Unit 2: Understanding the Importance of Mathematical Literacy
Welcome to Unit 2!
In order to understand the importance of Math Literacy, we must also understand the ways in which mathematical tasks are connected to assessment. This unit works to explore the connections between teaching, learning, and assessment in a way that makes sense of mathematical literacy. Additionally, this unit will look to explore how choosing the right task will help to more clearly identify the assessment necessary for the task.
What is Teaching? What is Learning?
An important understanding that all teachers should have is of the differences between teaching and learning. As far as teaching goes, teachers know what they taught in a lesson. They know the content they covered, the instructional strategies they used, and the methods of delivery. What is often unknown is what was actually learned. Each individual student comes out of the same lesson with a very different understanding and connection to the content covered. This is based on many different characteristics of the learner, including their predisposition to the material, learning style, their interaction with the content, and the deliver style of the lesson as well. In order to see what a student understands in mathematics, it is necessary to reveal this information through assessment. To truly understand how mathematically literate our students are we need appropriate assessments the truly look to make sense of the students' learning.
In the past, often learning is not revealed in assessment. Many teachers have assessments that work in in two ways. One way is that students would replicate the teachers answers. The other way is that they choose not to answer at all, at risk of getting the answer as what they perceive is "wrong." The view of assessment as question and response needs to be broken in order for us to truly see how mathematically literate our students are. Assessment must become the exchange of information. On the one hand, students are looking to show what they know, understand, and what skills they possess. Where as, on the other hand, teachers are attempting at communicating what is important mathematics and necessary understandings for the students to hold. This form of assessment must be viewed as a part of instruction, deeply embedded in the content, processes, skills, and understandings. Many informal dialogues and observations are not viewed as forms of assessment, but rather tests and assignments are viewed as assessing. However, assessment is much more than one assignment. Teachers are continually assessing in many ways: observing a student working on a project, engaging students in a classroom discourse, o reven talking to a student individually about their performance. These observational assessments provide great insights into the understandings and skills that each student possesses. Additionally, these informal observations can help to make sense of assessments later on, in terms of interpreting what the student has learned.
Fitting the Task to the Assessment
In mathematical literacy, there are many tasks, skills, and understandings involved. Since students must hold all these varying skills and understandings to complete mathematical tasks, it follows that there must be multiple different forms of assessment to make sense of student learning in these areas. The first step in figuring out the assessment is to know what type of performance is being elicited from the students. By using a teacher-generated task to assess students, rather than a standardized, normed test, the task will more accurately match individual student's needs and abilities. In turn, this will generate more useful information about the mathematical skills and understandings a student possesses. This important distinction should be made because assessment is the key to revealing insights into any one particular student's learning. To truly find out how mathematically literate our students are we must develop assessments similar to these discussed here.
Some characteristics of more constructive assessment are open-ended tasks, diverse contexts, the allowance for multiple modes of communication, and a range o modes to display understanding and expression. In an open-ended task, students are given the opportunity to answer a question through a variety of sophistications. This idea is important for many reasons. Importantly of all of those reasons is that the ability to respond in varying levels of sophistication allows for students to truly express their understandings and in turn allows the teacher to truly see how mathematically literate they are; what understandings and skills they have.
Consider the following example presented by Clarke:
- Students are asked to answer the following open-ended question on an assessment:
- A number is rounded off to 5.8. What might the number be?
- Students may respond at many varying levels of sophistication, including the following:
- No response or an incorrect response
- A single value: 5.81
- Several values: 5.77, 5.79, 5.81
- A systematic list: 5.75, 5.76, 5.77, 5.78, 5.79, 5.80, 5.81, 5.82, 5.83, 5.84
- A class of solutions (with varying levels of precision): The smallest it can be is 5.75 and the largest it can be is less than 5.85. (x: 5.75≤ x < 5.85)
- (Clarke, pg. 29)
Having diverse contexts for the task is also important for more constructive assessment. Teachers should ensure that their tasks are posed in a range of contexts with a range of detail surrounding those contexts. To figure out the richness and depth of the context used, teachers should consider how much they value the contextualization vs. the mathematical consistency for that given task. This is key to student performance considering that each student's personal interaction with contexts is just that, highly personal. The context used and the depth of detail can significantly change how the student understands and creates meaning from the problem. In the digital age, mathematical modes of communication have changed and developed. Students can now orally express understandings, visually express understandings, or even electronically express their understandings with relative ease. The many modes of communication link directly to the range of modes of expression. These modes of expression include, physical models, calculations, geometric constructions, narrative expression, demonstration, performance, or symbolic notations. Each has a great deal of value attached to it as to how it reveals student understandings and sophistication of thought. Though there are many modes of both expression and communication, not all need to be included in every task. However som teachers should be open to and allow students to use a different mode of expression if they believe that it will more effectively demonstrate their understanding, regardless of whether that mode was included in the task during task development.
One task and alternative form of assessment that is key to this mini-project is the mathematical project. Mathematical projects are an opportunity for extended problem solving and even investigative work. These tasks offer a great change in pace that many students can easily become engaged in and motivated to complete. Since mathematical projects are long-term (in comparison to other mathematical performance tasks), there are many other considerations to be made. A mathematics journal can be key in supporting mathematical projects and in revealing some of the learning done within them. As students employ self-reflection and document their learning process in their mathematics journals, student learning will become more apparent. This topic will be discussed further in the next unit.
- Clarke, D. (1997). Constructive assessment in mathematics: practical steps for classroom teachers. Berkley, CA: Key Curriculum Press.