Best Practices for Quantitative Reasoning Instruction
In her book Powerful Learning: What we Know about Teaching for Understanding, Darling-Hammond (2008: 5) argues that "looking across domains, studies consistently find that highly effectively teachers support the process of meaningful learning" by (1) creating ambitious and meaningful tasks, (2) engaging students in active learning, (3) drawing connections to students, (4) scaffolding the learning process, (5) assessing student learning continuously, (6) providing clear standards and constant feedback, and (7) encouraging strategic and metacognitive thinking. These approaches are also important for teaching quantitative reasoning skills."
Likewise, in his book Effective Teaching Strategies: Lessons from Research and Teaching, Roy Killen (2006) notes that, "In the past 30 years, our understanding of how people learn has changed dramatically. New approaches to cognitive research and developmental psychology suggest that learning is a much more individualised process than was previously thought. It is now generally accepted that most people learn best through personally meaningful experiences that enable them to connect new knowledge to what they already believe or understand. Such constructivist views of learning have led to a redefinition of effective teaching. It is now more widely accepted that teachers have to deliberately help learners construct their own understanding, rather than simply tell them things that they are expected to memorize. Good teaching is no longer about helping students to accumulate knowledge that is passed onto them by teachers; it is about helping students to make sense of new information (no matter what its source), to integrate new information with their existing ideas and to apply their new understandings in meaningful and relevant ways."
Several pedagogical approaches are especially important for teaching QR which are described in more detail on this page including:
- real world applications and active learning, including discovery methods;
- pairing QR instruction with writing and critical reading;
- using technology, including computers;
- collaborative instruction and group work;
- pedagogy that is sensitive to differences in students' culture and learning styles; and
- scaffolding the learning process and providing rich feedback and opportunities for revision.
Of course, these approaches are frequently overlapping.
A Video about Best Practices for Teaching QR
A video produced by the University of Colorado at Boulder Journalism and Mass Communications students that discusses the educational practices that contribute to quantitative illiteracy and proposes solutions. This video features an interview with Sanjoy Mahajan, author of Street-Fighting Mathematics.
(1) Real World Applications and Active Learning, including Discovery Methods
There is a famous Chinese proverb that states, "What I hear, I forget; what I see, I remember; what I do, I understand." This proverb is a fundamental principle of active learning. Extensive research has shown that students learn more rapidly, retain knowledge longer, and develop superior critical thinking skills when they are actively involved in the learning process (see, e.g., Himes and Caffrey 2003, Kain 1999; Kenny 1998; King 1994). For example, active learning that combines teaching & assessing graphing has been shown to have positive educational outcomes (McFarland 2010). When theory and data analysis are combined in an active learning setting, students often come to understand that quantitative reasoning skills are relevant to social issues.
Indeed, the premise that "students learn math by doing math, not by listening to someone talk about doing math" (Twigg 2005) provides the philosophical approach for NICHE (i.e., "students learn quantitative reasoning by doing quantitative reasoning"). Successful instruction in QL requires progressive pedagogy: "connecting content to real-life situations, lighter coverage of topics, an emphasis on understanding concepts rather than facts, integrating content across disciplinary boundaries" (Cuban 2001: 89). Indeed, the real-world applicability of numbers is one of the key criteria that defines quantitative reasoning and distinguishes it from traditional mathematics. For example, Marshall and Swan (2006) showed how working with M&Ms could promote statistical literacy among young children.
The importance of teaching QR skills within an applied (real-world) context has been emphasized by a number of authors. For example, Burkhardt (2008) argues in favor of Numeracy through Problem Solving (NTPS), which grew out of a concern that students see school mathematics as irrelevant to their present or future lives. Likewise, Karim and Wakefield (2007: 3) stress the importance of presenting real-world example before introducing more general theoretical concepts. These approaches to QR instruction are important, especially since empirical research has failed to demonstrate that traditional remedial math courses improve student performance (see, e.g., Lagerlöf and Seltzer 2008; Pozo and Stull 2006).
Discovery methods are also important for quantitative reasoning instruction, just as they have been successfully used in mathematics and science education. For example, "constructivists see learning as a form of understanding constructed by the learner", and they focus on ways in which the individual learner make sense of the subject matter (Cakin 2008; Tout and Schmitt 2002). By way of contrast, "in the transmissionist model, teachers act as the experts, and their role is to transmit knowledge directly to their students. This knowledge is seen as objective, and the learning is about receiving the information handed down, absorbing the facts, and reproductive them." Switzer (2004:93) outlined ten principles for designing constructivist learning environments including: "(1) Anchor all learning activities to a larger task or problem, (2) Create real-world environments that employ the context in which learning is relevant, (3) The instructor is a coach and analyzer of the strategies used to solve these problems; (4) Provide for authentic versus academic context for learning; (5) Stress conceptual interrelatedness, providing multiple representations or perspectives on the content; (6) Instructional goals and objectives should be negotiated and not imposed; (7) Learning should be internally controlled and mediated by the learner; (8) Support collaborative construction of knowledge through social negotiation; (9) Encourage ownership and voice in the learning process; and (10) Provide opportunity for and support reflection on both the content learned and the learning process.
Based on experience in the pre-college classroom, Brooks and Brooks (2001) provide guidelines for instructors interested in incorporating constructivist approaches (e.g., "constructivist teachers inquire about students' understanding of concepts before sharing their own understanding of those concepts") that are very useful to all those interested in these methods. A solid overview of constructivist approaches to education is provided by Asia E-University (2009). These techniques also need to be applied in a way that is sensitive to the variation in students' abilities within the classroom (see, e.g., Stern 2004).
Hatano (1996) described the implications of constructivist approaches towards mathematics education by asserting that "students should be given the opportunity to actively participate in the learning process rather than be forced to swallow large amounts of information." At the same time, teaching QR does not always necessitate active learning and constructivist approaches. For example, McLaughlin and Talbert (1993: 4) contend that "teachers need to learn when the interactive, constructivist forms of teaching are called for and when other less demanding, conventional strategies are appropriate."
The research on the physiology of learning also supports the integration of discovery methods. Caine and Caine (1994) have put forward several core principles for brain-based learning and identified several conditions that must be met for complex learning to occur. First, there must be relaxed alertness, that is, a low threat and rich learning environment. Second, there must be orchestrated immersion, or an education environment that includes complex and authentic experiences. And finally, active processing must occur, whereby the learning makes meaning through experiencing processing. As Leonard (2002) writes, "For active processing to take place, teachers must create a realistic context for learning, they must let students work in teams and work through mistakes themselves, and they must allow for students' continual self-assessment of how they learn and why they learn."
(2) Pairing QR Instruction with Writing and Critical Reading
Madison (2012) argues that there are several important reasons why quantitative constructs and language should be combined. In particular, such a merging serves to (1) strengthen academic arguments; (2) strengthen quantitative literacy/reasoning; (3) interpret and improve public discourse; (4) encourage quantitative reasoning across the curriculum; and (5) prepare for the workplace. The American Mathematical Association of Two-Year Colleges has also highlighted a number of important pedagogical approaches, such as pairing developmental mathematics with reading to enhance success in mathematics (Kirk and Lerma 2010).
Research has also shown that placing QR programs within the context of writing programs brings a number of benefits. For example, it improves writing instruction, challenges the notion that QR is only remedial math, and provides a route for the incorporation of QR into the curriculum (Grawe and Rutz 2009). Stressing the importance of connecting writing and QR, Lutsky (2008: 63) argues that "quantitative information may be used to help articulate or clarify an argument, frame or draw attention to an argument, make a descriptive argument, or support, qualify, or evaluate an argument. Quantitative analysis may also influence how arguments are marshaled and how exchanges of arguments are conducted."
(3) Using Technology, including Computers
Computer skills (operating systems, spreadsheets, etc.) are also essential to QL/QR (Collison et al. 2008; Jabon 2006; Steen 2004; Vacher and Lardner 2010; Wiest et al. 2007). Indeed, the use of computers can actively engage students in QR work, promote logical thinking and help students master QL/QR skills that are central to the research process (Fuller 1998; Markham 1991; Persell 1992; Raymondo 1996). Interactive computer software; personalized, on-demand assistance; and mandatory student participation have also been recognized as key elements of successful math instruction (NCAT 2005). The integration of spreadsheets across the curriculum has also been shown to successfully promote QR engagement in a variety of fields (Vacher and Lardner 2000).
Moseley and colleagues (1999) found that various strategies of using information and communication technology promoted effective instruction in literacy and numeracy in primary schools in Great Britain. Indeed, research has shown that active learning using computers helps promote students' QR skills (Wilder 2009), and that computer literacy is itself seen as a QR skill (Wilder 2010). For instance, the interdisciplinary, technology-infused approach to QR adopted by DePaul University had a number of positive benefits (Jabon 2006); students mastered technology tools by undertaking realistic analyses, and the computer-based activities created an active, lively learning environment that was engaging for students.
(4) Collaborative Instruction and Group Work
Interdisciplinary and collaborative approaches to QR instruction are also important. Previous research has shown that group work is an effective educational strategy for promoting mathematics and QR education. For example, in their research on teaching social science reasoning and quantitative literacy using collaborative groups, Caulfield and Hodges (2006: 52) reported, "Our data clearly reveal that most of our students work harder and learn more while working in groups." Indeed, Grouws and Cebulla (2000: 20) argue that "teachers must encourage students to find their own solution methods and give them opportunities to share and compare their solution methods and answers. One way to organize such instruction to have students work in small groups initially and then share ideas and solutions in a whole-class discussion."
At Macalester College, for example, the economics and mathematics departments jointly offered a course which taught students fundamental quantitative skills within an applied context (e.g., sampling issues and the interpretation of polling data) (Bressoud 2009: 8). More recently, Dingman and Madison (2010; see also Madison and Dingman 2010) taught a course that engaged students in collaborative small-group learning exercises in which they read and evaluated data from newspaper articles.
(5) Pedagogy that is Sensitive to Students' Differences in Culture and Learning Styles
Considerable research has been written about how women and minorities experience mathematical and quantitative disadvantages. The strategies for overcoming this inequality remain a topic of considerable interest and debate. Tout and Schmitt (2002) note that in the United States, considerable effort has been directed towards fostering the success of females in mathematics and such approaches to teaching have "challenged the traditionally male-dominated domain of math education and promoted alternatives that in many cases are attractive not only to girls but to the many boys who struggle with learning mathematics in the class. Such approaches include working cooperatively, promoting discussion and idea sharing, and using hands-on materials."
Teaching quantitative reasoning in a manner that is sensitive to different cultures also has the potential for improving student learning. For example, Ubiratan D'Ambrosio, the Brazilian educator who coined the term "ethnomathematics," described it as the study of different forms of mathematics that arise in different cultural contexts. Zaslavsky (1994: 7) argues that the incorporation of ethnomathematical perspectives calls for "a complete turn-around from the way mathematics is now taught in many classrooms." In a nutshell, she argues that:
(a) The entire mathematics curriculum must be restructured so that mathematical concepts and ethnomathematical aspects are synthesized. Rather than a curriculum emphasizing hundreds of isolated skills, mathematics education will embody real-life applications in the forms of projects based on themes and mathematical concepts.
(b) Teachers at all levels must be well-grounded in mathematics and at the same time familiar with the interface between mathematics and other subject areas. They will need the initiative and time to work with other teachers, with parents and the community in planning lessons that are relevant specifically to their students. Preservice and inservice education should incorporate these perspectives.
(c) The revised curriculum will require various methods of assessment – on-going assessment of projects, evaluation of portfolios, etc. Simplistic multiple-choice tests will be abolished or downplayed.
(d) Research must be conducted and the results made valuable to teachers on the ways in which underserved and underrepresented students, particularly females and people of color, can best learn mathematics.
Likewise, Rowlands and Carson (2002: 52) argues that "only through the lens of formal, academic mathematics sensitive to cultural differences that the real value of the mathematics inherent in certain cultures and societies [can] be understood and appreciated." Indeed, Tout and Schmitt (2002) argue that "functional math has much in common with ethnomathematics. Both argue for an approach that covers a wide range of math skills embedded within social contexts and purposes that values personal ways of doing math."
"Acknowledging the cultural component of mathematics will enhance our appreciation of its scope and its potential to providing an interesting, artistic and useful view of the world" (Barton 1996: 299) Just as an ethnomathematical framework can improve mathematics education, so too can an ethnonumeracy1 approach improve students' understanding of quantitative reasoning skills. For example, integrating culturally relevant QR exercises is important. At the same time, Orey and Rosa (2007: 15): caution that ethnomathematical work in the schools is not a simplistic presentation of cultural examples or simply situating mathematics in cultural contexts. Rather it requires considerable background work, complete understanding, and pedagogical sophistication . . . For example, it is convenient to state that teachers may interpret an ethnomathematical approach by starting with the students' outside socio-cultural-economic realities, but the students may refuse to study their own realities because they consider them to be oppressive."
Another example has to do with cultural modes of communication. As communication is one key component of quantitative reasoning, an ethonumeracy approach to QR instruction will need to respond to differences in language (particularly among non-native English speakers) and how words are used to describe data among different populations.
(6) Scaffolding the Learning Process and Providing Rich Feedback and Opportunities for Revision
Teaching QR for understanding involves a process whereby the instructor is an active facilitator of learning. Killen (2006: 21) notes that the teacher's goal should be "to encourage students to be both investigators and critics of the subjects they are studying, which providing them with sufficient scaffolding for them to be successful in their learning." Scaffolding entails "providing a student with enough help to complete a task and then gradually decreasing the help as the student becomes able to work independently" (Killen 2006: 7).
Moreover, throughout the learning process, faculty must be engaged in providing rich feedback to students and ensuring that students have ample opportunities to master the material, particularly if they are not successful the first time around. The Writing Across the Curriculum (WAC) model of instruction incorporates revision as an essential component in the process of learning how to write; this model is also critical for mastering quantitative reasoning skills.
I I (Esther Wilder) use the term "ethnonumeracy" to refer to an appreciation for the cultural context of quantitative reasoning skills and understanding.
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