• by Kathryn Brown • IDRA Newsletter • April 2006 •
Curriculum quality is a key element of IDRA’s Quality Schools Action Framework (Robledo Montecel, 2005). IDRA believes that this key element has to be in place to ensure a quality education for all students, in all content areas, in all schools and at all grade levels.
When you think of quality mathematics curricula, what do you envision? Massachusetts Institute of Technology professor and world-renowned mathematician and educator, Seymour Papert asks us to think of curricula in a new way, replacing a system where students learn something on a scheduled day, with one where they learn something when they need it in an environment that shows meaning and gives context as to why it is being learned. It is student-centered where students use what they are learning (Curtis, 2001).
Think for a moment what you would expect… teachers doing and saying; students doing and saying; and parents doing and saying. Reflect on the outcomes and possibilities that would unfold for students, families, teachers and the community if all schools had a quality mathematics curriculum in place.
Standard of Quality Math Curriculum
The National Council of Teachers of Mathematics includes in its Principles and Standards for School Mathematics the curriculum principle: “A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics and well articulated across the grades” (2000). This principle provides a framework in which to make instructional decisions and policies that impact student success and achievement in mathematics.
A quality mathematics curriculum must be vertically aligned, connecting and building upon concepts within and across grade levels, engaging students in meaningful mathematics where they see the value of learning the concepts, and facilitating the development of a student’s productive disposition toward mathematics (NCTM, 2000; Kilpatrick, et al., 2001).
Throughout Texas, school districts have invested many resources in creating a variety of curricula in attempts to meet national and state standards. A shift has occurred over the past decade from the optional use of course curricula to a more pervasive and monitored use. Although use of district mathematics curricula is more often the case than not, the quality of such curricula spans many levels:
- Mediocre test-driven curriculum where the only expectation for students is to pass a punitive, high-stakes standardized test;
- Scripted lessons and timelines detailing verbatim what teachers will say and dictating what materials will be used, leaving no room for teacher creativity or student investigation; or
- Highly challenging and engaging curriculum that is standards-driven and that values teacher’s professional expertise and values students as mathematics learners.
Sample Process Used in Math Smart!
IDRA models the development of highly challenging and engaging curricula through its Math Smart! program. Math Smart! integrates the Five Dimensions of Mathematical Proficiency with strategies for engaging students, dynamic technology tools for building and deepening student mathematical thinking, strategies for supporting English language learners, and strategies for engaging and valuing parents through a variety of methods. The process is outlined below.
Planning with Teachers
Planning sessions are an opportunity for mathematics teachers to reflect on math concepts and their teaching practice. In a planning session IDRA held with Math Smart! Algebra I teachers at one school, teachers reviewed the timeline and discussed how they were exploring the concepts of quadratic functions, finding roots, maximum and minimum values, and evaluating the functions with their students.
Teachers wanted to put into practice elements of the Math Smart! program in the curriculum and lead into polynomials and polynomial properties. What resulted was a deep discussion on how to bring to life quadratic functions, roots and maximums through kicking a soccer ball or football and using physics.
A plan for integrating non-traditional, brain-researched teaching strategies where students discover and present their own methods for simplifying polynomials, finding roots and real-life applications was also developed from the discussion among teachers.
Planning that reflects the teaching practice where teachers also explore the actual concepts is an integral part of building a quality mathematics curriculum. Curriculum development becomes a collaborative effort and parallels what we want to happen in the classroom, where communication and discovery is two-way: students and teachers participating in conversations about mathematical ideas.
Thus, quality curriculum development integrates the teacher and the reflection on the teaching practice and mathematics, where district content specialists and teachers participate in collaborative, curriculum development.
Curriculum that Engages Students
Taking what they had planned, teachers developed an activity that engaged students from the moment the bell rang. The following is a sample from one classroom.
Lesson Introduction: Engaging Students
The teacher began the class by telling students that if she knew how long a football they kicked was in flight, she could figure out exactly how high that ball went without having to chase the ball with a meter stick and ladder. None of her students believed her, and they asked her to “prove it.”
She proceeded to show them a video that she had downloaded from the United Streaming Video resource (that her school has a subscription to) of classic football games and soccer kicks. Students worked in groups of three, beginning with a warm-up activity (see box below) that included a timed brainstorm about quadratic functions in their everyday lives. She asked students to sketch a graph of the football in motion from the video.
As a closing to the introduction part of the lesson and to describe the next part of the lesson, she showed a humorous video of how “not” to kick the football. Humor, not sarcasm, is a highly effective strategy for engaging students. Students were eager to take on the task of finding their own quadratic functions to their kicks.
Experiencing Quadratic Functions
Using soccer balls and stop watches and working in groups of three outside, students kicked the ball and recorded the times the ball was in motion (see activity below). Many questions about how their graph would change surfaced as they were experiencing mathematics in motion. Students wondered about how the graph would differ if they kicked the ball straight up versus across the field and what if they kicked the ball off the ground versus as it is on the ground.
Every student was engaged in the activity. Part of the success of this is attributed to the physicality of the activity. Students were outside of their sterile classroom, and the soccer field became their lab. The act of doing something helps students remember properties of quadratics, what the roots mean, what the maximum/minimum mean and what happens when we change any of the parameters. They have something to tie it to.
When a student is taking a state-mandated test and comes across a problem asking about the change in a parameter, what will the student call upon – an exact equation that she worked on or the experience that explored what happens if the ball was kicked 0.5 meters off of the ground, how it would affect the graph equation, and the maximum value?
This was an activity that students found valuable. Many of the students were involved in sports and were able to relate their life experiences to quadratic functions.
Bringing it Back to the Classroom
After collecting the data and taking a much-needed water break, students went back to the classroom and began using a well-known quadratic function for finding vertical distance to find their own quadratic functions. Using cognitively-guided instruction techniques and building academic language from student’s natural language, the teacher was impressed and energized by how students were able to connect to the meaning of the coefficients and constants for initial velocity (v0), initial height (h0), and the dependent variable, vertical distance (d).
Students discussed in groups and shared with the whole group the meaning of the roots and the maximum in their own graphs, connecting them to their real-life application. Students said such things as: “The first root is where time and distance are both 0, or the origin, because I had not kicked the ball yet, and the second root is when the ball landed, and also the vertical distance is 0. This connects to when our teacher explained that the roots are where the parabola crosses the x-axis.”
Another student explained initial velocity as to how fast it is going at kick-off, but then the ball slows down because it is going up but gains speed as it is coming back down and will reach that velocity again right at the moment it lands.
These are highly complex mathematical ideas that students so readily explained as the meaning of the function d = -5t2 + V0t + H0 was being explored in conjunction with the graphs they had sketched.
It also enabled the teacher to bring in the idea of instantaneous rate of change, a concept formally presented in Calculus I, to her Algebra I students. This teacher has the expectation for all of her students to go on to Calculus I. It shows in statements she makes, such as, “When you get to calculus, you will hear the term ‘instantaneous rate of change’ to describe how fast the ball is going along the path.”
Finding the Functions and Making Conjectures
Students readily volunteered to present to and get guidance from each other in trying to figure out how they would first calculate the initial velocity as it was easy to find the initial height (which was 0 because the ball was on the ground when it was kicked). One student volunteered that even though he “didn’t know what to do,” he would “get help from the class.” The class eagerly helped him, justifying and bringing in ways that they knew how to “do the math” (i.e., solve equations to find the initial velocity given the time and the vertical distance after t number of seconds). Once students found the initial velocity, they were able to write their very own quadratic function describing their own kicks.
Students Challenging Students
The beauty of mathematics is in the “what if’s” – variables changing, parameters and coefficients changing, and analyzing what it all means and how it applies. Using an engaging activity paves the way for students to begin thinking of “what if” questions. It gives them the experience of mathematics.
As indicated above, students began asking the “what if” questions when they were out in the field collecting data. It was natural for them to do this, without being prompted by the teacher. Students were able to answer their questions using their graphing calculators, quadratic functions and natural mathematical reasoning abilities.
In closing the activity, students had to present one of the quadratic functions from the group, indicating the roots, the maximum height of the ball, why they chose that kick, and a what if question to their fellow classmates. Some of the questions included: what if we were on another planet where the gravity is not so strong, what do you think the graph would look like? And, what if I kicked the ball at a faster initial velocity and it was three feet off the ground, how would my equation change?
As a result of planning and of teachers’ experiencing with students a highly challenging and engaging activity that had them involved in mathematical conversations, these Math Smart! teachers wanted to continue to contribute to the district curricula and include collaboration and teaching practice reflections as an ongoing way of ensuring a quality mathematics curriculum for their students.
IDRA and the teachers were able to explore a model for creating a quality mathematics curriculum: reflecting on current curricula, sharing ideas on how to get students involved and appeal to their interests so they find mathematics valuable, using available resources, breaking out of traditional one-way conversations into two-way conversations with students about the mathematics, and realizing that as time and technologies change, so too will the curriculum.
Quality curriculum is dynamic; involves teacher practitioners in ongoing reflection, development and refinement; values students’ experiences and the knowledge they bring; and is rigorous and vertically aligned so that students are not only prepared to enter higher-level mathematics courses, but also experience higher-level mathematics within their current courses.
Are You Ready for Some Football?
This is a sample mathematics activity developed by secondary teachers during IDRA’s Math Smart! training series.
Brainstorm for three minutes. Where are parabolas (quadratic functions) in your everyday life?
After watching the video, sketch the graph of a football/soccer ball in motion. Identify on your graph the roots, maximum, when the ball is going the fastest, and when the ball is going the slowest.
a. holding the football,
2) Each “kicker” gets two kicks. Record the time in seconds for each kick:
3) Let’s find the quadratic function and graph for your group’s kicks. Work as a group to find and verify your calculations. Using what we know about physics and gravity on earth, the form of the quadratic function for a ball in motion is: d = -5t2 + V0t + H0
Fill in the values you know after the ball has hit the ground. Work together as a group to set this up so you can find the initial velocity (V0):
4) Now that you know the initial velocity, write your quadratic function. For example, if my initial velocity was 25 m/s2 and I kicked the ball from the ground, my equation would be:
Select one of the functions above and sketch the graph at right. Be sure to show the time on the x-axis, various heights of the ball while in motion, maximum height, roots, when the ball is going the fastest, and when the ball is going the slowest.
The “What If” question for your group: What if the ball was kicked 0.5 meter off the ground (like a punt) initially…. What effect would it have on your
5) Plan your presentation.
Source: Intercultural Development Research Association, 2006.
Curtis, D. Start With the Pyramid (San Rafael, Calif.: The George Lucas Educational Foundation, 2001),
Kilpatrick, J., and J. Swafford, B. Findell (Eds). Adding it Up: Helping Children Learn Mathematics (Washington, D.C.: National Research Council Mathematics Learning Study Committee, November 2001).
National Council of Teachers of Mathematics. Principles and Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 2000).
Robledo Montecel, M. “A Quality Schools Action Framework – Framing Systems Change for Student Success,” IDRA Newsletter (San Antonio, Texas: Intercultural Development Research Association, November-December 2005).
Kathryn Brown is the technology coordinator in the IDRA Division of Professional Development. Comments and questions may be directed to her via e-mail at firstname.lastname@example.org.
[©2006, IDRA. This article originally appeared in the April 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.]