Quantitative Literacy

“If we are going to have a successful democratic society, we have to have a well-educated and healthy citizenry.”
Thomas Jefferson

According to an article from The Mathematics Association of America (MAA), "a quantitatively literate person is a person who, with understanding can both read and represent quantitative information arising in his or her everyday life" (Richardson & McCallum, 2003).

Some elements that may factor in to a person being quantitatively literate are:

• confidence with mathematics
• cultural appreciation
• interpreting data
• logical thinking
• making decisions
• mathematics in context
• number sense
• practical skills
• prerequisite knowledge
• symbol sense (Richardson & McCallum, 2003).

The tricky thing about talking about quantitative literacy is the belief that it should be taught solely in the vacuum of a mathematics classroom. However, according to Richardson & McCallum (2003), "quantitative literacy is not simply a matter of knowing how to do the mathematics but also requires the ability to wed mathematics to content." In many ways, it can be likened to the belief that writing should be taught across curriculum in order for students to become better writers. Quantitative literacy taught across the different content areas will allows students to become more critical in their thinking.

Teaching quantitative literacy across the curriculum allows for teachers to incorporate matheracy, "the capability of inferring, proposing hypotheses, and drawing conclusions from data" into their lessons (D'Ambrosio, 2003). In addition, it allows students to have a conceptual understanding of mathematics in a variety of settings. Becoming someone who is quantitatively literate means he or she can understand how to stock market works, balance a checkbook, or understand the concept of risk (Colwell, 2003). All of these things are important in making a well-educated citizen who has the ability to think critically for themselves.

The definition of literacy from the Adult Education Admendments of 1988 reminds us that literacy "is an individual's ability to read, write, and speak English and compute and solve problems at levels of proficiency necessary to function on the job and in society to achieve one's goals and to develop one's knowledge and potential" (Colwell, 2003). In some ways, Congress was ahead of their time because what their definition of literacy looks a lot like what people are calling 21st Century Skills.

Recently, in a professional learning community, I shared a mini-lesson on inferences using a scientific procedure of sorts for inferring what was happening in a literature passage to show that quantitative literacy can happen anywhere. The worksheet that accompanied the min-lesson/conversation is below:

Passage

From: The English Patient by Michael Ondaajte

She moves backwards a few feet and with a piece of white chalk draws a rectangle onto the wood floor. Then continues backwards, drawing more rectangles, so there is a pyramid of them, single then double then single, her left hand braced flat on the floor, her head down, serious…
She drops the chalk into the pocket of her dress. She stands and pulls up the looseness from her skirt and ties it around her waist. She pulls from another pocket a piece of metal and flings it out in front of her so it falls just beyond the farthest square.
She leaps forward, her legs smashing down, her shadow behind her curling into the depth of the hall. She is very quick, her tennis shoes skidding on the numbers she has drawn into each rectangle, one foot landing, then two feet, then one again until she reaches the last square.

Step 1: Form a hypothesis (inference) about what is happening in this passage.

Step 2: List all information that supports your hypothesis.

Step 3: Explain what is happening in the passage above. (Information from passage + Your knowledge = Inference or explanation)

Step 4: What steps did you take to figure out what was happening in this passage?

This mini-lesson was a great way to talk about inference, critical thinking skills, metacognition, and active reading strategies all in one -- plus it connects to another content area if you really think about it. The formula used to come up with a solid inference or explanation looks a lot like a basic math problem. Plus, the way in which different people determined was happening showed how each of us thinks critically when faced with different types of information. The teachers who read this passage noted that they made connections, visualized, determined important details, used prior knowledge, etc.

In fact, I look forward to watching this lesson in action in a math class. The teacher plans to use this passage, as well as a few others, to show how logical thinking skills/steps are essential in all areas of life not just in explaining a geometry proof. This use of the lesson is just one way of showing how literacy spiderwebs across all content areas in a variety of different and creative ways.

Using quantitative literacy in a variety of content areas is a great way to bridge the divide between literacy and content areas because it gets at an area of literacy (critical thinking, inferring, etc.) that isn't always address in a conscious way. In addition, it can help have the conversation with teachers that literacy is much bigger than just reading and writing or rather the idea that reading and writing encompass a lot more than we give them credit for in all we do.

Some other ideas for working on quantitative literacy in the classroom are:

• Using Data -- graphs, charts, statistics, etc. to get students to analyze what it is really telling us. In addition, looking at how certain media or other places use data incorrectly OR to get their point across. Getting our students to be critical consumers of information is extremely important in them becoming good citizens.
• Using Writing -- focus on having students write to learn versus learn to write. Ask them to explore their ideas through writing prior to sharing in discussions or discourses in the classroom. Allowing this think time might produce students who rely on their own thoughts rather than wait for his or her peers to suggest the answer. Let students get messy with their thinking on paper so they have something to go back to at the end to see if what they thought initially still holds true. Also, maybe even have them use "Silent Discussions" or "Written Conversations" to get the conversation going with peers in the classroom as well. This writing can happen by hand or through a technology resource like moodle, blog, wiki, etc.
• Modeling how to make Inferences -- using material you might encounter in your life -- share with students how you make inferences, hypotheses, and analyze data in your life. Share how it connects with skills both at school and in the world outside of school.

Question-Answer Relationships and Three-Level Study Guides

With two weeks before April break and what seems like forever since I have had a good night's sleep, I am beginning to reflect on what works and what doesn't. Over the course of the last couple weeks, I have been working with different students on a range of different literacy tasks. As I walk them through the task and guide them along, I feel good that they are getting something out of it. When they forget to do the tasks I assign outside of my office or even to show up, I wonder who is working harder -- the student or me?


As many know, I am more than willing to give my all to any student or colleague who needs me, but in an ideal world I guess part of my job is to help provide guidance and resources so eventually I won't be needed. In working towards that theory, I have found two great strategies that focus on using a gradual release of responsibility model to help students learn higher level critical thinking and comprehension skills. In addition to the resources on this blog, there is a bulletin board with information, examples, and handouts outside my office.


Question-Answer Relationship (QAR)and Three-Level Study Guides are two strategies that focus on asking the student to know the difference between the literal, interpretative, and applied levels in the area of comprehension.

Question-Answer Relationships (QAR)


QAR is "a language for use in the classroom" for "thinking about and talking about sources of information for answering questions" (Raphael, Highfield, & Au, 2006). The two overarching categories are In the Book and In My Head. Questions that focus on information from In the Book ask for author information to develop an answer. Ones that focus on In My Head have answers that are based on a reader's experience, ideas, or background knowledge. Within these two categories, there are subcategories.


In the Book is broken down into Right There and Think and Search. Right There questions focus on answers that can be found in one sentence within a text; whereas, Think and Search require using different parts of a text to develop an answer.


In My Head is broken down into Author and Me and On My Own. Author and Me involves using information from the text and the reader's experience, ideas, and background knowledge to answer it; whereas, On My Own involves only the thoughts and ideas of the reader.


Example of Different Types of Categories:
Math Example --
Right There --
Q: "State the slide model for addition."
A: "The slide model for addition is..."


Think and Search --
The text shows an example of how to simplify
-3 + -x + 7
Q: "Simplify -2 + y + -9


Author and You --
After learning how to simplify (as in Think and Search example above), the author shows students how to solve simple equations like x + 6 = 10, and 17 - y = 12
Q: "Solve -3 + t - 4 = 0"


On My Own --
Negative numbers appear on television in many situations.
Q: "What real situation might each negative number represent? a.) -1.32 in stock market averages, b.) -9 in rocket launches, c.) -3 in golf."


Music Example --
Right There --
Q: What kind of music is this -- band, orchestra, piano?


Think and Search --
Q: What instruments do you hear throughout this piece?


Author and You --
Q: What feelings do you think the composer was trying to convey? How did the composer create feeling?


On Your Own --
Q: If you could play any instrument, what would it be and why?


Art Example --
Right There --
Q: What is the subject of the painting?


Think and Search --
Q: What colors did the artist use?


Author and You --
Q: How did the painter show perspective in this piece? What feelings do you think the artist was trying to convey?


On My Own --
Q: If you made a living as a painter, what medium would you use? Why?


From Project CRISS: Creating Independence through Student-owned Strategies by C.M. Santa, L.T. Havens, & B.J. Valdes (2004).


Social Studies Example
Science Example
English Example

Three-Level Study Guides
According to Janet Allen, Three-Level Study Guides are "instructional tools that guide students into higher levels of comprehension. The first part asks you to consider questions at the literal level. The second asks you to consider questions at the interpretative level. The third asks you to consider questions at the applied level. As a teacher, you can construct a guide that allows you to direct students' attention to the most important aspects of the text." (2008)

In creating the types of questions, it is important to use sentence starters that guide the reader to think either at the literal, interpretative, or applied level. For example, when creating a question at the interpretative level you may consider using summarize, predict, or determine; whereas at the literal level, you may consider using name, select, or define. In addition to using questions to guide the reading, you may also make statements that ask students to think at the literal, interpretative, or applied level. Check out the link at the top of the page for Three-Level Study Guides to see examples from math and science.

These strategies are just two ways to get students thinking about higher level critical thinking and comprehension skills by having them practice these strategies with you and on their own. Having students understand the language involved in this type of thinking might allow for them to move beyond the facts of a topic and really be able to apply the concepts they have learned. Guiding students through the process of learning this type of language and thinking will help them become independent in it as they move along. As always, contact me if you have questions or thoughts.

Critical Literacy and "Online" Text

Change is something that people tend to do either very well or not so well. I usually fall into the later category. However, since becoming a parent, I have found that I have to be extremely flexible to change. One of the biggest changes I have encountered is my use of technology. In the past, I have always been the book girl, the pen and paper girl, and the meet and talk about it girl. Now with a squirmy, wormy four month old, I find my time precious and limited. My family takes more of my time so I don't find time to pick up a book to find something, jot a note to someone, or readily be able to meet for coffee to connect about what is going on. However, I still want to connect with the world outside me and share the goings on of my family.


In the opening chapter of Handbook of Research on New Literacies the authors note that "Literacy is no longer a static construct from the standpoint of its defining technology for the past 500 years, it has now come to mean a rapid and continuous process of change in the ways in which we read, write, view, listen, compose, and communicate information (Coiro, Knobel, Lankshear, & Leu, 2008). As a mom-to-be on bedrest and a new mom, I found that the web was one of the only ways I could keep in touch with folks in a timely manner, as well as get information about topics I was interested in both personally and professinally. However, as I began to navigate all the different possibilities, I realized I needed to use a variety of critical thinking skills to evaluate the text in my "online" or digital reading that was different from the way I had approached my "offline" or print reading (Weigel & Gardner, 2008). As I looked up topics about my brand new son, I needed to make sure that the sources I was consulting were reliable, accurate, and had multiple perspectives.


According to Eagleton and Dobler (2007), there are two views of the reading process that give readers a background for comprehending print and digital texts -- Cueing Systems Theory and Transactional Theory. Although "offline" reading strategies, such as imagining or visualizing, making connections, analyzing text structure, making inferences or predictions, asking questions, determining important ideas, evaluating and synthesizing, and rereading or adjusting approaches to text (Boke, 2004) are important when reading "online" material, there are other devices to help us be more literate with "offline" text. The New Literacies Perspectives based on the work on Donald Leu identifies five "functions" of new literacies: (1) generating important questions or problems to be solved; (2) locating relevant information; (3) critically evaluating the usefulness of that information; (4) synthesizing information to address those questions or problems; and (5) communicating possible solutions to others (Leu, Kinzer, Coiro, & Cammack, 2004). Although each of these "functions" are important for looking at new literacies, I was most interested in being able to critically evaluate not only the usefulness of the information but also the relevancy, accuracy, reliability, bias (perspective), and commercial bias of the information (Coiro, 2008). Julie Coiro does a great job explaining the elements involved in critically evaluating text:

Relevancy: the information’s level of importance to a particular reading purpose or stated information need
Accuracy: the extent to which information contains factual and updated details that can be verified by consulting alternative and/or primary sources
Reliability: the information’s level of trustworthiness based on information about the author and the publishing body
Bias (perspective): the position or slant toward which an author shapes information
Commercial bias: the extent to which information appears to be influenced by commercial interests for or against a certain product (Coiro, 2008).

On her website for a workshop I took this summer, she gives a series of great lessons and activities to show students how to critically look at elements of a website by investigating relevancy, accuracy, reliability, bias (perspective), and commercial bias. Having students be able to critically evaluate "online" text is extremely important since these types of texts sometimes involve more layers than an "offline" text like a book or magazine article. I can't pretend to fully comprehend all that I encounter in my pursuit of "online" materials, but I can say that now I use a more critical eye. Hopefully, some of these resources will help spark some ideas that you can use to get students more involved in this process. I hope in future posts to continue the conversation of what some are calling Literacy 2.0 (Knobel & Wilbur, 2009) as I think we will be seeing a lot about it in the next couple years.


Resources


Coiro, J., Knobel, M., Lankshear, C., & Leu, D. (Eds.). (2008). Handbook of new literacies. New York, NY: Lawrence Erlbaum Associates.


Eagleton, M.B., & Dobler, E. (2007). Reading the web: Strategies for internet inquiry.New York: The Guilford Press.


Knobel, M., & Wilber, D. (2009, March).Let's Talk 2.0. Educational Leadership, 66, 20-24.