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Video (Graham Fletcher)

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The Progression of Fractions (Meaning, Equivalence, & Comparison)

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Kindergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5

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Beyond Pizzas and Pies (book)
Julie McNamara

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Math

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Mathematics Standards – Common Core State Standards Initiative

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For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, t…

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Grade PK      Grade K         Grade 1
Grade 2         Grade 3         Grade 4
Grade 5         Screening Cards

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EMDI Overview

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Counting, Computational Fluency, Story Problems, EMDI Reports

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Frequently Asked Questions (PDF)

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Report Templates
SAMPLE Reports
Gr 2 CCSS Alignment
Gr 2 CCSS Resources

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Definitions
Trajectory

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Video (Graham Fletcher)

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PK Counting and Cardinality (PK.CC.1-6)

K Counting and Cardinality
(K.CC.1-7)

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Virtual Rekenrek
Make a Rekenrek

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The rekenrek, or arithmetic rack, was designed by Adrian Treffers, a mathematics curriculum researcher at the Freudenthal Institute in Holland, to support the natural development of number sense in children. Smaller versions consist of two rows of 10 beads. Larger versions with ten rows of ten beads are also available. Each row is made of five red beads and five white beads. This allows students to make mental images of numbers. Using 5 and 10 as anchors for counting, adding and subtracting is obviously more efficient than one-by-one counting. This tool provides learners with the visual models they need to discover number relationships and develop a variety of addition and subtraction strategies, including doubles plus or minus one, making tens, and compensation, thereby leading to automaticity of basic facts.

Possible Rekenrek Activities

Possible Math Journal Tasks:

There were 8 children on a bunk bed. Some were on the top bunk and some were on the bottom bunk. How many children were on the top bunk? How many children were on the bottom bunk? Show as many different solutions as you can. Number Stories: Have children use individual arithmetic racks as a tool to solve various types of addition and subtraction number stories. This may be used as a journal or oral activity, with the focus on children explaining their strategy for solving the problem. Be sure to include open-ended problems that have more than one solution that children can model on the arithmetic rack, such as the following:

There were 12 passengers on a double-decker bus. Some passengers were on the top deck and some were on the bottom deck. How many passengers were on the top deck? How many passengers were on the bottom deck? Show as many different solutions as you can.

The above two work samples were completed by a Kindergarten student who represented her work very clearly on the unlined pages in her Math Journal using pictures, numbers and words. To begin with some Kindergarten students may find it difficult to represent their work on paper. Having recording paper available for children who choose to use it is one way to scaffold early attempts at drawing.

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The rekenrek, or arithmetic rack, was designed by Adrian Treffers, a mathematics curriculum researcher at the Freudenthal Institute in Holland, to support the natural development of number sense in children. Smaller versions consist of two rows of 10 beads. Larger versions with ten rows of ten beads are also available.

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Multi-Dimensional Fluency
Strategies for Developing Fluency
Fluency Without Fear

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Video (Graham Fletcher)

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Sense-making?

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Watch how 32 eighth graders responded to: "There are 125 sheep and 5 dogs in a flock. How old is the shepherd?"

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I'm sure we've all seen it happen at one time or another in math class. We give a student a story problem to solve and after a quick skim, the student pulls the numbers from the problem, computes with them, and writes down an answer.

If the answer is correct, we assume the student has a grasp of the concept. However, if it's incorrect, we're left with a laundry list of questions: Do they realize their answer doesn't make sense? Did they not understand the context? Did they simply pull the numbers and operate in order simply to finish or did they truly not know what to do with them? Most importantly, we ask ourselves, how can I help students make sense of what they're reading and to think about the logic of their answer in the context of the problem?

If we're lucky, we can identify the student's mathematical misconception and work with that. Oftentimes, though, the student's answer isn't even reasonable. Then what do we do?

This scenario has me reflecting on the Common Core Standard of Mathematical Practice 1:
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
The best way I've found to help students make sense of a problem is, ironically, to take the question out altogether. Inspired by the wonderful folks at The Math Forum, I do a lot of noticing and wondering with students in this fashion. Most recently, after reading Brian Bushart's awesome blog post, I've started taking the numbers out as well! Instead of students thinking about how they're going to solve the problem as they read, they are truly thinking about the situation at hand. It's been an amazing way to give every student entry into a problem and allows me to differentiate for all learners in the classroom. At the same time, I'm provided insight into my students' mathematical understanding.

Recently, I had the opportunity to work with a 3rd grade class. They had just finished their multiplication and division unit and would soon be starting work on fractions. In order for their teacher and I to see and hear how students apply the operations, make sense of contexts, and currently think about fractions, I thought it would be interesting to take a story problem from their Student Activity Book and take out the question and numbers.

The Planning

I chose the problem below and thought about what I would learn about a student's mathematical understandings and sense-making after they answered the questions.

Based on the idea of removing the numbers and the question, I changed the problem to this simple statement:
"Webster has boxes of granola bars to share with his class."
I anticipated the students would wonder about the missing mathematical pieces involved in an open-ended statement like this. I believed their wondering might lead them to develop questions that could be answered based on the very information they were wondering about. I knew the mathematical ideas of multiplication, division, and/or fractional sharing would arise and that I would learn so much more about their thinking than if I had given them the original problem.

In The Classroom

I launched the lesson by posting the sentence on the board and recorded the things they noticed and wondered.
They noticed:

"We don't know how many boxes of granola bars there are."

"There isn't enough information to know what's going on."

"We don't know if it is adding, subtracting, multiplying, or dividing."

"There are multiple people in the class, because it says boxes and share."

They wondered:

"How many granola bars are in each box?"

"How many boxes did he bring to class?"

"How many kids are in his class?"

"What kind of granola bars are they?"

Based on what they noticed and were wondering about, I felt everyone had a strong grasp of the context and a sense of where this was going. I picked up on the notion that "there isn't enough information to know what's going on" and asked what more they would want to know. They responded that they wanted the answers to the first three of their wonders: bars per box, number of boxes, and number of kids in the class.

I asked them what questions they could answer if I gave them those pieces of information, and they responded:

"How many bars does he have?"

"How many bars does each kid get?"

"How many does he have left?"

At this point, I could have given them the information they wanted. However, I thought it would be so much cooler to allow them to choose that information for themselves. I was curious how they would go about choosing their numbers! Would they strategize about the numbers to make it easier for themselves? Would they even think that far ahead? What would they do with the leftovers?

When I told them I was not giving them the information and that instead they were choosing their own numbers along with the question they wanted to answer, they were so excited!

Some partners chose their numbers very strategically to make it easier for themselves. To me, this demonstrated a lot of sense-making and forethought of what was going to happen in their solution path. And, as an added bonus, despite being asked to answer only one question, the group answered all three! (Teacher note: if students choose numbers strategically and therefore finish quickly, give them extra bars to factor into their problem to see how they deal with the leftovers.)

Other students chose the opposite route and strategically picked numbers to make it "harder for themselves." Check out the way these two students showed strong reasoning and perseverance through division of numbers larger than any they've ever worked with.

Others chose numbers without much forethought and dealt with some amazing leftovers. This was a great way to formatively assess students' thinking related to fractions before they began that unit.

And then there are always the surprises. Who would have thought 3rd graders would reason about the leftovers in terms of percentages?

Finding Out What They Know

Reflecting on the difference between what students would have done with the original problem versus the reasoning work they did related to the one simple sentence, I'm amazed at the outcomes. I learned so much more about what each of the students know beyond simply multiplying five and six. Taking out the numbers and the question allowed every student to think about the meaning of the sentence, the implied mathematical connections, and to plan a solution pathway before jumping into a solution attempt.

I highly recommend everyone try this strategy with a word problem from your current text. It's a wonderful way to give every student access to the math and the freedom to think beyond just getting an answer.
This work was made possible through support by the Leona M. and Harry B. Helmsley Charitable Trust. Kristin Gray is a National Board Certified fifth grade math teacher at Richard A. Shields Elementary School in the Cape Henlopen School District in Lewes, Delaware and a Teaching Channel Laureate. During her 19 years in education, she has taught 5th-8th grade math, and spent two years as a K-5 Math Specialist. She feels fortunate to be involved with Illustrative Mathematics and Teaching Channel on projects developing math tasks, facilitating professional development, and blogging about these experiences. She is always excited to share her love of teaching at conferences such as NCTM, NCSM, ISTE, as well as on her blog. Follow Kristen on Twitter: @MathMinds.
If you know me or have ever read my blog, you know I could talk for days about student math work! You can visit my blog for a more detailed description of the work shown in this post, as well as additional work captured from the lesson.

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Help Kids Understand Math Problems: Take Away The Numbers And the Question!

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I'm sure we've all seen it happen at one time or another in math class. We give a student a story problem to solve and after a quick skim, the student pulls the numbers from the problem, computes with them, and writes down an answer.

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The 3-Read Protocol

The Three Read Protocol is one way to do a close read of a complex math word problem or task. This strategy includes reading a math scenario three times with a different goal each time. The first read is to understand the context. The second read is to understand the mathematics. The third read is to elicit inquiry questions based on the scenario.

Why would I use this strategy?

The Three Read Protocol is designed to engage students in sense-making of language-rich math problems or tasks. It deepens student understanding by surfacing linguistic as well as mathematical clues. It focuses attention on the importance of understanding problems rather than rapidly trying to solve them. It allows for the use of authentic, instead of overly simplified, text. This strategy also allows for natural differentiation within a class of diverse learners.

When do I use this strategy?

This strategy can be used for math tasks that include complex language structures or language that lends itself to a variety of interpretations. While this is a particularly useful strategy for English Language Learners, all students can benefit from the deeper understanding of word problem structures and open-ended questioning.

How do I use this strategy?

The Three Read Protocol uses the " problem stem " of a word problem. This is essentially the word problem without the question at the end. The purpose of presenting the problem stem alone is to have students focus on the contextual and mathematical information before dealing with any question that is involved. This gives students the freedom to create their own questions for a given scenario, which is an excellent skill to develop both in math and in reading in general. It is important that the teacher choose the problem carefully and anticipate potential linguistic and mathematical roadblocks the students may encounter.

1. First Read: Teacher reads the problem stem orally.

The teacher may have visuals to accompany the oral read of the problem stem. Students listen to the story with the goal of turning to a partner and sharing what they remember of it. Memorizing it is not necessary. Students may act out the problem if that helps them grasp the context.

Key Question: What is this situation about?

After the Turn-and-Talk, the teacher asks students to volunteer information they remember from the story. Teachers and students ask clarifying questions about the vocabulary as needed.

2. Second Read: Class does choral read or partner read of the problem stem.

The teacher projects the problem stem so the whole class can see it. The teacher leads the class either in a choral read of the problem or has partners read the problem orally to each other. Choral read is preferable because it allows all students to participate without excessive pressure, but a partner read can work fine if that is a better fit to the classroom culture or age of students. The teacher explains that math stories usually have information about quantities (numbers) and the units that are being counted.

Key Question: What are the quantities in the situation?

An example is 25 cats, where "25" is the quantity and "cat" is the unit. Sometimes the quantities are implied. For example, "some cats" implies a quantity but we do not know what it is. There can also be implied units. An example is "I have one at home." The implied unit in this case depends on the context of the story. Bottom line : The discussion of quantities and units can be important for focusing student attention, but how deeply the teacher delves into the explicit and implicit information depends on the math and language objectives.

3. Third Read: Partner or choral read the problem stem orally one more time.

The teacher asks students to do one more read of the "story" and asks them to think, " What is missing to make this a good math problem? " Students volunteer their answers to that question. Responses will likely vary because many students assume there is a question without actually reading one. Without correcting student responses, the teacher probes until the class decides that a question is missing. The teacher asks, " Is there only one question that we can ask of this story? " Students responses may vary, but there are usually many different questions that can be asked of almost any scenario.

Key Question: What mathematical questions can we ask about the situation?

The teacher asks partners to determine at least two questions that can be asked using the problem stem. Students share their questions. The teacher writes a couple of the questions and clarifies language as appropriate. After each question, the teacher asks the class, " Can this question be answered with the information from this story? " and the class discusses why or why not.

4. Students work in collaborative groups on the problem.

Students work in groups to solve a question based on the problem stem. The teacher may assign a specific question for all groups to answer, or groups may choose a question from the list asked by the class. If groups are asked to choose their own questions, it is important that the teacher circulate and clarify expectations for the work. This can be an opportunity to differentiate the math work because the range of possible questions to a problem stem is broad.

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Download the description of the 3-Read Protocol from our Math Teaching Toolkit

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Grades K-2

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H_ Problem Solving Situations by Grade Level

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Grades 3-5

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J_3-5 Problem Solving Situations by Grade Level

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Video

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Ever Wonder What They’d Notice?: Annie Fetter

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Understand the Structure

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GregTangMath.com

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A revolutionary, online math program comprising games, animated books and downloadable materials.

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Expected fluencies in the Common Core

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Fact Mastery Through Games
Mastering Multiplication Facts
Assessing Basic Facts

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Shared Files on Google Drive

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Video (Graham Fletcher)

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Visual Multiplication

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Video (Graham Fletcher)

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EMDI scoring, Fact fluency, ultiplication

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1. Teaching Student-Centered Mathematics K-2
2. Teaching Student-Centered Mathematics 3-5

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Mastering Basic Math Facts +/-
Mastering Basic Math Facts x/÷

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Kathy Richardson (Books)

Book 1: Counting, Comparing and Pattern
Book 2: Addition and Subtraction
Book 3: Multiplication, Division and Place Value

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Computation Based on Place Value Understanding

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Computation Based on Place Value Understanding

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Addition, Subtraction, Multiplication, Division

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K-2 Content
3-5 Content
General Content

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Your feedback will help me know how this PD has been helpful to you and what can be improved in the future. Many thanks!

Click here to open.

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Fractions, "What's Next?"