Quantitative Reasoning on the Redesigned SAT
By Allyson Tobias
Allyson has more than 15 years of experience as a classroom mathematics teacher and curriculum developer/manager. She has served in various leadership positions focusing on curriculum development in educational companies (Kaplan K–12, the Grow Network/McGraw-Hill). Currently, she is collaborating on math instruction and assessment projects with Khan Academy and Student Achievement Partners.
What Is Quantitative Reasoning and Why Is It Important?
Experienced teachers realize that the development of strong quantitative reasoning skills is critical for a student’s college and career readiness. To help us understand what that means, we must start with a basic definition of reasoning. Reasoning is the act or process of drawing justified conclusions from facts and evidence. Strong reasoning skills allow people to make sense of complex situations in their everyday lives and make informed choices and decisions. For example, we figure out how much money to leave for a tip in a restaurant; whether we can afford to buy a computer after we factor in the sales tax; or how much faster we need to run in our next race in order to achieve a personal best time. We make decisions about how much of an ingredient to use if we’re tripling the recipe. We read critically through data found in current events, newspapers, and scientific articles to detect patterns, and we analyze the data and established patterns as evidence to make informed predictions or decisions.
To be best prepared to utilize quantitative reasoning skills, students need a strong background in numerous mathematical concepts so that they may apply that knowledge effortlessly when thinking critically about various situations and problems. Students also need many opportunities to practice quantitative reasoning in diverse settings in order to help prepare them for college and future careers.
The math section of the redesigned SAT will emphasize the use of quantitative reasoning by assessing a student’s ability to create representations for a wide array of problems.
What Does Quantitative Reasoning Look Like on the Redesigned SAT?
The redesigned SAT will provide students with questions that will require them to apply their reasoning skills in varied contexts, from the abstract to problems with real-life applications. The real-life application questions will strongly emphasize quantitative reasoning skills through problems that are set in science, social studies, and career-based contexts. The content and data for these problems will come from sources such as newspapers and science and history books, as well as from various career settings.
What is the math in these types of situational problems? The concepts include describing rates, using proportions, synthesizing data, evaluating measures of tendency, and interpreting patterns within a data set. These are the math concepts that are most useful in everyday life, and they are the building blocks for more advanced math concepts with rich applications in science and social studies. The concepts assessed on the redesigned SAT require the same quantitative reasoning skills that teachers emphasize and value in their classrooms in order to prepare students for subsequent course work in math, or for application in many different careers.
Reasoning in Varied Contexts
The redesigned SAT assesses student understanding of a concept by using questions set in the abstract as well as situated in real-life contexts. For example, we could assess a student’s understanding of the concept of weighted averages by using an abstract problem such as the one below:
There are two lists of numbers. One list contains 8 numbers, the average of which is 39. The second list consists of 12 numbers, the average of which is 64. If the two lists are combined, now totaling 20 numbers, what is the average of the numbers in the new list?
However, experienced teachers also include opportunities that require students to see how to leverage their mathematical knowledge and quantitative reasoning skills to solve real-life problems as well. Therefore, the redesigned SAT could also assess this same concept by using the real-life context of a chemistry experiment in which students must find the density of a solution. Students should be able to transfer their understanding of the concept of weighted averages and apply the same quantitative reasoning skills required to solve the abstract problem above to solve for the density. For instance, a student may be conducting a chemistry experiment where they are asked to do the following:
Combine one solution with a density of 1.26 grams per milliliter (g/mL) and a volume of 8.6 mL with another solution having a density of 2.1 g/mL and a volume of 14.2 mL. Then find the density, in g/mL, of the resulting solution, assuming that the volume of the resulting solution is the sum of the volumes of the original solution.
The mathematical reasoning that students should use to analyze the underlying concept of weighted averages in either of these scenarios is similar. Students can find the total score/density (multiply the average for each set by the number in each set, add the total for each set), and then divide by the total number of scores/volume (add the number in each set).
But if students are only given opportunities to think of weighted averages in abstract practice problems, they might not realize that the problem they face during their chemistry experiment is also a weighted average question, which conceptually isn’t any different than the abstract example. Students who experience many types of weighted average problems — including those found in science contexts and with “harder” numbers — are more likely to recognize one in a setting they haven’t encountered before, which is what they need to be able to do.
For students to correctly answer questions set in these varied contexts, they need to use their mathematical content knowledge and reasoning skills to read through the questions, make sense of what is being asked, pull out the information necessary to do the math, and perform the appropriate calculations. Students should have a broad experience with applications so they are able to recognize what math is needed and use it appropriately in unfamiliar settings.
Reasoning from data
We know students are faced with problems every day that may require the ability to analyze data and draw conclusions based on evidence. Students encounter these types of challenging problems in math, science, and social studies classrooms as well as in everyday activities, such as simply reading a news article online. In order to assess a student’s ability to reason from data, the redesigned SAT will have some questions that focus on using and analyzing data from various sources, such as scientific articles or current events. Students will be given data sets that will have corresponding questions that refer to and require analysis of the data provided. This format allows students to explore information in greater depth and across multiple mathematical topics.
For example, students could be given a table with demographic and voting pattern information from the Census Bureau and asked to draw conclusions based on analyzing the data provided. For students to succeed with this data analysis scenario, they need to use their reasoning skills and content knowledge to make sense of the data, figure out what the question is asking them to find, and pull out the necessary information in order to create and compare proportions. Students with strong reasoning and estimation skills might eyeball the data to estimate solutions so they can quickly eliminate some choices and perform only the necessary calculations. Their mathematical fluency and strong reasoning skills will give them an efficient edge on the SAT.
Making Connections to Classroom Instruction
Students in many classrooms already receive the rigorous instruction necessary for the type of reasoning and skills that are emphasized on the redesigned SAT. Good teachers expose their students to relevant math concepts in the context of rich applications and require that students learn to think for themselves.
However, many students may lack the motivation and perseverance to tackle challenging math problems, often because they don’t see the connection between math and “real life.” The use of data from current events helps students see the math around them. Using U.S. Census Bureau data, for example, helps students analyze and apply their math skills to a real and current situation. Students should have experience with these types of problems regularly in their classrooms. As teachers, we can motivate students by helping them see that the math concepts we focus on are not only the foundation for higher-level math courses but also necessary in countless everyday situations, and that these skills will be required for many jobs.
In addition to the math concepts, it is important for students to develop strong reasoning skills. For them to be able to do so, we need to give students numerous opportunities to practice challenging, real-life application problems and time to practice thinking about them and how they might go about solving them without teacher intervention. Let students be uncomfortable and frustrated. Allow them to develop strong reasoning skills by cultivating an atmosphere in which persistence is expected and rewarded. Let them know that it is okay to struggle. The more students engage with difficult problems, the less scary they will seem. By setting problems in real-life contexts, we are motivating students to go beyond just getting the right answer, or simply focusing on recalling a math procedure without thinking critically about what the problems are asking.
Settings for problems that build quantitative reasoning skills should be varied: abstract, real-life, and examples from science and history. If students are exposed only to abstract settings, they might have a strong command of mathematical procedures, but not the ability to apply them in other settings. For example, a student who can successfully complete the abstract weighted average question but doesn’t know what to do with the chemistry experiment question likely hasn’t had enough experience with varied types of applications for this concept.
The most rewarding and challenging classroom settings are ones in which students are empowered to think, argue, create models, and make presentations. The teacher is their coach to help them find the math and learn how to use it. It is important to keep in mind that effective quantitative reasoning is grounded in the development of a strong mathematical knowledge base, and that good classroom instruction also cultivates a student’s ability to find and apply those math concepts in any context. The development of strong reasoning skills builds strength in the ability to think through an unfamiliar problem and know how to break it down.
Classroom instruction that gives students opportunities to tackle mathematics in varied contexts will build a willingness and ability to apply math in diverse situations. These principles are strongly reflected within the redesigned SAT.