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Concept Map over Dr. Wathall's Concept Based Mathematics Book Chapters 1 & 2

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Chapter 1: Why Is It Important for 
My Students to Learn 
Conceptually?

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Chapter 1: Why Is It Important for 
My Students to Learn 
Conceptually?

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Chapter 2: What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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Chapter 2: What Are the Levels of the Structures of Knowledge and Process for Mathematics?

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According to Wathall, The Structure of Knowledge is a graphical representation of the relationship between the topics and facts, the concepts that are drawn from the content under study, and the generalization and principles that express conceptual relationships. The Structure of Process is a graphical representation of the relationship between the processes, strategies, skills and concepts, generalizations, and principles in process-driven disciplines. Applying these structures in teaching can help students not only know just the facts but help them by analyzing concepts, transferring them into generalizations and allowing to connect them to deeper understanding. Both of these structures are needed as the foundation for understanding conceptual knowledge and procedural skills in mathematics.

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Many educators around the world are familiar with H. Lynn Erickson's Structure of Knowledge. More recently, people are discovering the Structure of Process. What is the difference? Why is it important to know each Structure as a Concept-Based teacher? Let's look first at the two Structures side-by-side.

The graphics above represent the most essential components of Concept-Based Curriculum and Instruction (CBCI) and the relationships among them, explained more in Concept-Based Curriculum and Instruction for the Thinking Classroom, 2nd edition. A conceptually organized curriculum departs from traditional curriculum, which focuses on topics, skills, and facts. CBCI brings in another dimension-an intentional focus on the transfer of learning. Transfer occurs at the conceptual level of understanding, so why not design curriculum and instruction to support this goal? Erickson (2008) points out that conceptual structure for curriculum is important because conceptual understanding requires content knowledge but the reverse is not necessarily true.

Considering the two graphics, see how the Structure of Knowledge more naturally depicts elements of subject areas that are heavy in content knowledge (topics and facts)? These subjects include social studies and science. CBCI curriculum writers learn how to draw concepts out of a unit topic and use them to write transferable, conceptual understandings (generalizations). These unit generalizations become learning targets for instruction, along with the essential, supporting facts (and skills). The Structure of Process more naturally depicts the elements of subject areas that are organized by processes, strategies, and skills. If you examine the standards of process-heavy disciplines, such as English language arts, music, fine arts, and world language, you find the standards in these subject areas primarily identify what students are expected to be able to do vs. the topics and facts students are expected to know.

When teachers begin considering the concepts and generalizations to include in an English language arts curriculum, the dialogue moves beyond a list of books and skills to cover to the consideration of the essential generalizations we want to make sure students understand by the end of the unit. For example, the following generalization might be found in an English unit:
"Audience, purpose, situation or cultural norms can confine a writer's authentic voice."
When teachers guide students to this conceptual understanding, students become more capable of crafting, recognizing, appreciating, and evaluating an author's voice across different examples of their own writing and that of others. This raises the bar for teaching and learning in process disciplines!

Both the Structure of Knowledge and the Structure of Process illustrate moving curriculum and instruction from the lower cognitive levels (facts, skills) to a level of transferable understanding (generalizations). We need to consider unit generalizations in terms of the two Structures when designing curriculum in any subject area. Clearly, a more content-heavy (Structure of Knowledge) discipline, like social studies, also requires that students understand key processes (such as the research process). A more process-oriented (Structure of Process) discipline, like art, also requires students to understand key knowledg e about the culture embedded in the work being studied (such as Indigenous arts). There is definitely an interplay between the two Structures.

At the same time, depending on the core subject of the curriculum unit, one Structure requires more attention than the other so that the integrity of the discipline is not compromised. For example, if an English language arts or a world language curriculum unit is primarily populated with generalizations about text or culture (that serve as the context for learning), essential understandings about how language works are being ignored.

Let's take another look. The next three generalizations are part of a 3 rd grade unit titled: Mythical Characters. The title of the unit anchors the unit of study in a context. Throughout this 8-week unit, students read, respond to, critique, and write several fairy tales, folk tales, and fables. One generalization in this unit is drawn from the Structure of Knowledge-seen in the forefront of the graphic. This conceptual understanding represents a transferable idea relevant to all the books within the unit.

However, since this is an English language arts unit where the intent of the discipline is to teach students how to be capable readers, writers, speakers, viewers, and listeners, most of the unit's remaining 4-6 generalizations will reflect important understandings of the processes, strategies, and skills students are expected to understand. See two such examples below:

Notice that the Structure of Knowledge now takes a backseat to the Structure of Process. This is often the missing piece of CBCI in process disciplines. There may be wonderful, conceptual curriculum and instruction around content but the integrity of a process discipline falls back to the practice of learning skills by doing without raising the bar to the level of transferable, conceptual understanding. Becoming a master Concept-Based teacher requires using the Structures of Knowledge and Process to appropriately guide curriculum writing and instruction, knowing how the two Structures work together and why they are separate and important in their own right.

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author: Jeffrey D. Wilhelm, Michael W. Smith, Hugh Kesson, and Deborah Appleman

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What You Need to Know About the Structure of Process - Corwin Connect

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Many educators around the world are familiar with H. Lynn Erickon's Structure of Knowledge. More recently, people are discovering the Structure of Process.

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How to Help Students Build Deep Understanding of Math Concepts - Blog by Alyssa Buccella on Four effective ways to build Conceptual Knowledge in a Math Classroom

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Knowledge & Process Structures Application in Teaching

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Knowledge & Process Structures Application in Teaching

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When teaching students Math, Its important for students to build the foundation on how to understand mathematical concepts whether it be through direct or indirect instruction. Wathall reviews how Two-Dimensional Model of Instructions focuses on facts and content of the subject where as Three Dimensional Model of Instruction includes what the two dimensional model offers but encourages students conceptual thinking and allows them to fully understand effectively through the use of inquiry.

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How to Help Students Build Deep Understanding of Math Concepts

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It’s possible to help students build lasting knowledge of math concepts as well as procedural fluency. Armed with both, students can become confident and proficient with math inside and outside the classroom.

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The Levels of the Structure of Knowledge

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The Levels of the Structure of Knowledge

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There are four levels of the Structure of Knowledge:

1. The Factual Level

Basic Level of Structure of Knowledge

Base around Rote Memorization

Include "Specific Examples" (Facts)

2. The Topic/Conceptual Level (Conceptual)

Understanding Theories, Structures, and Models

Knowing Basic Elements of a Larger Structure

3. The Procedural Level

The "How to" Processing Level

Making Connections to what they know and how they can interpret the information

Practice!

4. The Metacognitive Level

Top Level - Usually not Acknowledged or Achieved

Metacognitive knowledge can be understood as (1) strategic knowledge, (2) knowledge about cognitive tasks (i.e. contextual, conditional), and (3) self-knowledge

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Dear companion,

Do you know the four knowledge types?

You should if you are interested in knowing how to close knowledge-based performance gaps in any area of life. According to Krathwohl (2002), knowledge can be categorized into four types: (1) factual knowledge, (2) conceptual knowledge, (3) procedural knowledge, and (4) metacognitive knowledge. It's important to know the distinctions and to understand your own knowledge strengths and areas of need to better meet your personal and/or organization's goals.
Factual Knowledge
You can define factual knowledge simply as the terminologies, specific details, and basic elements within any domain. This is the information that can and must be learned through exposure, repetition, and commitment to memory. Luckily, since our memories are not the best places to store facts, we can help ourselves by knowing where to access factual knowledge when we need it (i.e. where to find the information in our books, online, our notebooks or journals, or asking that person who you know knows it!).

It is common knowledge that to be successful in meeting a goal, you need to know the related "facts". A salesman better know the facts about the product or service he is selling! The CEO better know "the facts" about his core business if he or she wants to have credibility. A school principal better know "the facts" about good teaching methodology and pedagogy. How else can he or she be an instructional leader?
Conceptual Knowledge
Related to factual knowledge, conceptual knowledge can be understood as knowing the interrelationships and/or functions among the details and elements that make up a larger structure. This definition includes (1) knowing information classification and categorization, (2) knowing principles and generalizations, and (3) knowing theories, models, and structures. Basically, conceptual knowledge is knowing that facts can be organized in meaningful ways. Taking the example of a business marketer, it is not enough to know the details of his or her products or services and that of the competition. There must be conceptual knowledge of the differences and the meaningful competitive advantage of one over another.
Procedural Knowledge
This knowledge type is critical for success in goal attainment because it puts the "what" into action through the "how" process. Procedural knowledge can be understood as knowledge of (1) subject-specific skills and algorithms, (2) subject-specific techniques and methods, and (3) criteria for deciding when to use the right procedures. Many times, we see others performing wonderfully, and we ask ourselves: How do they do it? We can read their books or watch their videos to learn the needed factual and conceptual knowledge, however, knowing "how to" put that declarative knowledge into practice requires... practice! When you cognitively know "how to" do something, then you need to physically try it and pay close attention to both the process and outcome. If you are listening to your body, your mind, and your gut (using all your senses), you will gain information through multiple feedback loops, and those loops of information will guide your analysis and future actions in becoming better at "how to" do it.
Metacognitive Knowledge
This is probably the least paid-attention-to knowledge type because sometimes it feels uncomfortable to reflect on what is happening inside your world. We fear what we might find. Metacognitive knowledge can be understood as (1) strategic knowledge, (2) knowledge about cognitive tasks (i.e. contextual, conditional), and (3) self-knowledge. Because people are complex, and groups of people only add to the dynamic of complexity within a system, having a good measure of metacognitive knowledge (that is, engaging in this type of thinking) is critical to your performance, well-being, and success. For example, if you are meeting a client who shares vastly different cultural values and ways of knowing than you, then it behooves you to be paying attention to contextual clues. Like a dance, you move together, in sync, and there is no way of knowing beforehand what the next step is going to be! You must be aware of yourself, the person as he or she moves and speaks, and the situation as it unfolds. You bet that you better be listening and making the most of your information inputs.

If you have any goals in your personal or work life, pay attention to your knowledge needs- it will help you to increase success and goal attainment. Answer the question: Am I lacking factual, conceptual, procedural, or metacognitive knowledge? By addressing gaps in knowledge, you will be on your way to greater learning and success with your goals.

Your friend in learning,

Dr. Joel Tapia

-
Reference
Krathwohl, D. R. (2002). A revision of Bloom's taxonomy: An overview. Theory into practice, 41(4), 212-218.

Joel Tapia is passionate about learning, strategy, and helping others to succeed. View all posts by Dr. Joel Tapia

Published

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Dear companion, Do you know the four knowledge types? You should if you are interested in knowing how to close knowledge-based performance gaps in any area of life. According to Krathwohl (2002), knowledge can be categorized into four types: (1) factual knowledge, (2) conceptual knowledge, (3) procedural knowledge, and (4) metacognitive knowledge.

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The Levels of Strucrues of Processes include:

Organized by Strategies, Skills, and Mathematical Processes

Encourages Higher Thinking and thinking of other concepts or ideas by considering concepts and making generalizations

Fully Understanding Material and Applying it across different platforms

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Many educators around the world are familiar with H. Lynn Erickson's Structure of Knowledge. More recently, people are discovering the Structure of Process. What is the difference? Why is it important to know each Structure as a Concept-Based teacher? Let's look first at the two Structures side-by-side.

The graphics above represent the most essential components of Concept-Based Curriculum and Instruction (CBCI) and the relationships among them, explained more in Concept-Based Curriculum and Instruction for the Thinking Classroom, 2nd edition. A conceptually organized curriculum departs from traditional curriculum, which focuses on topics, skills, and facts. CBCI brings in another dimension-an intentional focus on the transfer of learning. Transfer occurs at the conceptual level of understanding, so why not design curriculum and instruction to support this goal? Erickson (2008) points out that conceptual structure for curriculum is important because conceptual understanding requires content knowledge but the reverse is not necessarily true.

Considering the two graphics, see how the Structure of Knowledge more naturally depicts elements of subject areas that are heavy in content knowledge (topics and facts)? These subjects include social studies and science. CBCI curriculum writers learn how to draw concepts out of a unit topic and use them to write transferable, conceptual understandings (generalizations). These unit generalizations become learning targets for instruction, along with the essential, supporting facts (and skills). The Structure of Process more naturally depicts the elements of subject areas that are organized by processes, strategies, and skills. If you examine the standards of process-heavy disciplines, such as English language arts, music, fine arts, and world language, you find the standards in these subject areas primarily identify what students are expected to be able to do vs. the topics and facts students are expected to know.

When teachers begin considering the concepts and generalizations to include in an English language arts curriculum, the dialogue moves beyond a list of books and skills to cover to the consideration of the essential generalizations we want to make sure students understand by the end of the unit. For example, the following generalization might be found in an English unit:
"Audience, purpose, situation or cultural norms can confine a writer's authentic voice."
When teachers guide students to this conceptual understanding, students become more capable of crafting, recognizing, appreciating, and evaluating an author's voice across different examples of their own writing and that of others. This raises the bar for teaching and learning in process disciplines!

Both the Structure of Knowledge and the Structure of Process illustrate moving curriculum and instruction from the lower cognitive levels (facts, skills) to a level of transferable understanding (generalizations). We need to consider unit generalizations in terms of the two Structures when designing curriculum in any subject area. Clearly, a more content-heavy (Structure of Knowledge) discipline, like social studies, also requires that students understand key processes (such as the research process). A more process-oriented (Structure of Process) discipline, like art, also requires students to understand key knowledg e about the culture embedded in the work being studied (such as Indigenous arts). There is definitely an interplay between the two Structures.

At the same time, depending on the core subject of the curriculum unit, one Structure requires more attention than the other so that the integrity of the discipline is not compromised. For example, if an English language arts or a world language curriculum unit is primarily populated with generalizations about text or culture (that serve as the context for learning), essential understandings about how language works are being ignored.

Let's take another look. The next three generalizations are part of a 3 rd grade unit titled: Mythical Characters. The title of the unit anchors the unit of study in a context. Throughout this 8-week unit, students read, respond to, critique, and write several fairy tales, folk tales, and fables. One generalization in this unit is drawn from the Structure of Knowledge-seen in the forefront of the graphic. This conceptual understanding represents a transferable idea relevant to all the books within the unit.

However, since this is an English language arts unit where the intent of the discipline is to teach students how to be capable readers, writers, speakers, viewers, and listeners, most of the unit's remaining 4-6 generalizations will reflect important understandings of the processes, strategies, and skills students are expected to understand. See two such examples below:

Notice that the Structure of Knowledge now takes a backseat to the Structure of Process. This is often the missing piece of CBCI in process disciplines. There may be wonderful, conceptual curriculum and instruction around content but the integrity of a process discipline falls back to the practice of learning skills by doing without raising the bar to the level of transferable, conceptual understanding. Becoming a master Concept-Based teacher requires using the Structures of Knowledge and Process to appropriately guide curriculum writing and instruction, knowing how the two Structures work together and why they are separate and important in their own right.

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author: Jeffrey D. Wilhelm, Michael W. Smith, Hugh Kesson, and Deborah Appleman

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What You Need to Know About the Structure of Process - Corwin Connect

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Many educators around the world are familiar with H. Lynn Erickon's Structure of Knowledge. More recently, people are discovering the Structure of Process.

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Macro, Meso, and Micro Concepts in Mathematics

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Macro, Meso, and Micro Concepts in Mathematics

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Concepts in Mathematics have to meet the following criteria:

• Timeless

• Universal

• Transferable

• Abstract to varying degrees

• One or two words, or a short phrase

• Have different examples that share common attributes (Erickson, 2007)

Macro Concepts - Broader Concepts that emcompasses a huge part of knowledge

Meso Concepts and Micro Concepts - More specfic levels that provide depth of understanding

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Learning analytics is "the measurement, collection, analysis and reporting of data about learners and their contexts, for purposes of understanding and optimizing learning and the environments in which it occurs". It has emerged to understand learning data, and how we could do interventions to improve the learning systems (i.e. MOOCs, LMS,...etc.)

Since learning analytics covers a wide range of analytics, Simon Buckingham Shum discussed the three levels of micro, meso, and macro of learning analytics as follows:

1. Macro-Level: enables analytics within the cross-institutional level. Macro-analytics can become more increasingly real-time and involves more fine data from the meso and micro levels. It is also important to note that at the macro level, learning analytics can be incorporated in non-educational sectors such as the government sectors.

2. Meso-Level: enables analytics at the institutional level. At the meso level, partners from the institution are involved like the faculty and the support staff. Simon Buckingham Shum stated that the business intelligence imperative to optimize processes to build better meso levels of analytics (ie. academic analytics).

3. Micro-Level: Might be the most popular level in learning analytics. it operates at the interpretation and tracking of individual/learner data. At this level, student success is strongly related where click-stream data are collected, analyzed, and translated to improve learning and increase the success of students.

According to Shum, the breadth and depth at the macro and meso levels add power to micro-analytics. Aggregation of millions of learners' interaction data creates a solid meso and macro levels. Therefore, effective learning analytics demands mutual enrichments between the three layers of analytics.

References:
http://simon.buckinghamshum.net/wp-content/uploads/2012/10/UNESCOIITE-LearningAnalytics.v4.pdf

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The Micro, Meso, and Macro of Learning Analytics

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Mohammad Khalil Learning analytics is "the measurement, collection, analysis and reporting of data about learners and their contexts, for purposes of understanding and optimizing learning and the environments in which it occurs".

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How To Assess Three-Dimensional Learning 

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Two-Dimensional Curriculum Model:

"Traditional View of Instruction", Focuses on knowing the information rather than deep understanding

Focuses on bottom tier levels of Structure of Knowledge and Processes

Starts by generalizations followed by direct instruction

Emphasis on Skills and Algorithims (Little Conceptual Thinking)

Three-Dimensional Curriculum Model:

More engaged view of the two-dimensional model but has a "third" stage where Generalizations are revisited and enduce conceptual thinking

Focuses on making concepts, generalizations, and principles to ensure that curriculum and instruction focus on intellectual depth, the transfer of understanding, and the development of conceptual brain schemata.

More difficult to generate into curriculum and assess on students than the two-dimensional model

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How To Assess Three-Dimensional Learning

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Explore three-dimensional learning assessments in science, how early assessment tasks were developed, and resources available for evaluating assessments.

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In order to have students develop the technique to critically think and think conceptually when learning, Teachers need to instill techniques such as inquiry thinking. Inquiry thinking allow students to think past what information their given by posing questions, problems, or scenarios and allowing them to navigate themselves to the truth of the correct answer.

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Inquiry-based learning is more than asking a student what he or she wants to know. It's about triggering curiosity. And activating a student's curiosity is, I would argue, a far more important and complex goal than mere information delivery.

Despite its complexity, inquiry-based learning can be easier on teachers, partly because it transfers some responsibilities from teachers to students, but mostly because releasing authority engages students.

Teachers who use inquiry-based learning combat the "dunno"-a chronic problem in student engagement.

When you ask a student something like, "What do you want to know about _____?" you're often met with a shrug or "dunno." Inquiry-based learning, if front-loaded well, generates such excitement in students that neurons begin to fire, curiosity is triggered, and they can't wait to become experts in answering their own questions.

What inquiry-based teachers do isn't easy at all; it's just hidden, and some people confuse the two. Teachers hide the strategies they use to encourage inquiry, and the students develop their own skills as content-area experts.

Learning Something New

Triggering inquiry is about learning something new, and triggering curiosity is no small feat. It takes modeling enthusiasm, and learning something new generates our own enthusiasm, even if it's something new about the content we've covered for years.

Let's say you're clicking through your Twitter or Facebook feed and you stumble on a link in your content area. You realize it's a new factoid, a new perspective on an age-old topic. Maybe it's a new TED talk or graph with statistics, something that makes a concept more concrete. Maybe it's an infographic or a photo, something that startles you and leads to furrow your brow and say, "Whaaa?!"

By the way, I think one of the reasons why the whole world seems to be losing its mind over the Broadway production of Hamilton is because it presents a fresh take on a story we've all heard before. The power of learning something new is undeniable.

You have to bring that "whaaa?!" into your classroom. You have to model your own curiosity quotient-that hunger to learn that defines how we advance our knowledge of the world. According to the Harvard Business Review, a higher curiosity quotient can indicate more flexibility and help build a greater ability to handle complexity.

So think about your content area. What is a new take on a topic that you can bring to your classroom? What new piece of information might help you trigger your own enthusiasm that can then trigger your students' curiosity?

The 4 Steps of Inquiry-Based Learning

So you've discovered something that generates your own inquiry, and you've recreated that moment for your students when your curiosity was triggered. So what comes next in inquiry-based learning? This can be answered in four basic steps that should represent the outline of a simple unit.

1. Students develop questions that they are hungry to answer. Have them develop a problem statement that requires them to pitch their question using a constructed response, further inquiry, and citation.

2. Research the topic using time in class. It's crucial to have some of this be classwork so students have access to the head researcher in the room-you. You aren't going to do the work for them, but you are going to guide them and model methods of researching reliably.

3. Have students present what they've learned. Students should create and present a culminating artifact. When I have my students present what they've learned, I use a rubric with "Able to Teach" as the acme of what to reach for. After all, many people can understand content, but can they communicate it? Students can develop a website using Weebly, or perhaps a slideshow using Google Slides.

4. Ask students to reflect on what worked about the process and what didn't. Reflection is key. And it isn't just about asking them to think back on their opinion of the topic. It's about reflecting on the process itself. That's where you can work in metacognition-thinking about thinking. Have students focus on how they learned in addition to whatthey learned.

In terms of your content area, imagine a classroom where different kids are presenting their findings on a single, simple aspect of the content. You'd have a classroom that, overall, learns deeper and wider than ever before.

In terms of student achievement, the power of their question should help drive the research, the writing, and the presentation. It should help motivate them to become experts in their self-described field. And the more often a student gets a taste of what it feels like to be an expert, in however small a concept, the more they will want that feeling later on in life.

It all starts with finding your own enthusiasm, your own excitement, and your own curiosity. Do that, and you'll be heading toward a classroom built on inquiry.

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What the Heck Is Inquiry-Based Learning?

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What the Heck Is Inquiry-Based Learning?

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What is Conceptual Learning?

Conceptual Learning is approach where one learn through getting little bits of information and relying on concepts and generalization to learn. This promotes critical thinking and a deep understanding at the topic at hand. According to UTeach, Conceptual thinking allows students to see the "Big Picture" and understand deeper meanings.

Why is Conceptual Learning Beneficial?

Conceptual Learning is Beneficial for students because it allows students to apply concepts and generalizations in order to learn instead of using a direct way to learn. This promotes the student to critically think and encourage deep understanding to learn. This method of teaching can be beneficial to teachers as it allows their students to stay engage with the curriculum and builds life skills in the students.