How many of us ask students to solve a problem and then stop when the student produces a correct answer? Give students a good rich task but don’t ask how they arrived at the answer or how they know it is correct? This is where reasoning and proof comes in.
What do we teach math for if it is not to have students use their mathematical knowledge and understanding to solve problems? I used to tell my students that their employer was not going to ask them to do a straight computational problem such as, 346,328 - 132,004. No, they were going to ask them how many widgets were available to ship after the Acme Company’s order of 132,004 widgets was filled.
In the book, Adding It Up: Helping Children Learn Mathematics, (National Research Council, 2001) (beginning in chapter 4, page 137) researchers talk about the components or strands of mathematical proficiency. They identify these strands as: