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Examples like Jigsaw, Round Robin, ect.

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Cooperative Learning has been proven to be effective for all types of students, including academically gifted, mainstream students and English language learners (ELLs) because it promotes learning and fosters respect and friendships among diverse groups of students. In fact, the more diversity in a team, the higher the benefits for each student.

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Co-operative Learning: What makes group work-work?

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Co-operative Learning: What makes group work-work?

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This book brings together the lessons of research on both the nature of learning and different educational applications, and it summarises these as se

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Just about everyone loves the idea of cooperative learning. Think of kids working productively and excitedly in groups, everyone getting along and enthusiastically helping one another learn. Think of kids completing great projects together, encouraging one another, and forming friendships. In this ideal scenario, all students are engaged, active, and minds-on. They are learning cooperation itself, an important life skill. Cooperative learning: how pro-social, how liberating-and how does anyone really accomplish this?

Cooperative learning can be wonderful. Students often love working this way. I've heard comments like, "An explanation is easier to understand if it's coming from another kid" or "My teammates ... make sure I understand the work." But for many teachers who've tried it, cooperative learning is a noisy mess rather than a stairway to heaven. One student might be doing all the work while the others socialize, for example. Kids may ignore or belittle group mates they perceive to be low achievers. Some students may think cooperative learning is party time rather than learning time. This is why studies find that most teachers don't use cooperative learning regularly, despite extensive research supporting it (Gillies, 2014; Roseth, Johnson, &amp; Johnson, 2008; Slavin, 1995, 2013; Webb, 2008).

It is the "learning" in cooperative learning that is too often left out. But it needn't be. Using these five strategies, teachers can get the greatest benefit possible from cooperative learning and ensure that collaboration enhances learning.

An effective cooperative group is not a collection of kids thrown together for a brief activity. It's a team composed of diverse students who care about helping one another learn-and about the success of the team itself. All members must know they can depend on one another for help.

In the forms of cooperative learning we've used for years at Johns Hopkins University and at the Success for All Foundation in elementary and secondary schools (Slavin, 1995), we group students into four-member teams (if the class doesn't divide evenly, we have a few five-member teams). A foursome provides flexibility. Some activities can be done in pairs and some with the whole team. Teams work together daily for 6-8 weeks; after that period, teachers assign new teams.

Team members move their desks together and choose a team name. At the beginning of the team's work, members engage in group bonding activities, like creating a motto or coat of arms to express their team personality.

In middle or high school, groups might investigate careers that interest them and colleges known for majors tied to those careers. Their research often culminates in each team giving a presentation about why they recommend a particular college, and teams sometimes name themselves after the school they recommend. Even if students choose Duke because of basketball fame, they've spent time helping one another learn something new, shared what matters to them, and met a deadline.

It's best if teams are composed of a cross section of the class: high and low achievers, boys and girls, students of different ethnicities. Groups definitely won't have such diversity if students choose their teams, so the teacher should make team assignments.

Too often in cooperative learning, students are put into teams and instructed to "help one another." Helping one another is a good thing, but kids will do a lot more of it if they share a team goal.

A team goal is a target, product, or indicator that shows a team has done a good job of getting every member to perform at his or her personal best. A team goal could be increasing the average score on a quiz that all students take individually after they help one another prepare. It could be one overall product with individual components that each team member clearly contributed to, such as by coding a specific part of a computer program or contributing essential data to a group lab report.

In each case, a teacher both looks at the team average and evaluates the individual products. Teams whose work meets certain criteria are awarded certificates or small privileges. This helps team members see their joint work as achieving something important. Teams that don't have a goal are like sports teams in a game where no one knows, or cares, what the score is-not the sort of game in which all players do their best.

This is the essential element most often left out of cooperative learning-and when it is, teachers lose a lot of cooperative learning's potential.

Individual accountability means that to reach the team goal, all team members must master the targeted content or skills. Team success should depend on the hard work-and therefore the learning-of all members.

To understand why this is important, consider a team working together without individual accountability. Imagine that a team studies together and then takes one quiz, on which all team members can help one another. This arrangement will likely produce two undesirable outcomes: the free rider and the know-it-all.

Free riders do little work. They may chat with others, but they don't try much to learn or help others learn. "The others will do it anyway," they reason. "Why should I try?" Students who have doubts about their own skills may be shy or reluctant to participate if they know others will do the work.

The know-it-all is another story. A know-it-all (or two) may dominate the group and tell others the answers. I once saw a group of middle schoolers trying to solve a complex math problem together. (Their task was to solve the problem and give the answer as a group.) Two kids were doing most of the talking. When one of their (usually quiet) teammates ventured an opinion, both know-it-alls shushed her and told her to wait for them to deliver the answer.

How can teachers ensure individual accountability? First, make sure the team goal requires the learning and participation of all team members. For example, if the team goal is to score an average of 80 percent or better on a quiz, all students need to do well. Or a goal might be to write an outstanding report on a given country, with each chapter signed by a team member. Never choose an outcome one student could do alone. In addition, frequently reinforce the idea that the purpose of the team is to make sure that all members are learning, not just to get the right answers or complete the project. Explain why individual accountability is fair; students will readily understand why no team member should get a free ride.

During lessons, use informal cooperative structures that reinforce the idea of individual accountability. Teachers we work with in Success for All programs, for example, make frequent use of a technique called "random reporter." Each student is given a number from 1 to 5. When teachers ask a question, they direct it to a team and then pick a number at random. The student on that team with that number has to respond for the whole group-and the team can earn points based on the answer.

Practices like random reporter communicate consistently that teams must make sure all their members learn. Potential free riders understand that they will be held accountable and that they can't wait for their teammates to make all the effort. Know-it-alls realize that if they don't engage, teach, and give feedback to every team member, their team won't succeed. One middle school math teacher noted, "Students ... know they need to work hard on math problems and be ready to explain them. If they aren't ready, they can't get off the hook easily."

Finally, it's important that the teacher lets the work be challenging and doesn't step in too early to do the work for students who struggle. Doing so not only undermines teammates' developing interdependence, but also teaches students that if they feign helplessness, display apathy, or work slowly, the teacher will soon step in and complete the tasks. If kids instead learn to make appropriate levels of effort and to persist, they will build confidence in their ability to improve and learn. They'll realize that asking for help-not the answers-helps them learn. Letting students struggle constructively and safely in their teams enables each student to reflect on his or her thinking, compare it with others' ideas, and refine that thinking before sharing it with the class-or on a test.

Setting up structures that promote effective cooperative learning is not enough. Team members need to know how to make good use of the opportunity to work with one another; this means they need to learn about, practice, and refine key interpersonal skills.

Active listening. Active listening skills are essential in good group work-and in life. When students are listening well, their eyes are on the speaker and they occasionally nod. Active listeners avoid interrupting but periodically summarize what they hear and ask for clarification when needed. These practices communicate respect for the speaker and enable the listener to learn as much as possible from that speaker's words.

Explaining ideas and opinions. Team members need this skill to communicate and persuade in cooperative interactions. Explaining must go beyond single-word answers; students must be able to identify sources or reasons for their personal opinions or conclusions. They must explain their ideas to others so that peers can understand them too. This demands metacognition, evidentiary thinking, summarizing, paraphrasing, and listening to others thoughtfully.

Encouraging teammates. Effective team members know how to encourage and support teammates, disagree with dignity, and help maintain a positive, prosocial tone within the group. Tell students you expect them to make sure that all team members are actively participating in the thinking parts of the group's tasks. Teach them how to make that happen.

Explaining your own ideas while also encouraging others is a complex skill that demands that students respect one another. Guide kids to make teamwork time a safe environment in which to speak, receive feedback, and admit what they don't understand. Once students establish a productive working relationship, they can set goals together, monitor their own progress, and solve learning problems together.

Completing tasks. Students should be expected to work on the group's tasks until they are finished to a high standard and to ensure that at the end of a team study session, all participants have learned the class objective.

When students know what constitutes great work within their team, and reach that standard, they can be proud of themselves and of any recognition their group receives.

Cooperative learning should be seen as a key part of each lesson, but not the whole lesson. Effective class lessons might also include teacher instruction, media- or computer-based activities, and individual assessments of various kinds. Informal cooperative-learning activities such as random reporter or think-pair-share may also be used, but these shouldn't be the only cooperative activity.

The best way to use cooperative learning is to replace individual work, which in traditional lesson cycles happens after lessons and before assessments. Individual, isolated practice is boring and ineffective for most students, especially if they struggle. Cooperative learning makes practicing to mastery engaging and social and gives all students "study buddies" to help them when they run into difficulties.

Sometimes team activities may come before teacher instruction, as when teams are conducting experiments in a discovery learning format. But at some point, a teacher needs to explain the essential objectives and give students parameters and guidance for their group work so they can move forward.

I once observed a class of 9th graders who started the school year reading significantly below grade level. Their first-year teacher worked hard to provide a research-based cooperative learning model in which they would read a wide variety of complex texts, discuss them in teams, and prepare one another to participate in class discussions. However, students weren't making the desired progress. As the teacher looked for what was missing, she noticed students were sloppy in their work and slow to complete tasks. Although they often talked to one another, they weren't helping one another.

The first change this educator made was holding each student individually accountable for the work. The second was providing and discussing with the class rubrics for high-quality answers or comments in discussions. Students and teachers practiced scoring sample answers and exchanged feedback.

Then she made the task more interesting, challenging, personal, and relevant. She asked students to read closely and compare several journal articles about the science of emotions. Team members were to help one another answer questions that required more than recitation of facts from the articles, but to individually form their own hypotheses about the author's intent in writing the articles-and support those hypotheses with evidence from the texts.

In a few days, when the students realized that the teacher wasn't going to dilute this work to save time, they dug into the articles. Discussions flourished. Well-structured cooperative learning and recognition of work that met the standards of the rubrics ultimately transformed this class from remedial to advanced. When the students were asked what they thought about how their class was changing, one girl summed it up: "When I work with my teammates and we use the rubrics and our brains, we know we have a good answer. We're proud to show what we know-we are ready for any of us to report out."

Used properly, cooperative learning is an exciting way for all kids to learn. Research finds that if teachers make the five elements discussed here part of group learning, students learn more, feel more successful, love school, and enjoy the subject they're studying-and like and accept one another (Roseth et al., 2008; Slavin, 2013; Webb, 2008). For outcomes like these, it's worth the effort.

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Five key practices bring out the tremendous potential of this approach.

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The Effect of Cooperative Learning On Middle School Math Students

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The Effect of Cooperative Learning On Middle School Math Students

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Using Cooperative Learning to Teach Mathematics to Students with Learning Disabilities

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Using Cooperative Learning to Teach Mathematics to Students with Learning Disabilities

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Using Cooperative Learning to Teach Mathematics to Students with Learning Disabilities | LD Topics | LD OnLine

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LD OnLine is the leading website on learning disabilities, learning disorders and differences. Parents and teachers of learning disabled children will find authoritative guidance on attention deficit disorder, ADD, ADHD, dyslexia, dysgraphia, dyscalculia, dysnomia, reading difficulties, speech and related disorders.

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Tips for implementing small group instruction in the middle school classroom.

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What do you do with the kids that "don't get it"? I think this is a constant battle that we face as teachers. Sometimes it even keeps us up at night. Earlier I shared about my Math Intervention Schedule and wanted to share tips for implementing math small group instruction.

If your classroom functions like mine, a typical class period might consist of a warm up, guided notes, a class activity that is simultaneously accessible and engaging, and finally, an exit ticket for students to demonstrate mastery. Whew! That is a lot for 60 minutes.

In my first year of teaching, math small group instruction wasn't even on my radar, yet the longer I have spent in the classroom, the more comfortable I am making small groups part of my daily routine. Like any classroom routine, you must train yourself as much as you must train your students. Today, I am going to give you some tips I have learned along the way, so you can set up your math small groups to be as successful as possible.

Before SMALL GROUP INSTRUCTION

Know Who You Are Pulling

This might seem like a no brainer, but there were several days that I would get to the time in class for a small group, and then I would not do it because I was not prepared. Sure, I knew who my strugglers were, but I did not have a plan on who to pull or even why I would pull them. Either make a list based on class groups ahead of time, or take mental notes on which students are struggling as you circulate while teaching.

Consider the personalities of your students as well as their academic needs when making small groups. You will most likely pull the same students on a consistent basis, so create standard small groups that will make your job easier. Typically, I had around 8 students who I would pull, so I split those 8 students into 2 groups of 4. Group 1 might be those students who need a little more guidance to reinforce the new concept while group 2 might be the students who need significantly more support and be pulled more frequently. I also split up any behavioral problems amongst the two groups. You can even give your groups names, so that when you transition from notes to classwork, you can just say, "Gryffindor, meet me at the horseshoe table" or "Ravenclaw, you will need a highlighter for small group."

Set Expectations for Rest of Class

What is the rest of your class doing during this time? How much time do they have to do it? Are they allowed out of their seat? What do they do if they need help? Your students should know the answers to all of these questions before you pull a small group. You will end up frustrated and annoyed if you're spending your time correcting students or getting up to answer questions from other students. At the beginning of the year, I explain to my whole class why I pull small groups and how important it is for us to support our teammates who need extra help. One way they can support those students is to maintain a reasonable volume and to complete their work regardless of who is watching them. During a small group, there is a 100% chance that students in your class will have a question, so assign a responsible student to field questions for you. You can also set the expectation that students may only ask a question if no one else at their table knows the answer and they have looked at their notes.

During SMALL GROUP INSTRUCTION

Gauge Understanding Before You Start

Before you begin, find out where your students are. You can do this a couple different ways. Informally, you can ask students on a scale of 1 to 5, how well do they understand what they just learned. Side note: make sure to explain how the scale works (1 = no clue and 5 = you could teach the class). I have found that most students are great at evaluating their needs. If a student has no idea what is going on, they will tell you 1. More formally, you can have students start a problem on their own, and watch to see where or if they begin to make a mistake. And on a rare occasion, be amazed that they can do it all by themselves, give yourself a pat on the back because you are the BEST teacher ever, and send them back to work on their own.

Keep It Simple

You have enough to do as a teacher. You do not have to create different material for your small group. Students can work on what the rest of the class is completing. If you need to make the material more accessible, ask yourself 'what skill are we practicing?' If the objective is to calculate volume, perhaps you can change the classwork to include only whole numbers while the rest of the class is finding volume with decimals and fractions. Sometimes, different material is appropriate, but remember to not bite off more than you can chew. When you are first starting, keep it simple.

Whiteboards Are Your Friend

Students love to write on whiteboards! Whiteboards keep students engaged, and as the teacher, allow you to easily see what the students are doing. This will enable you to correct misconceptions on the spot and provide immediate feedback. Plus, they are just so much more fun than paper and pencil.

After SMALL GROUP INSTRUCTION

Watch 'Em Work

Provide a problem for students to complete independently. Watch them as they work. Don't ask any questions and just allow them to show you what they have learned in your small group. If there are still misunderstandings after the student has finished, give another student from your small group who did it correctly an opportunity to explain what they did to the struggling student. Sometimes, students explain better than us teachers. 😉

Determine Next Steps

Remember, one small group pull out will not fix every math misconception, and some students need more time and more attention. Baby steps are still steps after all, and your students are lucky to have a teacher that cares so much. If you are looking for a way to track their progress, you can check out these forms I use here. Stay positive and make sure these students know that you, as their teacher, believe in them!

For more ideas and specifics on math small group instruction check out Guided Math Instruction. It is aimed towards elementary students, but has some great tips that transfer to middle school. Teachers, what other tips do you have?

For even more tips on organizing data and small group instruction, check out this follow up post.

Mrs. Brack is a sixth grade math teacher in Texas and my sister. She is super creative, yet practical and loves her students so well. She will be periodically posting on the blog this summer because she is the best sister and because she has great ideas that I know you will love. 😉

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Implementing Math Small Groups in Middle School

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What do you do with the kids that "don't get it"? I think this is a constant battle that we face as teachers. Sometimes it even keeps us up at night. Earlier I shared about my Math Intervention Schedule and wanted to share tips for implementing math small group instruction.

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-pulling struggling learners into a small group instruction

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Intensifying Interventions for Students by Identifying and Remediating Conceptual Understandings in Mathematics

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Intensifying Interventions for Students by Identifying and Remediating Conceptual Understandings in Mathematics

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http://web.b.ebscohost.com.library.marygrove.edu:2048/ehost/pdfviewer/pdfviewer?sid=f82291f7-a586-4c4e-b648-a207c07cabea%40sessionmgr102&vid=4&hid=124

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Strong Evidence: Intervention Strategies Video

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Strong Evidence: Intervention Strategies Video

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Instruction during the intervention should be explicit and systematic.

This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

https://ies.ed.gov/ncee/wwc/PracticeGuide/2

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Will pull-out instruction lead to increased understanding and performance?

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Would more time spent in cooperative learning groups results in increased conceptual understanding?

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Students do not understand the conceptual teaching method and look for procedural shortcuts.

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"Another tradition deeply ingrained in mathematics instruction is a focus on algorithms and shortcuts without first ensuring conceptual understanding" (p 3).

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Conceptual understanding in mathematics: What is it? Why do we need it? How do we get it?

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Conceptual understanding in mathematics: What is it? Why do we need it? How do we get it?

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Richard Andrew refers to a trig example in teaching conceptual understanding and explains how it saves time in the long run.

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Students do not have a strong conceptual understanding in math.

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5 Tips To Help Develop Conceptual Understanding In Math

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5 Tips To Help Develop Conceptual Understanding In Math

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

As my teaching methods have evolved, especially since embedding the math practices in every lesson, I've pushed myself to use inquiry-based tasks and hands-on manipulatives with my students as a way to help them develop their conceptual understanding. Yet there are still topics (such as solving inequalities with a negative coefficient) where I feel unprepared to teach in this way.

I recently got a sneak peak of Leah Alcala's new video, Concept First, Notation Last. While Leah is probably best known on Teaching Channel for her My Favorite No and Highlighting Mistakes strategy videos, this new example of practice is a chance for us to see how Leah helps her students develop conceptual understanding in math. After watching, I felt as if some of my own questions on the topic had been answered.

Here are my 5 takeaways, which are now reminders to myself and tips for you:

1. Belief
Leah Alcala has the utmost belief that her students will be able to access and attempt the task at hand. In the video, you'll hear her make the statement, "Everyone likes to learn." It's clear that her beliefs move students forward. Take inventory of yourself. Do you believe that ALL students in your classroom can access mathematical ideas without being told what to do and how to do it? To continue to deepen your beliefs, I recommend reading Jo Boaler's research on how all students are capable of developing mathematical ideas.

2. Sense Making
When my teaching started to evolve into one that focused more on conceptual understanding rather than rote memorization, I started asking one question of all my students: "How/Why does your solution make sense to you?" I found that by developing justification as a norm, I could understand exactly what students believed, why they believed it, and respond with more effective questions. Moreover, in order to have those conversations, the task they were responding to had to be a rich, inquiry-based task. That meant I was no longer teaching tricks or math shortcuts. Students had to be able to construct viable arguments and critique the reasoning of others. In the video, you'll see Leah posing carefully developed tasks alongside constant questioning, as opposed to teaching mathematical tricks. This allows her students to use sense making to come to a conclusion on their own.

3. Scaffolding
From the outset of this lesson, Leah keeps the numbers easy to manipulate while the mathematics remains demanding. If the numbers become too difficult at the outset, students can get overwhelmed and will often shut down. As Leah mentions, easier numbers help give an access point to all students. Over time, you can increase to more challenging numbers while maintaining challenging thinking as well.

4. Time
Conceptual ideas are not built in one day or even two. They are developed after repeated exposure to a particular mathematical idea in various contexts. Students have to struggle and resolve that struggle to internalize a concept. This process takes time. Leah mentions that the one day her class is being filmed will, by itself, not be sufficient for complete learning to occur. In my classroom, this means a learning objective may remain consistent for a week or two as we are developing the idea.

5. Multiple Representations
Leah allows her students to engage with the mathematical idea of solving inequalities through graphs, lists, and/or mathematical notation. When developing conceptual understanding, it's imperative to give students freedom of choice in how they might potentially respond. Narrowing to one representation too often causes students to try to find the one correct path towards a solution, rather than thinking expansively and for themselves.

[video_embed src=" https://www.teachingchannel.org/videos/teaching-inequalities/embed.js" width="480"]

Since I've begun to focus on the development of conceptual understanding, I've run into some challenges. Students and parents alike want me to teach short cuts and algorithms. Yet, when I hear students talking and thinking mathematically, I'm certain that this struggle will prepare them to be risk-takers, not only in my classroom but in their daily lives, too. My hope is that this will lead to self confidence and an internal belief that when faced with any problem, they have ideas worth exploring.

I encourage you to watch this video and think about how developing conceptual understanding is both doable and essential in your classroom. I would love to read your responses in the comments section below on how you were able to take these ideas and apply them with your students!
Crystal Morey is a middle school math teacher in Enumclaw, WA. She has served on the county's Common Core transition team, and is a teacher leader on a statewide project with Washington State's Office of the Superintendent of Public Instruction and the Mathematics Education Collaborative.

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5 Tips To Help Develop Conceptual Understanding In Math

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

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This article addresses how to catch struggle learners up by pacing lessons, building on math vocabulary and having students write out their thoughts or talk about them to other students.  "Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own".

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November 2007 | Volume 65 | Number 3
Making Math Count Pages 16-21

Marilyn Burns

Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty. Here's how our conversation began:

Marilyn: Can you tell me something you know about multiplication?

Paul: [Thinks, then responds] 6 × 8 is 48.

Marilyn: Do you know how much 6 × 9 is?

Paul: I don't know that one. I didn't learn it yet.

Marilyn: Can you figure it out some way?

Paul: [Sits silently for a moment and then shakes his head.]

Marilyn: How did you learn 6 × 8?

Paul: [Brightens and grins] It's easy-goin' fishing, got no bait, 6 × 8 is 48.

As I talked with Paul, I found out that multiplication was a mystery to him. Because of his weak foundation of understanding, he was falling behind his classmates, who were multiplying problems like 683 × 4. Before he could begin to tackle such problems, Paul needed to understand the concept of multiplication and how it connects to addition.

Paul wasn't the only student in this class who was floundering. Through talking with teachers and drawing on my own teaching experience, I've realized that in every class a handful of students are at serious risk of failure in mathematics and aren't being adequately served by the instruction offered. What should we do for such students?

Grappling with Interventions

My exchange with Paul reminded me of three issues that are essential to teaching mathematics:

It's important to help students make connections among mathematical ideas so they do not see these ideas as disconnected facts. (Paul saw each multiplication fact as a separate piece of information to memorize.)

It's important to build students' new understandings on the foundation of their prior learning. (Paul did not make use of what he knew about addition to figure products.)

It's important to remember that students' correct answers, without accompanying explanations of how they reason, are not sufficient for judging mathematical understanding. (Paul's initial correct answer about the product of 6 × 8 masked his lack of deeper understanding.)

For many years, my professional focus has been on finding ways to more effectively teach arithmetic, the cornerstone of elementary mathematics. Along with teaching students basic numerical concepts and skills, instruction in number and operations prepares them for algebra. I've developed lessons that help students make sense of number and operations with attention to three important elements-computation, number sense, and problem solving. My intent has been to avoid the "yours is not to question why, just invert and multiply" approach and to create lessons that are accessible to all students and that teach skills in the context of deeper understanding. Of course, even well-planned lessons will require differentiated instruction, and much of the differentiation needed can happen within regular classroom instruction.

But students like Paul present a greater challenge. Many are already at least a year behind and lack the foundation of mathematical understanding on which to build new learning. They may have multiple misconceptions that hamper progress. They have experienced failure and lack confidence.

Such students not only demand more time and attention, but they also need supplemental instruction that differs from the regular program and is designed specifically for their success. I've recently shifted my professional focus to thinking about the kind of instruction we need to serve students like Paul. My colleagues and I have developed lessons that provide effective interventions for teaching number and operations to those far behind. We've grappled with how to provide instruction that is engaging, offers scaffolded instruction in bite-sized learning experiences, is paced for students' success, provides the practice students need to cement fragile understanding and skills, and bolsters students' mathematical foundations along with their confidence.

In developing intervention instruction, I have reaffirmed my longtime commitment to helping students learn facts and skills-the basics of arithmetic. But I've also reaffirmed that "the basics" of number and operations for all students, including those who struggle, must address all three aspects of numerical proficiency-computation, number sense, and problem solving. Only when the basics include understanding as well as skill proficiency will all students learn what they need for their continued success.

Essential Strategies

I have found the following nine strategies to be essential to successful intervention instruction for struggling math learners. Most of these strategies will need to be applied in a supplementary setting, but teachers can use some of them in large-group instruction.

1. Determine and Scaffold the Essential Mathematics Content

Determining the essential mathematics content is like peeling an onion-we must identify those concepts and skills we want students to learn and discard what is extraneous. Only then can teachers scaffold this content, organizing it into manageable chunks and sequencing these chunks for learning.

For Paul to multiply 683 × 4, for example, he needs a collection of certain skills. He must know the basic multiplication facts. He needs an understanding of place value that allows him to think about 683 as 600 + 80 + 3. He needs to be able to apply the distributive property to figure and then combine partial products. For this particular problem, he needs to be able to multiply 4 by 3 (one of the basic facts); 4 by 80 (or 8 × 10, a multiple of 10); 4 and by 600 (or 6 × 100, a multiple of a power of 10). To master multidigit multiplication, Paul must be able to combine these skills with ease. Thus, lesson planning must ensure that each skill is explicitly taught and practiced.

2. Pace Lessons Carefully

We've all seen the look in students' eyes when they get lost in math class. When it appears, ideally teachers should stop, deal with the confusion, and move on only when all students are ready. Yet curriculum demands keep teachers pressing forward, even when some students lag behind. Students who struggle typically need more time to grapple with new ideas and practice new skills in order to internalize them. Many of these students need to unlearn before they relearn.

3. Build in a Routine of Support

Students are quick to reveal when a lesson hasn't been scaffolded sufficiently or paced slowly enough: As soon as you give an assignment, hands shoot up for help. Avoid this scenario by building in a routine of support to reinforce concepts and skills before students are expected to complete independent work. I have found a four-stage process helpful for supporting students.

In the first stage, the teacher models what students are expected to learn and records the appropriate mathematical representation on the board. For example, to simultaneously give students practice multiplying and experience applying the associative and commutative properties, we present them with problems that involve multiplying three one-digit factors. An appropriate first problem is 2 × 3 × 4. The teacher thinks aloud to demonstrate three ways of working this problem. He or she might say,

I could start by multiplying 2 × 3 to get 6, and then multiply 6 × 4 to get 24. Or I could first multiply 2 × 4, and then multiply 8 × 3, which gives 24 again. Or I could do 3 × 4, and then 12 × 2. All three ways produce the same product of 24.

As the teacher describes these operations, he or she could write on the board:

It's important to point out that solving a problem in more than one way is a good strategy for checking your answer.

In the second stage, the teacher models again with a similar problem-such as 2 × 4 × 5-but this time elicits responses from students. For example, the teacher might ask, "Which two factors might you multiply first? What is the product of those two factors? What should we multiply next? What is another way to start?" Asking such questions allows the teacher to reinforce correct mathematical vocabulary. As students respond, the teacher again records different ways to solve the problem on the board.

During the third stage, the teacher presents a similar problem-for example, 2 × 3 × 5. After taking a moment to think on their own, students work in pairs to solve the problem in three different ways, recording their work. As students report back to the class, the teacher writes on the board and discusses their problem-solving choices with the group.

In the fourth stage, students work independently, referring to the work recorded on the board if needed. This routine both sets an expectation for student involvement and gives learners the direction and support they need to be successful.

4. Foster Student Interaction

We know something best once we've taught it. Teaching entails communicating ideas coherently, which requires the one teaching to formulate, reflect on, and clarify those ideas-all processes that support learning. Giving students opportunities to voice their ideas and explain them to others helps extend and cement their learning.

Thus, to strengthen the math understandings of students who lag behind, make student interaction an integral part of instruction. You might implement the think-pair-share strategy, also called turn and talk. Students are first asked to collect their thoughts on their own, and then talk with a partner; finally, students share their ideas with the whole group. Maximizing students' opportunities to express their math knowledge verbally is particularly valuable for students who are developing English language skills.

5. Make Connections Explicit

Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own. They need help building new learning on what they already know. For example, Paul needed explicit instruction to understand how thinking about 6 × 8 could give him access to the solution for 6 × 9. He needed to connect the meaning of multiplication to what he already knew about addition (that 6 × 8 can be thought of as combining 6 groups of 8). He needed time and practice to cement this understanding for all multiplication problems. He would benefit from investigating six groups of other numbers-6 × 2, 6 × 3, and so on-and looking at the numerical pattern of these products. Teachers need to provide many experiences like these, carefully sequenced and paced, to prepare students like Paul to grasp ideas like how 6 × 9 connects to 6 × 8.

6. Encourage Mental Calculations

Calculating mentally builds students' ability to reason and fosters their number sense. Once students have a foundational understanding of multiplication, it's key for them to learn the basic multiplication facts-but their experience with multiplying mentally should expand beyond these basics. For example, students should investigate patterns that help them mentally multiply any number by a power of 10. I am concerned when I see a student multiply 18 × 10, for example, by reaching for a pencil and writing:

Revisiting students' prior work with multiplying three factors can help develop their skills with multiplying mentally. Helping students judge which way is most efficient to multiply three factors, depending on the numbers at hand, deepens their understanding. For example, to multiply 2 × 9 × 5, students have the following options:

Guiding students to check for factors that produce a product of 10 helps build the tools they need to reason mathematically.

When students calculate mentally, they can estimate before they solve problems so that they can judge whether the answer they arrive at makes sense. For example, to estimate the product of 683 × 4, students could figure out the answer to 700 × 4. You can help students multiply 700 × 4 mentally by building on their prior experience changing three-factor problems to two-factor problems: Now they can change a two-factor problem-700 × 4-into a three-factor problem that includes a power of 10-7 × 100 × 4. Encourage students to multiply by the power of 10 last for easiest computing.

7. Help Students Use Written Calculations to Track Thinking

Students should be able to multiply 700 × 4 in their heads, but they'll need pencil and paper to multiply 683 × 4. As students learn and practice procedures for calculating, their calculating with paper and pencil should be clearly rooted in an understanding of math concepts. Help students see paper and pencil as a tool for keeping track of how they think. For example, to multiply 14 × 6 in their heads, students can first multiply 10 × 6 to get 60, then 4 × 6 to get 24, and then combine the two partial products, 60 and 24. To keep track of the partial products, they might write:

14 × 6

10 × 6 = 60

4 × 6 = 24

60 + 24 = 84

They can also reason and calculate this way for problems that involve multiplying by three-digit numbers, like 683 × 4.

8. Provide Practice

Struggling math students typically need a great deal of practice. It's essential that practice be directly connected to students' immediate learning experiences. Choose practice problems that support the elements of your scaffolded instruction, always promoting understanding as well as skills. I recommend giving assignments through the four-stage support routine, allowing for a gradual release to independent work.

Games can be another effective way to stimulate student practice. For example, a game like Pathways (see Figure 1 for a sample game board and instructions) gives students practice with multiplication. Students hone multiplication skills by marking boxes on the board that share a common side and that each contain a product of two designated factors.

Figure 1. Pathways Multiplication Game

9. Build In Vocabulary Instruction

The meanings of words in math-for example, even, odd, product, and factor -often differ from their use in common language. Many students needing math intervention have weak mathematical vocabularies. It's key that students develop a firm understanding of mathematical concepts before learning new vocabulary, so that they can anchor terminology in their understanding. We should explicitly teach vocabulary in the context of a learning activity and then use it consistently. A math vocabulary chart can help keep both teacher and students focused on the importance of accurately using math terms.

When Should We Offer Intervention?

There is no one answer to when teachers should provide intervention instruction on a topic a particular student is struggling with. Three different timing scenarios suggest themselves, each with pluses and caveats.

While the Class Is Studying the Topic

Extra help for struggling learners must be more than additional practice on the topic the class is working on. We must also provide comprehensive instruction geared to repairing the student's shaky foundation of understanding.

The plus: Intervening at this time may give students the support they need to keep up with the class.

The caveat: Students may have a serious lack of background that requires reaching back to mathematical concepts taught in previous years. The focus should be on the underlying math, not on class assignments. For example, while others are learning multidigit multiplication, floundering students may need experiences to help them learn basic underlying concepts, such as that 5 × 9 can be interpreted as five groups of nine.

Before the Class Studies the Topic

Suppose the class is studying multiplication but will begin a unit on fractions within a month, first by cutting out individual fraction kits. It would be extremely effective for at-risk students to have the fraction kit experience before the others, and then to experience it again with the class.

The plus: We prepare students so they can learn with their classmates.

The caveat: With this approach, struggling students are studying two different and unrelated mathematics topics at the same time.

After the Class Has Studied the Topic.

This approach offers learners a repeat experience, such as during summer school, with a math area that initially challenged them.

The plus: Students get a fresh start in a new situation.

The caveat: Waiting until after the rest of the class has studied a topic to intervene can compound a student's confusion and failure during regular class instruction.

How My Teaching Has Changed

Developing intervention lessons for at-risk students has not only been an all-consuming professional focus for me in recent years, but has also reinforced my belief that instruction-for all students and especially for at-risk students-must emphasize understanding, sense making, and skills.

Thinking about how to serve students like Paul has contributed to changing my instructional practice. I am now much more intentional about creating and teaching lessons that help intervention students catch up and keep up, particularly scaffolding the mathematical content to introduce concepts and skills through a routine of support. Such careful scaffolding may not be necessary for students who learn mathematics easily, who know to look for connections, and who have mathematical intuition. But it is crucial for students at risk of failure who can't repair their math foundations on their own.

Marilyn Burns is Founder of Math Solutions Professional Development, Sausalito, California; 800-868-9092; mburns@mathsolutions.com.

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November 2007 | Volume 65 | Number 3 Making Math Count Pages 16-21 Marilyn Burns Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty.