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The Mathematics Revolution: Helping Children Learn and Love Mathematics with Jo Boaler

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Helping leaders thrive in our VUCA world with science-neuroscience, behavior change & lean communications®. www.connectconsultinggroup.com
Have you discovered the "power of yet"? This three-letter word packs a much bigger punch than its diminutive size suggests. The use of "yet" is particularly potent in these three situations: encouraging learning, maintaining engagement and increasing confidence when working with powerful jerks.

Dr. Carol Dweck, who's best known for her work on growth and fixed mindsets, has discussed the "power of yet" in a TEDx Talk and at a 2014 NeuroLeadership Summit talk that I attended. At the time, I had no clue what an outsized impact this tiny word would have on my clients and me.

Dweck and her former student Dr. Heidi Grant were previewing their new work on organizational growth mindset. Their main point was that those with predominantly a growth mindset - individuals and organizations alike - demonstrate a love of learning that keeps them developing and building resilience that's essential for great accomplishments.

Yes, organizations have mindsets, too, although it's still more common to talk about mindsets within individuals. Those with a growth mindset believe intelligence and talent are merely a starting point and that future growth requires hard work and dedication. By contrast, those with a fixed mindset view their intelligence and talent as traits that come naturally to them. They believe they can create their success without effort. As a result, they spend their time proving their intelligence and talent instead of improving and developing themselves.

None of us - neither individuals nor organizations - have a pure growth or fixed mindset, according to Dr. Dweck. We're on a spectrum, and our mindset can vary depending on the situation.

Nevertheless, when you're familiar with the concept of mindsets, it's easy to pick up clues as to the predominant mindset of either an individual or an organization. And if you want to support maintaining or nurturing a growth mindset, you can strategically use the word "yet" to encourage continual learning and development.

For example, those with a fixed mindset talk about "being good" while those with a growth mindset set goals for "getting better." This means for those who are striving to get better, you'll talk with them about improving skills they're not good at yet. To say it another way, those with a growth mindset consider themselves "works in progress." They have the confidence to admit there are skills they're not good at yet. Plus, you can help them embrace challenges they haven't yet faced before.

Because learning and development are so highly valued in a growth mindset environment, the concept of weaknesses takes on new meaning. As the two social psychologists explained at the conference, "weaknesses are defined as strengths you haven't developed yet." This phrasing acknowledges the current state while providing assurance and encouragement for continued personal growth.

You also can use "yet" to keep individuals engaged in a task, especially one that you're more enthusiastic about than they are. For example with my coaching clients, I've learned to ask as we near the set deadline: "How are you doing with X? Are you ready to review it yet?" The insertion of "yet" shows that I'm expecting them to talk with me about the task; however, I recognize that they may need more time with it.

As I've experimented with this technique, I've discovered that it puts individuals at ease and encourages them to acknowledge their true feelings. For example, they may say the assignment is harder than they expected, they're having trouble completing it due to other competing priorities, they forgot they agreed to do it, etc. Whatever it is, it gives us an opportunity to discuss the assignment, review its purpose, and adjust the task or timeline if necessary while keeping them engaged in their professional learning and development.

The word "yet" is also helpful when you're working with a powerful person who's acting like a jerk or you're advising someone in this situation. Rather than react and become a jerk yourself, step back. Start focusing on balancing the dueling social needs each human faces: our need to belong versus our need to rely on our uniqueness.

Then, remind yourself (or your colleague) that you have these specific skills (which you want to name, including the emotional intelligence that will help here) that make you a good team member who can fit in. Yet, you (or your colleague) also have this distinctiveness (which you also want to articulate) that's an important part of who you are. The powerful jerk may not appreciate this uniqueness or even know what it is; however, the fact that you know gives you your superpowers and you can rise above the jerk.

This constructive conversation can raise your self-confidence and your sense of personal worth to help you successfully manage the situation and the jerk. (I've lost track of how many times I've effectively used this technique with myself and others.)

Now it's your turn to try the power of yet to make a positive impact on learning, engagement and self-confidence. And you may find other successful ways to apply the power of this three-letter word. Please share what you learn.

Got any ideas yet?

Forbes Coaches Council is an invitation-only community for leading business and career coaches. Do I qualify?

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Council Post: How To Use The 'Power Of Yet' To Encourage Learning And Growth

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Helping leaders thrive in our VUCA world with science-neuroscience, behavior change & lean communications®. www.connectconsultinggroup.com Have you discovered the "power of yet"? This three-letter word packs a much bigger punch than its diminutive size suggests. The use of "yet" is particularly potent in these three situations: encouraging learning, maintaining engagement and increasing confidence when working with powerful jerks.

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RESOURCE:  Marzano's 6 Step Vocabulary Instruction

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RESOURCE:  Marzano's 6 Step Vocabulary Instruction

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RESOURCE:  Using Jamboard to Create Sketch Notes! These can be collaborative 

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RESOURCE:  Using Jamboard to Create Sketch Notes! These can be collaborative 

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RESOURCE:  Integrated and Designated ELD: 2nd/3rd Grade

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RESOURCE:  Integrated and Designated ELD: 2nd/3rd Grade

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Integrated and Designated ELD: 2nd/3rd Grade

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RESOURCE:  Collaborative Culture, Routines- How can routines help students take greater responsibility for their classroom, themselves, and each other?

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RESOURCE:  Collaborative Culture, Routines- How can routines help students take greater responsibility for their classroom, themselves, and each other?

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RESOURCES:  Using Visual Representations with Students Struggling with Math

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RESOURCES:  Using Visual Representations with Students Struggling with Math

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Using Visual Representations with Students Struggling with Math

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Creating routines that help hold students accountable

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Creating routines that help hold students accountable

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Students who are doing well-but could be doing much better-benefit from feedback that encourages self-monitoring.

There are a number of students who are commonly found to be in the place of high proficiency in their learning but demonstrate low growth in terms of their progress. They meet deadlines, support others in their work, and take the lead on projects, but their busyness doesn't always yield the growth one would or should expect for these children.

Such students require instruction that is, in many ways, different from the approach when learners are working to reach the basic facts and skills of a subject. Interestingly, students who are typically in the situation of high achievement/low growth don't require a separate class or shift in grade level. Moreover, they don't need more of the same-more homework and reinforcing their understanding by helping others.

What these students need are subtle shifts in tasks, instruction, and feedback. One of the best ways to start making the impact we desire for students when they are cruising along the high-achieving/low-growth phase of learning is to differentiate our feedback.

While feedback for emerging learners is found to be best when it's corrective and prescriptive, these approaches are less than ideal for students as they gain proficiency. Often, the feedback that people who have gained proficiency but show low growth need is designed to focus on self-monitoring performance, making and reflecting upon self-correction, and receiving feedback on how to take proactive steps for future challenges. They need feedback by approximation, not precision.

3 Strategies to Support Coasting Students

Strategy 1 Nudge. Provide just enough feedback for students to self-monitor and take action in their learning. As students gain proficiency, they need less direct feedback related to their performance on a task and more feedback tied to their own ability to detect errors, monitor their performance, and find ways to take initiative on their own learning.

The following daily teacher actions help nudge students toward those dispositions.

One of many is incorrect: In lieu of sharing with students which question is wrong and what to do, share that one of the questions is wrong or could be improved. Give the student a few minutes to inspect their own work and determine next steps. Have students share with you (or others) which answer was incorrect, why the error occurred, and next steps they will take to improve.

Sticky notes protocol: Provide corrective feedback on a few sticky notes for a group of students on a task. Give the group those sticky notes, and ask them to figure out which comment goes to which task. Listen to their level of clarity and feedback to each other.

Dots, not comments: Place a dot near an error on a task, and give the work back to the student. Once the student receives the work, share that the dots provide a clue for the proximity of a place to improve. Debrief with the student after 5 or 10 minutes to determine if they detected the error and how they fixed the problem. Focus your feedback on their efficacy of detection skills with their own work. (Did they analyze the task? Did they review work samples?)

Strategy 2: Establish social norms. Ensure that collaborative agreements and processes are in place so that peer-to-peer feedback is accurate and helpful in students' deciding next steps. With feedback, we need to shift away from the gradual release of responsibility to center on the need to cultivate the mutual responsibility of learning. Instead of shifting from teacher-centric feedback ("I do") to student-centric feedback ("you do"), we need to focus on the learning-centered environment ("we do").

One way to start this work is by refining peer-to-peer feedback. Graham Nuthall's research has shown that students often receive and use feedback from their friends. Interestingly, he found that most peer feedback is inaccurate. To rectify this, we need to prepare students to give and receive accurate feedback.

We can support students by engaging in the following actions.

Feedback protocols: Establish a structured way to have a feedback conversation. Examples include a tuning protocol, critical friends, and a Charrette protocol. As a teacher, our role is to monitor the quality and accuracy of feedback.

Consultancy protocol: Small groups review a student's current work and offer ways to help the student improve their learning. Establishing the norm of having students share their own challenges and having others use inquiry to enhance the students' learning is a great way to improve the whole group's achievement.

Game film: Record the student engaging in a presentation or discussion, and ask students to reflect on their strengths, opportunities for growth, and next steps.

Strategy 3: Highlight nuance. Focus on range and/or subtle shifts in feedback to promote self-direction and prime creativity.

Divergent feedback: Instead of corrective feedback, provide a set of questions that promote creativity. Ask: What would you do differently? How would your response change if the context or conditions changed?

Contextually different work samples: Give students examples of exemplary work in different contexts or task types. For example, if students are focused on persuading others via a podcast, give them the opportunity to evaluate other forms of persuasion, including essays, debates, TED Talks, etc.

Open success criteria: If students have met the success criteria on a rubric, give them options to go deeper within the criteria. For example, imagine that students need to include a conclusion when writing an essay. Go to those students who have completed this objective and state that the work would be much better if they used a cliffhanger or metaphor to end the essay.

When the criteria are opened, students see that work can always be improved. This is also an opportunity to ask students what they can do in the future to be proactive in going beyond the requirements of a task.

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3 Ways to Encourage 'Coasting' Students to Reach Their Full Potential

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Students who are doing well-but could be doing much better-benefit from feedback that encourages self-monitoring. There are a number of students who are commonly found to be in the place of high proficiency in their learning but demonstrate low growth in terms of their progress.

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I had a student once who was a bit of a perfectionist. One day, I watched as she put pen to notebook paper for her in-class writing assignment. But before she was done, she'd turn to a fresh page and start over. And then she did this again.

When I asked her what was going on, she told me simply, "I messed up."

Instead of crossing out sentences or adding words in the margins, she was rewriting the whole paragraph and falling further and further behind her classmates in the process. She wasn't learning anything new by rewriting the sentences that were fine, either.

For years, we've seen the same thing happening in education across many subjects. At the end of a school year, students who haven't yet mastered some concepts from that year's curriculum are required to redo coursework the following year. They're placed in remedial classes or their new teacher re-teaches past units. Some even repeat an entire grade. We call it remediation.

And these students fall further and further behind their peers in the process.

As we return back to school post-pandemic and urgently need to address unfinished learning, many experts and organizations are advising educators to forgo the remediation route and opt for learning acceleration instead. But what do we actually mean when we talk about accelerating learning?

The Difference between Learning Acceleration and Remediation

What accelerating learning is not: Feeding students the same curriculum and pressing 2x playback speed on their brains. (How neat would that be?)

Neither is it simply extra learning time, perhaps with an after-school tutor, to catch up on knowledge gaps.

Learning acceleration is an approach that gives students laser-focused instruction on the specific skills and content that they need in order to learn the new grade-level material at hand. We accelerate learning by moving students forward on grade level and setting them up for success with just-in-time training on required foundational skills. (Racecar drivers only pull over when they need fuel or tires replaced. Imagine how silly it would be to have them restart the race instead!)

Schools that implement acceleration programs replace remedial classes with acceleration ones that take place before or after school or in the place of electives. Teachers tailor-fit their lessons to complement core instruction so that in their regular classes, students have the tools they need to fully grasp grade-level material.

Other accelerated learning approaches utilize technology to help teachers make the most of instruction time. For example, MATHia uses artificial intelligence to offer just-in-time support unique to each student, just like a human tutor would. It simultaneously saves teachers time and gives students personalized, grade-level learning opportunities.

Remediation is entrenched in the past: what students missed last year and what they need to redo. On the other hand, acceleration focuses on the present and the future: what students need right now to excel this year and beyond.

The concept of accelerating learning is nothing new. Suzy Pepper Rollins' 2014 book, Learning in the Fast Lane: 8 Ways to Put All Students on the Road to Academic Success, provides a useful table comparing acceleration and remediation.
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The Impact of Accelerated Learning on Equity

"If a student receives below-grade-level instruction all day, where will they end up at the end of the year? Below grade level, of course," according to the authors of Taking Action: A Handbook for RTI at Work™.

Remediation purports to support students who have fallen behind, but we know these students can never catch up if they keep working below grade level. Learning acceleration makes it truly possible to get and stay caught up.

But accelerating learning is not just the smart thing to do. It's also a matter of equity.

Opportunity gaps have long disproportionately affected students of color, those impacted by poverty, and other vulnerable populations such as ELL students and those receiving special education services. School closures necessitated by COVID-19 only exacerbated these issues.

These are the students who are most at risk if they are assigned remediation. Let's empower them with an accelerated learning approach.

The Messiness of Learning

When I think about my student who felt compelled to write a perfect paragraph in one go, unwilling to muddle the page with cross-outs or add-ins, I think about how important it is to embrace the messiness of learning.

Our students master concepts and skills at varied paces, and they'll struggle with and need support on different topics. Some will need the boost of accelerated learning to achieve their potential.

But the messiness of learning-when one class is fully engaged but the same exact lesson flops in the next, the delightful surprise of "aha!" moments, a new way to get an answer that a student discovers on their own-what a wonderful thing!

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Learning Acceleration, Not Remediation, for a Fantastic School Year

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I had a student once who was a bit of a perfectionist. One day, I watched as she put pen to notebook paper for her in-class writing assignment. But before she was done, she'd turn to a fresh page and start over. And then she did this again.

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Editor's Note: In the developed world, we often approach international partnerships with a deficit mindset, says Jennifer D. Klein, author of the new book, The Global Education Guidebook. She shares ways we, and our students, can move toward a more positive, asset-based mindset. Join Jennifer for #Globaledchat on Twitter this Thursday, June 8 at 8pm Eastern time. Just type #Globaledchat into the search box to join the conversation!

Global education experiences are too often developed on a foundation of deficit thinking, particularly in the most developed parts of the world. While many global educators are aware of the problem and working hard to reverse it, examples still abound: global partnerships that always end in fundraisers, the assumption that classrooms in the developing world must be behind the developed world academically, community service experiences that feed savior complexes instead of undoing them, and the very use of terms like "developed" vs. "developing," which implies that one is behind the other. Even the lack of a common language between classrooms is sometimes perceived as a problem more than a gift, even by well-intentioned educators, because it makes communication more complicated. Most of these errors originate in deeply ingrained and often unconscious biases that come from a history of colonization and exploration that objectified, exploited, and stole from cultures being " discovered."

Global education shouldn't exacerbate power differences, and many global educators work to reverse the paradigm by building partnerships founded in a "learning from and with " mentality, in which all partners bring equal, if different, value to the table. When teachers work to create partnerships using such asset-based thinking, they can create collaborative relationships that work to undo our colonial past and establish new paradigms for global learning. Following are a few key strategies that can help.

1. Build the relationship with your partner classroom on a foundation of mutual benefit, respect, and power.

All of the suggestions below play into this most fundamental of equity elements: global educators who want to reverse dominance paradigms strive to ensure that both planning and learning processes emphasize consistent and equitable collaboration. They strive to honor all voices, all perspectives, and all experiences, to always assume the best intentions, and to always "share the well," a phrase used at Mount Vernon Presbyterian School in Atlanta to suggest that educators should always share power, time, and resources.

2. During planning and learning experiences, start with questions rather than assertions and look for points of intersection.

All teachers have goals and demands, and the most equitable partnerships meet the curricular needs of all teachers involved. That doesn't necessarily mean that both teachers have to be teaching the same material to the same age groups; they might share content, they might share age groups, or they may have neither in common. What makes their partnerships successful is their ability to ask each other probing questions, to explore each other's ideas, and to see their curricula as a Venn diagram, looking for points of intersection that benefit both classrooms.

3. Get students involved in recognizing deficit mindsets when they emerge, and in strategizing ways to flip the paradigm.

We often underestimate the role that students might play in ensuring equity in our classrooms. Global educators striving for equity can engage their students in the effort to think critically about authentic challenges in real time through global partnerships, getting them involved in determining the potential warning signs of a deficit mindset. Once students have been involved in establishing the criterion for equity, they will become an incredible filter for everything that happens in the classroom. My students were the first to point it out when I made deficit-based assumptions in my own classroom-and we all learned a great deal from our collective effort to define and ensure equity in our interactions with each other and the world.

4. Lean into discomfort when inequities emerge or partnerships become controversial.

I believe it is an inherently human instinct to lean away from discomfort and controversy, though that tendency can vary along cultural lines to a degree. The bottom line? We can only resolve what we engage; we can only ensure a more just and peaceful world if we lean into controversy and difficult conversations, if we become comfortable with discomfort and intentionally engage in the hard work that equity and social justice require. I could only learn from my students by being open to their correcting me. Likewise, we can only flip the paradigm if we can let ourselves really hear and try to understand perspectives, grounded in real experiences, which differ from our own. Keeping it easy means we float on the surface, missing the opportunity to dig into the deeper complexities of the human experience.

5. Beware of orienting solutions toward solving problems for your partners; instead, build partnerships in which students solve for the world they know and learn from the world they don't.

As a writing teacher, I always told my students that they should write what they knew, emphasizing that their own story was the one that was most authentically theirs to tell, whereas to tell the stories of others requires extensive investigation and even then may be rife with misrepresentation or generalization. In global development, the "solve-for" mentality has birthed many unsustainable and inappropriate solutions, all because one group assumed they knew better than the other. Equitable global education strives to foster change makers and problem solvers who know that learning from and with leads us all to better solutions in our own communities.
Connect with Jennifer, Heather, and Center for Global Education on Twitter. Image created on Pablo.
In global partnerships founded in equitable action, students learn from and with each other, but the solutions they build are for challenges in their own communities, informed by what they learned from their partners. In the best examples of global partnerships, students all over the world work on the same challenge as it manifests in their own backyards, and they collaborate so that their actions bring together multiple perspectives and experiences, which lead to sustainable, multilateral development. This also creates an important opportunity for local partnerships that can help humanize global challenges, ensuring that students recognize the connections between global issues and local realities-and making abstract concepts like poverty more concrete for younger learners. Such partnerships also help students recognize change makers and leaders in their own communities, giving them opportunities to participate in authentic, meaningful action alongside individuals and organizations working on the same issues being addressed in the students' school or program.

As Canadian anthropologist and National Geographic writer Wade Davis put it, "Other cultures are not a failed attempt to be you." If we can help our students recognize this through global partnerships that really equip them to collaborate on equal footing with their global counterparts, maybe they can begin to see other cultures as the gift we so urgently need.

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Five Ways to Build an Asset-Based Mindset in Education Partnerships (Opinion)

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Editor's Note: In the developed world, we often approach international partnerships with a deficit mindset, says Jennifer D. Klein, author of the new book, The Global Education Guidebook . She shares ways we, and our students, can move toward a more positive, asset-based mindset.

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RESOURCE:  Tier Two Words integrating Marzano's Six Steps, MS/HS

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RESOURCE:  Tier Two Words integrating Marzano's Six Steps, MS/HS

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RESOURCE:  Remember that Number Talks is a year long (and life long) journey. Do not expect perfection your first time or even your hundredth time.  Do expect that students progress in their ability to think about math in a flexible way and communicate their thinking with others. 

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RESOURCE:  Remember that Number Talks is a year long (and life long) journey. Do not expect perfection your first time or even your hundredth time.  Do expect that students progress in their ability to think about math in a flexible way and communicate their thinking with others. 

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RESOURCE:   Oral Language Development Through Integrated & Designated ELD

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RESOURCE:   Oral Language Development Through Integrated & Designated ELD

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Oral Language Development Through Integrated & Designated ELD

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RESOURCE:  Creating routines is important for minimizing transition time in classrooms and creating efficiencies. For ELLs, routines are an essential scaffold because they create predictable structures, a way of navigating through a lesson when the language or concept may be unfamiliar or complex.

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RESOURCE:  Creating routines is important for minimizing transition time in classrooms and creating efficiencies. For ELLs, routines are an essential scaffold because they create predictable structures, a way of navigating through a lesson when the language or concept may be unfamiliar or complex.

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RESOURCE:  Math Teaching Models: Universal Design For Learning

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RESOURCE:  Math Teaching Models: Universal Design For Learning

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Math Teaching Models: Universal Design For Learning

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Using Math Journals to elevate thinking and monitor thinking

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Using Math Journals to elevate thinking and monitor thinking

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By reflecting on their math work in writing and drawings, students see what they've learned and find areas where they need to improve.

I love math journals for my students. Many parents and teachers are familiar with these tools in other subjects. You might be surprised, however, to see how effective they can be in a classroom where numbers rule.

You may recall learning logs in English and humanities or lab notebooks in science. Math journals (or logs-you can use the terms interchangeably) can look a little different from those used in other subjects. Pages can include written reflections and explanations. Ideas might also be expressed with illustrations, diagrams and charts, bulleted lists, and other visuals. If a classroom is completely digital, technology can be used to create the journals. For example, a student can work a problem or draw a diagram with tablet tools and then take a photo to include it in their journal.

The journal is a work in progress, without expectation of perfection on every page. Typically, journals show evidence of a student's learning processes and reflections. The log records not only important ideas related to content but also thoughts about the learning itself. It is a record of personal experience showing what a student tried, what worked and what didn't, what practices should be continued, and what improvements a student should focus on going forward. Math journals can be used in early grades through grade 12, once students have developed sufficient writing and drawing skills.

I think of it as being like a child unwrapping one of those holiday balls made of crepe-paper streamers. When you pull and unwind a streamer, little trinkets fall out. Similarly, with journaling, the process may reveal something about what students know, even before they recognize and own that knowledge base.

Why Use Them?

As teachers, we know that students vary in prior knowledge and in how they best engage in mathematics. A math journal offers opportunities for a student to try out, explore, and express ideas as solutions evolve. Nonverbal and nontechnical expression with drawings or diagrams provides additional access to understanding content.

Consequently, journals can promote initiative and confidence, which we know are gateways to learning.

Each entry helps organize and clarify thinking processes for deciphering mathematical situations. Students begin to make connections across mathematical ideas, see and use a variety of strategies, monitor and reflect on processes, and understand and use multiple representations. They can reference earlier work in the journal and revisit similar tasks to internalize and refine knowledge of problem-solving.

Sharing solutions with other students by talking about or sharing journal entries can be less intimidating than impromptu explanations in front of the whole class. Discussions, either in pairs or in small groups, can introduce new ideas and encourage students to evaluate strategies-their own and those of others. In addition, students have to try to communicate mathematical ideas effectively and use mathematical language correctly and precisely. This can help with accessibility by making strategies explicit.

Math journals sometimes reveal unplanned learning insights. An early elementary child who is working addition problems may reveal that they know the fundamentals of multiplication as they illustrate several groups of the same number of items. In my home state of Louisiana, we call that lagniappe, a Cajun word that means an extra or unexpected bonus or gift.

Types of Math Journal Prompts

Reflective prompts: These help students think critically about what they are learning, providing opportunities to synthesize learning and address unanswered questions. Prompts can lead a student to analyze successes and challenges, draw parallels or find differences, and make recommendations for modifications.

Problem-solving notebook prompts: A student might work through a mathematical situation and write about the strategy. In reflections, the student might analyze how that solution worked, consider other approaches, and make suggestions for tackling future problems.

Topic development prompts: If there is a specific content focus, such as understanding fractions or solving equations, a journal might be a record of how students developed their understanding. Frequent look-backs and corresponding reflections might provide those aha moments at certain junctures.

Organizing Suggestions

Teachers should help students organize their journals from the beginning. This helps organize thinking, ensure that critical components are included, and provide a way to locate work. Here are some suggestions you can give students.

Create a title page: Include the name of the subject and the name of the student. It might also include the date the journal began. A concluding date can be added when the journal is completed or the course ends. I always let students decorate this page if they wish.

Add a table of contents: Leave several pages at the beginning to catalogue the work as it is created. Over time, students will be able to quickly locate a particular problem or reflection.

Include title and dates for each topic: A title helps the mind identify and define the topic covered. The date creates a timeline of learning and thinking evolution.

Leave a margin on at least one side of the page: Many composition books have a red line about an inch from the right side of the page. Remind students not to draw or write beyond that line-it's a good place for comments or questions. A teacher can use this space to write a note to the student asking a question, recognizing a particular strategy, noting a calculation, etc. The student might then go back, make a correction, add an explanation, illustrate an idea, etc. Students might reply or go back and rework a problem with a different strategy. That back-and-forth serves as a record of thought development.

Write only on one side of the page: I like to use only the front of the right-hand page for my work. Many times, I have rethought the problem, seen a better way to approach it, or obtained additional information. Having margins and the left-hand page available leaves room for reflective notes, feedback, or edits to be in the same location. Students can see how their thinking evolves.

Keep all pages: Even if a student makes an error, that page should remain in the book. Students can go back and see how they learned from errors.

Sometimes, as educators and parents, we look for that shiny new thing to help propel children's learning. With journaling, we're using something tried and true and applying it in math class. Give it a try with your students-you'll soon see the benefits.

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How Math Journals Help Students Process Their Learning

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By reflecting on their math work in writing and drawings, students see what they've learned and find areas where they need to improve.

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Zone of Proximal Development and Scaffolding EXPLAINED!

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RESOURCE:  Understanding and asset-based approach  https://youtu.be/4s3ZaqvgFgQ

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RESOURCE:  Understanding and asset-based approach  https://youtu.be/4s3ZaqvgFgQ

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RESOURCE:  Integrating a total physical response strategy to help students physically interpret and interact with new or complex vocabulary

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RESOURCE:  Integrating a total physical response strategy to help students physically interpret and interact with new or complex vocabulary

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RESOURCE:   Using gestures to scaffold language.

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RESOURCE:   Using gestures to scaffold language.

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Integrated and Designated: Math, Grade 9 and 10

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Integrated and Designated: Math, Grade 9 and 10

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Grade 9–10 Math Integrated ELD: Math Explanation

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RESOURCE:  Math Instructional Tools to Integrate Language and Math Concepts (Early Math Collaborative)

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RESOURCE:  Math Instructional Tools to Integrate Language and Math Concepts (Early Math Collaborative)

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Instructional Tools to Integrate Language and Math Concepts (Early Math Collaborative)

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Integrated ELD Strategies, GLAD Math Strategies

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Integrated ELD Strategies, GLAD Math Strategies

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Inviting Participation With Thumbs-Up Responses

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A quick assessment strategy replaces raised hands, encouraging more students to contribute by accommodating their different processing needs.

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RESOURCE:  Learning Acceleration for Planning for the Next Three to Five Years, June 2021

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RESOURCE:  Learning Acceleration for Planning for the Next Three to Five Years, June 2021

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RESOURCE:  The benefits of a bilingual brain 

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RESOURCE:  The benefits of a bilingual brain 

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The benefits of a bilingual brain - Mia Nacamulli

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RESOURCE:  SIOP Model for Teaching English Learners - Lesson Delivery including language and content objectives

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RESOURCE:  SIOP Model for Teaching English Learners - Lesson Delivery including language and content objectives

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SIOP Model for Teaching English Learners - Lesson Delivery

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RESOURCE:  Constructive Conversations Skills

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RESOURCE:  Constructive Conversations Skills

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Grade 4 Math Integrated ELD: Justify Mathematical Argument

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Creating Math Routines that promote language use

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Creating Math Routines that promote language use

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RESOURCE:  Tiered Instruction in a Middle School Math Classroom 

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RESOURCE:  Tiered Instruction in a Middle School Math Classroom 

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Tiered Instruction in a Middle School Math Classroom - Lory Schieler

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Formative assessment is a proven technique for improving student learning, and the strategies shared here by Jay McTighe work both in the classroom and remotely.

Bob Daemmrich / Alamy Stock Photo

Using formative assessments designed to check for understanding and provide students with feedback and support is one of the most effective ways to improve and enhance student learning. Yet because of the need to cover large amounts of information and develop many skills, teachers may not take time checking to make sure students understand a concept or can effectively apply a skill, and, if they don't, figuring out ways to improve their learning.

Thankfully, there are practical, proven formative assessment techniques that teachers can use as a quick "pulse check" to gauge students' understanding. The eight techniques here can be applied across grades and subject areas in virtual, hybrid, and in-person learning environments.

The results of these checks should not be graded since the purpose of formative assessment techniques is to obtain feedback to use in improving teaching and learning, not to evaluate learning. It's important that students understand the purposes of these techniques, that mistakes are OK-and even expected-and that they will not be graded on their responses.

Thumbs up: I understand _____ and can explain it in my own words.

Wave hand: I'm not completely sure about _____ and doubt I could explain it.

Thumbs down: I don't yet understand _____ and cannot explain it.

Ask students to display a designated hand signal to indicate their degree of confidence in their understanding of a concept, principle, or process. For example:

Self-assessment and self-reporting can be unreliable, so use a random calling technique to periodically select students with their thumbs up (e.g., pull names out of a bowl) and ask them to explain.

In virtual learning environments, students can signal on camera or post designated emojis to signal their understanding levels.

Present students with a few binary-choice statements or questions containing an understanding or a common misconception and have them select a response (e.g., True or False, Agree or Disagree) and share it via a whiteboard, cell phone app, or hand signal (e.g., thumbs up or down). This efficient technique is particularly effective to use in checking students' prior knowledge or potential misconceptions before beginning new instruction.

True/False: When dropped from the same height, a bowling ball will land before a marble.

Agree/Disagree: Is this an example of alliteration?

Here are a couple of "choosing" formats with examples:

In virtual learning environments, students can use the chat box feature to record their choices, or respond to a poll.

3. Picture It

Draw a visual web of factors affecting plant growth.

Develop a concept map to illustrate how a bill becomes a law.

Create a story map or sequence diagram showing the major events in the story.

Visual representations, such as graphic organizers and concept maps, are widely used to enhance learning, and can also be used as formative assessments. Have students create a visual or symbolic representation (e.g., a graphic organizer, web, or concept map) of information and abstract concepts and then be prepared to explain their graphic. Picturing techniques are especially useful to see if students understand how various concepts or elements of a process are related.

In virtual learning environments, students can post their visuals on a Google slide or Pinterest board, or in Nearpod or Jamboard.

4. Troubleshoot It

1. Identify the flaw or error, and (even better)

2. Correct it.

One of the most efficient and effective quick checks for understanding involves troubleshooting. Present students with a common misconception or a frequent procedural error. See if they can:

Present a rough draft of writing and ask students to serve as an editor to mark compositional and grammatical errors.

Have students review work on a multistep word problem to identify computational mistakes and reasoning errors, and correct them.

In a photography class, show photos reflecting common compositional errors or flawed exposure or lighting, and have students recommend needed corrections using photo editing software.

Their responses will provide a quick check of the depth of their understanding.

5. Summarize It

Compose a tweet in 280 characters or less answering the question: What is the big idea that you have learned about _____?

Record a one-minute podcast or vodcast using an app on a smartphone or tablet to summarize the key concepts from one or more lessons.

Prepare a weekly letter to your teacher (or parents) summarizing something that you now understand as a result of your learning during the past week.

Here are a few examples of this technique:

Create a "real life" word problem to see if other students understand how to calculate surface area.

Locate a news article or blog post that presents an example of the tension between individual rights and the common good.

Find examples of symmetry somewhere in our school or on the playground.

Understanding is revealed when students can transfer their learning to new situations. Accordingly, one of the best checks for understanding is to see if students can apply material in a somewhat novel context. This technique includes asking students to find or create new and novel examples to illustrate a newly learned concept.

Develop a five-minute lesson to teach a younger student about how supply and demand can affect the price of things. Use one or more specific examples that we have not discussed.

Your friend has been absent and missed the last two lessons where you learned about community helpers. Draw a picture of at least five helpers in our community to help them understand the concept of a community helper.

This is a more involved, but valuable, formative assessment technique. Ask students to teach a new concept or skill to someone else-a new student, a student who has just returned from absence, or a younger child. You'll be able to gauge their degree of understanding as you review or observe their lesson. Here are two examples:

In virtual learning environments, students can record a lesson using a laptop, tablet, or cell phone camera.

8. Analogize It

A more sophisticated technique invites students to develop an analogy or metaphor to illustrate a newly learned concept or skill. The effectiveness of their explanatory analogy or metaphor can give you insight into their understanding. However, be cautious when interpreting student responses to this technique-a student may very well understand a concept but be unable to generate an appropriate analogy. Asking students to explain their analogies will give you further insight into their understanding.

A fraction is a part of a whole like a wheel is a part of a bicycle.

Formative assessment is like tasting a meal while you're cooking because it provides feedback that a cook can use to make adjustments to improve the meal.

Here's a prompt for students: A _____ is like a _____ because _____.

Students can also create visual analogies (combining techniques 3 and 8). In virtual learning environments, students can post their analogies and metaphors in a chat box or on a Google slide or Pinterest board.

Note: Several of these techniques (especially 1, 2, 5, and 8) can be naturally used in conjunction with another popular formative assessment technique-an exit card-given to students at the end of a class period or end of the school day.

While these techniques can provide valuable information about the effectiveness of teaching and the quality of students' learning, they're not ends in themselves. Instead, they should be seen as the first step in a " feedback cycle." The next step is to act on that feedback-reteaching something that many students failed to learn; correcting misconceptions that may be revealed; and/or providing scaffolded support to students who need it. Moreover, when students are given feedback, they must also be given opportunities to use it, such as revising their work, practicing the skill, or correcting errors. (Getting feedback without a chance to use it is like eating without digesting!) Then, the formative assessment cycle reboots as revisions can be assessed again, with progress noted and new learning goals set.

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Formative assessment is a proven technique for improving student learning, and the strategies shared here by Jay McTighe work both in the classroom and remotely.

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Accelerating Learning vs Remediation with audio

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You Roc! With Asset-Based Thinking: Kathy Cramer at TEDxYouth@Indianapolis

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RESOURCE:  Sheltered Instruction, SPECIALLY DESIGNED ACADEMIC INSTRUCTION IN ENGLISH, Effective sheltered content instruction provides English learners with access to the core curriculum while concurrently developing their proficiency in academic English.

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RESOURCE:  Sheltered Instruction, SPECIALLY DESIGNED ACADEMIC INSTRUCTION IN ENGLISH, Effective sheltered content instruction provides English learners with access to the core curriculum while concurrently developing their proficiency in academic English.

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Effective sheltered content instruction provides English learners with access to the core curriculum while concurrently developing their proficiency in academic English.

Student engagement strategies are essential tools to implement in your classroom to help students internalize language and get "miles on the tongue". Structured Oral Language Practice routines help students internalize academic language patterns and content-specific vocabulary. We need to provide students multiple opportunities to practice in engaging and supportive settings.

There are a myriad of strategies for structuring interaction. Here are instructions for incorporating nine different routines in your classroom.
Structured Oral Practice Routines
Here are some templates for some Structured Language Practice Routines that assist in engaging your students in purposeful and extended discourse while also requiring students to turn in a brief written product to ensure individual accountability.
A-B Conversations Clock Appointments Conducting an Interview Give One, Get One Send a Question Ticket Out the Door Weekly Agenda

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Effective sheltered content instruction provides English learners with access to the core curriculum while concurrently developing their proficiency in academic English. Student engagement strategies are essential tools to implement in your classroom to help students internalize language and get "miles on the tongue". Structured Oral Language Practice routines help students internalize academic language patterns and content-specific vocabulary.

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RESOURCE:  Teaching Students to Prove Their Mathematical Thinking through Questions, Charts, and Discourse

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RESOURCE:  Teaching Students to Prove Their Mathematical Thinking through Questions, Charts, and Discourse

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Teaching Students to Prove Their Mathematical Thinking through Questions, Charts, and Discourse

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Grade 8 Math Designated ELD: Math Problem Explanation

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3 Ways to Ask Questions That Engage the Whole Class

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These techniques guide all students to process course content and then actively demonstrate their learning.

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RESOURCE:   Facebook, Be Glad Math Strategy images

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RESOURCE:   Facebook, Be Glad Math Strategy images

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Rethinking what accountability looks like: Using Inquiry

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Rethinking what accountability looks like: Using Inquiry

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5 ways to raise the rigor with your newcomers

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Jo Boaler - Enquiry Based Learning for mathematics.

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RESOURCE:  Virtual Math Talks! Facilitate math discourse during remote learning.  Follow along at bit.ly/mathtalk5820

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RESOURCE:  Virtual Math Talks! Facilitate math discourse during remote learning.  Follow along at bit.ly/mathtalk5820

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Virtual Math Talks! Facilitate math discourse during remote learning.

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RESOURCE:  Integrating Math applications to support ELs

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RESOURCE:  Integrating Math applications to support ELs

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.
Watch the video to learn how sharing works.

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.

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Erikson Institute Early Math Collaborative

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Diane Briars describes math tasks as "the vehicles for mathematical learning." It is important, therefore, to offer students daily opportunities to engage in the kind of tasks that require reasoning and problem solving.

Diane Briars is the president of the National Council of Teachers of Mathematics (NCTM), an organization made up of over 80,000 members, whose goal is to ensure that students receive mathematics education of the highest quality. Before her tenure at NCTM, Briars researched for the Intensified Algebra Project, served as mathematics director for Pittsburgh Public Schools, and taught as a secondary mathematics teacher.
This presentation took place on August 11, 2015, as a part of the Summer P12 Mathematics Institute, an event put on by Chicago Public Schools, DePaul University, and Erikson Institute. The purpose of the event was to build and expand capacity to provide high quality mathematics instruction in alignment with the Common Core State Standards for Mathematics (CCSSM) in networks, schools, and classrooms.

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Diane Briars describes math tasks as "the vehicles for mathematical learning." It is important, therefore, to offer students daily opportunities to engage in the kind of tasks that require reasoning and problem solving.

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RESOURCE:  GLAD Strategies a compilation

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RESOURCE:  GLAD Strategies a compilation

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RESOURCE:  Number Talks, Number talks were developed for classroom teachers to engage students in “mental math” through grappling with interesting mathematics problems. 

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RESOURCE:  Number Talks, Number talks were developed for classroom teachers to engage students in “mental math” through grappling with interesting mathematics problems. 

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

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

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RESOURCE:  Math Manipulatives, applications to use for live or online learning

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RESOURCE:  Math Manipulatives, applications to use for live or online learning

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Unleash your creativity with the world's best virtual manipulatives!

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RESOURCE:  Productive Struggle and the Mathematical Practices

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RESOURCE:  Productive Struggle and the Mathematical Practices

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RESOURCE:  An Introduction To Visual Note-Taking

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RESOURCE:  An Introduction To Visual Note-Taking

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RESOURCE:  Introduction to Math Talks in Grade 4

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RESOURCE:  Introduction to Math Talks in Grade 4

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Asking middle and high school students to sketch something they're learning can give teachers a clear sense of what they do and don't understand.

Formative assessments, when done regularly at key points during the learning process, are windows into what students misunderstand or question about a concept or topic. Inviting students to process and express their understanding with drawings offers teachers incredible insights about what instructional moves to make next.

Asking Students to Draw as They Learn

Drawing while we learn allows us to capture a concept, image, event, or object and frees up cognitive space so that we can begin to think critically. If I quickly sketch a few characters from a novel with a small but key detail distinguishing each one, along with arrows depicting their relationships across generations, I can now think critically about those relationships, how characters influence each other, and how they develop throughout the story. Similarly, if I roughly sketch elements of a scientific process, I can begin to think about their functions and how they are integrated.

How should I represent the relationship between these parts?

How large/small should I draw these parts?

What shape should they be?

Where do I place each part?

Drawing as we learn requires abstract and metaphorical thinking about the content, which helps us retain and understand it in more sophisticated ways. A student who draws as they learn considers the following:

Because students make many decisions as they translate content into visuals, teachers can uncover their preconceptions, their misconceptions, the depth of their understanding, and what excites them about the content.

Building Confidence in Drawing as a Learning Tool

Drawing has significant learning benefits, and it's important to integrate it into a full academic experience. We can equip students with basic tools to become confident in drawing and allow them to use various methods to process and express their thinking. We can ignite interest in a topic by offering variety in modes of learning. Also, leveraging intuitive thinking appeals to students' diverse skills.

Students don't need developed artistic skill to experience the cognitive benefits of drawing as they learn. But, it's also important to ease any anxiety and dispel myths about visual work.

1. Offer drawing as one of a few options for students to process and demonstrate their learning.

2. Model the drawing option you offer. Replace fixed-mindset language like "I can't draw" with narration describing your process: "I want to show that the elements build on each other, so I'll draw boxes on top of each other."

Dan Santat's series #DrawLikeAKid demonstrates how to draw a lot of things with just a few shapes and the letters of the alphabet.

How well did I recall the material after having represented it in my drawings?

When would drawing as I learn be useful? When is it not useful?

3. Prompt students to reflect on their experiences in drawings so that they begin to understand it as a helpful learning tool.

Using Students' Drawings to Inform Instruction

The feedback we offer on student drawings and the instructional choices we make next are as critical as the drawing.

Students mimic the jargon or vocabulary they read or hear, which can obscure what they actually understand. As a formative assessment, verbal communication can be problematic because it can be difficult for the teacher to grasp what students truly understand. When students draw as they explain, it's almost impossible to obscure their knowledge. Obvious misunderstandings, along with subtler mistakes, make their way into students' drawings.

Line-color-symbol: Draw a line, choose a color, or select a symbol that represents your current understanding of the concept we are learning. Be prepared to explain your choice.

Draw an image that represents (gravity, how an argumentative essay is structured, the relationship between the protagonist and the antagonist, the limbic system).

Using only simple shapes (circles, squares, triangles), represent the relationships between (characters, principles, events, laws, mathematical concepts).

Create a comic/sequential art to represent a system, to distill key elements of a story or event, or to order essential steps of a process and represent cause-and-effect relationships.

Here are some ways to integrate drawing as formative assessment:

What facts, relationships, sequences, etc., does the student understand?

What key elements have been overlooked?

What factors are somewhat misrepresented?

How were certain factors, relationships, details, etc., represented?

How do students' drawings compare? What patterns do you see? What is distinctive about some of them?

What insights do their drawings offer about the content?

To effectively use students' drawings as formative assessment data, consider the following as you examine their visuals:

While reviewing a middle school student's visualization of the scientific process of making bread (below), I notice that he understands some key concepts: He knows there are two proteins that bind to form gluten, and his simple drawings demonstrate why each wouldn't, on its own, result in glutenous fluffy bread. But I wonder if he understands the roles that amino acids, water, mixing, and kneading play in this equation.
Courtesy of Shveta Miller
In my feedback, I'd acknowledge the elements he correctly illustrates. Then, to determine if he understands the complexity of the process, I'd ask him: Are these the only elements needed for the two proteins to result in gluten? What could be missing? Where and how could you draw it?

When we see thinking represented in visual form, we-students and teachers-get valuable insights into the concepts involved. We learn from what others see, the choices they make to represent an idea, and even the misconceptions they might have.

Thinking is messy -we generally don't follow a rigid and consistent procedure to think through problems, though a steady step-by-step process is often used to solve a math problem, form a hypothesis, or write an argument. Offering students structured opportunities to represent their thinking with drawing allows them to demonstrate their unique processes for thinking about complex concepts and problems and arriving at creative solutions.

When we see their thinking, we discover the subtleties of what they do and do not yet understand, so we can better prepare to solidify their understanding and build on it in our next lesson.

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Asking middle and high school students to sketch something they're learning can give teachers a clear sense of what they do and don't understand. Formative assessments, when done regularly at key points during the learning process, are windows into what students misunderstand or question about a concept or topic.

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GRIT by Angela Duckworth | Animated CORE Message

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RESOURCE:  Sketch Note-taking for Students | Analog and Digital: This is a remarkable to MODEL how to interpret new or complex content

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RESOURCE:  Sketch Note-taking for Students | Analog and Digital: This is a remarkable to MODEL how to interpret new or complex content

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Sketch Note-taking for Students | Analog and Digital | Selah Soul Art

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Welcome friends! Today's guest post was contributed by Shametria L. Routt, The Routty Math Teacher, and it is about implementing effective math talk in the classroom. This post includes tips and strategies to get you started, as well as a variety of resources to use on your journey. Happy reading!

Using mathematical discussions in the classroom is a powerful way to increase our students' critical thinking and communication skills. However, what does effective math talk in the classroom look and sound like? This article explores ways to get your students thinking and communicating mathematically from the very first days of school.

The National Council of Teachers of Mathematics (NCTM) defines math talk as "the ways of representing, thinking, talking, and agreeing and disagreeing that teachers and students use to engage in [mathematical] tasks" (NCTM, 1991). Effective communication about mathematics is essential to help students develop the thinking, self-questioning, and explanation skills needed to master required skills and concepts.

A successful mathematics program emphasizes communicating mathematically frequently in the classroom. In addition to NCTM's standards, most state standards include competencies related to communicating effectively through mathematical language, justifying solutions, and evaluating the mathematical thinking of others.

The picture below illustrates the interaction between the teacher and the students during effective math talk and demonstrates how knowledge is constructed through these interactions and exchanged with others. For a free download of my Looks Like/Sounds Like Poster, click here.

In order for math talk to be successful, students must understand how to collaborate fairly and hold a respectful exchange of ideas. Before implementing math talk in the classroom, brainstorm a list of classroom norms for how community members will participate and behave during the discussion. (See the picture below for an example of some of the norms you may want to include.)

As with any lesson, it is essential to create a plan for using math talk in the classroom. During open strategy sharing, students discuss how they solved a particular problem. The students listen and share their ideas. The teacher probes the students with questions related to how they determined the solution and why they chose a particular solution path. In addition, the teacher highlights a variety of strategies and emphasizes the similarities and differences among them.

Use the following steps to help implement math talk in the classroom:

Before the Discussion

Define your goal: What do you hope to accomplish? Goals for math talk include wanting students to: listen and compare methods used to solve a problem, look for the most efficient way to solve a problem, generate explanations for why a particular solution works, or examine why one solution is correct and why another is not.

Choose a Problem: This depends on your goal. If you want to highlight a variety of strategies that can be used to solve a problem, then choose a problem with multiple solution strategies. If you are examining why one solution is correct over another, then choose a problem where students frequently make missteps in their solution strategy. Use my Back to School Math Problem Solving Pack to get you started! You can find it in my TpT store.

Anticipate Student Responses: In order to create the best discussion opportunities, think about how students may respond to a particular problem, and create a plan to address any misconceptions that may develop. In addition, if there is an obscure solution that you think may be missed, then plant the seed with a student group or be prepared to introduce the strategy yourself. For example, you can say, "When I looked at this problem, I thought that I could solve the problem like this (show the strategy). How does this strategy compare to the others we used today?"

During the Discussion

Monitor Student Responses: During this time observe the interactions of the groups and make notes about their solution strategies. Also, note any observed areas of concern to address at a later time.

Select Students to Present: Based on your observations, determine which solution strategies to highlight that will best help you accomplish your goal. For example, if you want to emphasize a variety of strategies to solve a particular problem, select solutions that vary from one another.

Sequence Student Responses: Order the presentation of the solution strategies in a manner that will allow you to maximize the students' learning experience. For example, beginning with the most widely used strategy and then moving to the more abstract strategies may draw students' attention to new methods. Similarly, beginning with the more concrete strategies will give students the opportunity to move from a more concrete to an abstract understanding.

Connect Student Responses: The most important aspect of using math talk in the classroom is the connections between solutions that you and the students make. In the beginning students will need your support to make these connections. For example, if you sequence the presentations from less sophisticated to more sophisticated, you can have students discuss the similarities and differences between the solutions. You can also discuss efficiency. Which process is more efficient?
The "Open-Ended Questions" chart above has a great list of questions both you and the students can use during math talk to make connections and analyze solution strategies. Find the download for my Open-Ended Questions for Accountable Math Talk in my TpT store.

Use the "Math Talk Moves" poster above to teach students how to make connections and respond to others during math talk. The poster describes moves made by the student. The teacher should teach these moves and encourage their use during discussion. You may even want to have a "Move of the Day/Week" to highlight ways to connect student ideas. Additionally, you may want to include these in the students' math notebooks or tape a copy to their desks to refer to during the discussion. Please note: Several resources describe "Turn and Talk" as a math talk move. Because I chose to focus my poster on student moves, I did not include it; however, it is a useful move to use during math talk so that students have another participant to share ideas with during large class discussions. From that point, they can choose a student move to keep the conversation going or choose one move for a class-wide share-out. To get your own copy of my Math Talk Moves poster, please visit my TpT store.

Participation is key! Sometimes it's hard to get all students to participate because they're afraid to take risks. One of the ways you can get your more reluctant students to take part in the conversation is through the use of hand signals, such as teaching students to use the American Sign Language symbol for Y (see picture below) to indicate that they agree with someone else's thinking. Once reluctant students see that they're not alone, they may be willing to participate more.

After the Discussion

After reviewing the students' solutions and listening to their presentations, use the information that you gathered to determine the next steps. The following activities will help extend and deepen the students' understanding of the intended content and skills.

Look for Areas of Concern: Reviewing the solutions for common errors or misconceptions may provide material for a mini-lesson or content for an additional problem solving task at a later date.

Check for Reasonableness: It is essential that students develop the ability to verify their solutions and check them for reasonableness. After the discussion you can ask each group to develop a method to check their strategy for reasonableness or choose a specific solution and ask students to determine a method to check for reasonableness. Be sure to have students share and compare their strategies.

Justify the Solution: In addition to being able to make sense of a solution, students should be able to explain why a specific solution strategy leads to the correct answer. For this activity have students use pictures, numbers, and words to make sense of the solution and explain why it works.

Look for What Went Wrong: One of the most powerful activities for students is to examine their mistakes. Use one of the solution strategies that did not lead to the correct answer (if available) and explore where the solution went wrong. After determining the misstep, allow students an opportunity to complete the remainder of the solution strategy. If you don't have an incorrect solution strategy to use, create one yourself and say, "What if someone had done (show the strategy)?" Then, allow the students to discuss the error(s).

Implementing and planning regular math talk sessions will support the development of strong communication skills and deepen the students' ability to reason and think critically about the intended math content and skills.

Resources

Books: There are several resources that can provide additional structures and methods for implementing effective math talk in the classroom. Check out the following resources for more insight:

Pinterest Board: There are many great resources for all grade levels on Pinterest. Click here to check out my Math Talk Pinterest board.

* Minds in Bloom, Inc. is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon properties including, but not limited to, amazon.com, endless.com, myhabit.com, smallparts.com, or amazonwireless.com.

Shametria is a newlywed and has been a Texas educator for 13 years. She's currently a Teacher Mentor for first- and second-year teachers and is working toward a Doctoral degree in Mathematics Education. She is also a blogger (emerging, anyway) and teacher-author for all things math! Check out her Routty Math Teacher blog and Teachers Pay Teachers store for some additional teaching tidbits and freebies!

Credits

Books and Web Resources

Kazemi, E. & Hintz, A. (2014). Intentional talk: How to structure and lead productive mathematical discussions. Stenhouse Publishers: Portland, Maine.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Retrieved from http://www.nctm.org/standards/content.aspx?id=26628

Smith, M. S. & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Corwin: Reston, Virginia.

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Getting Started with Effective Math Talk in the Classroom

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Welcome friends! Today's guest post was contributed by Shametria L. Routt, The Routty Math Teacher, and it is about implementing effective math talk in the classroom. This post includes tips and strategies to get you started, as well as a variety of resources to use on your journey. Happy reading!

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RESOURCE:  Teaching through the learning pit, teaching for productive struggle

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RESOURCE:  Teaching through the learning pit, teaching for productive struggle

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Rich Learning with Dan Finkel | Part 3: Productive Struggle

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Using Problem Based Learning with Math Lessons

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RESOURCE:  Introduction to Growth Mindset and the Power of Practice

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RESOURCE:  Introduction to Growth Mindset and the Power of Practice

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Curiosity is underemphasized in the classroom, but research shows that it is one of the strongest markers of academic success.

When Orville Wright, of the Wright brothers fame, was told by a friend that he and his brother would always be an example of how far someone can go in life with no special advantages, he emphatically responded, "to say we had no special advantages ... the greatest thing in our favor was growing up in a family where there was always much encouragement to intellectual curiosity."

The power of curiosity to contribute not only to high achievement, but also to a fulfilling existence, cannot be emphasized enough. Curiosity can be defined as " the recognition, pursuit, and intense desire to explore, novel, challenging, and uncertain events." In recent years, curiosity has been linked to happiness, creativity, satisfying intimate relationships, increased personal growth after traumatic experiences, and increased meaning in life. In the school context, conceptualized as a " character strength," curiosity has also received heightened research attention. Having a " hungry mind " has been shown to be a core determinant of academic achievement, rivaling the prediction power of IQ.

Yet in actual schools, curiosity is drastically underappreciated. As Susan Engel has documented in her book, The Hungry Mind, amidst the country's standardized testing mania, schools are missing what really matters about learning: The desire to learn in the first place. As she notes, teachers rarely encourage curiosity in the classroom-even though we are all born with an abundance of curiosity, and this innate drive for exploration could be built upon in all students.

Curiously (pun intended), curiosity is also virtually absent from the field of gifted-and-talented education. A recent survey of required identification methods across all states found that only three considered motivation a part of giftedness. IQ, on the other hand, is required by 45 states, while 39 require standardized tests of achievement.

A recent feature story in Scientific American further punctuates this point. Misleadingly titled " How to Raise a Genius," the article summarized the results of a 45-year study of children who at age 12 scored in the top 1 percent on the SATs and were subsequently tracked and then supported. At least 95 percent of the participants experienced some type of educational acceleration as a result of their identification, and most participated in enrichment programs such as Johns Hopkins University's Center for Talented Youth (CTY). The CTY program-which counts Mark Zuckerberg and Lady Gaga as alumni-was initiated to "find the kids with the highest potential for excellence in what we now call STEM," and to figure out how to support them to increase the chances of them reaching this expected potential.

Much to the researchers' delight, the results confirmed their expectations. Their "profoundly gifted children" indeed grew up to be an impressive group. The majority completed doctoral degrees from some of the best universities in the world (which require high test scores as a gating mechanism), and many boasted impressive literary and scientific-technical achievements, including patents and published books.

These findings suggest that early advanced test scores are an indication of one's readiness for more enhanced resources, and this should certainly be supported. But what other conclusions can be drawn from these findings? That if you're a parent and you want to "raise a genius" but your child isn't precocious on academic tests at an early age, you're out of luck? Or worse, that these are the kids, and only these kids, who the country should bank on? One of the lead authors of the study, David Lubinsky, was quoted as saying: "When you look at the issues facing society now-whether it's health care, climate change, terrorism, energy-these are the kids who have the most potential to solve these problems. These are the kids we'd do well to bet on."

But is this really true? The researchers selected students based on a single criterion-advanced test scores-and supported these precocious youth throughout their schooling, failing to select for some other variable and thus disregarding all the other children.

The Fullerton Longitudinal Study (FLS), a 30-plus year study of the development of giftedness across various points in time conducted by Adele and Allen Gottfried of California State University, takes a different approach. Instead of relying on teacher nominations- recent research suggests that nominations miss at least 60 percent of gifted students-the researchers started by assessing a group of 1-year-olds, long before any of them had a chance to be officially labeled as gifted. The only criteria for inclusion in the study were that the infants had to be full term, of normal weight, and free of visual and neurological abnormalities.
"Motivation should not be considered simply a catalyst for the development of other forms of giftedness, but should be nurtured in its own right."
They initiated their study in 1979, and have been assessing the participants based on a wide range of variables (e.g., school performance, IQ, leadership, happiness) across multiple contexts (laboratory and home) since. During infancy and the preschool years, the participants were assessed every six months, and then they were assessed annually from the ages of 5 to 17.

One of their findings supports the work of Lubinsky and colleagues: Cognitive giftedness matters. Using the standard cutoff of 130 IQ resulted in 19 percent of the 107 children identified as "intellectually gifted" at the age of 8. While intellectually gifted children were not different than the comparison group with respect to their temperament, behavioral, social, or emotional functioning, they did differ in regards to their advanced sensory and motor functioning starting at age 1.5, their ability to understand the meaning of words starting at age 1, and their ability to both understand and communicate information thereafter. They were also more goal-oriented and displayed a greater attention span. By the time they began kindergarten, they performed at a higher level across diverse subject areas. Teachers rated intellectually gifted children as more competent in the classroom.

Parents of intellectually gifted children reported similar observations and were more likely than those of average children to say that their kids actively elicited stimulation by, for example, requesting intellectual extracurricular activities. (Intellectually gifted students tended to come from families that valued intellectual and cultural pursuits.) Children who become intellectually gifted, the Gottfrieds concluded, "are more environmentally engaged and may benefit more from their environment."

These results provide a window into the development of intellectual giftedness in relation to cognition. But they only demonstrate part of the picture. The researchers also measured what they described as academic intrinsic motivation and identified the top 19 percent of the 111 adolescent participants as "motivationally gifted," displaying extreme enjoyment of school and of learning of challenging, difficult, and novel tasks and an orientation toward mastery, curiosity, and persistence.

Interestingly, they found very little correspondence between intellectual giftedness and motivational giftedness. While the intellectually gifted students tended to show greater intellectual curiosity from infancy through adolescence, only eight children were both intellectually gifted and motivationally gifted. Also, the overwhelming majority of the differences on the academic intrinsic-motivation test could not be explained by differences in IQ scores, and academic intrinsic motivation predicted high-school GPA independently of IQ. The takeaway: Those with gifted curiosity are gifted in their own right.

Students with gifted curiosity outperformed their peers on a wide range of educational outcomes, including math and reading, SAT scores, and college attainment. According to ratings from teachers, the motivationally gifted students worked harder and learned more.

These findings have deep implications for gifted-and-talented education, as well as for education more generally. For one, they suggest that gifted curiosity is a distinct characteristic that contributes uniquely to academic success. As the Gottfrieds put it, "motivation should be considered as a criterion in and of itself to augment the selection of students into programs for the gifted and talented." For another, they're evidence of the benefits of programs that engage all students in the learning process-not as a means to developing other forms of giftedness (e.g., IQ, standardized test scores), but as an important characteristic all on its own. "Motivation should not be considered simply a catalyst for the development of other forms of giftedness, but should be nurtured in its own right," note the Gottfrieds.

Stimulating classroom activities are those that offer novelty, surprise, and complexity, allowing greater autonomy and student choice; they also encourage students to ask questions, question assumptions, and achieve mastery through revision rather than judgment-day-style testing.

But these experiences happen outside of the classroom as well. The Gottfrieds investigated the role parents play in fostering in their children an affinity for science by exposing them to new experiences that make them curious, for example, like taking them to museums. They found that such activities helped children develop an intrinsic motivation for science (e.g., "I enjoy learning new things in science; I like to find answers to questions in science") and teacher ratings of student academic performance. In turn, both of those factors predicted the number of advanced courses taken and interest in a science career, among other outcomes. This finding has strong implications for the development of STEM considering that curiosity is a fundamental predictor of the aspiration to become a scientist.

All in all, the Fullerton study is proof that giftedness is not something an individual is either born with or without- giftedness is clearly a developmental process. It's also proof that giftedness can be caused by various factors. As the Gottfrieds write in their book Gifted IQ: Early Developmental Aspects, "giftedness is not a chance event ... giftedness will blossom when children's cognitive ability, motivation and enriched environments coexist and meld together to foster its growth."

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Schools Are Missing What Matters About Learning

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Research shows that curiosity isn't just a means to an end: It's a type of ability in its own right.