LEARNING DESIGN BLUEPRINT
Topic: Teaching fraction concepts to early learners by addressing misconceptions, anxiety and motivation
1. Overview
Description of the topic
For our Learning Design Project, I, Yiran, Jingyun and Isaac chose to focus on teaching fraction concepts to early learners by addressing misconceptions, anxiety and motivation. Fractions are a foundational concept in mathematics all over the world, yet many students struggle to understand them. According to Infinity’s article, Learning Fractions: A Scoping Review, there are three factors that contribute to difficulties with basic fraction concepts: ontogenic, didactical and epistemological obstacles. Ontogenic are issues related to students’ level of readiness and motivation towards learning. Didactical are problems through instructional design and teaching methods and epistemological are misunderstandings regarding fraction concepts (Suryadi et al., 2024). Success with fractions is often a tell-tale sign of success later in more challenging notions as they align with skills needed to solve ratios and algebraic equations. Without a strong foundation of fractions, difficulties in math can persist through higher grade levels and post secondary schooling, leading to increased anxiety and a decreased motivation to learn.
Part of the reason that fractions are difficult for younger learners to grasp is because they require students to think past everything being simply a whole number. The article by Science Direct, the natural number bias and magnitude representation in fraction comparison by expert mathematicians, discusses the natural number bias, which is a tendency for learners to apply the same rules to fractions as whole numbers. For example, students may believe that 1/4 is larger than 1/3 because 4 is larger than 3 (Halme et al., 2024). This type of misunderstanding indicates that students see fractions as two sets of whole numbers instead of one quantity that represents a part of a whole number. Along with misunderstandings with fractions, emotional factors can play a role in a student’s ability to learn. From the article math anxiety differentially impairs symbolic, but not nonsymbolic, fraction skills across development by The New York Academy of Sciences, anxiety towards mathematics can significantly impact students ability to perform operations (Starling-Alves et al., 2021). Learners who experience anxiety tend to perform worse on assignments and during tests, resulting in a cycle of anxiety and lack of motivation to learn. Some students may not realize that they are misunderstanding fractions which can lead to them not correcting their thinking.
Because of these challenges, it is important to not only teach fractions for conceptual understanding but to teach with a consideration of emotional engagement. From experience, using visual models, real-life examples and hands-on interactive activities can significantly increase student engagement and motivation which can improve understanding. By building conceptual learning and not simply having students memorize facts, we as teachers can reduce anxiety students have towards fractions and start building their confidence.
Misconceptions
Misconception 1: Treating fractions as two separate whole numbers
One common misconception among early learners is viewing fractions as two separate whole numbers rather than as a single quantity representing a part, whole relationship. Influenced by natural number bias, students often apply whole-number reasoning to fractions, assuming that larger numerators or denominators automatically indicate larger values. For example, learners may believe that one-quarter is larger than one-third simply because 4 is greater than 3.
This misunderstanding suggests that students are focusing on the symbolic components of fractions without grasping their underlying meaning. When fractions are perceived as disconnected numbers instead of representations of relative magnitude, learners struggle to compare, estimate, and operate on fractions meaningfully. If unaddressed, this misconception can persist and interfere with later mathematical concepts such as ratios, proportional reasoning, and algebraic thinking.
Misconception 2: Believing that difficulty with fractions reflects a lack of ability
Another significant misconception is the belief that struggling with fractions indicates a lack of mathematical ability. For some learners, repeated confusion or errors with fractions contribute to math anxiety and a reduced sense of confidence. As a result, students may disengage from learning tasks, avoid participation, or rely on rote procedures without understanding.
This misconception is particularly problematic because it frames difficulty as a personal failure rather than as a normal part of learning a concept that requires conceptual restructuring. When learners internalize this belief, anxiety can further impair performance, creating a cycle of misunderstanding, low confidence, and decreased motivation. Addressing this misconception is essential for supporting persistence and helping students view mistakes as opportunities for learning rather than evidence of inability.
Rationale
Our group intentionally chose to focus on fraction learning because it sits at the intersection of educational practice, learner cognition, and data-informed understanding of misconceptions. Our group has members in elementary education, statistics, and math; therefore, our topic of teaching fraction concepts to younger learners fit well with our future careers and is pedagogically foundational and widely associated with persistent learning difficulties.
Fractions are a critical early mathematical concept, yet they are also an area where misconceptions, anxiety, and disengagement frequently emerge. By examining fraction learning through both instructional and analytical perspectives, we aimed to design a learning experience that addresses not only conceptual understanding, but also learner motivation and confidence.
The rationale for this learning design is grounded in the need to address both conceptual misunderstandings and affective factors that influence early learners’ experiences with fractions. As identified above, misconceptions related to natural number bias and negative beliefs about mathematical ability can significantly hinder students’ understanding, confidence, and motivation. Simply providing direct explanations or procedural practice is often insufficient to challenge these deeply rooted ways of thinking.
To effectively address these misconceptions, learning experiences must create opportunities for learners to actively engage with fraction concepts, articulate their reasoning, and encounter alternative perspectives. When students are encouraged to explain their thinking and compare ideas with peers, misconceptions can surface naturally and be examined in a supportive environment. This process helps learners reconstruct their understanding of fractions as meaningful quantities rather than disconnected symbols.
In addition, addressing math anxiety requires more than correcting errors. Learners need learning environments that normalize struggle, reduce fear of being wrong, and emphasize growth over speed or correctness. By fostering collaboration, dialogue, and shared problem-solving, students can develop a stronger sense of competence and belonging. These conditions support motivation and persistence, particularly when learners encounter challenging concepts such as fractions.
Therefore, this learning design prioritizes approaches that support conceptual understanding, learner agency, and emotional safety. By intentionally responding to common misconceptions and affective barriers, the design aims to help learners build both mathematical understanding and confidence, laying a stronger foundation for future learning in mathematics.
Equity, Inclusion, and Diverse Learners
This learning design also emphasizes equity and inclusion by recognizing that students bring diverse cultural backgrounds, learning needs, language abilities, and levels of mathematical confidence into the classroom. Fraction understanding can be particularly challenging for English language learners and students with learning differences when instruction relies heavily on symbolic notation alone.
To promote accessibility, the design incorporates multiple representations of fractions, including visual models, concrete materials, verbal explanations, and digital tools. These approaches allow learners to engage with concepts in ways that align with their individual strengths. Collaborative activities provide opportunities for peer support, while low-stakes discussions and hands-on exploration help reduce pressure and normalize struggle.
By prioritizing emotional safety and flexible learning pathways, this design aims to ensure all learners—regardless of background or ability—have equitable access to conceptual understanding. Supporting diverse learners in this way strengthens confidence, encourages participation, and fosters a sense of belonging in mathematics learning environments.
Target audience
The target audience for this blueprint are students in grades 3-5 (aged 8-11) who are starting to learn about fractions. Students in elementary school have short attention spans; therefore, using a direct instruction supported by cooperative learning will ensure that students are understanding the necessary steps in a way that is easy to comprehend while also getting to work with peers, engaging them further in learning. The activities below support younger learners with their interests in games, stories and real life problem solving. For elementary students their needs are to be able to understand content in a clear step-by-step manner that is easy to follow and replicate. The goal of our resource is to give students a “how to” explanation of mathematical content that supports motivation through hands-on learning and the ability to see quick progress and results.
___________________________________________________________________________
2. Learning Design Plan
Big ideas
- Fractions are a type of number that can represent quantities (taken from the BC elementary grade 3 curriculum)
- Understanding fractions conceptually builds confidence and reduces anxiety in mathematics
Subtopics
- Understanding fractions and their relation to real life contexts
- Comparing the size of fractions
- Confidence and motivation of learning
Essential questions
- Where are fractions used in daily life and what do they represent?
- How do you know when a fraction is bigger or smaller?
- How can fractions become a fun learning experience that builds confidence and motivation to learn?
Learning outcomes
By the end of this learning experience, students will be able to:
- Demonstrate a conceptual understanding of fractions by representing fractions as parts of a whole using concrete materials, visual models, or drawings.
- Explain and compare simple fractions (e.g., 1/2, 1/3, 1/4) using appropriate mathematical language, visuals, or real-life examples.
- Distinguish between the numerator and denominator and what they each mean.
- Identify and correct common fraction misconceptions, such as treating fractions as two separate whole numbers, through discussion and reflection.
- Collaborate with peers to share reasoning and strategies, actively listening to different perspectives and explaining their own thinking.
- Develop increased confidence when working with fractions, demonstrating willingness to attempt challenging tasks and engage in problem-solving without fear of making mistakes.
Learning activities (Sub-ideas taken from the content of the BC elementary grade 3 curriculum)
- Real life activity: Students divide food or objects (ex. pizza, pie) into equal parts to represent fractions and identify the numerator and denominator.
- Provide opportunities to explore and create fractions with concrete materials
- Visual fractions: Use fraction circles, number lines, or digital tools to compare the size of fractions.
- Recording pictorial representations of fraction models and connecting to symbolic notation
- Misconception challenge: Learners look through a series of fractions and identify which is incorrect and explain why they are wrong.
- Fractions can represent parts of a region, set, or linear model.
- Interactive fraction game: Digital or physical game where learners match fractions with visual models or real-life scenarios. This would act more as a culminating activity after trying the other exercises.
- Provide opportunities to explore and create fractions with concrete materials
- Inclusive learning supports: Provide sentence starters, visual prompts, and flexible grouping to support English language learners and students with diverse learning needs.
Assessment plan
Outcome: Represent fractions visually
Assessment: Students create and label a visual model of 3/4 using a drawing or digital tool.
Formative assessment:
- Observation checklist
- Use visual models correctly
- Explain numerator / denominator meaning
- Compare fractions using reasoning
- Exit ticket
- Students draw a visual representation of 3/4 and explain what the denominator represents.
- Quick drawing of a fraction representation
- Teacher conferencing to check understanding and provide individualized feedback
- Small group discussion
- Students discuss:
- “How do you know 1/4 is larger than 1/8?”
- Students discuss:
Summative assessment:
- Short quiz with visual fraction problems
- Focus on: Visual representation, fraction comparison, explanation, not memorization.
- Assessment aligns with inquiry because students must explain reasoning, not just compute.
- Performance Task
- Students solve a real-world problem:
- “You ate 2/3 of a pizza and your friend ate 3/6. Who ate more? Show your reasoning using a visual model.”
- Students solve a real-world problem:
- Concept explanation task
- Real-world problem
___________________________________________________________________________
3. Resources
Textbooks
- BC curriculum math textbook
BC Ministry of Education. (2019). Mathematics curriculum K–7.
- Open educational math texts
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and middle school mathematics: Teaching developmentally (10th ed.). Pearson.
Scholarly articles
Some with strong research directions we can search in the Uvic Library related to Mathematics Education.
Hiroshima Daigaku. Department of Mathematics Education. (1993). Hiroshima journal of mathematics education. Dept. of Mathematics Education.
British Society for Research into Learning Mathematics. (2000). Research in mathematics education (British Society for Research into Learning Mathematics). [British Society for Research into Learning Mathematics].
National Council of Teachers of Mathematics. (1970). Journal for research in mathematics education (Online). National Council of Teachers of Mathematics.
Grey literature
In addition to peer-reviewed research, this learning design is informed by practice-oriented grey literature that reflects current perspectives in mathematics education and learning design.
A video published by the National Council of Teachers of Mathematics (NCTM) emphasizes the importance of conceptual understanding alongside procedural fluency in mathematics learning. The speaker highlights that when students rely solely on memorized procedures without understanding the underlying reasoning, they often struggle to apply their knowledge flexibly or assess whether their answers are reasonable. This issue is particularly relevant in fraction learning, where students may follow algorithms (such as cross-multiplication or fraction division rules) without understanding what the operations represent. The video argues that overemphasis on speed and procedural accuracy can obscure meaning, whereas conceptual understanding enables learners to reason, estimate, and make sense of fractional quantities. This perspective aligns with the goals of the present learning design, which prioritizes sense-making and flexible thinking to address common misconceptions in fractions.
Additionally, a TED talk by Luis von Ahn, co-founder of Duolingo, provides insight into how learning environments can be designed to support motivation, persistence, and engagement. Although the talk focuses on language learning, it offers broadly applicable principles for education, particularly in technology-mediated contexts. Von Ahn explains that accessibility alone is insufficient if learners are competing with highly engaging digital distractions. Instead, effective learning design must consider motivation, emotional engagement, and habit formation. He highlights strategies such as low-stakes practice, consistent feedback, and design features that encourage regular participation, all of which help learners persist over time.
These ideas are especially relevant to mathematics education, where anxiety and fear of being wrong can discourage participation and undermine learning. Together, the NCTM and TED sources support the importance of designing learning experiences that emphasize conceptual understanding, normalize struggle, and foster sustained engagement. By integrating these practice-based insights, the learning design aims to create a supportive environment in which learners can build confidence with fractions while developing a deeper and more flexible understanding of mathematical concepts.
Technology tools
- Slice fractions app
- Math tappers: Estimate Fractions app
- Canva
- Google slides
These technology based tools are used to engage younger learners in solving fractions. Growing up, my elementary classes would go to the computer lab and play Jet Ski Addition, a racing math game. I remember this being the first time I started to enjoy math because I got to use technology to competitively race against friends, while also learning how to efficiently add numbers. To this day, this technology mediated learning experience has stuck with me. For students who are just starting to learn fractions, there are apps such as Slice Fractions and Math tappers. In Slice Fractions, students solve fraction problems to help a woolly mammoth move through ice, making learning engaging and adventure based. In Math Tappers, students build an understanding of fractions by placing them on a number line in order to compare which are bigger and smaller, something many younger learners struggle with. Lastly, Canva and Google Slides are resources that can be used by both teachers and students to create worksheets, presentations and graphics that represent fractions using clip art, shapes and text boxes. Overall, these tools make learning fractions more engaging, helping students develop an understanding while decreasing their anxiety and increasing their motivation to learn.
___________________________________________________________________________
4. Project Plan
| TASK | WHO | WHEN DUE | COMPLETED |
| Brainstorm idea | Everyone | Done | |
| 1. Topic overview | ________________ | ________________ | ________________ |
| Description of the topic | Isaac | Feb. 9 | Done |
| 1-2 misconceptions | Yiran | Feb. 9 | Done |
| Equity,inclusion and diverse learners | Saksham | Feb 9 | done |
| A rationale | Yiran | Feb. 9 | Done |
| Target audience | Isaac | Feb. 9 | Done |
| 2. Learning design plan | ________________ | ________________ | ________________ |
| 1-2 big ideas | Isaac | Feb. 9 | Done |
| Subtopics | Isaac | Feb. 9 | Done |
| Essential questions | Isaac | Feb. 9 | Done |
| Learning outcomes | Yiran | Feb. 9 | Done |
| Learning activities | Isaac | Feb. 9 | Done |
| Learning assessment | Jingyun | Feb. 9 | Done |
| 3. Resources | ________________ | ________________ | ________________ |
| Textbooks | Jingyun | Feb. 9 | Done |
| Scholarly articles | Jingyun | Feb. 9 | Done |
| Grey literature | Yiran | Feb. 9 | Done |
| Technology tools | Isaac | Feb. 9 | Done |
| 4. A project plan – THIS | Isaac | Feb. 9 | Done |
___________________________________________________________________________
References
Sari, I., Suryadi, D., Herman, T., Dahlan, J., & Supriyadi, E. (2024). (PDF) learning obstacles on fractions: A scoping review. Learning Obstacles On Fractions: A Scoping Review.https://www.researchgate.net/publication/380316747_Learning_obstacles_on_fractions_A_scoping_review
Wortha, S. M., Klein, E., Lambert, K., Dackermann, T., & Moeller, K. (2023, January 31). The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data. PloS one. https://pmc.ncbi.nlm.nih.gov/articles/PMC9888716/
Halme, H., Van Hoof, J., Hannula‐Sormunen, M., & McMullen, J. (2024). Not realizing that you don’t know. Not realizing that you don’t know : British Journal of Educational Psychology.https://www.ovid.com/journals/bjep/fulltext/10.1111/bjep.12637~not-realizing-that-you-dont-know-fraction-state-anxiety-is
Fitousi, D., & Noyman R. (2024). Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults. Science Direct. https://www.sciencedirect.com/science/article/abs/pii/S0010027723002901?via%3Dihub
Obersteiner, A., Van Dooren, W., Van Hoof, J., & Verschaffel, L. (2013). The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Science Direct. https://www.sciencedirect.com/science/article/abs/pii/S0959475213000455
Starling-Alves, I., Wronski, M., & Hubbard, E. (2021). Ann. N.Y. Acad. sci. ISSN 0077-8923 annals of the New York Academy of Sciences. Math anxiety differentially impairs symbolic, but not nonsymbolic, fraction skills across development. https://nyaspubs.onlinelibrary.wiley.com/doi/pdfdirect/10.1111/nyas.13522
YouTube. (2015). Building Conceptual Understanding in Mathematics. YouTube. https://www.youtube.com/watch?v=W1eLt0Dz8Fk
Von Ahn, L. (2023). How to Make Learning as Addictive as Social Media. YouTube. https://www.youtube.com/watch?v=P6FORpg0KVo
BC Grade 3 Mathematics Curriculum. Building student success – B.C. curriculum. (n.d.). https://curriculum.gov.bc.ca/curriculum/mathematics/3/core