Instructional Design Document (IDD)

Mini-course

Course Title: Fractions Operations for 

Elementary School Math Educators

UPDATE: 9/19/2024

Because the scope of this project is larger than I anticipated, the course will be built with only addition and subtraction at this time. Course and module objectives have been revised accordingly. The information for the entire intended course will remain in this IDD until the rest of the operations are built.

  The revised course title is:

Fraction Addition and Subtraction for Mathematics Educators

Overview

Many mathematics educators struggle to understand and explain the underlying mathematics for the 4 basic operations with fractions (addition, subtraction, multiplication and division). They can usually explain the steps in the procedures or algorithms, but often do not know or can explain why, mathematically speaking, these procedures and algorithms work. This course serves to deepen educator's personal knowledge of fractions operations so that they might better teach their students.  

Knowledge Gap

When an educator's fractions content knowledge has gaps, it can be difficult for them to help their students create effective models and develop a strong conceptual understanding of the mathematics. It can be difficult for the educator to lead effective classroom discussions or field student questions or validate student thinking. It can also be difficult to know if a student has produced a correct answer and determine why and where a misconception lies. In her book, Knowing and Teaching Elementary Mathematics (1999), Liping Ma describes what she calls PUFM: Profound Understanding of Fundamental Mathematics which she describes as, "an understanding of the terrain of fundamental mathematics that is deep, broad, and thorough." (p. 120). She goes on to describe that a teacher with PUFM understands the underlying connections among math topics, appreciate multiple perspectives and solution paths, are aware of the simple but powerful basic concepts and principles of math, and have a fundamental understanding of the whole elementary math curriculum. (p. 122).  This mini-course serves to bring educators closer to achieving PUFM by examining the underlying mathematics behind the algorithms for fractions operations. It may challenge a teacher's content knowledge at times because it will guide them to explore mathematics at a deeper level while also helping them connect to content beyond the grades they teach. 

Target Audience & Learner Profile

The target audience for this mini-course is mathematics educators who specifically teach 3rd, 4th, and/or 5th grade mathematics and are responsible for teaching to standards about fractions operations. Teachers may be from any part of the United States or Canada. Many have had teacher preparation courses in college for teaching general math, but it's unlikely that a university program provided in-depth instruction on fractions operations. This course will be self-selected by educators who are interested in this topic or are aware of their own knowledge gaps. Because this course will be asynchronous, I won't have many specific details about my audience until the feedback and surveys come back from the course.  When surveys come back, I will have more information about the demographics of educators who are taking this course. 

Course Type-- Problem Solving 

Educators will learn the underlying mathematics behind fractions operations through a problem-solving based approach that leads them to see why the algorithms and step-by-step processes work and how certain models can support understanding. While the focus of this course is largely on the mathematics itself, the course will also address how the educator can implement their learning in the classroom.  Because this course will challenge educators' content knowledge, I will need to be cognizant of information and cognitive overload and ensure that the most challenging content is chunked appropriately and has an appropriate trajectory that will keep learners engaged. Adult learning principles, especially those that honor teachers' prior knowledge and experiences, must be baked into the design of this course for maximum effectiveness.

Course Modality

The most appropriate format of this mini-course is a largely online asynchronous work with a recommended follow up with classroom-based instruction in the form of instructional coaching, which can be either virtual or in-person. The asynchronous component, ideally housed on a Learning Management System (LMS), will give educators an opportunity to learn and explore the content and personally construct meaning. The instructional coaching will encourage educators to try something they learned in the asynchronous space and then debrief it with the instructional coach and/or grade level colleagues with the guidance of the instructional coach. This format will provide educators with access to an expert around any content that they might still find confusing or difficult to teach after they have completed the asynchronous work.  

This course format supports accessibility in a variety of ways. WCAG compliance criteria will be addressed at least level 2. Videos will be equipped with closed captioning and transcripts. Images will have well-written alt text attached and audio descriptions when it is a better option than alt text. Using comprehensive Course Authoring Software (CAS), the design will accommodate those using a keyboard to navigate by using such features as "click to place" instead of "drag and drop" that requires a mouse or trackpad. Instructional coaching will assist educators by providing them with a live mathematics education expert who will guide them through implementation in the classroom, provide ongoing support and training, and support their understanding when they feel challenged. 

Subject Matter Expert/Resources

• I will reach out to my friend, Danielle, who is a mathematician for our company to ask for her input on my course outline, learning objectives, and suggestions for models.

• Elementary Mathematics for Teachers by Thomas H. Parker and Scott J. Baldridge. One of the authors of this book is a former mathematician for our company. 

• Young Mathematicians at Work: Constructing Fractions, Decimals and Percents by Catherine Twomey Fosnot and Maarten Dolk. This book is s strong resource for models and pedagogy with fractions.

• Uncomplicating Fractions to Meet Common Core Standards in Math, K-7 by Marian Small. This is another strong resource for models and pedagogy with fractions.

• https://greatminds.org/eurekamathsquared. This is where my company's math curriculum is housed. For proprietary reasons, I cannot put more information about it in this forum, but I will review fractions operations lessons that are in our curriculum to help me with models and pedagogy. 

• https://mathematicalmusings.org/wp-content/uploads/2023/05/Progressions.pdf. This document is.  compilation of the Progressions Documents that explain the mathematics behind the Common Core Standards. They include models that help support the underlying mathematical concepts behind fractions operations and show the trajectory of learning across multiple grade levels including middle school and high school math courses. 

• Coherence maps such as https://tools.achievethecore.org/coherence-map/ and https://jeffbaumes.github.io/standards/# will help me provide coherent and accurate connections within and across grade level content for fractions operations. 

Instructional Design Model

My early opinion is that I would like to use SAM as the design model for this learning opportunity. I think that the iterative, cyclical nature of SAM would allow for the most comprehensive input from mathematicians, math auditors, and writing editors to ensure that the mathematical concepts and the pedagogical suggestions for educators are sound while writing editors will help me ensure all communication is clear. While I have quite a bit of content knowledge on this subject matter, there are still nuances or lenses I might not have considered that SMEs will, especially when it comes to connecting to higher levels of math. It would also help me to have multiple rounds of  stakeholder input on learning activities to ensure I design the learning in creative and useful ways that meet the learning objectives. Often, stakeholders have valuable input on adjusting a learning activity to make it stronger, more creative, or more effective. For example, I would develop a rough draft of the content then ask for SME input. Based on that input I might re-order content, remove content, or expand sections to make the learning more robust. A SME would also help me make sure that the math is correct and the trajectory of the learning is sequenced correctly to avoid inadvertent misconceptions or create learning gaps. 

UPDATE: The model I ended up using was a Rapid Design model. 

• Content was carefully analyzed with the help of  SMEs to set clear performance-based learning objectives. In this case, I acted as the SME based on my past experience and credentials with this content.

• Strategies were decided based on the content and audience. They are specifically designed for upper elementary classroom teachers.

• Adult learning principles (andragogy) have heavily guided the content, design, and assessment creation.

Key Stages I used:

• • Analysis and Needs Assessment:  My background in instructional coaching and delivering this session in an in-person, full day workshop guided the needs and analysis. 

  • Design and Planning: High-level design of the overall structure, content organization, and instructional approach, prioritizing learner engagement and meaningful interactions while aligning to learning objectives. This is detailed in the learning activities section of this IDD.

  • Content Development: Created engaging and relevant resources, which can include modules, videos, simulations, or microlearning opportunities. Every activity in this course directly ties back to the course and module learning obejctives.

  • Prototype and Testing: There is typically a rapid prototyping and testing stage. Designers create a prototype or a small-scale version to gather feedback from stakeholders which allows for quick adjustments and improvements. Submission to IDX400 serves as the opportunity for feedbck. For future iterations, I would like to get feedback from mathematicians as well.

  • Deployment and Iteration: Rapid instructional embraces an iterative approach. Designers collect feedback, monitor progress, and make ongoing edits to the content to remain responsive to learner needs. Based on user feedback, this course should be updated periodically.

  • Evaluation and Maintenance: Continuous evaluation is important to rapid design. Designers assess the effectiveness of  learning materials. The last section of the course provides Kirkpatrick model levels 1 and 2 evaluation opportunities.

Learning Theories

Learning theories that would most effectively drive the design of the learning activities and would be most helpful include:

• Andragogy: Adults learn differently than children. They want to know, "Why am I learning this?" and "How will this relate to or benefit my life or career?" Adults are looking for relevance right now.  Deepening their content knowledge of fractions operations will help them better teach fractions operations to students. Adult learning theory must be considered when designing this mini course. Activities must be engaging for adults, relevant to classroom practice, and leave the adult learner with new understandings that they find exciting or motivating to apply to their teaching. The course cannot just merely focus on teaching educators about fractions operations. The course must also provide direct application and practice to a grades 3-5 classroom. 

• Cognitivism and Constructivism: Constructivism hinges on the idea that one is constantly trying to make meaning of new information and relate it to past understandings. Cognitivism focuses on thinking and how humans process, organize, and make sense of new information.  Teachers need to be able to construct their knowledge of fractions operations based on what they already know. A mini-course must consider the knowledge an adult educator already possesses and honor it. Teachers will also need opportunities to construct new learning through digital interactions that are carefully sequenced in thoughtful ways to demonstrate the underlying mathematics that they may not have learned when they were students. Because educators need opportunities to construct and organize their thinking, opportunities for reflection must be built into the course. 

• Connectivism: Collective connections between people in a network create new forms of knowledge. Knowledge is created based on the group's collective knowledge, which is constantly shifting and changing. Knowledge in connectivism also flows across networks that are inter-connected with other networks. Teachers need opportunities to connect with other educators around the content. It can often be difficult for a teacher to be vulnerable and admit that the mathematics content may be challenging for them. Providing instructional coaching as well as discussion boards and networks comprised of other teachers at the same grade levels will provide teachers with a forum ask each other questions, and read posts and responses on discussion boards that might spark new understandings. 

• Experiential Learning Model: This mini course will provide an experiential learning opportunity. Educators will engage with a series of videos, digital interactions, professional readings, community discussion threads, and a live instructional coach to provide a well-rounded learning opportunity. In addition, a gamification element will be added to keep adult learners motivated by earning badges or points for completing tasks and encourage them to engage in beneficial optional tasks such as discussion boards. Gamificiation could also be tied to the assessment components as incentive for the educator to complete assessment tasks. The assessment tasks can be formative and summative and assist both the educator and an instructional coach for future "next steps."