Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. … —CCSS
The problems we encounter in the “real world”—our work life, family life, and personal health—don’t ask us what chapter we’ve just studied and don’t tell us which parts of our prior knowledge to recall and use. In fact, they rarely even tell us exactly what question we need to answer, and they almost never tell us where to begin. They just happen. To survive and succeed, we must figure out the right question to be asking, what relevant experience we have, what additional information we might need, and where to start. And we must have enough stamina to continue even when progress is hard, but enough flexibility to try alternative approaches when progress seems too hard.
The same applies to the real life problems of children, problems like learning to talk, ride a bike, play a sport, handle bumps in the road with friends, and so on. What makes a problem “real” is not the context. A good puzzle is not only more part of a child’s “real world” than, say, figuring out how much paint is needed for a wall, but a better model of the nature of the thinking that goes with “real” problems: the first task in a crossword puzzle or Sudoku or KenKen® is to figure out where to start. A satisfying puzzle is one that you don’t know how to solve at first, but can figure out. And state tests present problems that are deliberately designed to be different, to require students to “start by explaining to themselves the meaning of a problem and looking for entry points to its solution.”
Mathematical Practice #1 asks students to develop that “puzzler’s disposition” in the context of mathematics. Teaching can certainly include focused instruction, but students must also get a chance to tackle problems that they have not been taught explicitly how to solve, as long as they have adequate background to figure out how to make progress. Young children need to build their own toolkit for solving problems, and need opportunities and encouragement to get a handle on hard problems by thinking about similar but simpler problems, perhaps using simpler numbers or a simpler situation.
One way to help students make sense of all of the mathematics they learn is to put experience before formality throughout, letting students explore problems and derive methods from the exploration. For example, students learn the logic of multiplication and division—the distributive property that makes possible the algorithms we use—before the algorithms. The algorithms for each operation become, in effect, capstones rather than foundations.
Another way is to provide, somewhat regularly, problems that ask only for the analysis and not for a numeric “answer.” You can develop such problems by modifying standard word problems. For example, consider this standard problem:
Eva had 36 green pepper seedlings and 24 tomato seedlings. She planted 48 of them. How many more does she have to plant?
You might leave off some numbers and ask children how they’d solve the problem if the numbers were known. For example:
Eva started with 36 green pepper seedlings and some tomato seedlings. She planted 48 of them. If you knew how many tomato seedlings she started with, how could you figure out how many seedlings she still has to plant? (I’d add up all the seedlings and subtract 48.)
Or, you might keep the original numbers but drop off the question and ask what can be figured out from that information, or what questions can be answered.
Eva had 36 green pepper seedlings and 24 tomato seedlings. She planted 48 of them. (I could ask “how many seedlings did she start with?” and I could figure out that she started with 60. I could ask how many she didn’t plant, and that would be 12. I could ask what is the smallest number of tomato seedlings she planted! She had to have planted at least 12 of them!)
These alternative word problems ask children for much deeper analysis than typical ones, and you can invent them yourself, just by modifying word problems you already have.
Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. —CCSS
Certain mathematically powerful representations—in particular, the number line, arrays and the area model for multiplication/division, and tabular/spreadsheet forms—are valuable complements to the use of concrete objects in elementary school, and remain important and faithful images of the mathematics through high school.