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Strategies that allow students to engage in active learning through the ability to plan, monitor and modify.  There are two types of metacognitive strategies: self-talk (students ask themselves questions as they go through work i.e. "did I understand these directions?") and self-monitoring (ensuring all steps were taken and make sense). Teachers can also teach metacognitive strategies through supports given in the classroom from checklists during work or modeling "think alouds" of how to work out a problem.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/
Inclusive Schools Network. (2015, September 3). Metacognitive Strategies. Retrieved July 12, 2020, from https://inclusiveschools.org/metacognitive-strategies/

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Page 7: Metacognitive Strategies

As you have now learned, students who struggle with mathematics tend to be poor problem solvers. They approach every mathematics problems using only a small number of strategies, and even these strategies they apply inconsistently. Teachers can begin to address these issues by teaching the students cognitive strategies (e.g., schema-based instruction, mnemonics) that help students focus their attention on relevant information, identify a given problem's structure, and solve the problem.

However, teaching students cognitive strategies alone is not enough to ensure that those strategies will be implemented correctly or independently. This is especially the case for students with mathematics difficulties and disabilities, who tend to implement the same strategy for every problem, implement strategies without considering the problem type, or fail to use a strategy at all. If students are to be more successful, teachers should pair instruction on cognitive strategies with that of metacognitive strategies -strategies that enable students to become aware of how they think when solving mathematics problems. This combined strategy instruction teaches students how to consider the appropriateness of the problem-solving approach, make sure that all procedural steps are implemented, and check for accuracy or to confirm that their answers makes sense. More specifically, metacognitive strategies help students learn to:

How does this practice align?

High-Leverage Practice (HLP)

HLP14: Teach cognitive and metacognitive strategies to support learning and independence

CCSSM: Standards for Mathematical Practice

MP1: Make sense of problems and persevere in solving them.

Plan - Students decide how to approach the mathematical problem, first determining what the problem is asking and then selecting and implementing an appropriate strategy to solve it.

Monitor - As students solve a mathematical problem, they check to see whether their problem-solving approach is working. After completing the problem, they consider whether the answer makes sense.

Modify - If, as they work to solve a mathematical problem, students determine that their problem-solving approach is not working or that their answer is incorrect, they can adjust their approach.

Research Shows

When paired with cognitive strategies, metacognitive strategies have been shown to increase the understanding and ability of students with mathematics learning difficulties and disabilities to solve mathematics problems.
(Pfannenstiel, Bryant, Bryant, & Porterfield, 2015)

Middle school students who received cognitive and metacognitive strategy instruction outperformed peers who received typical math instruction.
(Montague, Enders, & Dietz, 2011; Pfannenstiel, Bryant, Bryant, & Porterfield, 2015)

Types of Metacognitive Strategies

Metacognitive strategies that help students plan, monitor, and modify their mathematical problem-solving include self-instruction and self-monitoring. Not only are these strategies relatively easy for students to implement, but they also help students to become better independent problem solvers.

Self-instruction
Talking one's self through a task or activity (also known as self-talk)

"Did I understand what I just read? No, I didn't. I need to reread the problem."

"What is this problem asking? What information do I have?"

"What is the next step?"

Self-monitoring
Checking one's performance; often involves a checklist

Checking to make sure all steps are completed

Checking for computational errors

Checking to make sure the answer is feasible

Teaching Metacognitive Strategies

Teachers should use explicit instruction to help students understand how to use self-instruction and self-monitoring during the problem-solving process. To do this, teachers can:

Provide students with a list of questions or prompts to ask themselves while they are engaged in the problem-solving process.

Example questions: What information is relevant? Have I solved a problem like this before?

Example prompts: Identify the relevant information. Use a visual to solve the problem.

Model working through a problem using "think alouds," during which the teacher verbalizes her thoughts as she demonstrates using self-instruction and self-monitoring throughout the problem-solving process.

Provide sufficient opportunities for students to practice these metacognitive strategies with corrective feedback.

Encourage students to use these strategies independently, once they have achieved mastery.

Examples of Students Using Metacognitive Strategies

The videos below illustrate students using metacognitive strategies to solve mathematics problems. In the first video, in addition to self-instruction, an elementary student uses an age-appropriate self-monitoring checklist that includes visual cues for each step. Note that the student was explicitly taught how to use this checklist before using it to solve problems independently. In the second video, a high-school student uses self-instruction and self-monitoring to solve a word problem.

Elementary School Example (time; 1:49)

Transcript: Metacognative Strategies: Elementary School

Narrator: In this video, an elementary student uses metacognitive strategies while solving an addition problem. More specifically, he uses self-instruction and a self-monitoring checklist to guide himself through the problem-solving process. By doing so, he actively plans and monitors his work.

Student: I can't figure out what 3 + 5 is. What is it? Well, let me look at my checklist. First, it says, "read the problem." The problem says 3 + 5, so I've checked that. Now what is...now it says...my checklist says, "What is the problem asking?" It's asking me to add 3 + 5.

Now, to draw a picture. One, two, three. One, two, three, four, five. Now it says, "Does my drawing match the problem?" Up here it says 3 + 5, so down here it says one, two, three, one, two, three, four, five. Now I have to solve it. So one, two, three, four, five, six, seven, eight. The answer to 3 + 5 is 8.

High School Example (time; 2:55)

Transcript: Metacognative Strategies: High School

Narrator: In this video, a high school student uses metacognitive strategies while solving a word problem. By using self-instruction and self-monitoring, she actively plans and monitors her work.

Student: First, I'm going to read the problem. "Mr. Smith, the principal, is standing on top of the high school. He is looking at a tree in the courtyard that is 30 feet away from the school. The angle from Mr. Smith's feet to the base of the tree is 43 degrees. Using this information, determine the height of the high school."

So what am I missing? The problem says that the angle from Mr. Smith's feet to the base of the tree is 43 degrees. I've noticed that, if you connect this point to this point, we have a right triangle. So, while this angle is 43 degrees, this angle right here is a right angle that's 90 degrees.

There's a trick that I'm going to use that's called SOHCAHTOA that you can use to find the sides and angles in a right triangle. The opposite side to 43 degrees is 30 feet, right here. So what do I need to find? I need to find the adjacent side. I'll label it with an "A." I look at SOHCAHTOA, and I know that I need to find the tangent, because tangent equals opposite over adjacent.

Now all I have to do is plug in the information that I have in order to find "A." Tangent of 43 degrees, the angle, equals 30-that's the opposite side-over
"A." And I find that 30 over 0.93 equals 32.25. So the height of this building is 32.25 feet.

Now that I've solved the problem, I ask does my answer make sense? Given the information from the problem, and with what I know about most buildings, 32 feet seems like a reasonable answer.

Diane Bryant discusses the importance of teaching students cognitive and metacognitive strategies and how they benefit students (time: 2:23).

Diane Pedrotty Bryant, PhD
Project Director, Mathematics Institute for
Learning Disabilities and Difficulties
University of Texas at Austin

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View Transcript
Transcript: Diane Pedrotty Bryant, PhD

It really is important to pair metacognitive strategies with cognitive strategies. Metacognitive strategies simply refers to thinking about thinking. It's beneficial and certainly validated in research that they have a series of cognitive steps to employ to solve problems, whatever the problem might be. The metacognitive strategies help students think about what steps that they're supposed to be using-that's the self-instruction-and then pausing to check about whether they are indeed using those various cognitive strategies steps, which really refers to the self-monitoring. For students with mathematics learning disabilities, we want them to become independent learners and to use strategies for solving various problems and to be able to pause and ask themselves questions about how they're proceeding and back up and check on a particular step. Through the use of cognitive strategies paired with metacognitive strategies, the goal is to empower them to be more independent learners, and that's definitely something we strive for when we teach students with learning disabilities. I think that there is difficulty in students learning how to implement metacognitive strategies independently, because they may not know how to approach the learning task. They may not be aware of their own ability to self-monitor, to self-instruct, to use self-talk, self-verbalizations for tackling tasks. Usually students with mathematics learning disabilities really need to be taught to use metacognitive strategies and to learn the metacognitive strategies to mastery before being able to use them independently.

For Your Information

Although teachers can provide students with a generic list of questions or prompts to guide them through the problem-solving process, some students, such as those with mathematics difficulties and disabilities, might need more individualized support to address their specific learning challenges. The teacher can identify the student's common error patterns by conducting an error analysis -a process by which instructors identify the types of errors made by students when working mathematical problems. Using this information, teachers can develop a list of questions or prompts that students can use to address their specific needs. To begin with, many of these students might require a self-monitoring checklist, such as the one below, to guide them through the problem-solving process.

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As you have now learned, students who struggle with mathematics tend to be poor problem solvers. They approach every mathematics problems using only a small number of strategies, and even these strategies they apply inconsistently.

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What is schema? Schema is based on how we, our children, our students develop understanding of concepts, relationships, ideas, activities with what we do and surround ourselves with. Now in schema based instruction, students are taught to understand or identify the schema of the mathematical word problem and in turn helps them with the solving process. Teachers lead students through a series of three steps to identify the schemas going through one word problem structures at time: additive schemas, multiplicative schemas, and combine schemas.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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Explicit Systematic Instruction, better known as explicit instruction is the process of teaching specific content that is structured and communicated in a sequential way. For instance, the teacher begins a lesson by identifying the content to be learned and setting the stage to connect previous learned skills and content with the new material. The material is taught with specific guidelines given to modeling and guided questioning and practice to independent practice. Student dialogue is encouraged to help facilitate comprehension of strategy approach as well as constant feedback to help re-teach or clarify.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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Schema-Based Instruction Part 3:   "Change" Problems and Wrap-Up

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When you hear surface learning, you think initial, not thorough, however surface learning is the initiation of new ideas with explicit instruction. Through surface learning students learn new information and are able to further develop their cognitive and metacognitive skills. Surface learning helps students understand vocabulary concepts and labels and can help identify students misconceptions and errors about the new content. Surface learning is the start of a student's mathematics learning kit that they can build on throughout their academic career.

Reference:
Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

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Deep learning is  the next step after a student has surpassed conceptual understanding in the surface learning level. Deep learning combines opportunities for collaboration with peers, use of academic terms, and small group work and discussions. In deep learning teaching moments,  students can take the surface knowledge of mathematical concepts like algebraic terms and then use that knowledge to interpret other aspects of the term and apply the algebraic concepts into action. Deep learning most often takes place when students are allowed to lead their learning through small group work and class discussions.

Reference:
Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

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Katrina Schwartz, writing for KQED's MindShift blog, on the new Deeper Learning Planning Guide by Monica Martinez and Dennis McGrath:

Many school administrators, teachers and parents want the education provided to children to be high quality, rigorous and connected to the world outside the classroom . Teachers are trying to provide these elements in various ways, but a group of schools calling itself the "Deeper Learning Network" has codified some of what its members believe are essential qualities of deep learning (check out how students lead parent teacher conferences in this model ). Some of the goals include learning designated content, critical thinking, communication skills, collaborating effectively and connecting learning to real-world experiences.

To better understand what schools in the Deeper Learning Network were doing differently, Monica Martinez and Dennis McGrath visited several schools and wrote a book about what they found: "Deeper Learning How Eight Innovative Public Schools Are Transforming Education in the Twenty-First Century." They've also put together a guide to help interested educators create the conditions necessary to make this model thrive. As the infographic below shows, the model requires a big shift from traditional school and rests on positive school culture and collaborative professional teams of teachers who are committed to the vision of the school.

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Creating the Conditions for Deeper Learning

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Katrina Schwartz, writing for KQED's MindShift blog, on the new Deeper Learning Planning Guide by Monica Martinez and Dennis McGrath: Many school administrators, teachers and parents want the education provided to children to be high quality, rigorous and connected to the world outside the classroom.

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Transfer learning is quite literally as it sounds, the transfer of information.  As students are introduced to new concepts in surface learning they then take that information and  delve deeper into its application and understanding with deep learning, and finally as they grow in their educational journey and life they use transfer learning to apply what they know and have learned into new situations. Transfer learning is truly the beauty of education and learning, as students are now at the helm of their learning growth. Education is not just memorizing facts, procedures,  study guides, or getting grades to just pass on to the next grade level. Education is about promoting students to think critically with their own self-direction and motivation to be forever learners and transfer learning is the epitome of that.

References:
Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

Ferlazzo, L. (2017, May 09). Response: Ways to Promote Transfer of Learning. Retrieved July 13, 2020, from https://blogs.edweek.org/teachers/classroom_qa_with_larry_ferlazzo/2017/05/response_ways_to_promote_transfer_of_learning.html

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Education Week - Classroom Q&A With Larry Ferlazzo

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The new "question-of-the-week" is:
What is "transfer of learning" and what can teachers do to increase the odds of it happening with their students?
In Part One, Douglas Fisher, Nancy Frey, Adeyemi Stembridge, Todd Finley, Kenneth Baum, and David Krulwich shared their ideas on the topic. You can listen to a 10-minute conversation I had with Nancy and Adeyemi on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.

Today, Cathy Beck, Heidi Pace, Anna Bartosik, Jenny Edwards, Josh Patterson, Ashley Roberts and Andrew Miller contribute their responses.
Response From Cathy Beck & Heidi Pace
Cathy Beck is the assistant superintendent in Summit County, CO. She is the author of " Easy and Effective Professional Development," and " Leading ELL Students to Success: Strategies for Providing Equity and Access for All," published by Routledge Publishing. Cathy is a regular contributing blogger for Bamradio Network. She is an adjunct professor at both Concordia University and the American College of Education. You can connect with her on Twitter @cathypetreebeck

Dr. Heidi Pace, former superintendent for Summit County, CO, has 33 years' experience in education. Dr. Pace is a professor for Colorado College and also an adjunct professor for Concordia University. She is the co-author or " Leading Learning for ELL Students," published by Routledge Publishing:
Transfer of learning can have several contexts. Learning can transfer one language to another, from one level of Bloom's Taxonomy to a deeper level, or across content areas. All are equally valuable and teachers must consciously plan for activities and questions that help students make this move. All content needs to be taught to the application and creation level or higher. Sheer instruction alone does not guaranteed transfer of knowledge. To get to this level students must be proficient in the prerequisite skills necessary for a solid foundation. There are several strategies to increase the likelihood that students will be able to deepen their understanding and apply knowledge to other areas. These strategies include:

Think Alouds-Model some examples of transfer for specific skills by doing a Think Aloud. As you give the students one example ask them to generate another. The more you model, the more they will begin to understand how to think at this level.
Education's purpose is ultimately to ensure that students can be successful in the real world, in a job or a personal setting. Today's job market requires application, creativity, and innovation. Rote memorization of facts will not provide success in this arena. Students must have practice applying and transferring of knowledge before they go out in the world. This practice should be a normal part of school every day. Take a hard look at your instruction. Are you offering opportunities for students to collaborate, reflect, and apply their skills? Are you delivering higher order thinking and open ended questions daily? Do your assessments offer questions that require transfer? If not make some changes. The learning will deepen along with the engagement.
Response From Anna Bartosik
Anna Bartosik is an ESL professor from Canada. She is a teacher/facilitator with an extensive background in ESL/ELT education. Her current interests include instructional design, assessment and rubrics, storytelling, motivation's role in memory, and thoughtful incorporation of educational technology in the classroom:
Imagine you have a vessel in your classroom that transfers knowledge directly to your students' brains. The process consists of opening a flap in their heads and pouring in the knowledge. The recipients have no leaks and no evaporation of knowledge occurs over time. We can all dream, can't we. Transfer of learning occurs when knowledge we acquire moves from our working memory to long-term memory, and is retrievable. There are many things which impede this process. As classroom teachers, we need to make the information we provide to students manageable, and learnable, so that our brilliant lessons remain with our students for as long as possible. Some of you might think that working memory is the most important factor in the transfer of learning. I'd ask you to take a step back and look at your learners first. What motivates them? Some of you may remember the TV show WKRP in Cincinnati, and the character Venus Flytrap explaining an atom to a young man who wanted to quit school. It is a classic example of the transfer of learning taking place. A good teacher will analyse students in her class and see what motivates them, then tailor the learning to subject matters which address their interests. Do you have a room full of soccer players? Using soccer examples in word problems will capture the interest of those students. This kills two birds: it captures their interest, and connects new learning to existing knowledge. Activating prior knowledge, whether it be in the form of a review or connecting lessons to what students are familiar with, decreases the cognitive load of new information. Let's return to working memory: according to Miller's Law, it is the magical 7 ± 2. If a learner is only capable of remembering such a small amount of new information, then we need to keep that working memory open to new information by moving learned information to the long-term bank. We make connections for students to previous learning by activating that knowledge and reducing the cognitive load of taking in new information. Imagine taking a sponge and squeezing it into a bucket. Lots of room left in the bucket, and the sponge is dry enough to absorb more water. How do I do that practically in the classroom?

Reduce the cognitive load of multiplication tables by showing students a month on a calendar. Look at the 7th, then look down the 4 lines on the calendar month. Four of the 7 times table down, only 6 more to go.

Mindmap the plot of a novel the class is reading. Use images and symbols instead of words where appropriate. More pictures means less cognitive load.

Transfer of learning occurs when the student is motivated by the topic, motivated to learn, has previous knowledge on the subject, and knows how to connect new information to existing information. The learner must then be able to retrieve this information and apply it to new learning. A lot of breaks can happen in this chain, but if we model this formula in our teaching and make it a conscious process for students, then our learners stand a better chance of transferring that learning.
Response From Jenny Edwards
Jenny Edwardsteaches doctoral students in the School of Educational Leadership and Change at Fielding Graduate University in Santa Barbara, Calif. Edwards is the author of the ASCD books Time to Teach: How do I get organized and work smarter? and Inviting Students to Learn: 100 Tips for Talking Effectively with Your Students:
What is "transfer of learning? Students are able to transfer their learning when they can take something that they learned in one context and apply it in another context. Costa and Kallick (2000) listed Transfer of Learning as one of 16 Habits of Mind that would be beneficial for students to learn. Costa (2008) suggested that people learn from their experiences. He added that when they solve problems or face challenges, they call on their prior knowledge in order to be successful. Strategies that teachers can use to help students to transfer information from one context into another include:

Teaching them to use graphic organizers in all subject areas--the more complex, the better (Dexter & Hughes, 2011)

Teaching them to use metacognitive strategies (Dean & Kuhn, 2003; Halpern, 1998)

Engaging them in Problem-Based Learning (PBL) (Dixon, 2012; Hmelo-Silver, 2004)

Inviting them to play digital games (Swanson, 2014)

Teaching them to take strategic notes on what they are learning (Lee, Lan, Hamman, & Hendricks, 2008)

Inviting them to take tests a week after they have been exposed to information, as well as a month later, and a year later (Carpenter, 2012; Rohrer, Taylor, & Sholar, 2010)

Encouraging them to activate their prior knowledge and set goals for learning (Malmberg, Järvenoja, & Järvelä, 2013)

Inviting them to share what they are learning with others by creating a video presentation (Hoogerheide, Loyens, & van Gog, 2014)

Helping them to feel confident in subject areas, which will make them more likely to transfer their knowledge to other areas (Keiler, 2007)

Encouraging them to use their prior knowledge in new contexts (Cleary, 2013)

Showing students how what they are learning links with what they have previously learned and with what they are learning in other subject areas (Bhatti, Battour, Sundram, & Othman, 2013)

Providing them with coaching on what they have learned (Joyce & Showers, 2002)

Engaging them in talking about contexts in which they might apply what they are learning (Nikandrou, Brinia, & Bereri, 2009)

When introducing lessons, showing students how what they are learning links with what they have previously learned (Costa, 2008)
Which of these strategies might you use in the coming weeks to help students to transfer what they are learning into a variety of contexts?

Response From Josh Patterson & Ashley Roberts
Josh Patterson, PhD is the principal of Oakland Elementary School in Spartanburg, S.C. He is co-chair of the project-based learning action team with TransformSC, an ASCD Emerging Leader, and president-elect of South Carolina ASCD. Connect with Patterson on Twitter @ACE_Patterson.

Ashley Roberts is the literacy coach and Response to Intervention Coordinator at Oakland Elementary School. She is an ASCD Emerging Leader and facilitates numerous workshops on the topics of inquiry, literacy, and differentiation. Connect with Roberts on Twitter @_my3boys:
As school and district leaders, we often find ourselves speaking out of both sides of our mouth. We believe in creating meaningful, authentic, inquiry-based opportunities for exploration and discovery yet recognize the significance and pressure of achieving high test scores. We advocate for best practice, understanding that deeper learning requires intentional cognitive connections through critical thinking and reflection. We know students need time and space to make authentic connections in their learning, yet are compelled to create urgent pacing to ensure quick coverage of content standards. We feel conflicted and double-minded, immobilized and confused, split between "schooling" and education. Is there an appropriate balance? How can we ensure that in a world of high accountability, assessment, and rankings our students experience a genuine transfer of learning? As elementary educators, how do we best prepare our students--who are eight to twelve years from high school graduation--to be socially responsible problem solvers who will make a positive contribution to their world? Curriculum standards alone are not sufficient in preparing students for tomorrow's world. Ultimately, we create instructional experiences to prepare our students to become productive participants in society, not simply to determine what they can or cannot do as a result of planned lessons or units. Therefore, to experience success in the "real world" students must have opportunities to transfer learning in a real, authentic context. Perkins and Salomon (1988) describe transfer of learning as the application of skills, knowledge, and/or attitudes learned in one situation to another cognitive situation. Project-Based Learning (PBL) and STEM (science, technology, engineering, and mathematics) provide ideal, integrated opportunities for students to transfer learned skills and content to create, construct, and communicate their learning. In a PBL unit entitled "Tread Lightly," students discuss the perspectives of environmentalists and loggers, participate in a Socratic Seminar, write action plans to consider how they promote sustainability, and create infographics to argue both sides of the discussion. Additionally, components of STEM are included to increase student interest and transfer of learning. Students are challenged to construct their own edible land field, research various eco-friendly inventions, and restore the damage done to the ocean and affected wildlife through an oil spill simulation. An inventors and inventions PBL unit immerses students in the concepts of persuasion, supply and demand, and the STEM design process. Students are challenged to investigate various real world problems and create inventions that serve as possible solutions. Throughout the unit, students study reputable inventors, noting the similarities and differences of their experiences, successes, and failures. The culminating event results in an invention convention, using QR Codes to showcase students' advertisement videos. This unique opportunity provides students with a real, authentic audience to showcase their inventions. During a weather PBL unit, students visit the school's science lab to examine and develop various weather tools. Students collaborate and research answers to their own questions. To support the student's transfer of learning, teachers set up a green screen for students to create videos of severe weather announcements. These performance tasks are not only used to support students' transfer of learning in one grade level, they are used by the school's administration in routine emergency preparedness drills for the entire school to learn from. There is a significant limitation between the learning that occurs during traditional classroom instruction and the application of learning in a real world situation. Without meaningful, authentic connections, a student's transfer of learning is almost always weaker. Providing quality instruction within a broader PBL framework promotes an increased level of responsibility, both socially and academically. When planning for authentic experiences, the proverbial playing field is leveled and a stronger transfer of learning will occur for all students. Perkins, D. N. & Salomon, G. (1988.) Teaching for transfer. Educational Leadership 46 (1): 22-32.

Response From Andrew Miller
Andrew Miller (@betamiller) is an instructional coach and educational consultant who focuses on project-based learning, assessment and student engagement. He is on the faculty for both ASCD and the Buck Institute for Education. He is the author Freedom to Fail and also writes regularly for Edutopia and ASCD:
Transfer goals focus on the big picture. They focus on almost philosophical questions of what do we want for our students in the long term. Although these goals may focus on long-term school goals, they also focus on goals outside of school and in the future. While they may focus on course-level content of skills, they can focus on other skills like good judgment, self-regulation, and collaboration. The first step to focusing on transfer goals is to practice creating them. Teacher teams across grade-level should meet to discuss these goals. This takes diligence and time and commitment. Once that that step is completed, then teachers individual and collaboratively need to create units and assessments that focus on these transfer goals. How will you a unit on genes or creative writing, for example, align and support the learning of the transfer goal? Do we need to create a new transfer goal or modify an existing one? This again highlights that transfer goals connect learning across content and also need to be refined and reflected upon. Transfer goals help us focus on daily work with students on larger, important goals that we people can value regardless of grade-level, content area, or experience.

Thanks to Cathy, Heidi, Anna, Jenny, Josh, Ashley and Andrew for their contributions!

Please feel free to leave a comment with your reactions to the topic or directly to anything that has been said in this post.

Consider contributing a question to be answered in a future post. You can send one to me at [email protected] .When you send it in, let me know if I can use your real name if it's selected or if you'd prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo.

Anyone whose question is selected for this weekly column can choose one free book from a number of education publishers.

Education Week has published a collection of posts from this blog, along with new material, in an e-book form. It's titled Classroom Management Q&As: Expert Strategies for Teaching.
Just a reminder--you can subscribe and receive updates from this blog via email or RSS Reader. And, if you missed any of the highlights from the first five years of this blog, you can see a categorized list below. They don't include ones from this current year, but you can find them by clicking on the "answers" category found in the sidebar. This Year's Most Popular Q & A Posts! Classroom Management Advice Student Motivation & Social Emotional Learning Relationships In Schools Instructional Strategies Author Interviews
I am also creating a Twitter list including all contributers to this column.

Look for the next "question-of-the-week" in a few days...

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Response: Ways to Promote Transfer of Learning

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The new "question-of-the-week" is: What is "transfer of learning" and what can teachers do to increase the odds of it happening with their students? In Part One, Douglas Fisher, Nancy Frey, Adeyemi Stembridge, Todd Finley, Kenneth Baum, and David Krulwich shared their ideas on the topic.

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Mnemonics are memory tools to help students retain and recall information. Students with learning disabilities have trouble recalling information, but with the help of mnemonics strategies, students are able to use the visual and auditory cues of the mnemonics to  pull on their prior knowledge in relation to the new information. Mnemonics strategies range from songs (Alphabet song) to names in mnemonics (ROY G. BIV), word mnemonics (PEMDAS) etc.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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How many lyrics to songs do you remember? How did you come to remember them? The same method you used to recall song lyrics also can work just as well in academics. Music can used to help students recall important details to main ideas and many learners have made songs out of information when a list of items must be learned.

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Graphic organizers are great tools for students who struggle with processing information. Graphic organizers help students organize information, see the relationships between ideas, and an easier ability to understand, remember, and apply information. There are a variety of graphic organizers ( cause and effect, classifying, comparing and contrasting, describing, sequencing)  for teachers to choose from and teachers must ensure they choose the right option in order for their students to best learn. Graphic organizers can help with reading comprehension, writing skills, understanding of linear equations, etc.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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Concrete-Representational-Abstract Framework (CRA) is a tool teachers use to help students transition from concrete or visual representations to the abstract mathematical equation. CRA can help students at any age/grade level as the CRA methods must reflect the math process by first showing the concrete component (students interact and manipulate three-dimensional objects), then the representational component (students use two-dimensional drawings to represent problems and these pictures can be shown to students by their teacher or through the course content) and finally the abstract component (students solve problems with numbers, symbols, and words without any concrete or representational assistance). This process allows for students to scaffold their math understanding.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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Guided questioning are like directions that lead students to better understanding what they are learning and directs them into making their own conclusions. Guided questioning requires practice and the ability to pick the right questions that elicit deeper though rather than providing students too much information and not allowing them to draw the conclusions. Guided questions allow our students to critically think and move into a higher order thinking process.

Reference:
Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

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Creating Standards-Based Guiding Questions

In addition to helping students ask better questions, you can create better guiding questions by aligning them with standards. Begin at the end. What do you want your students to learn? What standards are you trying to cover? Form questions based on the standard.

Scenario: Imagine that you need to teach the following social studies standards. The first is a performance standard, and the second is a content standard:

B.8.8 Identify major scientific discoveries and technological innovations and describe their social and economic effects on society.

8.5 Learn about the Civil War and Reconstruction, 1861-1877.

Building Guiding Questions

Now frame a guiding question that addresses both the performance and content standards. Also, make sure you ask questions that have many possible answers and require research and analysis.

"What invention had the biggest effect in the Civil War?"

"What kind of weapons did soldiers use in the Civil War?"

"How did non-military technology shape the Civil War?"

"What medical practices were used in the Civil War?"

"How did medicine improve during the Civil War?"

"What one technology from today would a Civil War soldier most want?"

Using Guiding Questions

By framing a guiding question that combines a content standard with a performance standard, you can make sure that the inquiry that students do stays on topic. You can also provide students the chance to choose which of the possible guiding questions they want to pursue. If you provide five questions, you can end up with five groups, each working to answer one of the questions and report what they find to the class.

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How can I form strong guiding questions?

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Creating Standards-Based Guiding Questions In addition to helping students ask better questions, you can create better guiding questions by aligning them with standards. Begin at the end. What do you want your students to learn? What standards are you trying to cover? Form questions based on the standard.

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Visual representations (also known as schematic representations and schematic diagrams) are strategies to help students solve problems and learn abstract mathematical concepts. Visual representations need to taught, students with learning disabilities do not know how to create their own without explicit instruction so teachers must model and provide examples to correlate to the learning concepts. For examples, there are visual representation options such as number lines, strip diagrams, pictures,  graphs/charts, and graphic organizers for teachers to choose from that best fit their learners. Another option of visual representations are manipulatives, concrete, hands-on materials and objects like coins, blocks.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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By: Judy Zorfass, Angela Han, and PowerUp WHAT WORKS

Introduction

All students can benefit from using visual representations, although struggling students may require additional, focused support and practice. Visual representations are a powerful way for students to access abstract mathematical ideas. To be college and career ready, students need to be able to draw a situation, graph lists of data, or place numbers on a number line. Developing this strategy early during the elementary grades gives students tools for engaging with—and ways of thinking about—increasingly abstract concepts. Over time, they will work toward developing Common Core Standards for Mathematical Practice:
WAYS TO SUPPORT STUDENTS
Helping students choose the "right" visual representation often depends on content and context. In some contexts, there are multiple ways to represent the same idea. Show your students a variety of examples in order to demonstrate when (and why) they should choose each one (see UDL Checkpoint 2.5: Illustrate through multiple media). Consider how you could use the following strategies to support your students:

Check for understanding to determine a starting point. For example, you could ask the following questions: Why do you think that? How do you know that is correct? How does that picture represent the problem? Can you explain your answer? Is there another way you could do that?

Ask students about features of the visual representation (including labels and scales, when appropriate).

As students create visual representations, ask questions to ensure that they understand all the features of the representations. Prompt students to focus on the information the visual representations provide.

When possible, include alternative visual representations and discuss the similarities and differences between them.

Vary the shapes and orientations of representations so that students focus only on the important features as they learn about the objects and situations represented.

Show your students a specific representation—a graph or a table—that is missing an important feature. Ask them to identify the missing feature.

Drawing on technology tools

New technologies are constantly expanding our ability to visualize data and explain mathematical concepts. For teachers looking to incorporate technology into the classroom, using virtual manipulatives (instead of physical ones) can be a good start. Students can begin with simple graphical representations of mathematical concepts and then work toward more complex modules that require them to create the data or work within a system of rules, like a game. Infographics—visualizations that are designed to communicate complex information effectively—have become increasingly popular. They can be used to "tell a story" with numbers, such as international democracy rankings or climate change impacts. Learning to create infographics gives students additional tools to communicate data and other quantitative information.

3D printing is a technology that, until recently, has been too expensive to make use of in a classroom. However, thanks to falling prices, they have now started to appear in high schools and it may not be long before elementary schools and middle schools also embrace this technology. 3D printing allows you to create solid, three-dimensional models from a digital design. You can explore what others have created to get a sense of what is possible. Imagine having students design and create their own mathematical models and manipulatives!

For more ideas on using technology to create visual representations, visit the Tech Matters blog or PowerUp's Pinterest page. You can also check out the " Virtual Manipulatives" video, which supports students' use of visual representations.

In the classroom

Geometry lends itself naturally to teaching with visual representations, as can be seen in Ms. Richardson's Grade 6 class. So far, students have learned how to classify different quadrilaterals and triangles, and they are beginning to decompose polygons. They have also started using software (e.g., GeoGebra) that can support their understanding by emphasizing the connections between mathematical language and visualization.

Ms. Richardson's lesson objective is to have students decompose polygons into triangles, rectangles, and trapezoids. She will address two s Common Core State in this lesson:

CCSS Math 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

CCSS Math MP4 Model with mathematics.

Ms. Richardson has students work on these standards within the context of a real-world example—a painting by the artist Sol LeWitt.

Sol LeWitt. Wall Drawing #1113. On a wall, a triangle within a rectangle, each with broken bands of color, 2003. Hirshhorn Museum and Sculpture Garden, Smithsonian Institution.

Students will build on their existing technology skills and create a model of this work, decomposing polygons and creating their own virtual LeWitt in the process. Ms. Richardson's lesson plan is organized into three sections: a warm-up exercise to review concepts, the main learning task, and a closing discussion and assessment.

Lesson plan

Launch

Warm up with a review problem.

Have students demonstrate and explain different solutions to the problem.

Compare and contrast students' visual representations.

As a class, identify additional solutions to the problem.

Learning Task

Introduce the learning task.

Connect the review problem to the real-world problem.

Have students begin work on the problem in pairs.

Circulate, listening to discussions and supporting students as necessary.

Have students view all the solutions during a gallery walk.

Closure

Lead a class discussion on students' different solutions.

Have each student complete an exit ticket.

Online resources for visual representations

This article draws from the PowerUp WHAT WORKS website, particularly the Visual Representations Instructional Strategy Guide. PowerUp is a free, teacher-friendly website that requires no log-in or registration. The Instructional Strategy Guide on visual representations includes a brief overview with an accompanying slide show; a list of the relevant mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; and links to resources that will help you use technology to support mathematics instruction. If you want to dig deeper into the research foundation behind best practices in the use of virtual manipulatives, take a look at our Tech Research Brief on the topic. If you are responsible for professional development, the PD Support Materials provide helpful ideas and materials for using the resources. Want more information? See PowerUp WHAT WORKS.

Judy Zorfass, Angela Han, and PowerUp WHAT WORKS (2014)

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Using Visual Representations in Mathematics

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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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Encouraging student dialogue is a great practice for students to express their mathematical reasoning. Not only is student discourse important for students to learn how to communicate mathematical concepts, but for students to hear and see how their peers process and communicate information. Discussions should be student led and teacher guided enabling students to facilitate their understanding which can be most effective in whole class discussions through small group activities. In order to ensure students know how to engage in discussion, the teacher has to establish guidelines and work to practice those behaviors with students so it becomes automatic.

Reference:
The IRIS Center. (2015). Early childhood environments: Designing effective classrooms. Retrieved from https://iris.peabody.vanderbilt.edu/module/env/

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Page 8: Effective Classroom Practices

A number of other classroom practices are supported by moderate levels of evidence, even if they have not yet met the requirements to be considered evidence-based. Implementing these kinds of effective practices in conjunction with an EBP is yet another way teachers can improve their students' mathematical understanding. Among these effective classroom practices is:

Encouraging student discussion

Presenting and comparing multiple solutions

Assessing student understanding

Research Shows

Students' mathematics achievement improved significantly when student discussion was an integral part of instruction.
(Ing, et al., 2015; Huinker, 1992)

When teachers presented multiple solution strategies for solving the same problem, students demonstrated significant increases in procedural flexibility, conceptual knowledge, and procedural knowledge.
(Durkin, Star, & Rittle-Johnson, 2017; Jitendra et al., 2011)

More than 30 years of research indicates that curriculum-based measurement (CBM), provides stable, accurate student screening and progress monitoring data in mathematics.
(Lembke & Stecker, 2007; Tindal, 2013)

Teachers have successfully used error analysis to identify the problem-solving difficulties and conceptual errors made by their students.
(Kingsdorf & Krawec, 2014)

For Your Information

When implementing the effective practices listed above, teachers will expect students to explore new concepts, attempt challenging problems, discuss their thought processes, or be open to corrective feedback. However, many students might not feel comfortable engaging in these activities, therefore, teachers need to establish a supportive and safe classroom environment. Within this type of environment, teachers can stress that making mistakes is not only acceptable but also valuable because doing so creates opportunities to identify and address faulty thinking or misconceptions.

Encouraging Student Discussion

How does this practice align?

CCSSM: Standards for Mathematical Practice

MP3: Construct viable arguments and critique the reasoning of others

Student discussion or discourse is a practice that encourages students to express their mathematical reasoning. It allows them to become aware both of their own problem-solving processes as well as those of others, and to refine their conceptual understanding. Additionally, student discussion allows the teacher to assess student understanding. This practice could be implemented during whole-group discussion or during small-group activities. To implement this practice, teachers should:

Establish discussion procedures (e.g., students justify answers by explaining their reasoning, students ask other students for clarification).

Establish behavioral expectations (e.g., respect others while they are talking).

Provide supports for students with disabilities (e.g., word wall of mathematics vocabulary, opportunities to discuss thinking with a partner before sharing in a whole-group).

Create a list of prompts to encourage discussion among the students (e.g., "What do you think about Shay's explanation?" "Can you add to Ramsee's explanation?").

Provide sufficient wait time so that students have an opportunity to formulate a response.

The video below depicts a teacher encouraging his students, during whole-group instruction, to discuss their thoughts and ideas about a series of problems they have solved regarding a V-pattern. As you watch, note that the students have difficulty articulating how they arrived at their answer, but the teacher continues to prompt and guide the discussion (time: 3:07).

Transcript: Patterns

Teacher: Does anybody see any pattern that's happening here? What seems to be happening? Oh, everybody seems to be seeing something. Ayesha?

Ayesha: Two is adding to every one.

Teacher: Two is adding to every one? What do you mean by that?

Ayesha: One plus two is three.

Teacher: So how does it go to the second one?

Ayesha: Two times two is four, plus one.

Teacher: And where are you getting that plus one from?

Ayesha: The middle.

Teacher: What do you mean? Can you show us? What do you mean by the middle? Josh, can you help her out?

Josh: The two on the side. The two on the one side, two on the other side, that's four, plus the one on the bottom.

Teacher: Does anybody see another pattern?

Sulanette: I see that they're all odd numbers.

Teacher: Why do you think they're all odd numbers?

Sulanette: Because it starts off with the V1, right? It shows 3. Well, 3 is an odd number. If it was an even number, I don't know where it should go in the V pattern, so they're all odd numbers.

Teacher: I think Sulanette was saying something pretty interesting. She was saying that she didn't know if it could be an even number. Can there be an even number? Can I say if there were 84 birds, what V pattern would that be? Could there be 84 birds?

Ashley: I think that it can never be an even number because as in the V pattern of number 1, that is actually a complete V, and an even number couldn't make that.

Teacher: Why not?

Ashley: Because the V is made up of three birds.

Teacher: Okay, the V is made up of three birds, yeah?

Ashley: And if you keep going and adding two, it's made up of all odd number of birds, the V patterns.

Teacher: So here's the first one right here. And then you say that two more birds come in like that, and so it's going to be odd. Why?

Ashley: It's going to be odd because if you add two more birds to three, that'll make five, and five's an odd number.

Teacher: Can someone tell me the number of geese that would be in the V number 10, the 10th pattern along?

Oscar: I got 21.

Teacher: And how did you find that information?

Oscar: I multiplied 10 by 2

Teacher: Why did you do that?

Oscar: Because I was thinking that every time the two extra birds come, they come and then they have a group. Then I multiplied it by 2 and it comes out to 20, and then I added a 1, the one in the middle, then it came out to 21.

Jenny: What they are saying is that there are two geese in a pair and there are 10 pairs, so he would try and multiply it by 2 times 10 and then add the leader would be 21.
Credits
This video is part of the Modeling Middle School Mathematics (MMM) project. If you want to order the DVD MMM series, contact bolstereducation@yahoo.com.

Presenting and Comparing Multiple Solution Strategies

How does this practice align?

According to CCSSM, comparing multiple solution strategies allows children to gain an understanding of the relationship between:

Addition and subtraction

Multiplication and division

Teaching multiple ways to solve a problem helps students to develop flexibility (i.e., understanding that a problem may be solved accurately using different procedures and being able to use efficient procedures) and might support conceptual understanding of the procedure. To do this, teachers should:

Demonstrate how to solve a mathematics problem using multiple strategies.
Present the strategies side-by-side.

Guide students through a process of comparing multiple strategies to solve a problem.

Use common labels to draw attention to similarities.

Prompt for specific comparisons tailored to your learning goals.

Be sure that students, not just the teacher, are comparing and explaining.

Include a summary of the main idea from the comparison, highlighting key points.

Reinforce the concept of being able to solve a problem using multiple strategies.

Encourage students to solve problems using a strategy of their choice.

Ask students to share their strategy with peers in a small-group or whole-group setting. By doing this, students have opportunities to see how other students solved the problem, which increases their exposure to multiple solution strategies.

Note: This does not mean that each student must solve every problem using multiple strategies, a common misinterpretation of the CCSSM requirements. Rather, students exposed to multiple strategies have a greater possibility of finding at least one problem-solving approach that they can understand and apply.

Watch the video below for an example of how a teacher can present and compare multiple strategies for solving a two-digit addition problem (time: 4:31).

Transcript: Comparing Multiple Solutions

Teacher: Today during math class, we are going to be adding two-digit numbers. And what we're going to do today is we're going to compare multiple strategies that you can use to solve problems, and in doing so you can compare the strategies for similarities and differences and really think about when you might use one strategy over the other.

So we're going to look at two strategies: breaking numbers apart, and we're also going to look at the vertical addition algorithm. And to start, I'm going to be over here with breaking numbers apart. Our problem is 34+28. So to break the numbers apart, you want to break the numbers apart into the tens and the ones. So 34 breaks apart into 30 and 4. And then the number 28 breaks apart into the number 20 and 8.

The next step is to think about how you can add the ones together and how you can add the tens together. So I'm going to draw lines to help me connect the numbers. So 4+8=12. And then I need to add the tens together. So I have 20, and I'm adding that to 30, and that gives me a sum of 50. So once we're here, the last step is easy. All you do is add the numbers together, and we know that 50+12=62. So 34+28=62 over here.

Next, what I'm going to show you is the vertical addition algorithm. So a different strategy, but we're still adding the same numbers, 34+28. And if you notice over here, I've stacked the numbers on top of each other so that they're in vertical columns. So to start, I'm going to start in the ones column. I'm going to add 4+8. 4+8=12. And when I'm solving with the algorithm, I need to think, "Only one digit can go here under my ones column, and 12 is a two-digit number." So I need to regroup, and regrouping means I'm thinking about how many groups of tens I have and how many ones I have.

So with the number 12, I know I have 2 ones, and I know I have one group of 10, so I'm going to write a 1 above my tens column. And now that we've regrouped and finished the ones column, I can then move on to add in the tens column. And here I have the numbers 3+2, but I can't forget about that one extra group of 10 that I regrouped. So 3+2=5, and if we have one more that equals 6. So here we solved 34+28=62, and if you notice 62 is the same as 62. So both of our strategies gave us the same answer.

Now, I want you to think what else is similar or the same about these two strategies besides just the same answer. Yes, Emma?

Emma: You added the ones first and then the tens.

Teacher: Oh, that's such great thinking, Emma! We did, for both of the strategies, I targeted thinking about my ones column or the numbers in the ones first, and then I moved on to my tens. And that's really important when you're thinking about addition is to always start in the ones. Now even though we started in the ones for both of them, I think you noticed that we regrouped for the algorithm, but we did not regroup when we were breaking numbers apart. Raise your hand if you can tell me why we had to regroup when using the algorithm.

So now that we've looked at both strategies, and we've had our discussion about similarities and differences, it's really important to think about in math that, while there are multiple ways to solve a problem, thinking about what is the best way to solve it in certain situations. So, again, if you're doing math in your head you might want to use a breaking apart numbers strategy, and if you have a pencil and a piece of paper you might want to use the algorithm if that's going to be more efficient for you.

So now that I've shown you one example, you are going to do this problem next, and when you're done you are going to choose...well, you're going to solve it using one of the strategies, and when you're done you're going to work with a partner and you're going to discuss the similarities and differences between how you solved this problem. I'll be walking around to answer questions or to provide help as needed.

Assessing Student Understanding

How does this practice align?

High-Leverage Practice (HLP)

HLP6: Use student assessment data, analyze instructional practices, and make necessary adjustments that improve student outcomes.

As we previously explained, assessing student understanding allows teachers to determine whether students have learned the mathematical procedures or concepts covered in class. Teachers can use different types of assessment data, including formative assessment and error analysis, to make instructional decisions (e.g., identifying what they need to revisit or reteach).

Formative Assessment

Formative assessment is the ongoing evaluation of student learning as a means of providing continual feedback about performance to both learners and instructors. By using formative assessment teachers can determine what students have mastered and what concepts they are struggling with. Teachers can use both informal and formal formative assessments. Informal assessments include exit tickets, quizzes, and class work samples. Formal formative assessments include curriculum-based measurement (CBM), sometimes referred to as general outcome measures (GOM), which is a type of progress monitoring.
exit ticket
A tool for assessing student understanding of a topic or skill introduced in class; sometimes referred to as an exit card. To assess the students' understanding of the day's lesson, teachers hand out blank index cards and ask their students to do such things as:

Answer a specific question about the lesson

Demonstrate a skill (adding two two-digit numbers)

List three things they learned

Ask a question about something they don't understand about the topic

Draw a picture of an item and label its parts

Explain a concept

Write down one thing they would like to know more about

The students write their names and responses on the cards and turn them in to the teacher. For older students, these cards are often turned in as students exit the class.
Example Lesson: fractions
The teacher writes on the board and reads aloud: "If Sue gets $4.00 this week for her allowance and she spends one fourth of it at the store, how much did she spend? Show how you solved the problem."

By reviewing these two exit tickets, the teacher realizes that Sara needs to use a graphic representation to solve the problem, whereas Nathan is able to use a mathematical representation to solve it.
progress monitoring
A type of formative assessment in which student learning is evaluated on a regular basis in order to provide useful feedback about performance to both learners and instructors. The resulting data can be graphed to observe change over time. Sometimes referred to as curriculum-based measurement (CBM) or general outcome measurement (GOM).

For more information on CBM for mathematics, view the following IRIS Module:

Error Analysis

Error analysis is the process by which instructors identify the types of errors made by students when working mathematical problems. It allows teachers to assess student understanding, or misunderstanding, and to identify and analyze the student's error patterns -errors a student repeatedly makes when solving a mathematical problem. The teacher can use the information from the error analysis to target instruction to help the student understand the correct procedure for solving the problem. If the reasons for the student's incorrect answers are not apparent, the teacher can ask the student to describe the procedure she used to solve the problem, as is illustrated in the box below.
Example: Error Analysis Student Solutions: Student Explanation:
For the first problem, I added 8+3 and got 11, so I wrote 11 down. Then I added 3+2 and got 5. I wrote the 5 after the 11. So I got 115. I did the same thing for the other problems.

Diane Bryant discusses the instructional implications for using formative feedback and error analysis (time: 2:11).

Diane Pedrotty Bryant, PhD
Project Director, Mathematics Institute for
Learning Disabilities and Difficulties
University of Texas at Austin

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View Transcript
Transcript: Diane Pedrotty Bryant, PhD

Formative assessment and error analysis are critical aspects of instruction and help teachers understand if students are benefiting from the math interventions that they're using. Formative ongoing assessment, really, is important so that teachers are aware of the skills that have been taught that students are still not understanding. If students are stuck on a particular mathematical concept or skill and don't learn it to mastery and don't understand how to generalize mathematical ideas, they will continue to struggle in mathematics as the curriculum becomes more advanced in later academic years. In terms of error analysis, it's important for teachers to understand where the breakdowns are occurring, whether it's a mathematical procedure, a set of steps, how solutions are arrived at. Error analysis can be really informative in terms of the mistakes that students are making or the misconceptions that students have. It's important for teachers to work with students individually, to have students articulate how they went about solving problems, because that is such an important part of error analysis, because then you could potentially tap the thinking process that students are using and really get at some of those misconceptions that students learn over the years. I think pairing formative assessment with error analysis can really help teachers figure out where the faulty thinking is and where the misconceptions might be, which can really help to inform instructional decision making about the next steps for math instruction.

To learn more about error analysis, visit the following IRIS Case Study Unit:

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IRIS | Page 8: Effective Classroom Practices

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A number of other classroom practices are supported by moderate levels of evidence, even if they have not yet met the requirements to be considered evidence-based. Implementing these kinds of effective practices in conjunction with an EBP is yet another way teachers can improve their students' mathematical understanding.