Reasoning is an important part of the mathematics curriculum in the early grades. Through reasoning, or logical thinking, we determine if and why our answers make sense. In the Difference Trains problem, you were asked to think about your reasoning as you worked toward a solution. That's what this Standard is all about: making a decision about a mathematical idea -- in this case, patterns -- and then justifying your thinking by explaining how you know that your decision is correct.

Questions for reflection:

• Describe the reasoning you used for each difference train. How could you justify your reasoning to prove that you actually have a one-, two-, or three-difference train?
• How would you prove to a friend that the pattern of your train is correct?
• There are many solutions to the Difference Train problem. How, do problems with multiple solutions enrich students' reasoning?
• Did you think of extensions to the activity, or wonder about additional questions you might ask your students (for instance, how many trains of each type could be created? Was there any limit to the length of each type of difference train? Would there be a way to represent the trains symbolically?)

Through experiences similar to those in this section, we learn to justify and explain our reasoning. The habit of providing explanations for our answers can begin as early as preschool -- we don't have to wait until high school geometry to incorporate reasoning and proof into the mathematics curriculum. It should be an integral part of every mathematical experience for all children!

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