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Mathematical Domain: Mathematical Practice Standards

Mathematically proficient students … justify their conclusions, communicate them to others, and respond to the arguments of others. They … distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

… Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. … Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. —CCSS

As every teacher knows, children love to talk. But *explanation*—clear articulation of a sequence of steps or even the chronology of events in a story—is very difficult for children, often even into middle school. To “construct a viable argument,” let alone understand another’s argument well enough to formulate and articulate a logical and constructive “critique,” depends heavily on a shared context, especially in the early grades. Given an interesting task, they can *show* their method and “narrate” their demonstration. Rarely does it make sense to have them try to describe, from their desks, an articulate train of thought, and even more rarely can one expect the other students in class to “follow” that lecture any better than—or even as well as—they’d follow the train of thought of a teacher who is just talking without illustrating. The standard recognizes this fact when it says “students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.” The key is not the concreteness, but the ability to situate their words in context—to show as well as tell.

To develop the reasoning that this standard asks children to communicate, the mathematical tasks we give need depth. Problem that can be solved with only one fairly routine step give students no chance to assemble a mental sequence or argument, even non-verbally. The inclination to “justify their conclusions” also depends on the nature of the task: certain tasks naturally pull children to explain; ones that are too simple or routine feel unexplainable. Depending on the context, “I added” can seem to a child hardly worth saying. And finally, *skill* at “communicating [a justification] to others” comes from having plentiful opportunities to do so. The way children learn language, including mathematical and academic language, is by *producing* it as well as by hearing it used. When students are given a suitably challenging task and allowed to work on it together, their natural drive to communicate helps develop the academic language they will need in order to “construct viable arguments and critique the reasoning of others.”

One kind of task that naturally “pulls” children to explain is a “How many ways can you…” task.

How many ways can you make 28¢? (Variant: How many ways can you make 28¢ without using dimes or quarters?)

How many different 5″-tall towers of 1″ cubes can be made, using exactly one white cube and four blue cubes? (Variant: How many different 5″-tall towers of 1″ cubes can be made, using exactly *two* white cubes and three blue cubes?)

The first time young students face problems like these, they tend to be unsystematic. But after they have worked problems like this two or three times, they tend to develop methods (not necessarily efficient or correct, though often so). Then, faced with the question “How can you be sure there are no more?” most children, even as young as second grade, are drawn to explain and do so readily.

Similarly, in the playful context of an imaginary island with two families—one that always tells the truth, and one whose statements are always false—students can hardly stop themselves from explaining how they get answers to questions like this:

You meet Adam and Beth. Adam says “We’re both from the family of liars.” Which family is Adam from? What about Beth?

Children (and adults) typically find it far easier to solve the puzzle than to say how they solved it, but it’s also typical for them, given the slightest “how’d you get that?,” to feel compelled to explain!

While young students can sometimes detect illogical arguments, it is not generally sensible to ask young students to critique the reasoning of others, as it is often too hard for them to distinguish flaws in the logic of another student’s argument from artifacts created by the difficulty all young students have in articulating their thinking without ambiguity.