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Number Talks Recordings in Distance Learning

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Number Talks Recordings in Distance Learning

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Number Talks in Distance Learning Videos 20-21

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Daily Number Talks with D200 Math Coaches

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Scores for Latino students more than doubled at Robbins Elementary since 2014-15.

Sydney Johnson

Sydney Johnson

California school districts have long struggled with a persistent gap in math test scores between racial and ethnic groups. But at one small rural school district, the gap between Latino and white students has narrowed more than it has at most districts in the state.

At Winship-Robbins Elementary School District, a single-school district in south Sutter County, the percentage of Latino students meeting or exceeding standards on the Smarter Balanced math test more than doubled over the last five years. And although their scores still slightly trail those of their white peers, the gap between them has narrowed by nearly 16 percentage points during that time.

One of the main reasons for the improved scores is a change in the way teachers talk to their students about math, said officials at Robbins Elementary, the district's one school. Starting last school year, teachers focused on creating more opportunities for students to talk through word problems with each other and build their ability to recognize and apply math terms, rather than relying on rote memorization.

The shift has made math at Robbins Elementary more social - and that's intentional. During a recent morning math lesson, students in Mayra Medina's 6th-grade math lesson were buzzing with group discussion about how scaling shapes up and down relates to size and proportion.

"If students aren't talking about their math learning, they won't have retention," said Kim Richter, curriculum and instruction coordinator for Winship-Robbins Elementary School District. "Our English learners particularly need that opportunity to think and read out loud and collaborate with a partner who can support them."

In the spring of 2015, California began administering Smarter Balanced assessments, annual standardized tests in math and English language arts aligned with the Common Core standards. Last school year, 39.7 percent of all California students met or exceeded state standards in math, an increase of about 1 percentage point from the year before.

While the average statewide math score was low, the scores for some student groups were even lower. Statewide, the scores of Latino and African-American students on the Smarter Balanced math test trailed those of their white and Asian peers, a chronic pattern known as the achievement gap. California adopted the Common Core standards in 2010 in part to close that gap and the changes Robbins Elementary teachers made as they implemented them point to practices that might be working.

Nestled between rivers and farmland north of Sacramento, Robbins Elementary serves about 120 students in grades K-8. Overall, nearly 80 percent of the overall student population is Latino. And about 4 in 5 students at the school are low-income, according to data from the Department of Education. About 40 percent are English learners and many others speak a language other than English at home, including Spanish, Russian and Arabic, according to Dawn Carl, principal and superintendent.

Creating more opportunities to share out loud and learn from one another while solving math problems is intended to support students from all language backgrounds, Richter said, not only the school's English learners. But many factors beyond the classroom create difficult hurdles for the school's low-income students.

"Our town is poor and rural. We don't have a public library and many of our students don't have internet at home," she added. "We have a store where you can buy beer and bait, but you can't buy fresh vegetables."

The new Common Core math standards require significant shifts in instruction, such as emphasizing math concepts over memorizing facts and procedures. After seeing a dip in math test scores a few years after the district began implementing the Common Core standards, staff met every other Wednesday last school year to review and refine its approach to math.

They noticed their math lessons were missing what Richter describes as "productive struggle," where students can work through challenging problems and mistakes, rather than providing each step and answer. "The rigor just wasn't there," she said.

Without the money to send teachers to professional learning events, teachers spent the afternoons working together through "Visible Learning," a book-based program with strategies for teachers to evaluate their own teaching practices. They also attended free math instruction workshops at UC Davis.

The work is starting to pay off. Among all students at the school, the number meeting or exceeding state standards for math went from 43 percent in 2017-18 to more than half last school year.

"By making sure we were teaching not only content but also academic language changed everything for us, especially our English learners," Richter said.

The new focus on language shows up in different ways. Medina started her recent math lesson by first introducing a word problem to students, who read the question out loud. Then students, who worked in groups of four, entered into a discussion about how they would solve the problem. After the students worked through their ideas, Medina then modeled how to approach the problem by walking through key steps. Finally, the students took turns sharing how they got to their answers.

"The way I learned, teachers would model how to solve a problem and there was a big focus on speed. But we didn't get a lot of time to work through problems together," Medina said. "This is much more collaborative-based learning. It prompts discussion so students can share strengths and experiences."

Also starting this year, each lesson has a language objective, or terms and phrases that students should know at the end of class. That is paired with a content objective, meaning the academic material that students should come away understanding and a social objective such as problem-solving with peers. Teachers outline the objectives for students during class and keep them visible on the whiteboard for reference.

"Every time we plan the objectives, we unpack the standards again and again," Richter said. "Before, it was easy for teachers to follow the book if they didn't know what they were doing. Now, they have to really get into the math concepts when lesson planning."

Medina said the Wednesday math meetings last year helped her learn how to facilitate discussions instead of giving students the answer. She also learned ways to integrate math vocabulary more seamlessly in lessons by letting students know when they are practicing mathematical concepts, such as ratios and proportional relationships.

Each morning, students who need extra help in math or writing also attend a small 30-minute class focused on those subjects. Every six weeks, staff review student performance data to determine if they need additional support, such as smaller breakout groups that focus on specific math concepts.

The school is still working through some lingering challenges. In January, the district purchased new math textbooks to better align with the shifts teachers were making in the classroom, and they are still getting familiar with the new materials.

And due to its small size, the school has uneven enrollment in grade levels, resulting in three classes that combine different grade levels at the school. Richter said, "It's hard to do this kind of specific instruction when you're dealing with two grades."

Since 2014-15, the first year of the Smarter Balanced test, the percentage of California's Latino students who met or exceeded standards on the math test increased from 21 to 28 percent, a larger rate of increase than the state's white and Asian students. However, a wide achievement gap persists.

"The gap for Latino students is still huge, and that's troubling," said Patricia Gándara, a research professor at UCLA and co-director the Civil Rights Project, a research and policy institute that analyzes racial inequities in public education.

The slight increase in math scores among Latino students could reflect positive trends in the economy, but it could also reflect negative trends in the student population, Gándara said.

"We have people leaving and being deported and the ones who leave are the ones who are most stressed," said Gándara, referring to students who experience trauma and threats of deportation and how that can impact academic performance. "That could change the composition of who takes the test."

Staff at Robbins say there's still more work to be done.

"It makes us proud that the things we have chosen to focus on are making a difference. But we aren't where we need to be yet," Richter said. "And we definitely aren't sitting back, that's for sure."
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Students talk through math in this California school. Now test scores are rising.

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Sydney Johnson Sydney Johnson California school districts have long struggled with a persistent gap in math test scores between racial and ethnic groups. But at one small rural school district, the gap between Latino and white students has narrowed more than it has at most districts in the state.

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As teachers and students ramp up for testing I was invited to visit a 5th grade class and asked to focus on fractions. Before the lesson, the teacher and I discussed the importance of a number line, which was an extension of a conversation we had a couple weeks back. We decided to start the lesson by throwing this on the board.

A = one-fourth B = one-half C = ten-elevenths, eight-ninths, fourteen-fifteenths

We accepted various answers for "C "and were thrilled that students could justify about the numerator and denominator being really close to one another. Then we showed the next number line and the rest of the lesson went to " hockey sticks and handbaskets. "

We blamed it on zero...or lack of it.

With the lesson plan out the window, we quickly improvised and went to a Number Line Number Talk. I don't know where it came from, we just made it up. Number Talks promote fluency and strategy and we needed both things immediately!

Knowing that the majority of student understanding hangs on zero, we started there.

They nailed one-half so we went here...

Students: The number line shifted a fourth and the "x" is still in the middle, so the "x" shifts a quarter greater than one-half to 3/4.

Then we moved on and the same reasoning was applied to locate the "x" at 1 whole.

So far so good. If SMP#3 was a person, he would have been proud of these students as they reasoned their way through the number line strings.

Then we dropped this one on them.

Students were given time to collaborate and discuss in groups about the location of "x".

Student: Well, it has to be greater than 1 whole, but it's not quite 1 and 1/4. It's halfway and half of a quarter is 1/8. So it's 1 and 1/8.

We went nuts. Kids went nuts. Math was happening and everyone was happy. " We want another one! " was the chant around the room.

Not a problem. So we moved on to this guy...

About half the class was immediately able to use their reasoning from the previous number string to locate the "x" at 7/8. We picked up about another quarter of the students through the explanation. Some students were completely lost a this point (note to self "see you in small group").

Then we showed this one...

I'm not going to lie, about half the students were completely lost by the time we arrived here. What I loved most about this process was that students were still discussing and persevering! That's a winner in my book.

I was hesitant... but we did one more because they wanted it.

Because these number strings were created on the fly, students found an unintended pattern.

when the number line began with 1/2 the "x" was located at 3/4

when the number line began with 3/4 the "x" was located at 7/8

when the number line began with 7/8 the "x" was located at 15/16

when the number line began with 15/16 the "x" was located at 31/32

The students identified that if the pattern continued the denominator doubles and the numerator is double plus 1. (Freakin' nuts)

So we tested our luck again with the number line from the beginning of class...

Me: So can we use what we just did to find out where "A" is located. Anyone want to change their answer?

Student: Halfway between 1/2 and 2 is 1 and 1/4. If we jump in one quarter on each side it would be at 3/4 and 1. Halfway between 3/4 and 1 is 7/8.

From the 24 students in the class, only 3 were able to make sense and repeat what the student shared. We started with zero understanding and 1-hour into the opener we had 4 kids understanding and a ton of work left to do!
My takeaways:

At most, this is a DOK 2 question but it really chewed brains.

Students need more exposure with number lines that don't start at zero. The number lines I encounter, typically begin with a zero which can give a false-positive of student understanding.

The number string highjacked the entire lesson and that's okay.

Understanding benchmarks and how to use them are the underpinnings to solid fractional reasoning.

These Number Line Strings are wicked awesome for building fractional reasoning. If they're out there please show me where I can find more. I want more. If you created them, I promise I didn't rip you off and want to give you full props. I've just never seen or heard of them.

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A number line, number string, and Number Talk walk into a class...

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As teachers and students ramp up for testing I was invited to visit a 5th grade class and asked to focus on fractions. Before the lesson, the teacher and I discussed the importance of a number line, which was an extension of a conversation we had a couple weeks back.

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Math.4.NBT.B.5

Common core State Standards

Math: Math

4: Grade 4

NBT: Number & Operations in Base Ten

B: Use place value understanding and properties of operations to perform multi-digit arithmetic

5:
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Download Common Core State Standards (PDF 1.2 MB)

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Number Talks encourage students to think flexibly about the operations and their properties as they solve problems mentally.

Discussion and Supporting Materials

School Details

Shields (Richard A.) Elementary School
910 Shields Avenue
Lewes DE 19958

Data Provided By:

Teachers

Math / Kindergarten 1 2 3 4 5 / Teacher

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Multiplication Number Talk: Fourth and Fifth Grade

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Instructional coach Kristin Gray engages a group of fourth and fifth grade students in a multiplication number talk. She places one problem on the board and records the strategies students have for finding the product. She uses talk moves to encourage student discourse.

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Math.3.OA.B.5

Common core State Standards

Math: Math

3: Grade 3

OA: Operations & Algebraic Thinking

B: Understand properties of multiplication and the relationship between multiplication and division

5:
Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Download Common Core State Standards (PDF 1.2 MB)

|

Math.3.NBT.A.3

Common core State Standards

Math: Math

3: Grade 3

NBT: Numbers & Operations in Base Ten

A: Use place value understanding and properties of operations to perform multi-digit arithmetic

3:
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Download Common Core State Standards (PDF 1.2 MB)

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Number Talks encourage students to think flexibly about the operations and their properties as they solve problems mentally.

Discussion and Supporting Materials

School Details

Shields (Richard A.) Elementary School
910 Shields Avenue
Lewes DE 19958

Data Provided By:

Teachers

Math / Kindergarten 1 2 3 4 5 / Teacher

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Teacher Jennifer Guido engages a group of third grade students in a multiplication number talk. She places one problem on the board and records the strategies students have for finding the product. She uses talk moves to encourage student discourse.

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100 Questions that promote Math Discourse

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100 Questions that promote Math Discourse

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Can the questions you're asking in the mathematics classroom be answered with a simple "yes" or "no," or do they invite students to deepen their understanding?
Asking better questions can open new doors for students, promoting mathematical thinking and classroom discourse. To help you encourage deeper discussions, here are 100 questions to incorporate into your instruction by Dr. Gladis Kersaint, mathematics expert and advisor for Ready Mathematics.

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100 Questions that Promote Mathematical Discourse

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Can the questions you're asking in the mathematics classroom be answered with a simple "yes" or "no," or do they invite students to deepen their understanding? Asking better questions can open new doors for students, promoting mathematical thinking and classroom discourse.

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Grassroots Workshops

Have you ever been a part of a process that just fascinated you? That's the only way to describe my summer project. With this one project, I learned so much. I was approached by Robert Kaplinsky (who I've known for several years) to plan and produce a Grassroots mini-workshop on the use of Clothesline Math. Wow! This was going to ...

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KristenAcosta.com | Where Math is an Active Subject

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Have you ever been a part of a process that just fascinated you? That's the only way to describe my summer project. With this one project, I learned so much. I was approached by Robert Kaplinsky (who I've known for several years) to plan and produce a Grassroots mini-workshop on the use of Clothesline Math.

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Math.4.NF.B.4a

Common core State Standards

Math: Math

4: Grade 4

NF: Number & Operations--Fractions

B: Build fractions from unit fractions

4a:
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
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Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.

Download Common Core State Standards (PDF 1.2 MB)

|

Math.4.NF.B.4b

Common core State Standards

Math: Math

4: Grade 4

NF: Number & Operations--Fractions

B: Build fractions from unit fractions

4b:
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
<br />
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.

Download Common Core State Standards (PDF 1.2 MB)

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A dot image gives students the opportunity to see connections between operations and visualize the properties of operations by counting a set of dots in many ways.

Discussion and Supporting Materials

School Details

Shields (Richard A.) Elementary School
910 Shields Avenue
Lewes DE 19958

Data Provided By:

Teachers

Math / Kindergarten 1 2 3 4 5 / Teacher

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Instructional coach Kristin Gray asks a group of fourth grade students to describe the ways they can count the number of half dots in an image. She records the students' strategies on the board and encourages students to reference their thinking in the dot image in order to make use of its structure.

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Math.Practice.MP4

Common core State Standards

Math: Math

Practice: Mathematical Practice Standards

MP4: Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Download Common Core State Standards (PDF 1.2 MB)

|

Math.5.NF.B.4

Common core State Standards

Math: Math

5: Grade 5

NF: Numbers & Operations--Fractions

B: Apply and extend previous understandings of multiplication and division

4:
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
<br />
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Download Common Core State Standards (PDF 1.2 MB)

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Discussion and Supporting Materials

Thought starters

1. How does Ms. Morey structure her class in order to provide tailored, small group instruction?

2. Why does Ms. Morey decide to use manipulatives with her small group?

3. How does Ms. Morey use this lesson as an opportunity to build her own skills?

School Details

Enumclaw Middle School
550 Semanski St
Enumclaw WA 98022

Data Provided By:

Teachers

Math / Kindergarten 1 2 3 4 6 / Coach

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Crystal Morey provides tailored instruction to a small group of students in her sixth grade math class. First, she formatively assesses all students. Next, she pulls together a small group of students who have similar misconceptions. In the small group, students use manipulatives and images to explore the math concepts.

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In the number talk, Arrillaga engages her students in a discussion of different answers to the addition problem 59 + 37. She gathers different responses from students, which vary but also demonstrate consensus around the correct response. She then asks students to talk about how they solved it, and develops their thinking through discourse, for example by asking, "Where does the 5 and the 3 come from?" One student shares how they "broke" the 7 into 6 and 1, and Arrillaga elaborates, "You broke, you decomposed the number?" Arrillaga attaches each student's method of solving to their name. Her student Cesar contributes that he wants to use the "Australian Method," which refers to a previous conversation in which a mathematics coach showed the students how they could use partial sums to solve in addition. The class discusses the "Australian Method" and compares how Noemi's method differs in its representation of the 16.

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Number Talk: 2nd Grade | Inside Mathematics

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In the number talk, Arrillaga engages her students in a discussion of different answers to the addition problem 59 + 37. She gathers different responses from students, which vary but also demonstrate consensus around the correct response. She then asks students to talk about how they solved it, and…

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Reasoning About Multiplication & Division

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Reasoning About Multiplication & Division

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Math.3.OA.B.5

Common core State Standards

Math: Math

3: Grade 3

OA: Operations & Algebraic Thinking

B: Understand properties of multiplication and the relationship between multiplication and division

5:
Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Download Common Core State Standards (PDF 1.2 MB)

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Discussion and Supporting Materials

Thought starters

1. How do Mr. Crandall's students learn from each other?

2. Mr. Crandall says it's important to give students time to make sense of their ideas. How does he do that?

3. How does Mr. Crandall give students opportunities to revise their thinking?

School Details

Lakeridge Elementary School
7400 South 115th St
Seattle WA 98178

Data Provided By:

Teachers

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Reasoning About Multiplication & Division

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Third graders in Drew Crandall's class look at patterns in multiples of four and compare strategies for solving a multiplication problem. Students begin by looking for patterns in multiples of four and explaining why the patterns are happening. The class then moves on to solving a multiplication problem using a variety of strategies.

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It's time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences.

All children are different in their thinking, strength and interests. Mathematics classes of the past decade have valued one type of math learner, one who can memorize well and calculate fast.

Yet data from the 13 million students who took PISA tests showed that the lowest achieving students worldwide were those who used a memorization strategy - those who thought of math as a set of methods to remember and who approached math by trying to memorize steps. The highest achieving students were those who thought of math as a set of connected, big ideas.

The U.S. has more memorizers than most other countries in the world. Perhaps not surprisingly as math teachers, driven by narrow state standards and tests, have valued those students over all others, communicating to many other students along the way - often girls - that they do not belong in math class.

The fact that we have valued one type of learner and given others the idea they cannot do math is part of the reason for the widespread math failure and dislike in the U.S.
Related: Common Core math experts say teachers need to stop using shortcuts and math tricks
Brain science tells us that the students who are better memorizers do not have more math "ability" or potential but we continue to value the faster memorizers over those who think slowly, deeply and creatively - the students we need for our scientific and technological future. The past decade has produced a generation of students who are procedurally competent but cannot think their way out of a box. This is a problem.

We don't need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.

Mathematics is a broad and multidimensional subject. Real mathematics is about inquiry, communication, connections, and visual ideas. We don't need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.
Related: Should we stop making kids memorize times tables?
Technology leaders are publically arguing that calculation is not math, and that math is a much broader subject. Conrad Wolfram, one of the leaders of one of the world's most important mathematics companies, Wolfram-Alpha, urges schools to stop emphasizing calculating and focus instead on problem solving, modeling, thinking, and reasoning as these are the mathematical abilities that students need in the workplace and their high tech lives. This broad, multidimensional mathematics is the math that engages many more learners and puts them on a pathway to life long success.
Related: Testing gives third-graders upset stomachs, tears, even fevers
Part of the problem in the U.S. is the desperation of many parents to advance their children in math, pushing them to higher levels of math faster and sooner, somehow believing that a resume packed with advanced math courses will guarantee their future. Bill Jacob, a mathematics professor at the University of California, Santa Barbara, speaks openly about the dangers of students being pushed to higher levels of mathematics too soon. "I know it is hard to persuade parents that their students shouldn't race to get calculus, but I really wish they wouldn't. Too much content and depth is left out when they do." said Jacob, who is not alone in saying that he would rather have students in his university mathematics courses that have breadth in their mathematical experiences than any additional Advanced Placement courses. Experts in England are giving the same advice to parents of high achieving students. Geoff Smith, chairman of the British and International Math Olympiads warns that accelerating children through the system is a "disaster" and a "mistake". He, like me, recommends that high achieving students explore the mathematics they are learning in depth, instead of rushing forward.

Mathematics is not a subject that requires fast thinking. Award winning mathematicians talk about their slow, deep thinking in math. Fields Medal winning mathematician Laurent Schwartz wrote in his autobiography that he felt stupid in school because he was one of the slowest thinkers in math. Eventually he realized that speed was not important - "What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant."

Some school districts, such as San Francisco Unified, are trying to slow down the math experience, requiring that advanced students go deeper rather than faster. Students still reach calculus but the pathway to calculus consists of deep understanding rather than procedures and memorization. This is an important move. There is no harm in students being introduced to higher-level mathematics earlier, as long as the mathematics is enjoyable and ideas can be explored deeply. Third graders can be fascinated by the notion of infinity, or the fourth dimension, but they do not need a race through procedural presentations of mathematics.
Jo Boaler is a professor of mathematics education at Stanford University, co-founder of This story was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Read more about Common Core. www.youcubed.org, and author of What's Math Got To Do With It: How Teachers and Parents can Transform Mathematics Learning and Inspire Success. (Penguin, 2015)
New brain science tells us that no one is born with a math gift or a math brain and that all students can achieve in math with the right teaching and messages. The classrooms that produce high achieving students are those in which students work on deep, rich mathematics through tasks that they can take to any level they want. No one is told what level they can reach and no one is held back by narrow questions that limit students' mathematical development and creativity.

Many people across the U.S. have gross misconceptions about mathematics learning, thinking that mathematics is a narrow subject of memorization and speed and that people who do not calculate fast or memorize well are not 'math people'. We need to change the conversations about mathematics, communicating to all children that they can learn. We also need to change the way math is taught, valuing the different ways of thinking that are so important to the subject. Mathematics, itself, needs this and although change is hard, it should be embraced.

We need to broaden mathematics and open the doors of mathematics to all students. When we do this we will see many more creative, energized young people equipped to think quantitatively about our ever-changing world. We all need this.

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Memorizers are the lowest achievers and other Common Core math surprises

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It's time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences. All children are different in their thinking, strength and interests.

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Ruth Parker & Cathy Humphreys: MAKING NUMBER TALKS MATTER

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Math Solutions Number Talks  K 1 Ten Frames and Dot Cards

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Online manipulatives bring all of the learning and fun without the mess. Our math virtual manipulatives help students grasp abstract concepts through modeling. Literacy virtual manipulatives create a fun space for practicing spelling, rhyming, and more!

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Virtual Manipulatives | Mathematical Modelling | Toy Theater

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Original teaching tools aimed at supporting teachers in the elementary school classroom. Two color counters, 3D dice, probability spinners, graph builders, fraction bars, base ten blocks, and more!

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The CCSS-M calls for students to use concrete and visual models to solve problems and make sense of number and operations. In the SFUSD Math Core Curriculum, these models are taught as tools and strategies to support students' development of conceptual understanding of operations and connections to algorithms.

The Math Department has compiled a progression of the concrete and visual models for addition and subtraction in grades K-2 and another for multiplication and division in grades 3-5.

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The CCSS-M calls for students to use concrete and visual models to solve problems and make sense of number and operations. In the SFUSD Math Core Curriculum, these models are taught as tools and strategies to support students' development of conceptual understanding of operations and connections to algorithms. The Math Department has compiled a progression of the concrete and visual models for addition and subtraction in grades K-2 and another for multiplication and division in grades 3-5.

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Steve Wyborney's Blog: I'm on a Learning Mission. - Sharing Thoughts About Education

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Math.2.NBT.B.7

Common core State Standards

Math: Math

2: Grade 2

NBT: Number & Operations in Base Ten

B: Use place value understanding and properties of operations to add and subtract

7:
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Download Common Core State Standards (PDF 1.2 MB)

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A number string is a series of mathematical expressions organized in a way that encourages students to build on their thinking as the string progresses.

Discussion and Supporting Materials

School Details

Shields (Richard A.) Elementary School
910 Shields Avenue
Lewes DE 19958

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Teachers

Math / Kindergarten 1 2 3 4 5 / Teacher

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A Number Talk lesson for 2nd grade students. See a complete lesson in action that begins with a simple addition equation. Lesson allows teachers to get as many answers as possible from students. At the end, there is also an individual assignment idea.

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powerpoint

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Slides with Number Talk Dots and Five & Ten
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Slides with Number Talk Dots and Five & Ten
Frames

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Math / Kindergarten Number Talk Resources

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New to Number Talks? Read the following article to find a summary of the Number Talks idea!(Content for slides below come from Number Talks by Sherry Parrish)Five and Ten FramesNumber 3(The slides above were compiled by Mary Elizabeth Mendenhall from Griffin RESA.)

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Apple Multiplication Array for Fact Fluency

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Apple Multiplication Array for Fact Fluency

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Intentional Talk Math Discussion Planning Templates

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Intentional Talk Math Discussion Planning Templates

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lesson planning NT 2.0

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Templates for all levels and types of Number Talk

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Templates for all levels and types of Number Talk

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Access Google Sites with a personal Google account or Google Workspace account (for business use).

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"Fluency in each grade involves a mixture of just knowing some answers, knowing some answers from patterns (e.g., "adding 0 yields the same number"), and knowing some answers from the use of strategies. It is important to push sensitively and encouragingly toward fluency of the designated numbers at each grade level, recognizing that fluency will be a mixture of these kinds of thinking which may differ across students." (CC/OA Progression, p. 18)

These documents provide a set of short activities extracted from Engage NY and Illustrative Mathematics, two open education resources. While initially created as a supplement to Go! Math, teachers who are using a variety of curricula have found these fluency resources helpful. The activities are designed to support students' progress toward the grade-level numeracy and fluency articulated in college- and career-ready standards. They are intentionally short, providing educators the flexibility to use them before or after a lesson or anytime during the school day. The activities are intended to be used throughout the school year.
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Fluency Resources for Grade-Level Routines

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"Fluency in each grade involves a mixture of just knowing some answers, knowing some answers from patterns (e.g., "adding 0 yields the same number"), and knowing some answers from the use of strategies.

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From Stanford Online's "How To Learn Math for Teachers and Parents": Number Talks

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ABCya! * Number Chart - Use to Learn Number Patterns & More

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The interactive number chart is a great way for students to discover number patterns, skip count, and learn multiplication tables. Color the interactive number charts to find patterns in numbers or use them for counting exercises. Children can print their charts too!

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Understanding Language: Mathematics Instruction for ELs

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Understanding Language: Mathematics Instruction for ELs

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There's a lot happening in elementary mathematics and it's really tough for teachers to keep up with it all...let alone parents and administrators.

I keep reminding myself that the turtle won the race.

It's my hope that in creating and sharing this video, we can all slow down a little bit more and focus on building fluency through conceptual understanding.

Hope it helps!

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There's a lot happening in elementary mathematics and it's really tough for teachers to keep up with it all...let alone parents and administrators. I keep reminding myself that the turtle won the race. It's my hope that in creating and sharing this video, we can all slow down a little bit more and focus on building fluency through conceptual understanding.

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So at a recent math workshop, I learned about a Rekenrek (which is basically an abacus with two colors). It really helps build number sense which is great since we are getting away from rote math problems and focusing on the understanding of math and how numbers work. It is genius to me and so I went about making a class set. As I was making them, my own kids started working with them and I loved listening to them talk through problems. They had so much fun with them they each requested I make one for them in their favorite color (which of course I did). To see a quick YouTube video of a teacher describing how to use them click HERE.

To make my Rekenreks, I used foam board (don't worry... there were all colors, not just the pink), string, beads (pony beads) a hole punch and scissors. My hole punch is small and fits the string nicely. I had it from my old scrapbooking days.

Punch four holes like this.

Then string the beads making sure you keep five of each color on either side.

I tied them different ways but this was the best (and fastest).

My kindergartner played with his (he wanted yellow) forever! I couldn't believe the way he was manipulating it after just minutes. I put a quick video of his review of how to use the Rekenrek that you can see too by clicking HERE or looking below. My bigger kids (going into 3rd and 4th) also really liked them. I made them Rekenreks that went to 30.

I also plan to make a more sturdy one for me on the harder foam board and bigger beads.

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So at a recent math workshop, I learned about a Rekenrek (which is basically an abacus with two colors). It really helps build number sense which is great since we are getting away from rote math problems and focusing on the understanding of math and how numbers work.

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1st Grade Teacher Website with Number Talk Ideas

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1st Grade Teacher Website with Number Talk Ideas

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I've had a quite a few people e-mail me about Number Talks. So I wanted to devote this post to how Number Talks operate in my classroom . W...

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Teacher Channel Division
Number Talk – 4th grade 

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Teacher Channel Division
Number Talk – 4th grade 

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Number talks were developed for classroom teachers to engage students in "mental math" through grappling with interesting mathematics problems. Educators can use number talks regularly as introductions to the day's mathematical practice, as "warm ups" for other lessons, or as stand-alone extended engagements with mathematical concepts. The teachers in these videos are from schools participating in the Silicon Valley Mathematics Initiative.

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Number talks were developed for classroom teachers to engage students in "mental math" through grappling with interesting mathematics problems. Educators can use number talks regularly as introductions to the day's mathematical practice, as "warm ups" for other lessons, or as stand-alone extended engagements with mathematical concepts.

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A few weeks ago I (Jo Boaler) was working in my Stanford office when the silence of the room was interrupted by a phone call. A mother called me to report that her 5-year-old daughter had come home from school crying because her teacher had not allowed her to count on her fingers. This is not an isolated event-schools across the country regularly ban finger use in classrooms or communicate to students that they are babyish. This is despite a compelling and rather surprising branch of neuroscience that shows the importance of an area of our brain that "sees" fingers, well beyond the time and age that people use their fingers to count.

In a study published last year, the researchers Ilaria Berteletti and James R. Booth analyzed a specific region of our brain that is dedicated to the perception and representation of fingers known as the somatosensory finger area. Remarkably, brain researchers know that we "see" a representation of our fingers in our brains, even when we do not use fingers in a calculation. The researchers found that when 8-to-13-year-olds were given complex subtraction problems, the somatosensory finger area lit up, even though the students did not use their fingers. This finger-representation area was, according to their study, also engaged to a greater extent with more complex problems that involved higher numbers and more manipulation. Other researchers have found that the better students' knowledge of their fingers was in the first grade, the higher they scored on number comparison and estimation in the second grade. Even university students' finger perception predicted their calculation scores. (Researchers assess whether children have a good awareness of their fingers by touching the finger of a student-without the student seeing which finger is touched-and asking them to identify which finger it is.)

Evidence from both behavioral and neuroscience studies shows that when people receive training on ways to perceive and represent their own fingers, they get better at doing so, which leads to higher mathematics achievement. The tasks we have developed for use in schools and homes (see below) are based on the training programs researchers use to improve finger-perception quality. Researchers found that when 6-year-olds improved the quality of their finger representation, they improved in arithmetic knowledge, particularly skills such as counting and number ordering. In fact, the quality of the 6-year-old's finger representation was a better predictor of future performance on math tests than their scores on tests of cognitive processing.
Many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible.
Neuroscientists often debate why finger knowledge predicts math achievement, but they clearly agree on one thing: That knowledge is critical. As Brian Butterworth, a leading researcher in this area, has written, if students aren't learning about numbers through thinking about their fingers, numbers "will never have a normal representation in the brain."

One of the recommendations of the neuroscientists conducting these important studies is that schools focus on finger discrimination -not only on number counting via their fingers but also on helping students distinguish between those fingers. Still, schools typically pay little if any attention to finger discrimination, and to our knowledge, no published curriculum encourages this kind of mathematical work. Instead, thanks largely to school districts and the media, many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible. Kumon, for example, an after-school tutoring program used by thousands of families in dozens of countries, tells parents that finger-counting is a "no no" and that those who see their children doing so should report them to the instructor.

Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development. Fingers are probably one of our most useful visual aids, and the finger area of our brain is used well into adulthood. The need for and importance of finger perception could even be the reason that pianists, and other musicians, often display higher mathematical understanding than people who don't learn a musical instrument.

Teachers should celebrate and encourage finger use among younger learners and enable learners of any age to strengthen this brain capacity through finger counting and use. They can do so by engaging students in a range of classroom and home activities, such as:

Give the students colored dots on their fingers and ask them to touch the corresponding piano keys:

Give the students colored dots on their fingers and ask them to follow the lines on increasingly difficult mazes:

(The full set of activities is given here.)

The finger research is part of a larger group of studies on cognition and the brain showing the importance of visual engagement with math. Our brains are made up of "distributed networks," and when we handle knowledge, different areas of the brain communicate with each other. When we work on math, in particular, brain activity is distributed among many different networks, which include areas within the ventral and dorsal pathways, both of which are visual. Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing.

A striking example of the importance of visual mathematics comes from a study showing that after four 15-minute sessions of playing a game with a number line, differences in knowledge between students from low-income backgrounds and those from middle-income backgrounds were eliminated.

Number-line representation of number quantity has been shown to be particularly important for the development of numerical knowledge, and students' learning of number lines is believed to be a precursor of children's academic success.

Visual math is powerful for all learners. A few years ago Howard Gardner proposed a theory of multiple intelligences, suggesting that people have different approaches to learning, such as those that are visual, kinesthetic, or logical. This idea helpfully expanded people's thinking about intelligence and competence, but was often used in unfortunate ways in schools, leading to the labeling of students as particular type of learners who were then taught in different ways. But people who are not strong visual thinkers probably need visual thinking more than anyone. Everyone uses visual pathways when we work on math. The problem is it has been presented, for decades, as a subject of numbers and symbols, ignoring the potential of visual math for transforming students' math experiences and developing important brain pathways.

It is hardly surprising that students so often feel that math is inaccessible and uninteresting when they are plunged into a world of abstraction and numbers in classrooms. Students are made to memorize math facts, and plough through worksheets of numbers, with few visual or creative representations of math, often because of policy directives and faulty curriculum guides. The Common Core standards for kindergarten through eighth grade pay more attention to visual work than many previous sets of learning benchmarks, but their high-school content commits teachers to numerical and abstract thinking. And where the Common Core does encourage visual work, it's usually encouraged as a prelude to the development of abstract ideas rather than a tool for seeing and extending mathematical ideas and strengthening important brain networks.

To engage students in productive visual thinking, they should be asked, at regular intervals, how they see mathematical ideas, and to draw what they see. They can be given activities with visual questions and they can be asked to provide visual solutions to questions. When the youcubed team (a center at Stanford) created a free set of visual and open mathematics lessons for grades three through nine last summer, which invited students to appreciate the beauty in mathematics, they were downloaded 250,000 times by teachers and used in every state across the U.S. Ninety-eight percent of teachers said they would like more of the activities, and 89 percent of students reported that the visual activities enhanced their learning of mathematics. Meanwhile, 94 percent of students said they had learned to "keep going even when work is hard and I make mistakes." Such activities not only offer deep engagement, new understandings, and visual-brain activity, but they show students that mathematics can be an open and beautiful subject, rather than a fixed, closed, and impenetrable subject.

Some scholars note that it will be those who have developed visual thinking who will be "at the top of the class" in the world's new high-tech workplace that increasingly draws upon visualization technologies and techniques, in business, technology, art, and science. Work on mathematics draws from different areas of the brain and students need to be strong with visuals, numbers, symbols and words-but schools are not encouraging this broad development in mathematics now. This is not because of a lack of research knowledge on the best ways to teach and learn mathematics, it is because that knowledge has not been communicated in accessible forms to teachers. Research on the brain is often among the most impenetrable for a lay audience but the knowledge that is being produced by neuroscientists, if communicated well, may be the spark that finally ignites productive change in mathematics classrooms and homes across the country.

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Using Fingers to Count in Math Class Is Not 'Babyish'

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A few weeks ago I (Jo Boaler) was working in my Stanford office when the silence of the room was interrupted by a phone call. A mother called me to report that her 5-year-old daughter had come home from school crying because her teacher had not allowed her to count on her fingers.

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