Let’s be clear, the Standards for Mathematical Practice are not new. According to the Common Core State Standards for Mathematics:
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. (CCSS 2010)
The practices are based on the NCTM process standards and the strands of mathematical proficiencies from the National Research Council’s report Adding It Up. In Texas, we call the practices “process standards”, and they’ve been part of our TEKS for years.
It may be helpful to understand the purpose of both the practice and content standards The content standards define the procedural skills students must master at each grade level. For example 2.NBT.3 and 2.NBT.4 tell us a second grader must be able to read, write, and compare three-digit numbers written in a variety of formats (base-ten numerals, number names, and expanded form). The practice standards, on the other hand, define the habits that mathematicians employ when doing their work. Let’s look at how some of the practices come into play in relation to that same standard:
- Students who have been working with 3-digit numbers with a digit in each place (214, 567, 982, etc.), are asked to make meaning of the number 302. This would be considered a problem solving situation (Practice Standard 1), because they have not encountered numbers with a 0 in either the tens or ones place before. Students use base-10 blocks to build the new number and describe it using expanded form (300 + 2) and number name (word form).
- A student says the number four hundred twenty-five should be written 400205 (very common misconception!). The student is asked to defend his answer (Practice Standard 3) while other students listen and ask questions for clarification. By listening to the arguments of fellow students, the first student comes to realize his mistake and is given another number, which he writes correctly.
- While comparing the numbers 234 and 219, the teacher reminds students to use good place value language (Practice Standard 6) to defend their comparison in writing (Practice Standard 3). Two students write the following explanations:
“I saw that both numbers had two hundreds, so I had to look at the tens place. 234 has 3 tens and 219 has only one ten, so 234 is greater.”
“I wrote the numbers in expanded form, 200 + 10 + 9 and 200 + 30 + 4. The 2s in both numbers mean 200, so they start out the same. But one number has 30 in the tens and the other has 10, so the one with 30 is greater.”
Do you see the difference? We often teach specific strategies for comparing numbers, resulting in kiddos who might master the content skill, but don’t necessarily understand the process. Even though we ask them to “show their work”, they don’t necessarily develop a deep understanding of the concepts and their “work” doesn’t always show us what they do and don’t understand. But when the Mathematical Practices are thoroughly embedded into all areas of instruction and kiddos are asked to explain and defend their solutions on a regular basis, they make sense of the math they are doing. And when they understand the math they are doing, they become better problem solvers. That is a huge shift in thinking about instruction, and it takes lots of practice and modeling!
Here’s another example. My newest unit on 3-digit place value includes an I Can card and three Math Talk cards. Do you see the difference in the I Can card and the Math Talk cards?
Embedding the mathematical practices into our instruction is our greatest challenge in teaching mathematics, but it is absolutely essential if we are to develop our students as mathematicians. Start this week! Do more with less. What I mean by that is have kiddos do fewer problems, but do more with them–more explaining, more defending, more TALKING!
I would love for you to add comments throughout the week about small changes you have made to your instruction to begin emphasizing the mathematical practices–or what you’re already doing! We can all learn from hearing what others are doing. 🙂