• by Kathryn Brown • IDRA Newsletter • March 2006 •
Mathematics is amazing. Seeing a student’s eyes focus on an internalized visualization map of mathematical concepts that are being unearthed and connected as she is detailing her intricate, mathematical thoughts is a breakthrough experience.
Having this experience yourself goes beyond the step-by-step kind of success one often comes across in carrying out the steps of solving an equation. That set of steps to solving the equation is just one of the pieces to this awesome mathematical experience. Others involve making connections to other ideas and seeing value in the reasoning and solutions to your own.
If you never had the opportunity to have this feeling in math class as a student, recall when you tackled a problem as an adult using your reasoning skills in your work or home and were able to find and implement solutions. Think of the journey you took to get to that point: drawing from other problems you may have solved of this nature, communicating with others to get their feedback or input, or maybe sketching out a map in your mind or on paper that helped you reason out your thoughts that spiraled you to an “aha” moment.
This feeling of accomplishment and success is what IDRA envisions for all students to experience in mathematics, throughout their lives.
Not surprisingly, many people do not have pleasant memories of math class. In survey after survey, adults note that mathematics was their least favorite subject in school. In a world where job opportunities are increasing three times faster for those that involve mathematics and computer science as compared to jobs that do not, we must break down barriers and create opportunities for access and success for all students in mathematics. Their future depends on it, and so does ours.
How do we accomplish this? How do we ensure that students enter college without having to take remedial math courses before they are prepared for the ones that give them real credits toward their degree? How can we exponentially increase the number of students, especially minority and female students, who enroll in higher-level mathematics courses and who have been traditionally left out? How can we engage students and use tools from their world to explore mathematics in meaningful ways? How can we, together, move beyond what has always been a challenge for this nation: teaching and learning mathematics? Most importantly, how do we help students feel successful, eagerly challenged and feeling those “butterflies in their stomach” that you get when you are energized about learning something that inspires you to search and forge ahead in the quest for knowledge?
IDRA believes that we, as teachers, parents, administrators and community members, should provide the means so that the focus of instruction and opportunities for learning are in place so that all students are mathematically proficient. IDRA is finding solutions to these questions through its work with several school districts in its MathSmart! secondary mathematics program (see Math Smart!).
First, let us explore this question: What does it mean to be mathematically proficient? Is it being able to easily do mental math, create and solve equations from situations, or even memorize formulas? Merriam-Webster Online defines proficiency as being “well advanced in an art, occupation or branch of knowledge.” What we are saying is that all students need to be on the road to mathematical proficiency beginning in pre-kindergarten.
What is an area that you are proficient in? Think about the process involved in reaching that level of proficiency. It is complex and rich and takes time to develop.
The Committee on Mathematics Learning was established by the National Research Council in 1998. In its report, Helping Children Learn Mathematics, the committee chose the term “mathematical proficiency” to capture what it means to learn mathematics successfully (NRC, 1998).
Mathematical proficiency has five strands:
- Conceptual understanding – understanding mathematical ideas and making connections to previously learned math concepts;
- Procedural fluency – carrying out effectively and efficiently procedures, such as addition, subtraction, division, when solving problems, not limited to written but including mental and hands-on strategies;
- Strategic competence – being able to formulate, represent and solve math problems (often referred to as problem solving) using various strategies;
- Adaptive reasoning – reflecting on, justifying and explaining mathematical ideas; being able to think logically and reason; and
- Productive disposition – seeing oneself as a successful learner of mathematics and its application to one’s own life situations.
These five dimensions of mathematical proficiency work together. Think of them as a rope that is comprised of five strands. One is not more important than the other, and they work in conjunction with one another.
You can also think of these strands as five musicians in a band. Each one plays its part in creating the music. Imagine Barry White’s “Can’t Get Enough of Your Love” without his deep melodic voice. There would be something missing; something inside of you guiding the ear to search for the essential sound of his voice. Barry White’s voice takes the song to another level, one of feeling and containing soul.
Such is the case with these five dimensions: having a productive disposition is a key element that makes possible the other four dimensions.
Math in the Real World
The committee’s report states: “The teacher of mathematics plays a critical role in encouraging students to maintain positive attitudes toward mathematics. How a teacher views mathematics and its learning affects the teacher’s teaching practice, which ultimately affects not only what the students learn but how they view themselves as mathematics learners” (NRC, 1998).
Having a productive disposition is essential to building any of the other strands, such as adaptive reasoning. If I value the mathematics I am learning and see how it connects to my world, I will have the confidence and the means by which I can think logically and communicate with my peers my own mathematical reasoning.
Students long for this type of a connection where math meets their real world. So many times we hear students asking “When will I ever use this?” in mathematics classrooms across the nation and across grade levels. So many times teachers and parents respond: “When you become an engineer” or “When you become a carpenter.” Students want to know how they are going to use it today.
IDRA’s Parent Information and Resource Center works in partnership with ARISE, a women-led, faith-based organization based out of six colonias in the lower Rio Grande Valley in Texas committed to local leadership and delivery of social services. Through this partnership, one focus has been to build student leadership where students are the bridge for connecting their parents and community to technology – an engaging effort in breaking down the digital divide. This is done through the ARISE centers that are equipped with a shared cable connection and computers that provide students and their parents access to information about their community, schools and world (see article).
When students and parents were analyzing school data on the Texas Education Agency web site and exploring why their school was listed as not meeting Adequate Yearly Progress (AYP) standards, one of the ninth-grade girls said that they were doing similar work in her math class, however, they were using “fake” data that did not really pertain to them. She wished that her teacher would have taken them to the TEA web site to look at data about their own school so that the students could analyze that data and make some calculations that enable them to make “real recommendations” for improving their own school and what students and teachers need.
This is an example of teachers creating an environment that nurtures a productive disposition and makes mathematics meaningful where students have a context for weaving the concepts of mean, median and mode (conceptual understanding); have a practical reason for calculating them (procedural fluency); and formulate strategies and share with one another solutions to questions like, “Which measure of central tendency best represents our school and why?”
People gasp when they hear, “How horrible! He graduated from high school and he could not even read.” Would the same hold true if we said, “How horrible, he graduated from high school and he could not even do algebra”? The gasps do not come. But they should.
Yet, in this era of punitive, high-stakes assessments, where meaningful, student-oriented teaching is sacrificed to test preparation, it is no surprise that schools emphasize one dimensional, prescription curricula that ensure mathematical mediocrity rather than proficiency. It is up to us as educators to ensure that every student is given every opportunity and that systems that value all students as mathematical learners are set in motion so that every student graduates from high school mathematically proficient.
Merriam-Webster Online. http://www.webster.com/dictionary/proficient.
National Research Council, Mathematics Learning Study Committee (Corporate Author) and Kilpatrick, J. J. Swafford, B. Findell (Eds.). Adding It Up: Helping Children Learn Mathematics (Washington, D.C.: National Academies Press, November 2001).
Kathryn Brown is the technology coordinator in the IDRA Division of Professional Development. Comments and questions may be directed to him via e-mail at email@example.com.
[©2006, IDRA. This article originally appeared in the March 2006 IDRA Newsletter by the Intercultural Development Research Association. Permission to reproduce this article is granted provided the article is reprinted in its entirety and proper credit is given to IDRA and the author.]