Over the summer, I had the opportunity to attend the Intel Mathematics Training provided by the Northeast Florida Regional STEM2 Hub in Jacksonville at the University of North Florida. Here is a personal reflection of the overall experience and how it impacts my instructional and content pedagogy in the classroom.
How did what I learn in math deepen my ability to integrate math into science instruction?
I have always been profoundly aware of the importance of learning concepts in a sequence to be able to make critical connections in a spiraled learning model and to develop cognitive ability. As a science teacher, I am immensely focused on understanding the conceptual development of a “Big Idea” and the knowledge scaffolding that is required in order to fully synthesize the learning of a concept. What I did not fully realize is that much of this same model holds true for the practice of mathematics.
The practice of science requires a careful construct of grammatical usage: so does the practice of mathematics. Both science and mathematics must be able to be taught at the level where students understand how to synthesize information, not just an algorithm or acronym for recall. This teaching skill allows students to apply a conceptual model to scenarios with which they may have not come into contact in their previous experiences. In other words, when students have practiced an algorithm until proficient this proficiency has traditionally translated into “success” in the academic construct. However, this application to a problem solving scenario may not hold true. Without understanding at a conceptual level, application is not possible. Again, this skill has been required in scientific conceptual understanding; however, I had not realized before the Intel Training that the same level of critical thinking is required in mathematics.
Importance of Math Vocabulary
One of the simplest, yet impactful examples of the mathematical conceptual model is using appropriate vocabulary. The use of appropriate vocabulary in science is well recognized. In mathematics, the same holds true. However, in class, when we began to explore the concept of the simplest level of subtraction, I realized the concept, which should recognize that this is a difference or a change was diluted with other vocabulary. This included teachers describing subtraction as “taking away from” or the “opposite of addition”. Another common statement was that subtraction was the inverse operation of addition. The challenge for the teachers is to identify and correct their own vocabulary as their understanding deepened. The Intel Mathematics trainers allowed for the gradual change in thinking until the thinking became the teachers own, just as we should want to happen in our classrooms with students. The same practice and pedagogy that we would like to demonstrate in our work was modeled by the trainers.
Examples: Graphing and Reporting Data
When students are presented with data in the form of visual representations such as graphs, studies suggest this is an area of weakness in mathematical practice. This disconnect can result in a poor foundational understanding of basic functions and data analysis. Slope is one example of an area where students struggle in Algebra. In scientific claims, many points are made by using the slope functions to validate the claim. This “rise over run” systematic practice is used widely when data are extrapolated to inference rather than observable in graphs. Students can make connections to actual experiences or context examples when marrying the mathematical practice with the scientific validation of claims. It is vital that students understand the foundational application and purpose of slope in science and mathematics.
In mathematics, a reiteration of the tabular, graphical and algebraic representation of inverse relationships are very closely tied. So this critical piece of learning will help support the understanding that when columns of tables are interchanged, the axes of the graphs are interchanged.
Use of Grammatical Relationships
Applying grammatical concepts into mathematics was a practice that I had not utilized until the Intel mathematics training.The practice of applying that adjective-noun relationship to mathematics was an eye opener for me! While this component of literacy is almost second nature in the literacy block, I would never have made the connection to mathematics. By applying the grammatical structure of adjective-noun relationship, the relationship between numbers became abundantly clear. While this may seem a simple construct, it has not been widely used in instructional or content pedagogy in the education systems in which I have been involved.
In addition, understanding that numbers modify nouns was a new concept for me. While a certain amount of precision to detail and attention to structure is required in a mathematical computation, it is not always automatic to think of the process as a grammatical relationship. However; with the ability to identify the relationship, it becomes a much easier process to complete the task. For example, when you say 6 is greater than 2 it is simply numbers. If you impose the noun, such as six bananas are greater than 2 bananas then it is much easier to relate to as a relationship. Another example of this process would be to identify a denominator as a noun, which clarifies its likeness or unlikeness. Using this structure, students can relate to the numerical representational model with greater ease than without the use of nouns.
One of the areas that students struggle with in mathemaitcs is the concept of place value. Utilizing the understanding of literacy with the noun as representative, the concept of zero as a placeholder becomes much more concrete. I recognize that students strugglewith the concept of place value, particularly in long division. This thorough explanation and use of literacy helps to define the understanding of zero.
Another simple, yet impactful moment in the course was how numbers, namely fractions, were represented visually. This information built upon the previous work we had done in subtraction and allowed us to parallel that work. We incorporated the adjective-noun relationship and the transition from subtraction to division was much more understandable when building scaffolds for students.
How this Supports Math Instruction AND Science Instruction
Intel mathematics broadens the educator’s perspective in the conceptual understanding of the usage and application of mathematics and mathematical practice. When educators are given the time and resources to thoroughly study the development of student learning, the task of appropriate instruction is much more effective. Intel Math Training allowed us to understand that the changes in instruction are minor in most cases. The training has been tailored to what have been larger issues in math instruction, so many common issues are easily identified. The work that I experienced was based on conceptual underpinnings and was not designed to support the existing text resource, but rather the mathematical concepts and practices.
Connections Between the Nature of Science and Mathematical Literacy
Obviously, science and mathematics are partnered in their practices and applications. Mathematics often stands as a quantitative substantiation for scientific claims. It is crucial that we identify the appropriate mathematical displays, data and models to support the claims within the processes of scientific claims.
Productive Struggle and Learning Moments for Educators
In my experience in the Intel Math Training, there were a few times when I struggled with grasping the concept in the instructional model. For example, working in base five, with representational models rather than integers was unfamiliar and took a substantial amount of time to grasp. This experience led me to the understanding that thinking can become linear and rigid and that such a narrow approach limits the understanding of student’s perceptions of a concept.
As we work to identify students’ level of understanding within a mathematical concept or practice, it is vital that we understand the concept ourselves so thoroughly that we can recognize the level of understanding in our students.
Additionally, there were times when I struggled with a physical model to represent a function rather than a rote memorized algorithm. This disconnect helped clarify my thinking in that area where students had not yet developed the algorithm but instead had a model constructed to represent their thinking. As educators advance in mathematical practice, they may have forgotten the journey that they took to develop their personal conceptual understanding. This work with Intel Math reconstructs the conceptual model from a rudimentary level, allowing teachers to reconnect with the processes that students experience and to gain a much more thorough understanding of their level of mastery.