"Wait, this isn't language time, is it?" Expressing Mathematical Thinking

Mathematical success comes from deep understanding, so teachers (and parents) need to set clear expectations for students: Do what makes sense to you. If children see math as a collection of tricks they must learn, that can soon learn to commit recurring errors - when subtracting, for instance, they may subtract the smaller from the larger rather than regrouping. In this instance, they arrive at answers that make no sense and they rarely know why. We need to encourage them to explain the logic of their procedures, and the reasonableness of their solutions.

By taking the time to have students explain their reasoning, they are forced to organize their ideas, and they are given the opportunity to extend their ideas. Workbook time is quiet time, yes, but at other times math time is definitely a time for talk. Communication is essential for learning. We take time nearly every day to go over our work from previous days - did what we do make sense? What did we get right? It is a time to respectfully share ideas and clarify points of view.

Let me give you an example from this week. I'll share three children's work. Two of these children got the correct answer, and one was able to articulate the "why" pretty clearly. The other (a younger child) gave an answer typical for their age. The third example shows an incorrect answer, but you can see where they were going. The group talk we had the following morning was extremely helpful, and that child was able to do the next similar problem correctly AND could articulate why their answer now made sense.

The problem was: Sam picked the numbers 2, 7, 5, 0, 6 and 8. How should Sam arrange the numbers to get the largest sum? Write the numbers in the boxes below and then find the sum. Explain how you know this is the largest sum that Sam can make with these numbers.

This child wrote 862 + 750 = 1612, which is correct. They then wrote the explanation, "Well 8 and 7 are the largest numbers here. Then comes 5 and 6, then 2 and 0. So the hundreds shud get the 7 and 9, the tens shud get the 6 and 5, then the ones 0 and 2." 

This child used the same configuration of numbers, also got the same correct answer. However, they are not yet at the stage where they can clearly articulate the "why". Instead, they answered, (rather adorably), "Becaua I tinct in my head." With a little encouragement, he added, "I choose the largest nubers and add." 

This student wrote the numbers from highest to lowest, but didn't quite understand that it made more sense to write the two highest values in the hundreds place, then the next two highest values in the tens place. After the classroom conversation went on, and we looked and talked about all different strategies and ways of thinking (all with names hidden for privacy purposes), it was like a lightbulb went off.

All children were able to successfully complete the next (more difficult, but similar) challenge.