UCLA

Many years of K-12 math education are dedicated to the learning of numbers (e.g., counting, different number forms: whole numbers, decimals, fractions). A common challenge is the integration of learned number forms (e.g., whole numbers) with new number forms (e.g., decimals) as children tend to carry the “rules” they learned to the next number concept. Research on number representation has covered numbers up to proper fractions (e.g., magnitudes less than 1), but little research has been conducted looking at number forms beyond this point, like improper fractions (e.g., magnitudes more than 1, 5/3), squares and square roots. In my dissertation, I ask the question of whether adults also rely on more familiar/known number forms when processing these more complex number forms. The overarching hypothesis is that people use more familiar/known number forms as references for assessing magnitude of these complex number forms because the number representation of the former is more precise and reliable than the latter. I use a combination of cognitive and perceptual tasks such as magnitude comparison of numbers and visuals, and magnitude estimation across four chapters. People were better at assessing the magnitudes of improper fractions when they used mixed fractions and decimals as reference points rather than improper fractions (Chapter 1). Performance on magnitude comparison of improper fractions worsened as (whole number) magnitude increased for both symbolic/numerical and non-symbolic/visual tasks (Chapter 2). The reference numerical range for squaring numbers anchored and restricted people’s estimates of squares (Chapter 3). Finally, the natural numbers within the square root sign predicted magnitude estimation of square roots better than its actual magnitudes (i.e., mental number line hypothesis) and distance from perfect-squares predicted magnitude estimation as well. Altogether, we found that people use more familiar numbers – whole numbers, proper fractions, decimals (Chapter 1); multiples (Chapter 2); numerical range (Chapter 3); and natural numbers and perfect-squares – when processing more complex numbers – improper fractions (Chapters 1-2), squares (Chapter 3) and square roots (Chapter 4). My dissertation fills in major gaps in the numerical cognition literature and its chosen number forms have implications for algebra and calculus readiness in students.