# data[].attributes.body
What resources would be most helpful to you for our next session?

# data[].attributes.body
What resources would be most helpful to you for our next session?

# data[].attributes.body
What resources would be most helpful to you for our next session?

# data[].attributes.body
Share your next-step ideas for learning about or incorporating number talk routines into your math instruction.

# data[].attributes.body
Share your next-step ideas for learning about or incorporating number talk routines into your math instruction.

# data[].attributes.body
Share your next-step ideas for learning about or incorporating number talk routines into your math instruction.

# data[].attributes.headline
Number Sense Routine - Same But Different

# data[].attributes.subject
Number Sense Routine - Same But Different

# data[].attributes.body
Algebra — SAME BUT DIFFERENT MATH

# data[].attributes.headline
Number Sense Routine - Same But Different

# data[].attributes.subject
Number Sense Routine - Same But Different

# data[].attributes.body
Fractions, Ratios, Etc — SAME BUT DIFFERENT MATH

# data[].attributes.headline
Number Sense Routine - Same But Different

# data[].attributes.subject
Number Sense Routine - Same But Different

# data[].attributes.body
Measurement — SAME BUT DIFFERENT MATH

# data[].attributes.attachment_link.title
Identify Opportunities to Strengthen Number Sense

# data[].attributes.headline
Using Number Talks to Build Procedural Fluency

# data[].attributes.subject
Using Number Talks to Build Procedural Fluency

# data[].attributes.attachment_link.title
Using Number Talks to Build Procedural Fluency Through Conceptual Understanding

# data[].attributes.attachment_link.title
Grades Kto2_Build and Extend Number Sense with Number Talks

# data[].attributes.attachment_link.title
Grades 3to5_Build and Extend Number Sense with Number Talks

# data[].attributes.attachment_link.title
Grades 6to8_Build and Extend Number Sense with Number Talks

# data[].attributes.attachment_link.title
Four Strikes and Youre Out Number Sense Routine

# data[].attributes.attachment_link.title
Tell Me All You Can Number Sense Routine

# data[].attributes.headline
Number Sense, What It Is and How to Develop It with Students

# data[].attributes.subject
Number Sense, What It Is and How to Develop It with Students

# data[].attributes.attachment_link.readable_text
Number sense, like commonsense, is easy to recognize, but not always easy to describe or support. In a world of computers and handheld devices, good number sense is more important today than at any time. As Keith Devlin, Stanford mathematician, wrote recently:

The most basic of today's new mathematical skills is number sense. (The other important one is mathematical thinking.) But whereas the latter is important for those going into STEM careers, number sense is a crucial 21st-century life skill for everyone.- (Huffington Post, 2017)

Moreover, we now know that number sense begins in the first years of life; if not developed in school, a lack of number sense can cause students to struggle in the upper grades and in college.

What do people with good number sense do?

compose and decompose numbers to solve calculations

compute mentally using a variety of strategies

make reasonable estimates

understand quantity and magnitude

reason with numbers

What does it mean to decompose and compose numbers to calculate? Let's look at an example. In a fifth grade class students were solving a word problem that required the product of 25 x 26. Instead of assuming everyone would find the product in the same way, students are encouraged to find a variety of methods.

Before you read on, think of some ways you might multiply 25 x 26. Students had a variety of responses:

25 x 26 is the same as 26 x 25

Think of 26 as 20 + 6 20 x 25 = 500 6 x 25 = 150 26 x 25 = 650

Another student said he did it similarly but instead of 20, multiplied by 10 twice.

Still another suggested that beginning with 30 x 25 was easier than 26 x 25:

30 x 25 = 750, but that is four 25's too many 4 x 25 = 100 750 -100 = 650

Some other solutions included the following:

Think quarters: 4 x 25 = 100 26 ÷ 4 = 6.5 6.5 x 100 = 650

or 100 x 26 = 2600 2600 ÷ 4 (since there are 4 25's in 100) = 650

One student suggested that 25 x 26 was too difficult, but it was the same as 13 x 50 because she doubled the 25 and halved the 26.

13 x 50 = (10 x 50) + (3 x 50 ) = 500 + 150 = 650

Finally one student said she studied a table of square numbers in the room and saw that 25 x 25 = 625 and 1 more 25 is 650.

Classrooms that nurture number sense Of course this variety and number of strategies doesn't occur naturally, but only in classrooms where students are encouraged to take risks, to look for structures like doubling and halving factors, to use known facts to figure out unknown ones, and most of all to share and examine each others' thinking. Students work in pairs to come up with solutions, first with easier problems (think 11 x 15) then with those of increasing complexity.

Student solutions are posted, discussed, and whenever possible, generalized. (26 = 20 + 6 or knowing that 4 x 25 equals 100 enables you to multiply by 100 and divide by 4 when multiplying by 25.)

Finally, providing visual models for procedures deepens understanding. For instance an area model helps students understand the distributive property.

20 x 25 = 500 6 x 25 = 150 26 x 25 = 500 + 150 = 650

Is the answer the same as one derived with the paper pencil algorithm? Of course, but which develops number sense? Which encourages students to use properties like the commutative and distributive property? Which enables students to solve more complicated problems mentally?

Think how different it would be if students merely wrote 25 x 26 vertically and then did something like this:

26 x 25

5 x 6 = 30, carry the 3, put down 0, 5 x 2 = 10, add 3 = 13. And so on. No number sense. No place value. No idea if the answer is reasonable.

Taking the time to thoroughly examine one or two calculations in depth, similar to the above, encourages students to focus on thinking-not merely on getting the correct answer. It leads both to number sense and to underlying math concepts.

Andy Clark will be presenting at the 2017 Math in Focus Institute, July 26-28, in Lafayette Hill, PA.

# data[].attributes.attachment_link.title
Number Sense, What It Is and How to Develop It with Students

# data[].attributes.attachment_link.description
Singapore Math thought leader Andy Clark discusses the importance of developing number sense and shares classroom strategies.

# data[].attributes.headline
Place Value: How to Assess Students' Understanding

# data[].attributes.subject
Place Value: How to Assess Students' Understanding

# data[].attributes.attachment_link.provider_name
MARILYN BURNS MATH | Marilyn's current thinking about math education and her ongoing classroom experiences and learning.

# data[].attributes.attachment_link.readable_text
If you're familiar with Cooks Illustrated magazine, you know that they use their test kitchen to tinker with recipes and improve on them. In each issue, they report on all of the attempts they make and then finally present a recipe they recommend. In a way, I feel as if my search for a way to assess students' understanding of place value has gone through a similar math teaching test kitchen.

By the end of first grade, students are expected to understand that the two digits of a two-digit number represent amounts of tens and ones. In second grade, they extend their understanding to three-digit numbers and the meaning of the digits in the hundreds, tens, and ones places.

First and second graders are generally familiar with numbers up to 100. Many can write the numbers from 1 to 100. Many can compare numbers-for example, 54 and 45-and tell which is greater. Many can identify which digit in two-digit numbers is in the tens place and which is in the ones place. These used to be the assessments I relied on for checking students' understanding of place value, typically using worksheet assessments on which students filled in missing numbers, circled the greater numbers, and indicated whether digits were in the tens or ones place.

But once I began to spend time asking students questions, face to face, I found that while worksheet assessments might be helpful, interviews often produced surprising and sometimes disturbing evidence of shaky understanding.

For example, while students could fill in missing numbers correctly, when I asked them to count to 100, some were confused when changing from one decade to the next-for example, stumbling when they reached 29, not sure what came next. When I asked students to write a number like 38, some incorrectly would write 308-writing 30 as they said "thirty" and then writing 8 as they said "eight." While they could tell which digit was in the tens place and which digit was in the ones place, when comparing 54 and 45, for example, some could only offer the explanation that 54 is greater because "it comes after 45"-their explanations didn't relate to the place values of the digits.

A More Reliable Way to Assess
Convinced that worksheet assessments were not sufficient, I collaborated with a team of colleagues (special thanks to Mallika Scott and Lynne Zolli) to create a litmus test for finding out whether students have the necessary understanding of place value. We wanted the interview to be short (because the information is important to have for all students in the class), efficient (requiring a minimum of materials), and informative (giving clear information about students' understanding). I'm not including here all the interview options we tried and eliminated, as the writers of Cooks Illustrated do for their recipes, but instead to cut to our current final version.

Last spring, when I spent a day in an elementary school in Brooklyn, NY, I used this as a baseline assessment to check second graders' place value understanding with two-digit numbers. View how a few of the students responded.

First watch Anna. After asking me to repeat the question, Anna responds automatically that the 1 in the number 16 means 10.

Daryl responds correctly, counting out the remaining cubes to show that the 1 in the number means 10.

Jonathan also demonstrated his understand of the meaning of the digits.

However, other students' responses indicated their lack of understanding. As an example, Gabriela shows one cube to represent the 1 in the number 16.

Dennis also responds incorrectly.

Now What?
My hope is that you'll try this assessment with your students. My suggestion: Predict first for each of your students how you think he or she will respond. Then ask and see what you learn. I'd like to hear from you, and for all of you to hear from each other, so please post comments, results, surprises, and any questions you have.

After Interviewing, Then What?
Incorporate into your class instruction investigations that help students build understanding of place value. In the new fourth edition of About Teaching Mathematics, I include a collection of suggestions for teaching place value on pages 344-358. Also, I rely regularly on the ideas presented in Teaching Arithmetic: Lessons for Introducing Place Value, which I wrote with Maryann Wickett. And for students who would benefit from additional intervention, modules Addition/Subtraction A and B are extremely useful.

# data[].attributes.attachment_link.title
Place Value: How to Assess Students' Understanding | MARILYN BURNS MATH

# data[].attributes.attachment_link.description
If you're familiar with Cooks Illustrated magazine, you know that they use their test kitchen to tinker with recipes and improve on them. In each issue, they report on all of the attempts they make and then finally present a recipe they recommend.