Chapter 3
Chapter 3 of Rigor by Design, Not Chance by Karin Hess focused on the importance of building schemas. Hess describes the role of schemas as that they allow students “to categorize a problem, construct mental representations, search for appropriate problem-solving strategies, evaluate the strategies, and store information for later use” (Hess, 2023, chp 3, para 3). A teacher's role is implementing strategies that help students build schemas (Hess, 2023). The two crucial components of building schemas are combining declarative knowledge and procedural knowledge (Hess, 2023). The teacher must be intentional with declaring the concepts and skills of a unit, connecting parts to the whole, building off students' prior knowledge, and addressing misconceptions (Hess, 2023). Teachers can specifically build off students' prior schemas by utilizing structural schemas and procedural schemas (Hess, 2023). For structural schemas, teachers can use an activity such as a SWBS chart that showcases the structure of content so that students have a framework to draw upon in the future (Hess, 2023). However, for procedural schemas, teachers can use an activity such as an anchor chart that showcases the process of tackling a problem so that students can apply the process to different problems (Hess, 2023). Teachers must also plan where to build schemas in the actionable assessment cycle (Hess, 2023). Stage one of the actionable assessment cycle is when the teacher introduces the essential concepts and skills of the unit and asks questions to probe deeper thinking (Hess, 2023). Stage two allows for building schema activities such as anchor charts and paint the parts to draw connections (Hess, 2023). Stage three is when students showcase their connections and understanding through activities such as word splash and find the outlier (Hess, 2023). Stages four and five build off of the previous stages with the goal of students using a structural or procedural schema to deepen their understanding (Hess, 2023). Overall, teachers have to deliberately deepen students understanding by guiding them in building schemas.
After reading Chapter 3, I became very interested in procedural schemas in mathematics. As a math educator, understanding procedural schemas is very important because solving math problems always involves a process. I found the article “Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning” fascinating as it explains the difference between different types of schemas (Powell & Fuchs, 2018). Two common strategies teachers use when teaching math word problems are teaching specific words to signal a mathematical operation and labeling a word problem as a specific operation (Powell & Fuchs, 2018). However, both of these strategies cause students not to have to mathematically reason what operation to use when solving a math problem (Powell & Fuchs, 2018). Additionally, the article points out that teaching that specific words signal a mathematical operation often causes even more misunderstandings since this strategy is not foolproof (Powell & Fuchs, 2018). The article explains that there are two effective teaching strategies. The two strategies are attack strategies and building schemas (Powell & Fuchs, 2018). The attack strategy “is an easy-to-remember series of steps students use to guide their approach to solving word problems” (Powell & Fuchs, 2018, p. 32). There are many variations of attack strategies; however, the steps usually include reading, identifying, and understanding the question and problem type (Powell & Fuchs, 2018). The article addresses two main mathematical schemas, additive and multiplicative schemas (Powell & Fuchs, 2018). The additive schemas include combine, compare, and change problems (refer to illustration 1) (Powell & Fuchs, 2018). The multiplicative schemas include equal groups, comparison, and proportions or ratios (refer to illustration 2) (Powell & Fuchs, 2018). All of the schemas are procedural schemas as they build students' knowledge of how to process and understand different mathematical problems. Additionally, the schemas often are used together to solve multi-step problems (Powell & Fuchs, 2018). Teachers must build students' schemas by modeling attack strategies and types of solution strategies, such as the combine strategy (Powell & Fuchs, 2018).
The book and article explain the importance of building procedural schemas. The book mentions using an activity called an anchor chart for students to understand the process of completing a math problem. The anchor chart allows the students to break the problem down into steps, including “understand (a visual of eyes reading); plan a strategy (visuals of models and symbols for possible operations); and show evidence (phrases telling what you did)” (Hess, 2023, chp 3, para 14). The article focuses on the importance of teaching students to use an attack strategy when completing word problems (Powell & Fuchs, 2018). With both sources acknowledging the effect of strategies to deepen understanding of a math problem, I understand I must teach and utilize these strategies with my students. I would use the RUN (read, underline, and name) attack strategy mentioned in the article since it utilizes a mnemonic device (Powell & Fuchs, 2018). Teachers must model strategies such as anchor charts and RUN before students can use them autonomously (Powell & Fuchs, 2018). Therefore, teachers should continuously use strategies throughout each unit (Hess, 2023; Powell & Fuchs, 2018). Once students feel comfortable with the strategy, they can deepen their understanding by making connections between mathematical concepts (Hess, 2023). The article allowed me to understand further the types of procedural schemas that teachers must help students build.
Teachers must address common misconceptions throughout their lessons. The book explains why students struggle with fractions when teachers explain that there are two numbers in a fraction instead of explaining it as being a single number with a numerator and a denominator (Hess, 2023). The article explains why students struggle with word problems when teachers teach that certain words equivalate to a mathematical operation (Powell & Fuchs, 2018). Therefore, teachers can not build schemas when misconceptions are roadblocks to forming connections. The teacher's responsibility is to anticipate misconceptions before the misconceptions become concrete understandings for students.
Both sources consistently reiterate that teachers need to aid students in building schemas to deepen their understanding. Schemas allow students to transfer knowledge to new concepts (Hess, 2023). Thus, teachers can create more rigorous opportunities for students to deepen their understanding by having students use multiple schemas to solve a problem (Hess, 2023). The more connections students can make, the easier they can apply prior knowledge to new content (Hess, 2023; Powell & Fuchs, 2018). Overall, “designing lessons that build schemas, that create opportunities for productive struggle, and that require students to transfer their skills and knowledge to new situations is what truly makes learning—and instruction—rigorous by design” (Hess, 2023, chp 1, para 20).
Hess, K. (2023). Rigor by design, not chance. Association for Supervision and Curriculum
Development (ASCD).
Powell, S. R. & Fuchs, L. S. (2018), Effective word-problem instruction: Using schemas to
facilitate mathematical reasoning. TEACHING Exceptional Children, 51(1), 31-42.
https://doi.org/10.1177/0040059918777250
After reading Chapter 3, I became very interested in procedural schemas in mathematics. As a math educator, understanding procedural schemas is very important because solving math problems always involves a process. I found the article “Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning” fascinating as it explains the difference between different types of schemas (Powell & Fuchs, 2018). Two common strategies teachers use when teaching math word problems are teaching specific words to signal a mathematical operation and labeling a word problem as a specific operation (Powell & Fuchs, 2018). However, both of these strategies cause students not to have to mathematically reason what operation to use when solving a math problem (Powell & Fuchs, 2018). Additionally, the article points out that teaching that specific words signal a mathematical operation often causes even more misunderstandings since this strategy is not foolproof (Powell & Fuchs, 2018). The article explains that there are two effective teaching strategies. The two strategies are attack strategies and building schemas (Powell & Fuchs, 2018). The attack strategy “is an easy-to-remember series of steps students use to guide their approach to solving word problems” (Powell & Fuchs, 2018, p. 32). There are many variations of attack strategies; however, the steps usually include reading, identifying, and understanding the question and problem type (Powell & Fuchs, 2018). The article addresses two main mathematical schemas, additive and multiplicative schemas (Powell & Fuchs, 2018). The additive schemas include combine, compare, and change problems (refer to illustration 1) (Powell & Fuchs, 2018). The multiplicative schemas include equal groups, comparison, and proportions or ratios (refer to illustration 2) (Powell & Fuchs, 2018). All of the schemas are procedural schemas as they build students' knowledge of how to process and understand different mathematical problems. Additionally, the schemas often are used together to solve multi-step problems (Powell & Fuchs, 2018). Teachers must build students' schemas by modeling attack strategies and types of solution strategies, such as the combine strategy (Powell & Fuchs, 2018).
Illustration 1: (Powell & Fuchs, 2018)
Illustration 2: (Powell & Fuchs, 2018)
The book and article explain the importance of building procedural schemas. The book mentions using an activity called an anchor chart for students to understand the process of completing a math problem. The anchor chart allows the students to break the problem down into steps, including “understand (a visual of eyes reading); plan a strategy (visuals of models and symbols for possible operations); and show evidence (phrases telling what you did)” (Hess, 2023, chp 3, para 14). The article focuses on the importance of teaching students to use an attack strategy when completing word problems (Powell & Fuchs, 2018). With both sources acknowledging the effect of strategies to deepen understanding of a math problem, I understand I must teach and utilize these strategies with my students. I would use the RUN (read, underline, and name) attack strategy mentioned in the article since it utilizes a mnemonic device (Powell & Fuchs, 2018). Teachers must model strategies such as anchor charts and RUN before students can use them autonomously (Powell & Fuchs, 2018). Therefore, teachers should continuously use strategies throughout each unit (Hess, 2023; Powell & Fuchs, 2018). Once students feel comfortable with the strategy, they can deepen their understanding by making connections between mathematical concepts (Hess, 2023). The article allowed me to understand further the types of procedural schemas that teachers must help students build.
Teachers must address common misconceptions throughout their lessons. The book explains why students struggle with fractions when teachers explain that there are two numbers in a fraction instead of explaining it as being a single number with a numerator and a denominator (Hess, 2023). The article explains why students struggle with word problems when teachers teach that certain words equivalate to a mathematical operation (Powell & Fuchs, 2018). Therefore, teachers can not build schemas when misconceptions are roadblocks to forming connections. The teacher's responsibility is to anticipate misconceptions before the misconceptions become concrete understandings for students.
Both sources consistently reiterate that teachers need to aid students in building schemas to deepen their understanding. Schemas allow students to transfer knowledge to new concepts (Hess, 2023). Thus, teachers can create more rigorous opportunities for students to deepen their understanding by having students use multiple schemas to solve a problem (Hess, 2023). The more connections students can make, the easier they can apply prior knowledge to new content (Hess, 2023; Powell & Fuchs, 2018). Overall, “designing lessons that build schemas, that create opportunities for productive struggle, and that require students to transfer their skills and knowledge to new situations is what truly makes learning—and instruction—rigorous by design” (Hess, 2023, chp 1, para 20).
References
Development (ASCD).
Powell, S. R. & Fuchs, L. S. (2018), Effective word-problem instruction: Using schemas to
facilitate mathematical reasoning. TEACHING Exceptional Children, 51(1), 31-42.
https://doi.org/10.1177/0040059918777250
Hi Lucy,
ReplyDeleteI do not know much about becoming a math teacher, but I do know there have to be procedural schemas, as you said, in order to solve math problems. I must admit that seeing you explain some of these strategies makes me almost want to do math (but not quite..). I like the way you approached the “attack strategy,” as that's how I learned how to solve word problems that include reading, identifying, and understanding the question. Building these schemas also stacks student knowledge for different mathematical problems. The illustrations shown give a breakdown of example questions and graphic organizers that could be used to break off the problem. You also state that teachers must address misconceptions, which can relate to any content area. The examples of using fractions in the classroom is an excellent point of how confusing some things can be. You said it best, teachers cannot build schemas with roadblocks in the way of instruction. Your post was insightful, and I look forward to hearing from you more. Great work.
Really well done, Lucy. You show some very strong connections. I wonder if you think that we may sometimes focus too much on procedural schema, as it seems that math educators might fall into the trap of sharing "steps" of a process for problem solving but not necessarily focusing on the concepts?
ReplyDeleteI loved the depth of your post. Your thoroughly explained the importance of building schemas and how they can help the students grow and apply the procedures to other concepts. You were able to identify that teachers need to teach and continuously teach and support the schemas for them to be effective to students. I think that it is so important for the teacher to explain to the students what the schema is and practice it with them. I wonder what content is most effectively taught by using schemas. I plan to implement schemas when I become a teacher myself as they will help deepen my students' knowledge and critical thinking.
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