy, students fail to independently apply the GRASP method. It is best to model as a whole class demonstration. Here, one can employ the cognitive apprenticeship. This pedagogical design helps foster problem-solving success by providing independent practice time. Moreover, this pedagogical awareness and experience helps promoting lifelong mathematical learning and success by installing a plausible approach to solve word problems.
Another pedgagoical consideration is about differentiation. The GRASP method enables all learners to find success in solving word problems. Some students find that advance operations (i.e. multiplication and division) are easy. In contrast, some students require direct operational signals to solve problems. This problem-solving approach allows educators to differentiate for slow or excelling learners. For example, teachers can modify the wording, include computation choices and provide sentence starters to fit the needs of their learners. Nonetheless, this depends on how well educators know about their students’ needs and strategies to support the needs.
Content knowledge
Shulman (1987) suggests that teachers serve “as the primary source of students understanding of content knowledge” (p.9). In the case of GRASP, teacher’s knowledge of signal words (i.e. math words and phrases that signals operations) and age-appropriate problems solving strategies influence how successful GRASP is. Explicit teaching of math vocabulary and computation strategies is key. Educators should be able to help students develop a strong repertoire of this knowledge. In this instant, educators need to develop content rich anchor charts to supplement students’ understanding of computation strategies and words that signal specific operations. Students also benefit when they are front-loaded with this information. These ideas are consistent with Shulman’s (1987) emphasis on the importance of teacher’s content knowledge. Without building a strong foundation of words and strategies, students fail to employ the GRASP method on more challenging word problems. Learners may also lack skills to develop their own problems.
GRASP & Technology
Although GRASP can be used without technology, simple word-processing knowledge would be beneficial. Educators can make set word problems more organized and attractive by printing off typed word problems and adding pictures. Additionally, to demonstrate that some problems are differentiated, teachers can place a picture on the opposite side.
Another key advantage of using the GRASP method is that educators can analysis the isolated parts of the word-problem solving process. Educators can digitally document students’ area of strength and or weakness. More specifically, teachers require higher digital competency to code an Excel sheet to automatically map student data and the evolving performance trends. For example, after values are imputed, the Excel can use color to highlight consistent errors and populate other visuals to show trends for student performance.
For learners, the computation part the GRASP can be digitized. For example, as students use GRASP, they can replace writing and drawing via technical means. More specifically, students can use Excel to make a graph or use Google drawing to visualize and manipulate variables to improve computation efficiency and accuracy. This is consistent with Jonassen’s (2000) claim. He believes that “that the most productive roles for media are as computation and memory tools for off-loading unproductive cognitive task that may interfere with knowledge construction by the learner.” (Jonassen 2000, p.33) Furthermore, students can also discuss their GRASP methods on student forums to extend their and extend their thinking.
Food for thought
In Mishra & Koehler (2006), although the overlapping circles appeal to be a useful method to talk about integrating these different types of knowledge, it may not accurately represent the extent of each type of knowledge. Rather, a power map may provide more information. In a sense, each example has its own power map. Here is what the GRASP model may look like:
Visibly, the success of GRASP relies heavily on pedagogical and content knowledge.
A follow up question is how educators evaluate whether this is a ‘good’ mix of Msihra & Koehler’s idea of TPACK? Are these variations dependent of content?
References
Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers college record, 108(6), 1017.
Jonassen, D. H. (2000). Computers as mindtools for schools, 2nd Ed. Upper Saddle River, NJ: Merrill/ Prentice Hall. Retrieved from Google Scholar: http://scholar.google.com/scholar?q=Jonassen+mindtools&ie=UTF-8&oe=UTF-8&hl=en&btnG=Search
Linn, M.C. (2000). Designing the knowledge integration environment. International Journal of Science Education, 22(8), 781-796.
Shulman, L.S. (1987). Knowledge and teaching. The foundations of a new reform. Harvard Educational Review, 57(1)1-23.
Hi Alice,
I enjoyed reading about GRASP in your post. The math teachers at our school have similarly implemented various problem-solving strategies to help students successfully navigate math word problems. The flavour of this year is a strategy called “Part, Part, Whole”. Last year, it was the use of the I-Chart (using the form of the capital I that has two vertical lines top and bottom, not this one that looks like a lowercase l) ;), before that there was a mnemonic acronym like GRASP (but not that word) which I can’t quite recall now.
I liked what you said in the Content section of your post about using anchor charts to help them recognize key math signal words. It reminded me of what Allison Greig said in her post about a specialized kind of reading that’s required for STEM subjects.
Are you familiar with any of their strategies or have you used other strategies other than GRASP to help students with word problems?
I liked the idea of having students using technology to answer and reflect on their solutions with others. I didn’t quite understand your follow up question. Were you asking about whether GRASP would be considered a good example of pedagogical tasks planned using TPCK or whether it was too content-specific?
Dear Jan,
Thank you for your suggestions and questions.
I have yet to explore more word problem strategies. As of now, GRASP is my favourite strategy for word problems. There are others I yet to try. In math, knowing the math vocabulary is key to solving problems.
I feel that GRASP is heavily reliant on educator’s content and pedagogical knowledge. Limited technological knowledge is required. More over, I am curious to see if the success of TPACK is dependent on how balance each field is.
Alice