The purpose of the following response questions is to support you as you make sense of the assigned readings and synthesize key ideas across them – specifically, the interconnected nature of mathematical concepts through conceptual understanding and mathematical meaning, broadly and within K-12 math classrooms (CLOs 1 & 3). They also provide a space for us to share thoughts, ideas, and wonderings so we can learn with (and from!) one another.

As you respond to your peers across this course, be sure you are thoughtfully engaging with the ideas we are working with in this course (and other education/mathematics courses) to answer questions and continue the conversation.

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Instructions
After completing the assigned readings for Week 3, choose two of the following response questions and answer them via this Canvas Discussion Board.

Initial Post
Choose two of the following Response Questions. Identify the numbers of the questions you answer in the initial post. Those numbers will be important when it comes to responding to peer posts.

One of the main arguments Su puts forward in this chapter is the idea of mathematical meaning found in the stories we build about relationships between objects or ideas. We might consider geometric, significance, explanatory, or experiential stories, among others, as ways to convey relationships between mathematical objects. What kinds of stories have you heard or built yourself across your mathematical journey? What kinds of stories do you think are easiest to construct? most difficult? most meaningful? Why? Provide examples or reasons to support your responses.
The notion of mathematical meaning as discussed by Su (2020) and the strand of conceptual understanding (NRC, 2001) may seem closely related as you read the selections for this week. Choose a quote from Su (2020) Chapter 3 that, to you, captures the relationship between these two ideas. What pieces of conceptual understanding does this quote magnify or clarify for you? What characteristics of conceptual understanding does it not attend to? What other reasons, features, or connections with this quote made it stand out to you? Feel free to reflect on your own experiences in mathematics (in or outside the classroom) in your response.

Conceptual understanding can be described as understanding how a cluster “interrelated facts and principles” can be unpacked, condensed, or linked with other clusters of mathematical ideas (NRC, 2001, p. 120). These clusters can sometimes be hard to identify, which makes it challenging to support students in seeing the relations between concepts. The reading mentions clusters such as place value, operations with single and multi-digit numbers, the numbers, etc. Propose another “knowledge cluster” or set of interrelated concepts (or choose one of the examples from the text). Suggest how you see this cluster unfolding across the K-12 spectrum, including an example from each grade-band, K-5, 6-8, and 9-12. State in 2-3 sentences why you see this cluster as important outside of the math classroom and pose one question to your peers.
Lischka & Stevens (2015) highlight one tool – the rectangular area model – that can support students’ development of conceptual understanding. Provide a 2-3 sentence argument to support this claim. If you disagree with this claim, argue why it is incorrect. Draw upon evidence from Lischka & Stevens (2015) and NRC (2001). Then, identify at least one benefit and one challenge or worry of using such a tool across grade levels. Provide an example of each to illustrate your understanding.
Respond to two peers who answered at least one question differently than you. For example, if you answered Questions 1&3, you could respond to someone who answered 1&2, 1&4, 2&3, 2&4, or 3&4 ). Remember to post a thoughtful response.Guidelines
Here are some suggested guidelines for constructive and generative responses to peers. Thoughtful responses include:

Answering a question posed by a peer. This might require you to look for additional information.
Responding to a peer’s post in a way that deepens the conversation and shows that you have been thinking about the idea(s) involved (perhaps in a different way).
Responding to a peer’s post in a way that allows you to make connections between your response and theirs.
Responses that are not sufficiently thoughtful or in-depth include responses such as “yes or no,” “I agree or disagree,” “I didn’t know that,” “I don’t know,” “that is cool,” “I like your idea,” etc. It is ok to say you do not know something. Use the resources at your disposal, such as the library, the web, and our class readings, to make a good-faith effort to find useful information that speaks to the question or statement. For more information on Reading Response check out: https://www.encyclopedia.com/social-sciences-and-law/education/education-terms-and-concepts/reading

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