Best Practices for Quantitative Reasoning Instruction
In her book Powerful Learning: What we Know about Teaching for Understanding, Darling-Hammond (2008: 5) argues that meaningful learning is accomplished through a number of key approaches including: "(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."
Several pedagogical approaches which are especially important for teaching QR 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.
Videos about Best Practices for Teaching QR
For a collection of excellent videos about the importance of and strategies for teaching numeracy, see the numeracy videos that were produced in Australia as part of National Literacy and Numeracy Week, an initiative that "represents a collaborative approach by the Australian Government and school communities to highlight the importance of literacy and numeracy skills for all children and young people, with a specific focus on school-aged children."
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.
A summary of Alistair McIntosh's (formerly Associate Professor at the University of Tasmania, Australia) ideas about the key ingredients of successful mathematics teaching and promoting numeracy.
Real World Applications and Active Learning, including Discovery Methods
Involving Students in the Learning Process
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).
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). 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.
The importance of teaching QR skills within an applied (real-world) context has been emphasized by a number of educators. 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, particularly since the contextualized use of data is central to QR and 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).
Other researchers have demonstrated how active learning with real applications can improve statistical literacy skills among young children (Marshall and Swan 2006) as well as graphing skills among college students (McFarland 2010). Moreover, at Macalester College, 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).
One approach to active learning that is valuable for teaching QR is the application of constructivist and discovery methods. Indeed, Killen (2006) contends:
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.
These approaches see learning as a form of understanding constructed by the learner and focus on ways in which the individual learner makes sense of the subject matter (see, e.g., Brooks and Brooks 2001; Caine and Caine 1994; Cakin 2008; Hatano 1996; Leonard 2002; Tout and Schmitt 2002; Switzer 2004). At the same time, teaching QR does not always necessitate such 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."
Pairing QR Instruction with Writing, Storytelling and Critical Reading
Below is a video which features John Allen Paulos discussing stories and statistics:
Pairing quantitative constructs with language 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 students for the workplace (Madison 2012).
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." Adding numbers to language does not only strengthen the latter, but the reverse is also true. Indeed, research has shown that pairing developmental mathematics with reading can enhance success in mathematics (Kirk and Lerma 2010).
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.
Collaborative Instruction and Group Work
Interdisciplinary and collaborative approaches, including group work, are effective educational strategies 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."
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.
Pedagogy that is Sensitive to Differences in Students' Culture and Learning Styles
Considerable research has shown that 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." Even within the classroom, teachers need to be sensitive to the variation in students' abilities (see, e.g., Stern 2004).
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) argue 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, "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" (Tout and Schmitt 2002).
"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 ethonumerical 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 numbers and data among different populations.
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, while 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.
Asia E-University. 2009. "Metacognition and Constructivism." Chapter 6 in Education Psychology (online course). Pages 142-169.
Barton, Bill. 1997. "Making Sense of Ethnomathematics: Ethnomathematics is Making Sense." Educational Studies in Mathematics 31 (1/2): 201-233.
Borich, Gary D. and Martin L. Tombari. 1997. Educational Psychology: A Contemporary Approach. Longman.
Bressoud, David. 2009. "Establishing the Quantitative Thinking Program at Macalester." Numeracy 2(1): Article 3.
Brooks, Jacqueline Grennon and Martin G. Brooks. 2001. "Becoming a Constructivist Teacher." Chapter 9 in In Search of Understanding: The Case for Constructivist Classrooms. Revised Edition. New York: Prentice Hall.
Bruner, Jerome. 1967. On Knowing: Essays for the Left Hand. 2nd Edition. Cambridge: Harvard University Press.
Burkhardt, Hugh. "Quantitative Literacy for All: How Can We Make it Happen." Pp. 137-162 in Calculation vs. Context: Quantitative Literacy and Its Implications for Teacher Education, edited by Bernard L. Madison and Lynn Arthur Steen. Mathematical Association of America.
Caine, Renate N. and Geoffrey Caine. 1994. Making Connections: Teaching and the Human Brain. Menlo Park, CA: Addison Wesley.
Cakin, Mustafa. 2008. "Constructivist Approaches to Learning in Science and Their Implications for Science Pedagogy: A Literature Review." International Journal of Environmental & Science Education 3(4): 193-206.
Caulfield, Susan L. and Caroline Hodges Persell. 2006. "Teaching Social Science Reasoning and Quantitative Literacy: The Role of Collaborative Groups." Teaching Sociology 34(1): 39-53.
Collison, Joe, Catherine Good, Sonali Hazarika, Matt Johnson, Jimmy Jung, Anita Mayo, Will Millhiser, Dahlia Remler, and Laurie Beck. 2008. "Report of the Provost's Task Force on Quantitative Pedagogy." Baruch College, the City University of New York.
Cuban, Larry. 2001. "Encouraging Progressive Pedagogy." Pp. 87-91 in Mathematics and Democracy: the Case for Quantitative Literacy, edited by Lynn Arthur Steen. Princeton, NJ: National Council on Education and the Disciplines.
Darling-Hammond, Linda. Editor. 2008. Powerful Learning: What we Know about Teaching for Understanding. San Francisco: Jossey-Bass.
Dingman, Shannon W. and Bernard L. Madison. 2010. "Quantitative Reasoning in the Contemporary World, 1: The Course and Its Challenges." Numeracy 3(2): Article 4.
Fuller, Theodore D. 1998. ''Using Computer Assignments to Promote Active Learning in the Undergraduate Social Problems Course.'' Teaching Sociology 26(3): 215-21.
Grawe, Nathan D. and Carol A. Rutz. 2009. "Integration with Writing Programs: A Strategy for Quantitative Reasoning Program Development." Numeracy 2(2): Article 2.
Grouws, Douglas A. and Kristin J. Cebulla. 2000. Improving Student Achievement in Mathematics. Geneva, Switzerland: International Academy of Education.
Hatano, Giyoo. 1996. '"A Conception of Knowledge Acquisition and its Implications for Mathematics Education. Pp. 197-217 in Theories of Mathematical Learning, edited by Leslie P. Steffe and Pearla Nesher. Mahwah, NJ: Erlbaum.
Himes, Christine L. and Christine Caffrey. 2003. "Linking Social Gerontology with Quantitative Skills: A Class Project Using U.S. Census Data." Teaching Sociology 31(1): 85-94.
Jabon, David. 2006. "Quantitative Reasoning: An Interdisciplinary, Technology Infused Approach." In Current Practices in Quantitative Literacy, edited by Rick Gillman. Washington, DC: Mathematical Association of America. Pp. 111-117.
Kain, Edward L. 1999. "Building the Sociological Imagination Through a Cumulative Curriculum: Professional Socialization in Sociology." Teaching Sociology27(1): 1-16.
Karim, Nakhshin and John Wakefield. N.d. "Experiences in Developing a Quantitative Reasoning Program for Students at Zayed University." Paper presented at Learning Technologies and Mathematics, Middle East Conference, Sultan Qaboos University, Muscat, Oman.
Killen, Roy. 2006. Effective Teaching Strategies: Lessons from Research and Practice. 4th Edition. Cengage.
King, Kim M. 1994. "Leading Classroom Discussions: Using Computers for a New Approach." Teaching Sociology 22(2):174-82.
Kirk, Honey and Diane Lerma. 2010. "Reading your Way to Success in Mathematics: A Paired Course of Developmental Mathematics and Reading." MathAMATYC Educator1(2): 10-14.
Lagerlöf, Johan, and Andrew Seltzer. 2008. "Do Remedial Mathematics Courses Help Economics Students?" VoxEU.org.
Leonard, David C. 2002. Learning Theories: A to Z. Westport, CT: Greenwood.
Lutsky, Neil. 2008. "Arguing with Numbers: Teaching Quantitative Reasoning through Argument and Writing." Pp. 59-74 in Calculation vs. Context: Quantitative Literacy and Its Implications for Teacher Education, edited by Bernard L. Madison and Lynn Arthur Steen. Washington, DC: Mathematical Association of America.
Madison, Bernard. 2012. "If Only Math Majors Could Write." Numeracy 5(1): Article 6.
Madison, Bernard L. and Shannon W. Dingman. 2010. "Quantitative Reasoning in the Contemporary World, 2: Focus Questions for the Numeracy Community." Numeracy 3(2): Article 5.
Markham, William T. 1991. ''Research Methods in the Introductory Course: To Be or Not to Be?'' Teaching Sociology 19(4): 464-71.
Marshall, Linda and Paul Swan. 2006. "Using M&Ms to Develop Statistical Literacy." APMC 11(1): 15-24.
McFarland, Jenny. 2010. "Teaching and Assessing Graphing Using Active Learning." MathAMATYC Educator 1(2): 32-39.
McLaughlin, Milbrey W., and Joan E. Talbert. 1993. "New Visions in Teaching." Pp. 1-10 in Teaching for Understanding, edited by D.K. Cohen, M.W. McLaughlin and J.E. Talbert. New York: Jossey-Bass.
Moseley, David, Steve Higgins, Rod Bramald, Frank Hardman, Jen Miller, Maria Mroz, Harrison Tse, Doug Newton, Ian Thompson, John Williamson, Jean Halligan, Sarah Bramald, Lynne Newton, Peter Tymms, Brian Henderson, and Jane Stout. 1999. Ways forward with ICT: Effective Pedagogy Using Information and Communications Technology for Literacy and Numeracy in Primary Schools. Report commissioned by the Teacher Training Agency, Great Britain.
Persell, Caroline Hodges. 1992. ''Bringing PCs into Introductory Sociology Courses: First Steps, Missteps, and Future Prospects.'' Teaching Sociology 20(2): 91-103.
Pozo, Susan, and Charles A. Stull. 2006. "Requiring a Math Skills Unit: Results of a Randomized Experiment." American Economic Review, Papers and Proceedings 96(2): 437-441.
Raymondo, James C. 1996. ''Developing a Computer Laboratory for Undergraduate Sociology Courses.'' Teaching Sociology 24(3): 305-09.
Rowlands, Stuart and Robert Carson. 2002. "Where Would Formal, Academic Mathematics Stand in a Curriculum Informed by Ethnomathematics? A Critical Review of Ethnomathematics." Educational Studies in Mathematics 50(1): 79-102.
Steen, Lynn Arthur. 2004. ''Everything I Needed to Know about Averages I Learned in College.'' Peer Review 6(4): 4-8.
Stern, Frances. 2000. "Choosing Problems with Entry Points for All Students." Mathematics Teaching in the Middle Schools 6(1): 8-11.
Switzer, Jamie S. 2004. "Teaching Computer-Mediated Visual Communication to a Large Section: A Constructivist Approach." Innovative Higher Education 29(2): 89-101.
Twigg, Carol A. 2005. "Math Lectures: An Oxymoron?" The National Center for Academic Transformation.
Tout, Dave, and Mary Jane Schmitt. 2002. "The Inclusion of Numeracy in Adult Basic Education." Chapter 5 in Review of Adult Learning and Literacy (3). National Center for the Study of Adult Learning and Literacy.
Vacher, H. Len and Emily Lardner. 2010. "Spreadsheets Across the Curriculum 1: The Idea and the Resource." Numeracy 3(2): Article 6.
Wiest, Lynda R., Heidi J. Higgins, and Janet Hart Frost. 2007. "Quantitative Literacy for Social Justice." Equity & Excellence in Education 40: 47-55.
Wilder, Esther Isabelle. 2010. "A Qualitative Assessment of Efforts to Integrate Data Analysis throughout the Sociology Curriculum: Feedback from Students, Faculty and Alumni." Teaching Sociology 38(2): 226-246.
Zaslavsky, Claudia. 1994. "'Africa Counts' and Ethnomathematics." For the Learning of Mathematics14(2): 3-8.