Instructional Strategies
When my students tell me they do not understand a concept or they do not know where to start the process of solving a problem, I often use the phrase, “When in doubt, draw it out!” This is a means for students to bring what is on their mind into action by drawing visuals or writing down words and phrases that come to mind on a topic of discussion or when approaching a problem. This is in alignment with encouraging students to engage in divergent thinking where they are asked to think differently to come up with alternative ideas through mentally processing.[16] By asking themselves questions such as “Why…?” when they encounter a problem or situation, then “What if…?” to process possible improvements or solutions to the problem, leading to “How…?” to identify ways to implement improvements or solutions, move to action and perhaps start a new set of “Why…?” question process. “Questions are more relevant than answers. Questions are bigger than answers. One good question can give rise to several layers of answers…Answers, on the other hand, often end the process.”[17] The thought process for students should focus on thinking through asking questions in a non-evaluative process that develops curiosity that supports the productive struggle.
An instructional strategy that will encourage students to grapple with problems independently and then through collaborative processing is a questioning exercise.[18] Teachers are often the ones asking prescription or pre-designed questions as they develop lesson sequences and the questioning exercise provides the much-needed shift in having students asking questions. Students will be encouraged to take ownership of their learning journey and spark inquiry and curiosity within themselves and other students.[19]
Students will start by being provided a topic of discussion, a content focus, or a prompt that engages them with content knowledge and content vocabulary. Students will be learning how to develop points of confusion while they learn about isometry with geometric figures and allometry with living organisms and geometric figures. I anticipate that many of the questions that are posed as points of confusion for students will be around using a scale factor for enlargement or reduction of a geometric figure. I want students to recognize the importance of knowing the dimension of their original figure and then determine whether those dimensions increase or decrease (non-rigid transformation), or stay the same (rigid transformation). When there are concepts or skills that become confusing to students, they will be asked to write down points of confusion within their lecturer notes. Then students will be provided some independent time to continue thinking in questions and write only questions about the topic - generate as many questions that come to mind regarding the given topic of discussion.
Students will have peer processing time in collaborative study groups with the questions they develop to identify questions that can be answered immediately and questions that will require more critical thinking and processing to arrive at a next step. Students will then get into pairs and begin the process of narrowing and expanding on questions. They will narrow questions to short, simple answers and expand questions to become open ended and spark curiosity. Students will then transition to whole class processing to hear the points of confusion and questions developed by students and then grapple with them to determine any similarities in the question process. This process provides students with the skills to recognize effective questioning and then prioritize questions (either in groups of 4 or whole class processing) where students figure out the most important questions that are on topic and move discussion forward.
These prioritized questions will eventually lead to students writing paragraph proofs using geometric concepts, postulates, and theorems, which is also an inquiry and writing skill students will learn and use in their biology courses through writing claim-evidence-reason (CER) paragraphs. The structure of the paragraph proofs starts with students stating a claim or given information about a topic, and in this curriculum unit it will be with respect to triangles and other polygons. The paragraph will then ask students to provide evidence of the topic with respect to a sequence of knowledge and information that was previously learned or known. Lastly, the paragraph concludes by providing reasons why the evidence strongly supports the initial claim. The focus for this paragraph proof will be to have students grapple with the questions and points of confusion they initially develop during the lessons, in small groups, and during whole class discussions. Developing questions strengthens student understanding and use of mathematical content vocabulary and better understanding of why definitions, postulates, and theorems are used in mathematics as evidence and reasoning for supporting a claim.
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