The Third Guiding Principle for Mathematical Practice

The third standard reads: Construct viable arguments and critique the reasoning  of others. 


The explanation is that mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, 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. Later, students learn to determine domains to which an argument applies. Students at all grade levels can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 


To develop the reasoning that this standard asks children to communicate, the mathematical tasks we give need depth. If students are used to working on simple, single step problems with one finite answer, it is hard to get them to explain, so you tend to get one word answers like, I added.  Problems that require thought, discovery and multiple attempts almost make it easier for children to talk about. In order for students to be able to communicate a process they need to be able to give a clear articulation of a sequence of steps or the chronology of a problem or strategy. The more "action" they have in solving, the more articulation is possible. 


"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.” (Thinkmath)


This process becomes more defined when you put it into the context of collaboration and group work.If given an open ended or discovery question, materials and resources to aid them in their exploration and a group or learning partner to work with, children begin to discover and "talk" about what they learn. These "learning conversations" are the backbone to reasoning and communicating their thoughts both verbally and in writing. 


Acknowledging the work that another group has done and critiquing their thoughts is difficult for most young children; constructing an argument that challenges their work, and proving or justifying the challenge is very difficult at any level. Math students must not only be proficient, but able to deconstruct and reconstruct the problem at hand and then explain their reasoning behind it. The ramifications of this standard are huge, but, if the inquiry and collaborative pieces are in place and if Accountable Talk is embedded in student practice then "prove it" becomes the norm.


Can you put your students into this situation? If given the opportunity, the tools and the time can you see this becoming a normal part of your math discoveries?