How Mathematicians Think

Mathematicians don't just calculate, they think in five particular ways, no matter what strand of math they're working in. They make sense of a problem and stick with it even when their first plan fails (Problem Solving). They notice that fractions, decimals, and percentages are really the same idea wearing different clothes, and connect new math to real life (Connections). They explain their thinking with precise language, models, and tools so someone else can follow exactly what they did (Representation & Communication). They check whether an argument, theirs or someone else's, actually makes sense, and revise their thinking when new evidence shows up (Analyze & Justify). And they hunt for patterns in repeated work, because a pattern is often a shortcut in disguise (Structure & Patterns). These five habits, the Mathematical Process Standards, are not a unit you finish. They're a lens you'll use in every strand, all year.

Those five habits will run through four strands this year. In Data, Probability, and Statistical Reasoning (DPSR), you'll analyze data sets, build box plots, and get your first taste of probability. In Measurement, Geometry, and Spatial Reasoning (MGSR), you'll calculate area, surface area, and volume, work with angles and a protractor, and graph in all four quadrants of the coordinate plane. In Numerical Reasoning (NR), you'll deepen your understanding of fractions, decimals, percentages, and negative numbers. And in Patterns, Algebra, and Functional Reasoning (PAFR), you'll meet functions, formal algebraic vocabulary, and your first equations. All four strands lean on real-world problems and expect you to explain your thinking, not just land on an answer.

We're starting in DPSR, with Standard 6.DPSR.1, because data is the strand where "explain your thinking" is easiest to see and hear out loud. Today's indicator, 6.DPSR.1.1, asks a deceptively simple question: how many pieces of data do you have (your sample sizeThe number of data values collected from a population to represent it in a data set.), and is that enough to represent the whole group you're studying? Box plotsA graph that displays a numerical data set using five key values: the lower extreme, first quartile, median, third quartile, and upper extreme. themselves are a brand-new tool for you this year, so we'll build the first one together, slowly. Once your data is ordered, you'll label its lower extremeThe least value in a data set, used instead of the word minimum in 6th grade. and upper extremeThe greatest value in a data set, used instead of the word maximum in 6th grade., then find the medianThe middle value of a data set once it's ordered from least to greatest. and check whether there's a modeThe value or values that occur most frequently in a data set.. The rangeThe difference between the greatest and least values in a data set. tells you how spread out everything is. And the shape of your data, whether it's skewedA distribution where data is bunched toward one side with a tail extending in one direction. with a tail off to one side, symmetricA distribution that is evenly balanced on both sides of its center. and balanced, uniformA distribution where values are spread roughly evenly throughout the range. and roughly flat, or bimodalA distribution with two distinct modes, or peaks, in its shape. with two peaks, helps you decide whether the median or the mode gives the more honest picture, especially if a stray outlierA data value that is unusually far from the rest of the data set, which can pull the mean away from what's typical. is quietly pulling things off balance.

Adapted from Studio Aletheia's The Center for Quantitative Studies curriculum library, drawing on mathematical resources and the SC CCR Mathematics Standards.