Unit 1: Defining Mathematical Literacy
Welcome to Unit 1!
Welcome to the first unit in the mini-course on Mathematical Literacy: Teaching with, developing, and utilizing Math Journals. To start off this unit, I want you to think in your head (or write out) words that come to mind when you think of Mathematics and Literacy. This can be about the two terms separately or even together as the idea of Mathematical Literacy. I imagine there are many ideas that come to mind about both. As we continue to uncover the ideas that are deeply rooted in the study of Mathematical Literacy, see how some of these words and ideas are related to the topics discussed. Perhaps the connections you see will help to better relate your current beliefs to the new ideas and understandings presented.
What is Mathematical Literacy?
Though this unit may be called "Defining Mathematical Literacy," I will not be handing you out a definition of the term. My purpose in this unit is for you to come to a conclusion on what you believe this definition truly is. Mathematical Literacy encompasses a large number of tools, skills, and understandings, as well as ways of thinking, doing, and teaching. Take a look at the following cartoon:
What does this cartoon truly communicate, beyond the student's misinterpretation of the idea of a variable? What underlying math literacy issue is at work here? By the end of this unit, you will have a better idea of what ideas and understandings impact this students ability to make sense of the term 'x.'
In order to understand and in turn define mathematical literacy, we must also make ourselves aware of the skills and methods for teaching mathematics. Mathematical literacy is about engaging in the real world conversation that surrounds mathematical thinking. For students to engage mathematical thinking and perspectives there are many skills, tools, and mathematical understandings that students must have. On the flip side of the coin, teachers must also consider how their own methods and practices impact the fostering of mathematical literacy in students, and how they can better help their students to engage in the deeper conversations that are necessary for mathematical thinking.
A first step in understanding Mathematical Literacy is to consider the skills involved in mathematical problem solving. There are several skills identified as new and important skills for mathematical thinking, which in turn help students to become more mathematically literate. These skills help students to process and solve their way through mathematics in novel, real world situations, an important implication for the study of mathematics.
Students should be able to:
- Apply mathematical skills in both familiar and unfamiliar situations.
- Learn how to select and combine various skills for the purpose of solving problems.
- Work in collaborative groups to solve problems and talk about mathematics.
- Write about mathematics and the problem solving process (A crucial skill for mathematical literacy).
- Use mathematical ideas and ways of thinking to communicate about the world (A second crucial skill for mathematical literacy).
- Reflect on their own learning process (A component of Mathematics Journals).
- Use technology to support problem solving in math.
- Utilize their skills appropriately and effectively in combination with research.
Once students develop many of these important skills they will better be able to participate in the three major domains of mathematical literacy. These three domains are comprised of Reading, Interpreting, and Expressing. When these three modes of thinking and communicating about mathematics are combined, students will start to become more mathematically literate.
Reading, in the context of math, is about learning to deal with the language specific to math. Within mathematical understanding, students encounter a math-specific set of technical language that they must comprehend to work and solve their way through math. Additionally, students must learn to interpret components of a language that is very different and new. Students will encounter symbols, algebraic equations, diagrams, and graphs, each communicating about the real world in a mathematical lens. In order to be literate in mathematics, students must learn to interpret and make meaning from this new and unfamiliar language.
This leads to the next domain important for mathematical literacy of Mathematical Interpretation. In making meaning from mathematics, students must learn to transform their everyday language into mathematical language, in a way that is accurate and meaningful. This is the interpretation step for literate mathematics students. There are a number of ways that this can look as well. Students may represent the real world and situations they encounter in it through numbers, symbols, diagrams, and graphs. (Which are all components of math language).
A third domain of mathematical literacy is expressing. In expressing mathematical understandings and ideas, students must learn to use math language for means of talking to others about every day occurrences and situations. This can occur in many ways, whether it be simply talking in mathematical language, modeling a process or skill, problem solving in math, or performances, such as a math demonstration or proof. When students are able to communicate about math, they add on the third crucial component to their own mathematical literacy. In a way, the three domains are simply, Reading, Comprehending/Understanding, and Writing/Speaking about mathematics.
Try working through one of these examples in order to practice using mathematical language and for purposes of self-reflection.
Example #1: Write a story about the information shown in the graph.
Example #2: Create a new representation of the article utilizing graphs and/or tables.
(Clarke, pg. 10)
Having completed one of these exercises, how did the exercise help you to better understand the domains of Mathematical Literacy?
There are many mathematical tools that a math literate student can use. However, more importantly, students must learn how, when, and why to use the tools that they have learned. Teachers help to shape students fluency in using tools to make meaning out of the situations that they encounter in problem solving through mathematics.
- Tool Possession: Does the student have the tool to solve a problem? (For example, can a student take an average of a set of numbers).
- Tool Understanding: Does the student know how to use the tool? (For example, if a student is given an average of a set of 5 numbers, can they develop a set of data that results in the same average).
- As a Concept: Is the student able to use a real world situation to demonstrate an understanding of the concept? (For example, using a string to measure and display the average height of a group of students).
- Tool Application: Can the student apply the tool using a real world context? (For example, finding averages related to expenses, incomes, or sales).
- Tool Selection: Does the student know when to use a tool? (For example, if a student is given an open-ended question that can be solved using averages, can they appropriately select that tool to solve the problem).
When students are fluent in the usage of mathematical tools they can better work their way through the problem solving process. In addition, when they can do this, they will be able to better communicate and express how they solve problems and what tools they decided to use when/how to solve those problems.
In Mathematics, there are a many different ways for students to think about mathematics in order to create the most meaning out of what they are learning. Sometimes, it is more important for students to be aware of their own thinking about a problem then to simply attempt the problem. When students are able to think about the best way to look at a problem, they can better generate a solution pathway into working through the mathematical ideas presented.
- Abstraction: a student’s ability to decontextualize a problem; recognize the mathematical ideas outside of the situation of the problem.
- Contextualization: a student’s familiarity with mathematical ideas in a variety of contexts/situations.
- Interconnectedness: a student’s ability to change how the information is represented and still see the similarities in mathematical ideas throughout.
Explore the following example to see how you think about the problem. Is it necessary for you to identify solely the mathematical ideas? Or do you need to understand something about the situation to solve the problem? Is it necessary for you to represent the situation in a different form to make meaning of the mathematical ideas?
- A photograph is 4 inches wide by 6 inches long. If an enlargement of this photo is 10 inches wide, how long is it? If the enlargement needed to fill a billboard is 8 feet wide, how long would it be? Use a graph to represent the photo's length for any width up to 12 feet. (Clarke, pg. 14)
How did your understanding of the context help you to understand the problem? How did you use that understanding in combination with the mathematical idea of proportions to solve the problem?
Good Educational Practice in Mathematics
Though many of the preceding ideas about math are very student-centered, they help teachers to better understand what is necessary for good instruction and assessment of important mathematics. Teachers must begin to think about what ideas and messages they truly want to convey about mathematics, and not just what the students need to know. Sometimes a teacher’s intentions for students’ learning in mathematics are not accurately matched to the way they assess. Teachers should be careful to maintain a sense of sensitivity towards how they convey ideas about math. Since each individual constructs their own personal meaning and understanding of mathematical ideas, teachers must prepare to work with and develop these meanings amongst their students. Furthermore, teachers must develop respect for diverse forms of learning existent in their classrooms. To do this teachers should utilize a form of assessment that looks at what students attach meaning to and how they use that to aide their problem solving process. This practice will be explored further in later units in the course.
Next, in order to ensure good educational practices teachers should recognize the two important messages that are conveyed when instructing and assessing students. These two messages are that the teacher is interested in what the student knows as well as that the teacher wants the student to know that what they are teaching is important mathematics. One way to do this is by monitoring the forms of questions that are asked. A teacher’s questions should be more than a simple search for the right answer. The questions asked should be about conveying the importance of learning for understanding of mathematics. When this occurs, students can truly focus on the important understandings in math and become more mathematically literate.
- Clarke, D. (1997). Constructive assessment in mathematics: practical steps for classroom teachers. Berkley, CA: Key Curriculum Press.
- Mathematics Cartoons. (n.d.). Home. Retrieved November 16, 2012, from http:// simonsingh.net/books/fermats-last-theorem/wacky-fermat-stuff/mathematics-cartoons/