Learning A Framework Summary

“Learning: A Framework” thoroughly explained the concept of “mathematical proficiency,” and the strategies teachers can employ to ensure that students with learning difficulties (and/or behavioral disorders) can achieve mathematical proficiency. First, chapter two began by illustrating the story of a second grade student named Aaron, who struggled in mathematics. Though Aaron received numerous supports from the special education teacher, this intervention was ineffective for many reasons. One of such reasons was that, “The special education teacher believed that, if children were empty vessels, then learning-disabled children were leaky vessels who needed to be overfilled—needed to “overlearn” material by massive amounts of practice—to retain

This chapter explained that, “conceptual understanding is knowledge of facts, generalizations, or principles underlying the comprehension of concepts (categories), relations (between categories), or operations.” That is, conceptual understanding demands that students recognize the interconnectivity between many aspects, in addition to knowing isolated facts and methods. Next, procedural fluency is the skill of carrying out routines/procedures accurately, flexibly, and efficiently. The third aspect of mathematical proficiency (i.e. strategic and adaptive mathematical thinking) is actually two competencies that are interrelated. This chapter explained that strategic competence is the ability to formulate, represent, and solve mathematical problems, while adaptive reasoning focus on logical thought, reflection, explanation, and justification. The final strand of mathematical proficiency is productive disposition. This strand stresses the importance of viewing mathematics as sensible, useful, and worthwhile. Additionally, this final strand emphasize that mathematics require diligence, and that we are all capable of significant mathematical learning. Moreover, though these four aspects of mathematical proficiency have distinctive features,

But before exploring this idea of mathematical proficiency for students with learning disability, this section explained how the goal of mathematical instruction has shifted from memorizing basic skills by rote (or direction instruction and drill) to an approach that foster the four aspects of mathematical proficiency. This new approach is called the investigative approach; this approach recommends, “mathematics instruction be meaningful, inquiry based, and purposeful” (Fennell, 2011, p. 25). After explaining that the investigative approach can foster the four components of mathematical proficiency, this chapter goes on to argue that all students can and should attain mathematical proficiency, even the students with learning or behavioral difficulties. This chapter strongly emphasized that, “mathematical proficiency is necessary for any one to live and work semi-independently or independently. Only with such proficiency can individuals with moderate mental retardation achieve the necessary self-reliance to manage their personal affairs, particularly their finances” (Fennell, 2011, p. 35). In other words, in order to be a functional member of this ever-changing world, students with learning or behavioral difficulties need to achieve mathematical