{'fraction_non_alphanumeric': 0.013431833445265278, 'fraction_numerical': 0.0, 'mean_word_length': 5.535087719298246, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Overcoming Errors in Fraction Computation by Emphasizing Unit Fractions, Length Drawings, and Student Explanations

This paper addresses the perceived opacity of fraction computation by summarizing results of classroom design experiments focused on making fraction concepts and computation meaningful to students and teachers. A nurturing Math Talk classroom environment, in which students made and explained math drawings supported sense-making by students and teachers. Students were able to make drawings of length models connected to symbolic computations for all fraction operations, and most students were able to explain their thinking. Correct performance on computations was considerably higher than that of U.S. students using traditional textbooks and was more comparable to performance of East Asian students. The paper shows the length drawings for each fraction operation and summarizes the nature of students' errors and how to overcome these. Length drawings (bar and number line models) connected to fraction symbols and words can focus students on unit fractions in all operations. Important difficulties to overcome are that fraction notation does not differentiate the roles of the numerator and denominator, and the notation shows the number but not the size of unit fractions. Multiplying fractions by multiplying each unit fraction and not the whole fraction magnitude is a general method, and division of fractions can begin by dividing numerators and denominators.