Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning. Specifically, Chapter 1 examined the debated role of integers in the development of fractions, and whether the comparison of integers and fractions on number lines facilitates a better schema for fraction magnitudes representation. Study 1 explored the developmental trajectory of 177 3rd-to-5th graders' representation of integer and fraction estimates. With age, children's estimates of integers and fractions became more coherent, and representational coherence predicts fraction proficiency. To further investigate whether comparison leads to schema generation, Study 2 assigned 100 third-to-fifth graders to an Alignment condition, where children estimated fractions and integers on aligned number lines, or to a No Alignment condition. Results showed that spatial alignment between integers and fractions on number lines facilitated a better understanding of fractional magnitudes, and increased children's fraction estimation accuracy to the level of college students. Chapter 2 further examined whether better schemas of fraction knowledge representation help explicit analogical reasoning. Study 3 examined whether knowledge of terms matters in a simple classical 4-term analogy task by asking 321 adults and children to complete 4-term analogy tasks featuring letters, lines, integers, or fractions. Performance was lowest for fractions, and strongly impacted by educational background. To further establish the causal link that training conceptual knowledge of fractions facilitates proportional analogies, Study 4 administered an educational intervention focusing on either Conceptual Knowledge, Procedural Knowledge, or Both to 343 3rd-to-5th graders. Children with poor pretest magnitude knowledge were more likely to fail analogical reasoning, and training on conceptual knowledge that fractions denote magnitudes improved children's analogies. These findings indicate that knowledge of fractional magnitudes is important to proportional analogy. Analogies require domain-specific knowledge, not merely the maturation of domain-general cognitive abilities. Finally, Chapter 3 explored whether supports that help children's quantitative reasoning (number lines) might also help political partisans. Political ideology leads educated adults - especially the highly numerate - to selectively reason about numbers that support their beliefs ('motivated numeracy'). Study 5 asked 1000 adults to interpret fictional data, in the table or number-line format, about politically neutral or controversial issues. We found data presented in number-line formats yielded greater accuracy than table formats controlling for numeracy skills, regardless of whether the true interpretation of data affirms, neutral to, or disaffirms participants' political outlooks. Solving table problems after number-line problems yielded greater accuracy compared to solving table problems first, suggesting the number-line practice is educational. Results imply that researchers should use number-line to communicate policy-relevant data, especially on politically divisive issues. Together, these studies indicate that analogies can play an important role in rational number learning, with the help of number lines as schemas. These studies aim to shed light on best practices in STEM education curriculum and science communication. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]