https://padlet.com/cecarpe4/SED512_Chp3Wathall A brief summarization of Chapter 3 from Wathall's Concept-Based Mathematics en-us 2022-09-24 20:40:53 UTC 2026-01-29 11:07:21 UTC hello@padlet.com cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311827376 Wathall describes generalizations to be the overarching umbrella that represents "the big ideas in a unit of work" (Wathall, p. 82). These generalizations are "statements of conceptual understanding" (Wathall, p. 82) that are used to guide us towards our specific learning goals within each unit that is covered in the course. Generalizations are used to help students understand ideas using two or more concepts. It is important that when we focus on generalizations that as teachers we are actively planning to address them in each unit and relate them back to the overall purpose of the course. Most courses have five to six generalizations they cover over a year-long course. ]]> 2022-09-24 20:50:55 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311827376 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311832432 Generalizations are often referred to as essential understandings, enduring understandings, and statements of inquiry. There are two different types of generalizations that Wathall discusses in his chapter: Overarching Enduring Understandings and Topical Understandings. Principles are "foundational truths that hold all the attributes of generalizations and commonly describe real-life situations" (Wathall, p. 64). Given the definition provided by Wathall, it seems that principles are a subgroup of generalizations that are specific. As generalizations are transferable across disciplines, principles are specific to their discipline and only support certain concepts. Principles appear as theorems within the mathematics discipline.]]> 2022-09-24 21:01:18 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311832432 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311834156 Topical generalizations are generalizations that "are subject or topic-specific" (Wathall, p. 64). These types of generalizations are used to help students focus on specific ideas within a unit of study.]]> 2022-09-24 21:05:05 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311834156 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311834556 Overarching generalizations are "the larger, transferable insights we want students to acquire" (Wathall, p. 63). These generalizations are to help students understand the concepts that span disciplines rather than the concepts that are math specific. In Figure 3.1 Wathall gives an example of this by saying "Mathematics reveals patterns that might have remained unseen" (Wathall, p. 64). Patterns that may remain unseen can appear across disciplines, times, and situations as opposed to only appearing in a singular situation within mathematics. ]]> 2022-09-24 21:05:47 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311834556 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311839038 Generalizations usually use qualifiers like often, can, or may which demonstrates that they are not absolute truths that hold in every case. Principles hold always which is why they never contain phrases that refer to exceptions. There are three different levels that generalizations fall under.]]> 2022-09-24 21:14:57 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311839038 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842367 Level 1 generalizations should start with "Students will understand that..." to provide clarity on what students will know and do for this concept. Level 1 is the most basic of the generalizations as it keeps students away from verbs like are, have, and impact. These verbs are weaker and the generalizations do not give much help for actually understanding the concept and why it is important (Wathall, p. 72). Most of the time we should strive to write generalizations that surpass the bare minimum that exists at Level 1.]]> 2022-09-24 21:22:08 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842367 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842505 Level 2 generalizations ask students "How?" and "Why?" to help them understand a concept deeper (Wathall, p. 72). This helps students gain clarity and adds specificity to the generalization. Since these generalizations need specificity to connect to the topic they should be classified as topical understandings. The Level 2 generalizations should be sufficient for enabling efficient student understanding. These should be what teachers strive to give their students as far as conceptual understanding of mathematics.]]> 2022-09-24 21:22:27 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842505 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842612 Level 3 generalizations should "be used to express the real-world value of mathematics or to mathematically extend the Level 2 understanding with greater conceptual specificity (Wathall, p. 72). These generalizations can be used to help students understand why these concepts are important to their lives outside of school, yet can be too advanced for some situations. Teachers should be aware if their students are able to be exposed to Level 3 understanding as it could distract from their primary understanding of the concept. Level 3 understanding could be extremely helpful in a class that focuses on the application of mathematics in the real world.]]> 2022-09-24 21:22:42 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2311842612 cecarpe4 https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2312620457 First, I will introduce the topic conceptually and try to relate the concept to their lives to peak interest. Once I introduce the topic I will teach the process behind the concept and continually check for understanding with guiding questions. In the next step, I will provide them the opportunity to work individually on problems that use the concept to further their understanding. After individual work time, I will allow students to work in small groups to facilitate peer corrections while I circulate the room to work with each group. Lastly, I will provide the students with an appropriate exit ticket to further reinforce the concept that we are focusing on and give them an opportunity to demonstrate their understanding. ]]> 2022-09-25 22:13:53 UTC https://padlet.com/cecarpe4/SED512_Chp3Wathall/wish/2312620457