Student engagement is a common part of most instructional delivery approaches. Teachers know well the lure of high-impact videos, hands-on activities, and charming games—each engineered to create excitement and generate student interest. Graphic organizers are also effective tools teachers can use to help stimulate learners’ thinking about prior knowledge while simultaneously evoking interest. Organizers help students engage in reflective practice. As part of reflective practice, teachers can employ graphic organizers to help students make faith connections as they learn.
The 5E Instructional Model is used to teach science and mathematics. At its core, the model seeks to engage students in discovery, building on their natural curiosity while strengthening critical thinking and problem-solving skills. The five “E’s” are Engage, Explore, Explain, Elaborate, and Evaluate.1 With "Engagement” being the first stage of the 5E Instructional Model, an unintended consequence arose: Adherents were likely to omit activating prior learning, concentrating only on piquing learner interest.2 Its later expansion, the 7E Model, purposed giving equal weight to that easily overlooked step. The solution, which carefully avoided making the second model appear to be a successor to the first, deemed the activation of prior understanding an intimate part of the first stage (Engage)3 (see Chart 1).
This drawing out of the student’s academic memory, an almost universally agreed-upon element of effective teaching, is present in learning-process theory, cognitive-development theory, attribution theory, and cognitive perspectives of learning.4 Embraced by so many schools of thought, it has become the recognized precursor to introducing new ideas. Why? It prepares the mental framework, the schema—that cognitive structure the mind needs to make sense of new information by relating known ideas to old concepts.
An Internet search reveals many creative strategies for activating this recall. Not a few involve versions of brainstorming, anticipatory guides, collaborative sharing, and the ever-popular, KWL chart (Know, Want to Know, Learned).5 But in my admittedly less-than-exhaustive search, among that family of graphic organizers renowned for its broad application and visual-verbal blending, why should the KWL chart seemingly stand alone as the tool to prod prior knowledge? Or, to orient the question toward our discussion, which graphic organizers specifically address students’ previous learning in mathematics? Since graphic organizers have been used to help both students with limited English proficiency and students with learning disabilities, a branch of graphic organizers that help target each student’s prerequisite capabilities in mathematics will likely address the needs of a broad audience of learners.6
In my attempt to create several of these, I established the criteria for the final products using these guidelines:
The organizers would do the following:
- Map the prior skills needed to learn a new topic;
- Relate old problems to new problems;
- Identify previously learned problem-solving steps;
- Include elements of reflective learning; and
- Make a faith-based connection.
I wanted these organizers to be flexible enough to accompany other strategies for stimulating student recall, while also doubling as stand-alone tools if needed. Two categories of products will be discussed in this article: organizers for mapping component skills, and organizers for identifying related problems and necessary problem-solving steps.
Mapping Needed Skills
Every mathematics topic has a set of component skills. Students will know some of these, while others will be new to them. Ideally, teachers will want to introduce only one novel skill at a time, thereby establishing a stepwise learning progression that sets a tolerable pace for most learners. In order to identify these skills, the Identify Component Skills organizer offers suggestions for creating organizers suitable for the teacher and student, respectively.
The Teacher Resource
The Identify Component Skills organizer presents an instructor’s tool that allows for the mapping of associated prior skills and new skills expected for a specific topic at a particular grade level. One way to do this would be to search preceding standards to identify building blocks relevant to the current topic. For example, adding fractions with like denominators is a precursor to adding fractions with unlike denominators. The foundational standards for these topics may extend across grade levels, or they may be imbedded within previously encountered standards for a single grade level.
Another approach would be to study the steps leading toward solving the new problem, flagging both those that students should already have learned, as well as the new skills to be introduced. Here is one example: Adding fractions with unlike denominators could reveal these prior skills: (a) finding equivalent fractions, (b) reducing fractions to their simplest forms, (c) adding whole numbers, and (d) adding fractions with like denominators. The new skill might be to find the lowest common multiple, or, if that skill has already been mastered, combining all these prior skills in proper sequence to produce the sum.
The Student Resource
The blank student version prompts the learner to brainstorm skills he or she thinks would be useful in solving the new problem. Comparing this process to the imagery of building a brick wall, it becomes clear that knowledge builds on itself, brick by brick. Because these strategies are used in tandem with cooperative learning and sharing strategies, the teacher does not intervene in the brainstorming but rather encourages teams of students to provide reasons for their choices.
Relating Old Problems to New
The three types of organizers discussed below can be used to help learners activate prior knowledge by revealing what they already know about a topic. Teachers can use these approaches to help learners relate old (previously learned) strategies for problem-solving to new ones. They can also be used to help identify and correct misconceptions.
1. “Six Things I Think I Know” Organizers
Having learners reveal what they think they know about a topic is the first part of a KWL chart. The graphic organizers titled Six Things I Think I Know ask students to tell what they know about a problem, whether that means relating the problem to a related previously studied topic, or sharing a fact relating to the new topic. Because these organizers are specifically tailored for mathematics instruction, they include prompts to draw pictures of what the problem looks like, describe the problem’s component parts, suggest a real-world context for the problem, and show how to solve the problem.
2. “Reminds Me of” Organizers
Often, but not always, new problems look strikingly similar to old counterparts except for a singular twist; thus, they are also solved similarly except for a singular twist. Exploiting this as a learning advantage is the purpose of the Reminds Me of . . . . organizers.
Consider, for example, the new problem, -1⁄3+2⁄5. The student might list the related old problems as 1⁄3+2⁄5, -1+2, or -3+5. If the teacher provides additional prompts to encourage the students to solve the old problems they identify, he or she can identify possible misconceptions in the problem-solving steps that might be replicated when students solve the new problem. Addressing misconceptions is crucial to the learning process, since prior knowledge may include application errors that can impede the student’s ability to move forward with new learning.
Conditionally, if students show proficiency in the old steps, this mode of engagement has the corollary benefit of relating so closely to the Explain stage of the learning cycle (where the student receives direct how-to instruction) that it can maximize the time available to learn the newly introduced problem-solving step. The significant difference between these stages is the limit on teacher intervention. When done in groups, it is the students who collaboratively clarify the steps as they share knowledge; the teacher facilitates the engagement and records misconceptions for later clarification.
3. “U Should Know” Organizers
By showing how to relate old problems to new ones, we have already seen how to identify students’ knowledge of previously learned problem-solving steps. The approach above, however, relies heavily on each student’s ability to perceive the connections, a feat possible only because the problems are visual twins. It will not always be obvious to the learner what prior knowledge he or she might need to solve a new problem.
Consider, for example, the problem, 3⁄4×6⁄15. Seen for the first time, it does reveal the need to reduce 3⁄15 and 6⁄4 to their respective simplest forms. The use of the distributive property to generate rules for multiplying integers is an even more obscure example. It will likely not be apparent that the new problem, 3×(-2) is related to 3(5-2) from which we develop the rule, “a positive multiplied by a negative yields a negative.” For these, the expected prior learning would have to be made explicit. The U Should Know organizers permit the teacher to give that direction, then step back to observe students’ approaches to solving old problems, recording misconceptions to address at a later time.
As students interact with the graphic organizers, instructors will want them to monitor their own experience so that the meaning of what they did becomes the topic of discussion.7 If teachers accept that recall is the motif of each organizer, it becomes reasonable to have the students reflect on what they have recalled. For this reason, each organizer includes two prompts: “Here’s what I do well:” and “I still struggle with . . . .” This provides valuable qualitative data not only for the student, but also for the teacher.
Integration of Faith and Learning
Reflective learning also extends to creating opportunities for students to reflect on and recall God’s Word. When it comes to seeking opportunities to integrate the Word of God, Christian teachers are rarely shy. Their ability to use curriculum as the vehicle to carry out this biblical mandate (Deuteronomy 6:6, 7) is the distinguishing feature of the Adventist ministry of education. Any document presented to a learner occasions the opportunity to integrate principles from the Bible. But many teachers wonder how to choose a text or passage of Scripture that relates to a mathematical topic. Table 1 offers five approaches. The choice of an approach should be guided by the goals of delivering an organic connection, one that is unforced because it easily relates to the topic at hand, using it in a way that provides insight into God’s character or the plan of salvation.
Each graphic organizer concludes with a Bible text (which can be repeated for emphasis), and these texts can be presented in a variety of formats. They can be partnered with a teaser question, a fill-in-the-blanks prompt, a puzzle, or other creative approaches. The key is to stay within the mode of engagement, stirring wonder and interest in an age-appropriate manner
Getting students to reflect on how their faith relates to their learning is not limited to mathematics instruction; the methods described here can be adapted to graphic organizers of any content area. But mathematics instruction, having its own nuanced requirements, can benefit from organizers that relate old problems to new ones, assess how well students understand previously learned problem-solving steps, and that map the component skills necessary to learn a new topic. Teachers of mathematics are encouraged to use and adapt these tools to assess their usefulness and ultimately to create their own (blank versions are included at this link). Whatever we do, it is crucial that we do not skip the stage of the learning cycle that accesses prior understanding but instead, view it as part of the engagement process.
This article has been peer reviewed.
Elvis Agard, “Using Graphic Organizers to Activate Prior Learning in Mathematics,” The Journal of Adventist Education 81:2 (April-June 2019): 45-49.
NOTES AND REFERENCES
- Lakeena Chitman-Booker and Kathleen Kopp, “The 5Es of Inquiry-based Science: Professional Resources for K-12 Teachers” (Huntington Beach, Calif.: Shell Education, 2013): 9, 10; Milwaukee School of Engineering, “Summary of the 5E Instructional Model” (n.d.): http://cbm.msoe.edu/teacherWorkshops/mspResources/documents/day1/5eSummary.pdf.
- Kimberly D. Tanner, “Order Matters: Using the 5E Model to Align Teaching With How People Learn,” CBE Life Sciences Education 9:3 (Fall 2010): 159-164. doi: 10.1187/cbe.10-06-0082.
- Arthur Eisenkraft, “Expanding the 5E Model,” The Science Teacher 70:6 (September 2003): 56-59: http://emp.byui.edu/firestonel/bio405/readings/learning%20models/expanding%205e.pdf.
- Margaret E. Gredler, Learning and Instruction: Theory Into Practice, 6th ed. (New York: Pearson, 2008).
- Rebecca Alber, “Are You Tapping Into Prior Knowledge Often Enough in Your Classroom?” Edutopia (July 2011): https://www.edutopia.org/blog/prior-knowledge-tapping-into-often-classroom-rebecca-alber; Jennifer Gonzalez, “Retrieval Practice: The Most Powerful Learning Strategy You’re Not Using,” Cult of Pedagogy (September 2017): https://www.cultofpedagogy.com/retrieval-practice/.
- Sara LaJeunesse, “Graphic Organizers Aid Students With Learning Disabilities,” PennState College of Education (December 2011): https://ed.psu.edu/news/news-items-oct-dec-2011/graphic-organizers-aid-students-with-learning-disabilities.html; Kara Wyman, “To Help English Learners, You Need Ways to Reduce Their Affective Filter,” A Blog by Concordia University, Portland, Oregon (February 2017): https://education.cu-portland.edu/blog/classroom-resources/affective-filter-english-learners/; Jennifer Gonzalez, “The Great and Powerful Graphic Organizer,” Cult of Pedagogy (October 2017): https://www.cultofpedagogy.com/graphic-organizer/; Douglas D. Dexter, Youn J. Park, and Charles A. Hughes, “A Meta-Analytic Review of Graphic Organizers and Science Instruction for Adolescents with Learning Disabilities: Implications for the Intermediate and Secondary Science Classroom,” Learning Disabilities Research and Practice 26:4 (November 2011): 204-213. doi.org/10.1111/j.1540-5826.2011.00341.x.
- Arthur L. Costa and Bena Kallick, eds., “Learning Through Reflection,” in Learning and Leading With Habits of Mind: 16 Essential Characteristics For Success (Alexandria, Va.: ASCD, 2008), 221-235: http://www.ascd.org/publications/books/108008/chapters/Learning-Through-Reflection.aspx.