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Download our Multiplication Chart Printable! 1-12 Multiplication Chart: Divided into 3 Sections to Memorize Fast Color Guide Times Tables Method for 1-12 Multiplication Facts Learning the multiplication chart doesn't have to be a struggle! In this article you'll learn how...

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Times Tables: Memorize Multiplication Facts Fast

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With Times Tales, memorizing the times tables has never been so easy! Our award-winning multiplication program will help your child memorize the times tables fast!

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Freckle Math - for grades K-12, similar to Lexia?

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Freckle is flexible, captivating every learner with differentiated practice, and is designed to be both teacher-led and student-paced. You can assign specific math exercises based on grade-level standards, and for students who need practice outside of their grade level, Freckle can provide self-paced practice that continuously adapts as they progress. Real-world lessons encourage students to work in groups to solve fun, practical challenges with the skills they've learned.

K-12 coverage of standards and skills

Aligned to state standards and to Common Core

Over 70,000 unique math questions available

As students work, supports such as audio and hints guide them through their practice and help them solve problems without giving up

Additional math fact practice and inquiry-based resources help drive deeper understanding

Learn more about how students who used Freckle for 100 days avoided the COVID-19 Slide and actually exceeded expectations.

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Freckle is flexible, captivating every learner with differentiated practice, and is designed to be both teacher-led and student-paced. You can assign specific math exercises based on grade-level standards, and for students who need practice outside of their grade level, Freckle can provide self-paced practice that continuously adapts as they progress.

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Imagine Math - for grades K-12

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Provide targeted, data-driven, and age-appropriate online intervention that accelerates struggling learners in math and reading toward grade-level achievement.

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For grades 6-12, similar to IXL?

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Problems are randomized, allowing students multiple opportunities to get the practice and feedback they need on the road to mastery.

Go beyond right and wrong. As soon as a student submits an answer, they get a detailed age-appropriate explanation.

DeltaMath allows teachers to mix and match problem-sets, control rigor, vary due dates, and, with PLUS or INTEGRAL, create tests and problems of their own.

Use DeltaMath's modules to create high-leverage assignments and track student learning. With DeltaMath PLUS or INTEGRAL, students also get access to help videos. Create and assign tests, assign specific problem-types, even create your own problem.

Learn More

Problems solved with DeltaMath

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Number sense screeners for grades K-6, can be administered 3x per year.

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Achievethecore.org :: Instructional Content Nav - Mathematics: Focus by Grade Level

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The Common Core and other college- and career-ready (CCR) standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, CCR standards require us to significantly narrow and deepen the way time and energy are spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. The following documents illustrate these concepts using the Common Core State Standards, but Focus, Coherence, and Rigor are integral to all college- and career-ready standards.

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Interventions are sorted based on what they have been developed for, with categories for reading, math and behavior interventions as well as a section for general academic interventions. Click on the categories below for specific evidence based interventions developed for those situations.

Reading Interventions

Math Interventions

Math - Generalization
The student has not done the academic task that way before.

Behavior Interventions

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Interventions are sorted based on what they have been developed for, with categories for reading, math and behavior interventions as well as a section for general academic interventions. Click on the categories below for specific evidence based interventions developed for those situations. Math - Generalization The student has not done the academic task that way before.

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Resources for Early Numeracy and Computation Skills

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Resources for Early Numeracy and Computation Skills

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Determine which skills in the MIND: Facts of Fire scope and sequence the students have been taught and assess the class in each of the skills for 1 minute. Identify the lowest skill for each student where he/she fails to surpass the mastery criteria. Place each student in his/her identified skill, print out the corresponding skill booklet, and the class is ready to begin the MIND: Facts on Fire. Please resist the urge to move students through skill areas when the mastery criteria is not met. Computation mastery in early skills is essential to quickly moving through later skills. (e.g., fluent subtraction will benefit students practicing multi-digit division).

Research has shown that daily practice is necessary to obtain desired growth rates, so it is imperative that every class and every student participate each day in the Facts on Fire program. Second, teachers must be active participants by walking around the room and providing behavior specific praise to students for task adherence and effort. Students should be acutely aware that their teacher sees fact fluency as an essential math skill. It is also important that administrators conduct spot-checks to provide performance feedback to classroom teachers on treatment integrity.

To ensure that students are practicing skills that are instructionally matched to their needs the classroom teacher scores two probes each week to obtain each students DC/M performance. These scores are used to inform educational decisions about skill placement (stay on current skill or move to the next) and communicate what skill packet needs to be printed for the upcoming week. In addition, if a student fails to progress within or across skills the MIND: Facts on Fire program contains a series of assessments to pinpoint potential causes and recommend solutions.

A person needs to be trained to oversee and manage materials. This will include knowledge of the website to locate necessary materials for printing, taking print orders from teachers, printing materials, getting booklets to classrooms, and organizing data to support incentive plans (if applicable) and report outcomes for consumers. This person is vital to keeping the MIND: Facts on Fire working week to week and allowing teachers to focus on implementing the program.

When administering group-wide interventions at the Pre-Kindergarten, Kindergarten and First Grade level there are factors that will uniquely impact administration.

1. Variety of Target Skills

At this level it is anticipated that student's skill deficits can vary tremendously . In this situation, where some students will be building gateway skills while other students are focused on numeracy skills, class-wide administration can be assisted by breaking the student's into smaller groups so that directions are specific to their intervention. At the grade-wide level, students can be broken into skill groups and each teacher can administer a different intervention in their classroom. Once students are familiar with the procedures associated with their target skill more generic directions can be read aloud and the intervention can occur class-wide.

2. Requires Substantial Monitoring

Group-wide administration at the early numeracy level requires school personnel to ensure student's are adhering to the presented task and understand task demands. It is important to address student's questions before beginning the intervention. When reading the scripted directions it is necessary to monitor student's because they may try to work ahead. Once the timer has gone off and the intervention is completed student's may also try to keep working so it is important to be aware of student's during this transition.

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Determine which skills in the MIND: Facts of Fire scope and sequence the students have been taught and assess the class in each of the skills for 1 minute. Identify the lowest skill for each student where he/she fails to surpass the mastery criteria.

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Same But Different is a powerful routine for use in math classrooms. The images can be used for teaching a way of thinking: grayscale thinking, categorical thinking; helping students to build a network of ideas and an approach for learning and thinking about mathematics.

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Many children and adults struggle with math. Teaching multisensory math techniques as well as incorporating multisensory math activities into lesson plans is not only best practice for students with dyscalculia, but can be helpful for all learners.

I'm thrilled to welcome a guest blogger today who can explain what multisensory math is to you. Adrianne Meldrum will offer links and references to multisensory math curriculum for dyscalculia as well.

"Hands shoot up all around me, a forest of arms straining and connected to minds that understand, to minds that know. A slight twinge of shame flushes my cheeks.I feel anxious and exposed.This feeling has become as familiar as water. Every day the nervous flutter, the twinge inside, the blank, the sinking reminder that I don't get it, that I don't belong here, that I am alone ."

When I read this passage from My Thirteenth Winter, a memoir of a student with a math learning disability, I wept for my students. This is what they feel when they go to school. Many of them with high IQs but are faced with the deep shame that they can't do math.

Math is a shameful topic for many. Just this week, I had a dear friend confide in me that she can't do basic math. She cried as she recounted how as an adult she still uses her fingers to do basic counting. Her lip quivered as she talked about how she calls her husband to help her figure out 20% off sales at the store.

Like many of you, I am on a mission to bring an end to students feeling this way. I've been witness to watching students who think they "don't belong here" rapidly become aware that in fact they DO belong here and even find joy in doing it!

So how do we help our students purge the shame they feel around math? We teach them using multisensory math!
(The following post contains affiliate links.)
What is Multisensory Math?
Multisensory math is using a structured approach and rich in sensory experiences. We use touch, movement, and specific language to help students become math-minded.

Did you know that 40% of students with dyslexia also demonstrate difficulty with math? This percentage also shows up in our student roster at Math for Middles.

I've been investing the last few years in learning from my mentor, Marilyn Zecher. For much of her career, she's dedicated her life to teaching struggling students using the similar multisensory techniques found in OG based reading but in a math setting.

For math instruction to be effective, it must work through the CRA instructional sequence (method) : . Concrete to Representational to Abstract.

Dr. Joyce Steeves saw a great need to use a similar format from OG reading with her math students. She developed a lesson plan that allows us to teach explicitly, practice skills so they are not lost, and introduce new concepts using CRA.

I wanted you to see what multisensory math looks like with the CRA approach in action. Watch this short clip to see how I teach place value to one of my students.The footage you are seeing are actual lessons I've done in the past. There are no views of the student visible or audio of the student speaking to protect identity.
How OG Reading and Multisensory Math Lessons Matchup
Emily did a great job summarizing the 5 basic parts to an OG reading plan. I used her same structure to show you how Dr. Joyce Steeves lesson plan also follows a similar format. The lesson is all done in less than an hour.Each part of the lesson plan moves at a quick pace which keeps the learner engaged and using that math knowledge they've mastered and are still practicing.
Math Concepts that Must Be Mastered
There are four conceptual horizons that Marilyn Zecher believes are necessary for all students to be successful in math.
1 - Pattern Recognition and Subitizing
The ability to quickly identify a quantity is an important concept and begins our math foundation. Patterns are the best way to see quantity and understand the five-ness of 5.Counting objects by starting at one each time is slow and creates an dependence that doesn't serve our students. This is one reason that I'm not a fan of using dot patterns on math numerals.Each time the student is forced to tap and count the dots one by one!

Recognizing dice patterns help us to visually see and know a quantity without having to count each dot. We also need to know various number bonds.That we can decompose 7 into 2 and 5 or 1 and 6.

We use craft sticks to teach place value because as the quantity gets larger, it has weight to it. The magnitude is easily felt! Base-ten blocks are great, but lack the weight that's needed for sensory input.

Our students need to understand the difference between expanded and standard form.

Expanded Form (tells us what it's made of)

Standard Form (tells us its name)
3 - Distributive Property
The ability to act on larger quantities. Consider 13 x 4.Students need to understand that we can break apart (decompose) 13 into a 10 and 3. We can act on those numbers and distribute (multiply) the 4 and add those quantities together to create the product of 13 x 4.
4. What is ONE and all its many names
One concept that has unlocked fractions for my students is grasping that one has many names. Essentially, any number written over itself is equivalent to one.Some teachers call it magic one, but I like to call it Copycat One. Most kids know what it's like to have a copycat friend or sibling.

It's important to understand that multiplying or dividing by some form of one does not change the quantity; it only changes the configuration of the quantity. I love this image because it shows the progression from the most simple concept of 2/2 to the more complex algebraic concept of (2x - 12)/2(x - 6). The skills we teach student in the primary grades actually do impact them in high school! If you want to learn more about the rules I teach my students for fractions,

These conceptual horizons open up the higher levels of math. Without these skills, math will continue to be difficult.
How to Get Started with Multisensory Math
The easiest way to begin is to revisit how you plan your lessons. It's best to start with skills the student has mastered for review and then introduce all new concepts using the CRA method. The goal will always be to remove the need for manipulatives because they do slow us down.We're striving for efficiency and fluency so that we can tackle bigger numbers!

Manipulatives do not need to be expensive! Many of the items we use are available online in bulk, at the dollar store, and in the craft section. Here are some of the items we use every day:

Lastly, I highly recommend you invest in training to support your students! There are quite a few options, but Marilyn has been such a great mentor to me. I highly recommend starting by following her .

(online Multisensory Math Level 1 and 2 with Marilyn Zecher)

Dr. Joyce Steeves OG Math (Only available through Nook)

Don't let your own shame of math stop you from learning! The best math teachers and tutors are often those who struggled. These trainings will help you learn math how you best learn and also unlock the hidden skills you didn't know you had to support your struggling math students.

PIN ME!

Author Bio: Adrianne Meldrum is the owner of Made For Math , an online multisensory math center.
She's a certified teacher with over 15 years of experience tutoring middle and high school students in
math. She received training in Multisensory Math 1 & 2. Adrianne's currently working on becoming
a certified trainer with Marilyn Zecher's Multisensory Math Approach. Connect with her on or listen
to her podcast, On Middle Ground .

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Multisensory Math Activities That Really Work - The Literacy Nest

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Many children and adults struggle with math. Teaching multisensory math techniques as well as incorporating multisensory math activities into lesson plans is not only best practice for students with dyscalculia, but can be helpful for all learners. I'm thrilled to welcome a guest blogger today who can explain what multisensory math is to you.

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The Perceptions Math Program is a Multi-Sensory, Video-Based Curriculum Offering Curriculum, Instruction, and Amazing Math Manipulatives!

Mastery is Teaching Back what You Have Been Taught!

Our Testimonials

What People Are Saying About the MasterPieces! "You literally took all of my tools that I have in my room - Unifix Cubes, base 10 blocks, Cuisenaire Rods, counters of all sorts, and Legos and replaced all of them with a simplified and fun tool."

- Missy Schaller, Jemicy School Teacher

What People Are Saying About MasterFractions! "I am sure that you have heard this before but we were dreading the training... great... another guy teaching us how to teach... yay... and then you blew our minds over and over."

- Robert May, High School Math Teacher

What People Are Saying About Master eTools!

These are amazing! Since my students don't have access to their physical manipulatives, these work wonderfully. We all know, virtual manipulatives don't meet the kinesthetic need of the student, but in these days where education is online, these are the next best thing! Thank you for making these available!

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Perceptions Math - Changing the way you look at math

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Mastery is Teaching Back what You Have Been Taught! What People Are Saying About the MasterPieces! "You literally took all of my tools that I have in my room - Unifix Cubes, base 10 blocks, Cuisenaire Rods, counters of all sorts, and Legos and replaced all of them with a simplified and fun tool."

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We provide resources related to effective math instruction. Our goal is to ensure that all students, regardless of background or status, have equitable access to high-quality math instruction. To guide the Science of Math, we rely on well-researched instructional strategies and research about how st

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Improving Mathematical Problem Solving in Grades 4-8

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Improving Mathematical Problem Solving in Grades 4-8

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WWC | Improving Mathematical Problem Solving in Grades 4 Through 8

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This practice guide provides five recommendations for improving students’ mathematical problem solving in grades 4 through 8. This guide is geared toward teachers, math coaches, other educators, and curriculum developers who want to improve the mathematical problem solving of students.

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Teaching Strategies for Improving Algebra Knowledge in MS Students

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Teaching Strategies for Improving Algebra Knowledge in MS Students

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practice guide; wwc; teaching; algebra; middle school; high school

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WWC | Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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This practice guide provides three recommendations for teaching algebra to students in middle school and high school. Each recommendation includes implementation steps and solutions for common roadblocks. The recommendations also summarize and rate supporting evidence. This guide is geared toward teachers, administrators, and other educators who want to improve their students' algebra knowledge.

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Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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WWC | Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades

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This practice guide provides evidence-based practices that can help teachers tailor their instructional approaches and/or their mathematics intervention programs to meet the needs of their students.

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Introduction

Word problems are described as "verbal descriptions of problem situations wherein one or more questions are raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement" (Verschaffel, Greer, & De Corte, 2000). Solving word problems involves:

The well-organized and flexible use of both conceptual and procedural knowledge

Strategies and metacognition

Positive affect and beliefs (De Corte, Greer, & Verschaffel, 1996; Schoenfeld, 1992)

Is Solving Word Problems the Same as Mathematical Modeling?

Solving word problems is not considered to be the same as mathematical modeling. Mathematical modeling tends to be a more complex process involving identifying questions to answer about the real world, making assumptions, identifying variables, translating a phenomenon into a mathematical model, assessing the solution, and iterating on the process to refine and extend the model (COMAP & SIAM, 2016). The process to solve a word problem isn't necessarily as complex, as the problem itself usually gives the reader the question to answer and the information necessary to answer it, and doesn't require modeling's level of meaning-making and interpretation. These differences are relative, however, depending on the abilities of the student and the nature of the solution required to answer the problem.

Understanding the Challenge

What Makes Word Problems Difficult for Students?

Students' primary difficulty in solving word problems is attributed to their "suspension of sense-making" (Schoenfeld, 1991; Verschaffel, Greer, & De Corte). Instead of thinking through the context of the word problem to understand it, many students simply seek a simple application of arithmetic needed to produce an answer, whether it makes sense or not. In the following video, Kaplinsky (2013) reproduces a result of early 1980s research conducted at the Institut de Recherche sur l'Enseignement des Mathématiques in France.

Math teachers are often concerned about students' abilities to transfer classroom learning into the world beyond the classroom, but this "suspension of sense-making" shows that the reverse is also difficult - students struggle to apply their knowledge and understanding of the world back into a mathematics classroom. Having been conditioned with years of arithmetic, almost always involving obvious operations and the expectation that each problem has a correct answer, students develop a "compulsion to calculate" (Stacey & MacGregor, 1999) that can interfere with the development of the algebraic thinking that is usually needed to solve word problems. Some (but not all) research findings suggest that "compulsion to calculate" worsens as students age and develop beliefs that math is a collection of rules (Radatz, 1983; Stern, 1992, both as cited in Verschaffel, Greer, & De Corte, 2000, p. 5).

Students can also struggle with word problems because they have difficulty with academic vocabulary, mathematical vocabulary, or both. Due to these difficulties, English language learners and students of low socioeconomic status score lower on standardized assessment items than proficient speakers of English (Abedi & Lord, 2001).

What Makes Word Problems Difficult for Teachers?

Some teachers ignore or struggle to apply their real-world knowledge when solving word problems, just like students (Verschaffel, De Corte, & Borghart, 1997). During instruction, teachers often try to help students "strip away the stuff we don't really need" (Chapman, 2006, p. 219) and reduce the problem to the numbers and keywords or phrases that indicate operations or relations. This dismissal of the real-world aspects of word problems can contribute to students' suspension of sense-making and their compulsion to calculate.

Most teachers believe or assume that students will have more difficulty solving a word problem than solving an algebraic equation that represents the same mathematics without the words. Because of this, they believe in teaching word problems only after students master solving similar problems as equations. Traditional math textbooks reinforce this belief by placing word problems at the end of practice sets. This belief or assumption has been shown to be false, at least under some conditions. When tested, students have shown that they can be more successful with word or verbal problems than they are with equivalent problems that are purely symbolic (Nathan & Koedinger, 2000a, 2000b). Other research suggests that skill in algorithmic computation may not correspond to students' ability to conceptualize the relationship between numbers in word problems (Fuchs et al., 2006).

Recommendations

Use Word Problems to Teach Students Mathematics

Word problems are not just for applications of already-known mathematics. In fact, the most powerful way to use word problems in the classroom is as a means to help students learn math. By situating mathematics in contexts that are understandable for students, word problems encourage students to pursue solution strategies that make sense to them and lead more often to correct answers (Koedinger & Nathan, 2004). These strategies can then be made more formal and symbolic with additional instruction.

This is obvious for teachers of young children. In early mathematics, problems are almost always situated in realistic contexts that children can make sense of. There is no reason that this should end in early childhood. Students at all levels should engage in mathematics in a sensible context before it is made formal and symbolic.

Engage Student Reasoning

Instead of dismissing the context of word problems, teachers should take time with students to make sense of word problems and their supporting context. Teachers should push back against students' compulsion to calculate by focusing on the relationship between the knowns and unknowns in word problems, and not rush to find an answer (Kieran, 2014). Some types of word problems might be particularly useful for promoting reasoning because they either lack an obvious strategy, don't have one right answer, or could be "tricky" for students who assume the problem is straightforward. Some examples:

1. Pete organized a birthday party for his tenth birthday. He invited 8 boy friends and 4 girl friends. How many friends did Pete invite for his birthday party?

2. Carl has 5 friends and Georges has 6 friends. Carl and Georges decide to give a party together. They invite all their friends. All friends are present. How many friends are there at the party?

3. Kathy, Ingrid, Hans and Tom got from their grandfather a box with 14 chocolate bars, which they shared equally amongst themselves. How many chocolate bars did each grandchild get?

4. Grandfather gives his 4 grandchildren a box containing 18 balloons, which they share equally. How many balloons does each grandchild get?

5. A shopkeeper has two containers for apples. The first container contains 60 apples and the other 90 apples. He puts all the apples into a new, bigger container. How many apples are there in that new container?

6. What will be the temperature of water in a container if you pour 1 jug of water at 80 degrees F and 1 jug of water at 40 degrees F into it? (Nesher, 1980)

Verschaffel, De Corte, and Lasure (1994) used these word problems to see if students would reason differently with the odd- and even-numbered items. Their research and subsequent studies have shown that the vast majority of students - sometimes more than 90 percent - will calculate and produce answers for the even-numbered items just as they do for the odd-numbered items, without any additional reasoning about real-world considerations. Giving students a general warning, such as "these problems are not as easy as they look," did not significantly help students. Instead, teachers can promote student reasoning by providing supports specific to each problem, such as encouraging students to explain their answer and why it makes sense, to draw a picture of their solution, or to consider a hypothetical but contrasting solution from another student. While these strategies can increase the number of students who reason with these problems correctly, in multiple studies they rarely produced correct answers for much more than 50 percent of students (see Chapter 3, Verschaffel, Greer, De Corte). In other words, these strategies are helpful but unlikely by themselves to ensure success for all students.

Numberless Word Problems

Perhaps the most direct way of pushing back against students' compulsion to calculate is to give them word problems without numbers. Brian Bushart, an elementary teacher and mathematics curriculum coordinator from Texas, popularized the idea of "numberless word problems" after a colleague tried the approach with some third-grade students. Numberless word problems aren't entirely new, as the book Problems Without Figures (Gillan, 1909) presented something vaguely similar in the early 20th century. Bushart's approach goes much further by focusing on the instructional moves and opportunities for student discourse that century-old approaches did not. Bushart's blog post (2014) and subsequent collection of resources (n.d.) describe both his process for numberless word problems and numerous examples for a range of content and grade levels.
Example 1: A Single-Step Word Problem
For an example of a numberless word problem, consider this item released from the Grade 4 PARCC test (PARCC, 2016a):

A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How many milliliters of juice are left in the pitcher after filling the glass?

Instead of giving students this problem as written, teachers can present a numberless version of this problem on a series of slides:

Slide 1: A pitcher contains some juice.

Slide 2: A pitcher contains some juice. A glass is filled with some of the juice from the pitcher.

Slide 3: A pitcher contains some juice. A glass is filled with some of the juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 4: A pitcher contains 2 liters of juice. A glass is filled with some of the juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 5: A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 6: A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How many milliliters of juice are left in the pitcher after filling the glass?

Teachers can adjust the number of slides depending on student ability, the difficulty of the problem, and the amount of time the teacher wishes to dedicate to building and sharing student sense-making. In this example, a teacher might stop after Slide 1 to make sure students know what a pitcher is and to have students estimate the capacity of a pitcher. After Slide 2, the teacher can ask students to explain what will happen to the amount of juice in the pitcher, and the relationship between the juice in the pitcher and the juice in the glass. With each subsequent slide, the teacher can continue to probe students' sense-making and understanding of the relationships between the quantities described by the problem.
Example 2: A Multi-Step Word Problem
Teachers can also use numberless word problems with multi-step word problems. Consider this multi-step item released from the Grade 5 PARCC test (PARCC, 2016b):

Dana is making bean soup. The recipe she has makes 10 servings and uses 3/4 of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She has 1/16 of a pound of beans in one container and 1/4 of a pound of beans in another container. How many more pounds of beans does Dana need to make 5 servings of soup?

Just as with Example 1, a teacher could present a numberless version of this word problem on a series of slides.

Slide 1: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount beans.

The teacher could use Slide 1 to make sure students understand the basic context. Students may observe that "If she wants to make more soup she'll need more beans," or have other insights that help establish the relationship between the amount of soup and the amount of beans.

Slide 2: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount of beans. What amount of beans does she need to make a smaller number of servings of soup?

With Slide 2, students should observe that less soup should need less beans. Students might begin to conjecture with statements like, "If she wants half as much soup, she'll need half the beans."

Slide 3: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount of beans. What amount of beans does she need to make a smaller number of servings of soup? She already has a small amount of beans. How much more beans does Dana need to make her soup?

After seeing Slide 3, students should now grapple with an even more complex relationship: Not only does Dana need less beans because she's making less soup than called for by the recipe, but she already has some of the beans that she'll need. Student observations and conjectures at this point should suggest two operations, such as a first step involving division to find the amount of beans needed in the reduced recipe, and a second step to subtract the amount of beans Dana already has. With this multi-step complexity, it would be useful for students to draw or otherwise illustrate their thinking.

Slide 4: Dana is making bean soup. The recipe she has makes 10 servings and uses an amount of beans. What amount of beans does she need to make 5 servings of soup? She already has a small amount of beans. How much more beans does Dana need to make 5 servings of soup?

Slide 4 specifies the numbers of servings, but not the beans. With this information, students can revise their observations to make clear that the recipe's amount of beans needs to be divided by 2, with an additional amount subtracted, to find how many more beans Dana needs.

Slide 5: Dana is making bean soup. The recipe she has makes 10 servings and uses 3/4 of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She already has a small amount of beans. How many more pounds of beans does Dana need to make 5 servings of soup?

With Slide 5, students should have already decided that the amount of beans in the recipe needs to be divided by 2, and now they can focus on finding 3/4 ÷ 2. Students may also notice that the problem now makes clear that all measurements of beans is in pounds.

Slide 6: Dana is making bean soup. The recipe she has makes 10 servings and uses 3/4 of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She has 1/16 of a pound of beans in one container and 1/4 of a pound of beans in another container. How many more pounds of beans does Dana need to make 5 servings of soup?

Slide 6 presents the word problem in its original form, and students may be surprised that the small amount of beans Dana already had is represented as two amounts that need to be considered together. Because of the complexity of this problem, there may not be an ideal time to introduce this information, but a 5th grader who has reasoned through the problem and gotten this far should be able to reason with the two small quantities of beans, either by adding them first and then subtracting the sum from 3/8 or by performing two subtractions.

Add Authenticity

Students have shown to be more successful with word problems when they are required to engage with the context in authentic ways. For example, DeFranco and Curcio (1997, as cited by Verschaffel, Greer, De Corte) gave a group of 20 sixth grade students the following word problem: "328 senior citizens are going on a trip. A bus can seat 40 people. How many buses are needed so that all the senior citizens can go on the trip?" Later, the researchers gave the students a similar problem, except they presented it in the form of a fact sheet with the numbers of people and the size of vans, and instructions for making a phone call that simulated placing an actual order for the number of required vehicles. In the first scenario, only 2 of the 20 students answered correctly and properly reasoned with the remainder left by the division. In the second scenario, using a more authentic setting, 16 of the 20 students answered correctly and reasoned appropriately with the remainder, either by rounding up to the next whole van or requesting "like a car or something" to transport the small number of remaining passengers.

Address Language Complexity

Language complexity can be addressed either by reducing the complexity or providing students with additional support. For students who struggle with the language of word problems, it can be helpful to rewrite the problem using simpler, more familiar language, or a student's native language (Bernardo, 1999). These kinds of modifications tend to help English learners and low-SES students more than their English-proficient and higher-SES counterparts, meaning this strategy could help reduce achievement gaps (Abedi & Lord, 2001).

Teachers can also provide extra support. Recommendations for teaching English language learners include focusing on student reasoning and discourse, rather than correctness of language use, and using language learners' knowledge and experiences as resources (Moschkovich, 2012). The Understanding Language website ( ell.stanford.edu) is a recommended resource for understanding how to support language learners in mathematics.

References

Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14(3), 219-234. https://doi.org/10.1207/S15324818AME1403_2

Bernardo, A. B. I. (1999). Overcoming obstacles to understanding and solving word problems in mathematics. Educational Psychology, 19(2), 149-163. https://doi.org/10.1080/0144341990190203

Bushart, B. (n.d.). Numberless word problems. Retrieved November 16, 2017, from https://bstockus.wordpress.com/numberless-word-problems/

Bushart, B. (2014, October 6). Numberless word problems [Blog post]. Retrieved November 16, 2017, from https://bstockus.wordpress.com/2014/10/06/numberless-word-problems/

Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62(2), 211-230. https://doi.org/10.1007/s10649-006-7834-1

COMAP, & SIAM. (2016). GAIMME: Guidelines for assessment & instruction in mathematical modeling education. Bedford, MA. Retrieved from http://www.comap.com/Free/GAIMME/index.html

De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics teaching and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 491-549). New York, NY: Lawrence Erlbaum Associates.

DeFranco, T. C., & Curcio, F. R. (1997). A division problem with a remainder embedded across two contexts: Children's solutions in restrictive vs. real-world settings. Focus on Learning Problems in Mathematics, 19(2), 58-72.

Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., ... Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98(1), 29-43. https://doi.org/10.1037/0022-0663.98.1.29

Gillan, S. Y. (1909). Problems without figures. Milwaukee, WI: S. Y. Gillan & Company. Retrieved from http://www.schoolinfosystem.org/pdf/2008/10/problemswithoutfigures.pdf

Kaplinsky, R. (2013). How old is the shepard? Retrieved November 3, 2017, from https://www.youtube.com/watch?v=kibaFBgaPx4

Kieran, C. (2014). What does research tell us about fostering algebraic reasoning in school algebra? Reston, VA. Retrieved from http://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Algebraic-Reasoning-in-School-Algebra/

Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129-164. https://doi.org/10.1207/s15327809jls1302_1

Moschkovich, J. N. (2012). Mathematics, the Common Core, and language. Understanding Language: Language, Literacy, and Learning in the Content Areas. Retrieved from http://ell.stanford.edu/publication/mathematics-common-core-and-language

Nathan, M. J., & Koedinger, K. R. (2000a). An investigation of teachers' beliefs of students' algebra development. Cognition and Instruction, 18(2), 209-237. https://doi.org/10.1207/S1532690XCI1802_03

Nathan, M. J., & Koedinger, K. R. (2000b). Teachers' and researchers' beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168-190. https://doi.org/10.2307/749750

Nesher, P. (1980). The stereotyped nature of school word problems. For the Learning of Mathematics, 1(1), 41-48. Retrieved from http://flm-journal.org/Articles/flm_1-1_Nesher.pdf

PARCC. (2016a). Math Spring Operational 2016 Grade 4 Released Items. Partnership for Assessment of Readiness for College and Careers. Retrieved from https://parcc-assessment.org/content/uploads/released_materials/05/Grade_04_Math_Item_Set.pdf

PARCC. (2016b). Math Spring Operational 2016 Grade 5 Released Items. Partnership for Assessment of Readiness for College and Careers. Retrieved from https://parcc-assessment.org/content/uploads/released_materials/05/Grade_05_Math_Item_Set.pdf

Radatz, H. (1983). Untersuchungen zum Lösen eingekleideter Aufgaben. Zeitschrift Fur Mathematic-Didaktik, 4(2), 205-2017. https://doi.org/10.1007/BF03339231

Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311-343). Hillsdale, NJ: Lawrence Erlbaum Associates.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). Reston, VA: National Council of Teachers of Mathematics.

Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18(2), 149-167. https://doi.org/10.1016/S0732-3123(99)00026-7

Stern, E. (1992). Warum werden Kapitänsaufgaben "gelöst"? Dav Verstehen von Textaufgaben aus phychologischer Sicht. Der Mathematikunterricht, 28(5), 7-29.

Verschaffel, L., De Corte, E., & Borghart, I. (1997). Pre-service teachers' conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems. Learning and Instruction, 7(4), 339-359. https://doi.org/10.1016/S0959-4752(97)00008-X

Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4(4), 273-294. https://doi.org/10.1016/0959-4752(94)90002-7

Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger.

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Word Problems - Research and Practice Guide

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Word problems are described as "verbal descriptions of problem situations wherein one or more questions are raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement" (Verschaffel, Greer, & De Corte, 2000).

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Evidence Based Specially Designed Instruction in Mathematics

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Have your students fallen behind in grade level work? Use these checklists and pre-assessments (with answer keys) to determine if MathLinks Essentials is a program that will help your students catch up on the major work of the middle grades.

Checklists, pre-assessments, and pre-assessment answer keys for incoming 6 th graders ( Grade 5 work)

Checklists, pre-assessment answer keys for incoming 7 pre-assessments, and th graders ( Grade 6 work)

Checklists, pre-assessment answer keys for incoming 8 pre-assessments, and th graders ( Grade 7 work)

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Center for Mathematics and Teaching Inc. | Checklists and Pre-Assessments

# data[].attributes.attachment_link.description
Have your students fallen behind in grade level work? Use these checklists and pre-assessments (with answer keys) to determine if MathLinks Essentials is a program that will help your students catch up on the major work of the middle grades.

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The Primary Numeracy Assessment System (FREE) is a math intervention curriculum and instructional approach that includes assessments, intervention frameworks, leveled activities, lesson plans, workbooks, curriculum maps, and countless other resources. To learn how to use this free assessment system and free resources please view the online training webinar. There are optional supplemental materials that can be purchased as well. To view the training click on the link below. Please scroll to the bottom of the page for free curriculum resources.

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Math intervention curriculum, assessments, lesson plans, teacher guides, math intervention strategies, workbooks.