Stem Integration in Mathematics Standards

The National Council of Teachers of Mathematics (NCTM) was founded in 1920 and has become the largest mathematics education organization in the world. Its board of directors established a commission to help improve school mathematics. In 1987 they drafted a manuscript that resulted in the Curriculum and Evaluation Standards for Teaching Mathematics (NCTM, 1989). The document was intended to create an extensive framework to lead reform in school mathematics in the future and to serve as a foundation to improve the teaching and learning of mathematics. Several updates followed, culminating in the Principles and Standards for School Mathematics in 2000. In 2006, NCTM published the Curriculum Focal Points to identify important topics at each grade level pre-K-8 to further implement the Standards (NCTM, 2014).

The State Board of Education is responsible for providing curriculum and instructional materials for public schools in Texas. The Texas Essential Knowledge and Skills (TEKS) were developed on the basis of the NCTM Principles and Standards as standards for what students should learn and be able to understand at particular grade levels (Texas Education Agency, 2014) and were most recently updated for adoption for the school year beginning in 2012.

The Common Core State Standards (CCSS) for English language arts and mathematics were established through a state-led initiative and were “designed to ensure that students graduating from high school were prepared to enter credit bearing courses in two or four year college programs or enter the workforce” (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 1). Development of the standards included consideration of scholarly research, surveys on skills needed for students entering college or workforce training programs, and findings from Trends in International Mathematics and Science. The middle school and high school standards require students to apply mathematical thinking to real world situations and to think and reason mathematically. Adoption of the standards by individual states was voluntary.

The Texas Career and College Readiness Standards (TCCRS) include the areas of English/language arts, mathematics, science and social studies. The TCCRS were designed to represent a full range of knowledge and skills that students need to succeed in entry-level college courses as well as in a wide range of majors and careers. The implementation and dissemination of TCCRS has taken a long time. The TCCRS were created to embody the total scope of knowledge and skills that students need to be successful in beginning-level college classes in addition to an extensive array of careers and majors. According to research, over 80% of jobs in the 21st century will require some post secondary skills (Matthews, 2012). The TCCRS was intended to address the needs for preparing students for postsecondary experiences and careers in addition to high school graduation standards, which basically focused on mastery of basic skills. Even high-quality college prep courses will not prepare a student to pursue certain majors in college. The TCCRS separate themselves from high school standards because they do not focus on content knowledge as an end—the content encourages students to become involved in more profound levels of thinking. These standards focus on “keystone” knowledge and skills. The assumption was that students had already mastered state standards. The final section of the TCCRS contained cross-disciplinary skills that were as important as the four content areas because they contained skills like problem solving across content areas.

There are many different definitions and interpretations of the meaning of “STEM” and what STEM principles include. For the purposes of this study, we confined STEM principles to any content or teaching strategy that involved the integration of two or more areas of science, technology, engineering, and mathematics. Because PBL and/or solving real-life problems almost always contained instructional strategies integrating two or more subject areas, we also included all indicators of these instructional strategies.

The four different middle school mathematics standards (see Appendix) that were examined for STEM integration were the NCTM Principles and Standards for School Mathematics, TEKS, Common Core State Standards for Mathematics (CCSSM) and TCCRS.

The coding followed a systematic and organized process. As we read through the four documents from cover to cover, we looked for use of the phrases “real-life problems,” “problems … in everyday life,” “technology as appropriate,” “with and without technology,” “connecting mathematics with other domains such as science and art,” “with computer,” “using materials such as . dynamic geometry software,” and other parallel statements. We copied the statements into a spreadsheet. We counted the statements that we selected as indicators of STEM integration.

After examining each of the four documents to locate instances of the integration of two or more STEM subject areas, a table was created and samples of each were included in the Appendix. The complete table, located at aggi-estem.tamu.edu/STEMstandards.pdf, demonstrated how STEM learning was infused into the various standards. Phrases from statements that were counted as an instantiation of STEM learning were displayed in Table 1. Table 1 also contained the number of statements that were found referring to real or everyday problems, integration of science (or possibly other subjects), and technology for each of the standards analyzed.

Sometimes the key phrases were embedded in the content, sometimes in the examples, but most often they were located within process standards. Additionally, there were no instances where integration was specifically required in the documents, but there were a few strong statements such as

• “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology” (TEKS, Grade 7, 2014, p. 1).

• “Finally, many measurement concepts and skills can be both learned and applied in students’ study of science in the middle grades” (NCTM Principles, Measurement, 2000, p. 241).

• “Many investigations in middle-grades geometry can be connected to other school subjects. Nature, art, and the sciences provide opportunities for the observation and the subsequent exploration of geometry” (NCTM Principles, Geometry, 2000, p. 238)

• “Teachers should regularly ask students to formulate interesting problems . both within and outside mathematics” (NCTM Principles, Problem Solving, 2000, p. 258).

• “Clearly, rich problem contexts involve connections to other disciplines (e.g., science, social studies, art) as well as to the real world and to the daily life experiences of middle grades students” (NCTM Principles, Connections, 2000, p. 274).

• “If all the middle-grades teachers in a school do their best to connect content areas, mathematics and other disciplines will be seen as permeating life and not as just existing in isolation” (NCTM Principles, Connections, 2000, p. 278).

Having been published in 2000, the NCTM Principles and Standards for Mathematics was the oldest document examined and analyzed for integration of STEM areas. The most obvious application of STEM integration was in relation to the numerous references to the use of technology, such as graphing calculators and dynamic geometry software or other computer utilities. Simulations were suggested for use in exploring data analysis and probability. In the Introduction to the Standards for Grades 6-8, it was noted, “The understanding of proportionality should also emerge through problem solving and reasoning, and it is important in connecting mathematical topics and in connecting mathematics and other domains such as science and art” (NCTM, 2000, p. 212). In the Number and Operations Standard and the Algebra Standard, there was no specific mention of integrating other subject content. However, in the Geometry strand, “nature, art, and the sciences” were listed as possible arenas in which to experience geometry in our world. In the Measurement Standard, “maps, blueprints, science, and even literature” (p. 245) were given as sources for creation of problems for students to learn more about similarity, ratio, and proportionality. It was suggested that mathematics teachers collaborate with science teachers for students to experience real-life data collection for the Data Analysis and Probability Standard. In the Problem Solving Standard, teachers were encouraged to ask students to “formulate interesting problems based on a wide variety of situations, both within and outside mathematics” (NCTM, 2000, p. 258). As might be expected, the Connections Standard was replete with advice on using other disciplines to find appropriate problems for students to solve and even working with other teachers to develop “integrated units of study” so that academic subjects “will be seen as permeating life and not as just existing in “isolation” (p. 279). The representation standard alluded to collaborative lesson planning and integration of mathematics and science with the statement, “Drawing on what they have learned in science and social studies, they might then….” (NCTM, 2000, p. 284).

The version of the TEKS we analyzed was implemented in September 2012 and is subject to change periodically. In the Introduction to the TEKS for each of the grade levels 5, 6, 7, and 8, technology use was listed as a choice of tools that students utilized to solve problems. One significant difference from the previous version (2009) was that the process standards were more detailed and explicit. For Grades 6, 7, and 8, the underlying processes for the 2009 version specifically stated that students were to “solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school” (p. 2) as well as again listing technology as one of the tools students may choose to use. The 2012 implementation implied such connections by stating that students should be “successful problem solvers” and “use mathematics efficiently and effectively in daily life” (p. 1). Because problems in daily life generally require knowledge of both science and mathematics and/or were best solved with the aid of a technology tool, clearly integrated STEM teaching was not only appropriate but also most effective in helping students develop conceptual understanding.

Although the CCSSM was a relatively new set of standards, the integration of STEM subjects was not mentioned within the content strands. There were instances in the standards for mathematical practice such as “[students] reason inductively about data, making plausible arguments that take into account the context from which the data arose” (mathematical practice 3) (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 1). In the modeling practice standard it is stated that students will be able to apply the mathematics they have learned “to solve problems arising in everyday life, society, and the workplace” (p. 1). The practice standard for using tools appropriately lists technology such as graphing calculators, spreadsheets, and dynamic geometry software. Although the content standards do not address integration of STEM subjects, most of the examples of problems given were framed in a science context. The phrase “solve real-life and mathematical problems” was found in a variety of places, and we counted those instances (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). We chose to include references to “real-life” problems because such problems almost always involve more than one subject area. For example, when people use mathematics in their lives it is almost always in the context of some scientific, technological, or business application (THECB, 2014).

The TCCRS applied career and college readiness to mathematics in a way that supported STEM integration. The statements related to integration of other subject matters were stronger in this document than the previous documents examined. The following assertions were examples that showed the intent for students to experience integration of STEM content:

• “Mathematics cannot be viewed solely as a series of stand-alone courses or a set of specific skills. It must also be considered as a source of cross-disciplinary knowledge that is essential for success in numerous areas of study” (THECB, 2014, p. 8).

• “Mathematical thinking never occurs in a vacuum; it is always embedded in appropriate content” (p. 8).

• “Use mathematical models to solve problems in areas such as science, business, and economics” (p. 22).

• “Use geometric concepts and properties to solve problems in fields such as art and architecture” (p. 22).

• “Use knowledge gained from other subject areas to solve a given problem” (p. 60).

The NCTM Principles and Standards for School Mathematics laid the foundation over 10 years ago for integration of technology into mathematics. In addition, the Principles and Standards introduces the ideas of solving problems embedded in context and content from other subject areas, integrating mathematics and science, and even teacher collaboration for integrated units of study. These were novel ideas to many educators at the time they were written, and movement in that direction has been slow. The CCSSM were disappointing in that they fell short of any real encouragement to integrate subject matter within the content standards, although the process standards alluded to integration and specifically mentioned technology use. To learn more about middle school mathematics teachers’ opinions about CCSSM consider Factors Underlying Middle School Mathematics Teachers’ Perceptions about the CCSSMand the Instructional Environment (Davis, Chop-pin, & Drake, in this issue). The TEKS have been evolving ever slowly in the direction of full alignment with the NCTM standards. The TEKS were stronger in their STEM language than were the CCSSM, which are used in 43 states in the United States. The TCCRS were even stronger in their endorsement of integration of subjects and even non-STEM subjects. In addition, they presented a different perspective by fitting content into a real-life problem-solving context rather than putting real-life situations into the subject content. These curricu-lar documents and standards seem to be moving in the right direction.

There were some curricular materials and textbooks published more than 10 years ago, about the same time the NCTM Principles and Standards were released, that included activities that integrated science concepts and knowledge into mathematics content through real-life problems. Texas Teachers Empowered for Achievement in Mathematics and Science materials for algebra, geometry, and precalculus provide appropriate activities for integrating science concepts. The materials were completed with teacher notes that (a) suggest student collaboration, (b) promote effective questioning for deep conceptual understanding, and (c) support problem solving. One of several high school textbooks with similar characteristics is Mathematical Models with Applications (Lege, Bell, Davis, & Nite, 2001). In that book students are encouraged to work collaboratively to present solutions to real-life problems. Two examples of these are (1) determining attributes of an individual whose bones are discovered, using investigation and analysis of data on long bones and height, and (2) presenting information on whether or how restricting water craft might save endangered manatees after analyzing data about water craft registrations and manatee deaths and investigating related issues (Lege et al., 2001). Although they are not slated as such, these problems are examples of STEM PBL, which ideally integrates all four STEM subject areas.

Many stakeholders are seeing the need for full integration of all four areas of STEM to prepare students for the work force and for college. While no one would argue that every teacher should be an expert in every subject, every teacher can benefit from working with other subject area experts in their school to improve the educational setting and curriculum that will foster STEM teaching and learning. There were many references in the TEKS to problem solving and using a problem solving process. The NCTM standards have a problem-solving strand with examples of projects that could be facilitated in the classroom. In the Connections Standard from NCTM and the TCCRS, collaboration of teachers to integrate interdisciplinary studies is briefly discussed. The characteristics of problem solving by students with teacher facilitation as described in the NCTM standards almost describe STEM PBL, which effectively integrates engineering process design with mathematics, science, and technology (Capraro, Capraro, & Morgan, 2013). In order to find and present a solution to an authentic problem, students must work collaboratively and be able to effectively communicate their findings and recommendations. These are among the 21 ^st century skills that students need in order to function successfully in a constantly changing high-tech world for which problems to be solved tomorrow are not even possible to imagine today.

In summary, students exposed to CCSS are lacking in opportunities to experience STEM integrated coursework unless teachers modify their lessons and identify compatible objectives. For ideas of using STEAM to effectively address the Standards for Mathematical Practice, consider Using Inquiry Principles of Art to Explore Mathematical Practice Standards (Conley, Douglass, & Trinkley). However, students who are exposed to NCTM standards based curriculum fare much better. NCTM standards contained the greatest number of opportunities for students to experience STEM teaching and learning while not prescribing exact lessons or a STEM dogma. Finally, the TEKS and TCRS are examples of state documents that foster the opportunity for STEM teaching and learning while mostly focusing on real world experiences.