# Standards for Mathematical Practice…The Missing Link for Deep Understanding

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. ย 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)ย 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 3 tens and the other has only 1, so the one with 3 tens 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. ย 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!

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1. Ms. Wilkerson says:

I love your blog. As a new Math Coach your ideas and posts are so helpful. Thank you!

1. Donna Boucher says:

Congratulations on the new position! It’s one I highly recommend. ๐

2. Julie Sawyer says:

Working hard with my third graders on math talk — listening to each other, restating someone else’s reasoning, explaining and justifying reasoning! I’m trying to get better at asking a question and facilitating the discussion without giving away whether an answer is correct or incorrect.

Julie

1. Donna Boucher says:

Love it, Julie! Sounds like you’re creating a great mathematical environment in your classroom. Those are powerful words–listen, restate, explain, and justify. ๐

3. Kelley Cirrito says:

I love this! I am a math instructional facilitator for my district and we have been really working hard with our teachers to change the way they teach math by using the math practice standards. I also created some student friendly poster for students. Thanks for all your great resources!

1. Donna Boucher says:

It’s a steep learning curve, Kelley! I do find, however, that teachers who embrace the new role of teacher as facilitator, rather than giver of knowledge, really love the results in their classroom.

4. Qiana Harris says:

I love when I log on and see a new post on your blog. The way you put things just makes so much sense! I always find things I can use from your blog posts. I will start this week with doing more with less! Thank you

1. Donna Boucher says:

I really appreciate your comment, Quiana! I’m excited to hear how your kiddos respond to the new approach. ๐

5. Kelli Ruley says:

I see lots of Math Practice Standards posters for the classroom around but I would love to know what teachers are doing to introduce them to their students. Or are they? I don’t want to just put them on the wall, I want my students to understand them but I haven’t found anyway of doing this that I like. What is everyone else doing? Please give me some ideas!

1. Donna Boucher says:

I think part of the value of the posters is that they act as a visual reminder for the teacher as well as the students and they set the expectations for “doing” math in the classroom. For example, if a student uses great mathematical vocabulary, I can point to the poster and compliment the student for attending to precision. If a student doesn’t give up on a difficult problem and finally solves it, I can comment that they persevered and it paid off. The more students hear the verbiage of the mathematical practices being spoken in the classroom, the more the practices becomes a habit.

6. Mandi Palmer says:

What a great reminder!

1. Donna Boucher says:

Something we all need to keep in mind! It’s a big shift in thinking. ๐

7. TheElementary MathManiac says:

Having kids explain and defend their reasoning can have a huge impact on students’ ability to reason mathematically. I would also add using think-pair-share and having kids share their partners strategy adds another new dimension of mathematical reasoning to classroom discussions. When kids have to explain another students’ ideas, it helps them see another way of thinking of the same problem and ensures that kids are listening during think- pair- share.

Tara

1. Donna Boucher says:

Absolutely, Tara! I loved Chapter 5 in Math Sense and the chart on pages 79 and 80 featuring Rephrase, Rewind, Review, and Recharge. Great stems for initiating student conversations!

8. Dana Lester says:

I love your example on why we have the two sets of standards. I am presenting in a few weeks at the TN 1st Grade Teachers Conference on first grade math and plan on sharing your blog with my participants as a great math resource!
Dana
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