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Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Vi Hart likes triangles! AND APPLES! All videos are available to download, or play at full quality without ads, at vimeo.com/vihart Feel free to share, show,...

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Enjoy learning math -- or learning HOW to teach math! I always emphasize conceptual understanding & number sense, because I feel that is how students learn math best. Check out also the accompanying Math Mammoth books at my site www.mathmammoth.com.

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education math k12 "math potential" algebra geometry worksheets "video lessons" "state standards"

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TenMarks' web-based math program is used by students across 25,000 schools and 7,000 districts. Each day, students in all 50 states use TenMarks to solve mil...

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Videos about numbers and mathematics. Videos by Brady Haran since 2011.

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math drezac mathademics tutorials education gcf edu

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Math tutorials by qualified, creative individuals.

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This channel has the BEST math tutoring videos you will find anywhere. I have here my complete series of "Basic Math" lessons (i.e. Pre-Algebra) plus many he...

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MDTP grade-level and course-level diagnostic assessments are designed to measure students' mathematical preparation in foundational topics of the course students are entering, and/or to inform students' preparation for the mathematical content in the course they are promoting to.

The diagnostic results should be used formatively to understand students' strengths and areas of unfinished learning, inform instruction and potential interventions, evaluate course readiness, measure program growth, and identify appropriate professional learning.

MDTP tests can be administered any time throughout the school year and are available in both paper form and on the MDTP Online Testing platform.

MDTP has a full selection of released aligned to the California Common Core State Standards for Mathematics (CA-CCSSM) curriculum expectations.

MDTP continues to field-test two tests: the Assessment for Preparedness for Grade 6 Mathematics and the Mathematical Analysis Readiness Test for students entering precalculus or math analysis coursework.

In Fall 2019, MDTP released the Algebra 1/Integrated 1 Readiness, Geometry Readiness, and Second Year Algebra Readiness.

Order Tests

All MDTP testing is free for mathematics educators in California K-12 schools.

Grade-Level Assessments of Preparedness

Assessment of Preparedness for 6th Grade Mathematics Field Test

(6M35D19)

Assessment of Preparedness for 7th Grade Mathematics

Renaming the Grade 7 Mathematics Readiness Test (7R40A15)

*Available in Spanish online and on paper

Assessment of Preparedness for 8th Grade Mathematics

Renaming the Grade 8 Mathematics Readiness Test (8R40A15)

*Available in Spanish online and on paper

Course-Level Readiness Tests

The Prealgebra Readiness (PR40A04 ) has been replaced by the MDTP Grade-Level Assessments for students enterring grades 6-8. Teachers may continue to use the PR with the understanding that it is not aligned to California Common Core curricular expectations for middle school mathematics courses.

The Algebra Readiness (AR45A10 ) has been replaced by the Algebra 1/Integrated Math 1 Readiness Test (AMR45A19) which is aligned to the CA-CCSSM. Teachers may continue to use the AR45A10 with the understanding that it is not aligned to California Common C ore curricular expectations for entry-level high school mathematics courses.

Refer to Recommendations for Testing and Suggested Use to inform decisions about which tests to use at each secondary level.

Download complete information on Effective and Appropriate Uses of the MDTP Assessment System and Recommendations for Testing and Suggested Use for assessment planning and professional training.

Refer to Understanding MDTP Readiness and 9th Grade Assessments to learn more about MDTP grade-level and course-level tests and MDTP 9th Grade Assessments.

Order Tests

All MDTP testing is free for mathematics educators in California K-12 schools.

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Assessments of Preparedness and Readiness

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MDTP grade-level and course-level diagnostic assessments are designed to measure students' mathematical preparation in foundational topics of the course students are entering, and/or to inform students' preparation for the mathematical content in the course they are promoting to.

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Using self and peer feedback as assessments for learning

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Using self and peer feedback as assessments for learning

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Mindset Kit - Using self and peer feedback as assessments for learning, Assessments For A Growth Mindset

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Learn how self and peer feedback can be used as formative assessments, and see how to use them in the classroom.

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Understand how college- and career-ready assessment is different

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Teachers share stories of how their students are responding to shifts in assessment. New college- and career-ready standards are asking more of our students; here, teachers reflect on how aligned assessments support the hard work happening in classrooms and how students are rising to meet these higher expectations.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress.

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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The California Department of Education (CD), California Mathematics Council (CMC), California Mathematics Project (CMP), and CCSESA's Mathematics Subcommittee of the Curriculum and Instruction Steering Committee (CISC) have joined together to provide needed information to California's teachers on California's Version of the Common Core Content Standards Mathematics (CaCCSS-M).

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Inside Mathematics is a professional resource for K-12 teachers, coaches, and administrators.

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Desmos offers best-in-class calculators, digital math activities, and curriculum to help every student love math and love learning math.

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Provide relevant texts that consider complexity, academic success, cultural competence, and critical consciousness.

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CALIFORNIA COMMON CORE STATE STANDARDS FOR MATHEMATICS in SPANISH

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CALIFORNIA COMMON CORE STATE STANDARDS FOR MATHEMATICS in SPANISH

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The Estándares Estatales Comunes de Matemáticas are a translation of the Common Core Mathematics Standards for the State of California and follow the exact outline of the Common Core State Standards:

Standards define what students should understand and be able to do.

Clusters summarize groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

We are currently updating the Estándares Estatales Comunes de Matemáticas for California for grades 9-12 to reflect and correlate with the changes approved by the California Department of Education in January 2013.

Grades 9-12 coming soon.

The Estándares Estatales Comunes de Matemáticas are a translation of the Common Core Mathematics Standards for the State of California and follow the exact outline of the Common Core State Standards:

Standards define what students should understand and be able to do.

Clusters summarize groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

We are currently updating the Estándares Estatales Comunes de Matemáticas for California for grades 9-12 to reflect and correlate with the changes approved by the California Department of Education in January 2013.

Grades 9-12 coming soon.

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The Estándares Estatales Comunes de Matemáticas are a translation of the Common Core Mathematics Standards for the State of California and follow the exact outline of the Common Core State Standards: Standards define what students should understand and be able to do. Clusters summarize groups of related standards.

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Mathematics Resources for Parents and Guardians

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Mathematics Resources for Parents and Guardians

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Mathematics Resources for Parents and Guardians - Common Core State Standards (CA Dept of Education)

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Information and resources for parents and guardians that explain mathematics instruction used today and provide suggestions for supporting their child’s academic success.

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California Common Core State Standards: Professional Learning Modules

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California Common Core State Standards: Professional Learning Modules

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Content Home :: CCSS Mathematics: K-12 Standards for Mathematical Practice :: Digital Chalkboard

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California Common Core State Standards: Professional Learning Modules

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California Common Core State Standards: Professional Learning Modules

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California Common Core State Standards: Professional Learning Modules

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Content Home :: CCSS Mathematics: K-8 Learning Progressions :: Digital Chalkboard

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California Common Core State Standards: Professional Learning Modules

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When teachers think differently about themselves as math learners, students benefit

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When teachers think differently about themselves as math learners, students benefit

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Like a lot of students, many elementary school teachers dread math class. But, according to a new study from researchers at Stanford Graduate School of Education, when teachers reexamine how they were taught math and their perceptions of their ability, student test scores and attitudes about math dramatically improve.

The research, published this month in the peer-reviewed journal Education Sciences, found that fifth-grade teachers who took an online class designed to give them a different approach to mathematics teaching and learning achieved significantly higher test results for their students compared with a control group of teachers in the same schools who did not take the class.

Student achievement increased when teachers changed their mindset from believing only some students could learn math well to believing that all students could succeed, said study co-author Professor Jo Boaler. The increase was particularly significant for girls, English language learners and students from economically disadvantaged backgrounds.

"When teachers adopt a 'growth mindset' about maths, the gains are notable especially in light of the fact that the effects of successful educational programs are often quite modest," said Boaler, who uses the plural form of math to reflect her British roots and belief in the subject's depth and diversity. "As teachers reevaluate their own potential as learners, they are more likely to embrace new forms of teaching. This helps their students build confidence, develop positive attitudes and, ultimately, achieve better test scores."

The new research is one of the few studies to look at the impact that a growth mindset, taught through an online class, can have on teachers' perceptions of their own skills and those of their students.

It also provides evidence to the benefit of mindset interventions that incorporate changes to teaching and approach, not just changes in what is said in a classroom.

"Mindset interventions will never achieve their full impact if they remain only as words, and put the onus on students to change, while teachers continue to transmit fixed ideas through their teaching," said Boaler, the executive director of Youcubed, a GSE center that provides math content, courses and materials for educators and students.

In a related analysis published this spring, Boaler found significant improvements in test scores, engagement and attitudes among middle school math students who took a student online class with similar messages, and who switched from a "fixed" to a "growth" mindset.

The power of a 'mathematical mindset'

To hear Boaler tell it, elementary school math instruction in the United States and throughout the western world is "stuck in the Victorian age." Procedures and rote memorization still dominate, despite ample evidence that the exploration of ideas, concepts and creativity works. Studies show that math teachers are more likely than other subject-specific instructors to think their students' abilities are static.

"The idea that only some people can be successful in maths is at the root of widespread anxiety about the subject, in the United States and elsewhere," said Boaler. According to the most recent comparisons available, the United States ranked 40th out of 72 countries in math achievement in 2016.

Boaler and her collaborators - Robin Anderson, a GSE doctoral student and research associate at Youcubed, and Jack Dieckmann, director of research at Youcubed - wanted to know what would happen if math educators looked inward and challenged their own ideas about who can learn math and how. As part of their study, which was funded through the Technology for Equity in Learning Opportunities ( TELOS) initiative at GSE, teachers would also revisit the messaging they internalized as young math students.

The researchers recruited 40 fifth-grade teachers from eight school districts in Central California to take an online course, "How to Learn Math," through Stanford's OpenEdX platform. The participants also engaged in a series of in-person meetings to share their experiences in the course and strategies for putting its ideas into practice. In both contexts, teachers learned about new brain science and effective teaching methods as part of what the researchers are calling a "Mathematical Mindset Approach."

To measure results, the scholars relied on a combination of interviews with the teachers, written observations of their classroom work, surveys of students and the teachers, and results from a statewide math test.

Statistically significant improvements

In follow-up surveys and interviews, teachers described how the course and in-person meetings fundamentally changed their approach to math lessons. They shared the neuroscience findings about brain plasticity with their students. They encouraged them to share ideas and alternative ways of solving math problems. They countered notions that best in math means fast in math. They taught students to embrace their mistakes.

Students, for their part, reported feeling more engaged and positive about math - and their test scores reflected that newfound optimism. Among all students whose teachers participated in the study, math scores on a key California assessment test rose nearly 8 points - the equivalent of almost 3 1/2 months of additional lessons - compared with a control group. For girls, English learners and economically disadvantaged students, the payoff was especially large: Girls' performance on the test translated to six months' worth of additional instruction, English learners gained nine months and economically disadvantaged students advanced nearly five months.

Boaler attributes the three groups' rapid progress to the fact that they typically underperform in math and harbor views that they cannot achieve, so changing those ideas had a larger impact.

Boaler says the teacher study - and the earlier one focused on students and their attitudes about math learning - offers hope for education reformers at home and abroad. "Change in math performance can happen," said Boaler. "But it's critical that damaging mindsets shift, among students and their teachers."

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When teachers think differently about themselves as math learners, students benefit

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Like a lot of students, many elementary school teachers dread math class. But, according to a new study from researchers at Stanford Graduate School of Education, when teachers reexamine how they were taught math and their perceptions of their ability, student test scores and attitudes about math dramatically improve.

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Social & Emotional Learning (SEL)

Math doesn't have to be serious. One teacher believes play is the key to math comprehension for all ages.

The concept of play is often limited to younger students and less academic endeavors, but play can be a useful strategy in an unlikely discipline: math.

Mathematics is known as cold, logical, and rigorous, but the subject doesn't get enough credit for its true mischievous spirit, which is well-hidden from the world. The K-12 mathematics curricula can involve time and space to play with concepts and ideas, even in the rush of required topics.

In my research for a 2017 talk I gave at TEDxKitchenerEd called " Math Is Play," I found few materials on the joy of math in upper grades. Much of the literature on play as a learning approach is based on the early years, particularly kindergarten, where it is an accepted pedagogical mode.

Young children at play often reach a state that psychologist Mihaly Csikszentmihalyi calls "flow," an elusive state of mind where time seems to disappear as they focus deeply on what they're doing. Getting to this ideal state in the classroom requires more than the freedom to play-teachers must also react to students' ideas and guide them through concepts like counting and numbers. This type of guided play requires decision-making about how and when to give direct instruction. Creating freedom while at the same time offering direction allows for productive play that opens students' minds to better understand difficult mathematical concepts.

The amount of play in "serious" academic topics like mathematics is inversely proportional, it seems, to the age of students, but this does not have to be the case. A playful pedagogy of mathematics can be codified and made real, rigorous, and authentic.

Embrace and Incorporate Play

In my book Teaching Mathematics Through Problem-Solving in K-12 Classrooms, I wrote about the need to accept that humans are born to play. Play is irrepressibly human, and we can play to learn. Playing and thinking are not at odds with each other. Though play is usually associated with turning off thinking and giving oneself to a pleasurable activity, working on interesting problems can be a trigger for flow. There is a sweet spot, often about 30 minutes into working on an interesting problem, where ideas start to become solutions.

Create a culture where mathematical ideas are not just formulas on a page but instead are concepts to be discussed and reasoned through. Play moves math instruction beyond rote memorization to a more expansive understanding of mathematics. Encourage students to talk, think, reason, and wonder as they move through problems. Creating a sense of curiosity, even for simple concepts, engages students in a playful way.

Simple strategies like turn-and-talk can create opportunities for collaborative, playful learning. Using prompts as part of the daily classroom routine can make mathematical concepts fun. Sites like Visual Patterns, Fraction Talks, or Estimation180 offer easy, quick ways to make mathematical concepts entertaining.

Lean Into the Unknown

Math is full of surprises that can be interesting and fun. There is no single path or strategy to the solution in many problems. Be receptive to surprises in how your students think about and solve problems. An openness to the unexpected can foster a culture of playful curiosity in the classroom. A playful mathematics learner is hopeful and optimistic-elements of a mindset that helps students improve their understanding of complex concepts.

Embrace the mess of the problem-solving process. Thinking is messy. We don't always get things right the first time. Course corrections, revisions, and even total transformations of the work are necessary.

Observe your students as they work. Where are the roadblocks? How are they adapting to specific challenges? Listen to your own self-talk while you work, and use your challenges to think through ways your students might be challenged. Solutions are important, but so is the process along the way. Listening and talking to your students while they work allows you to give good feedback and receive assessment data.

Play creates open spaces for thinking where teachers can guide students to engage with big and interesting ideas of mathematics.

Use Concrete Methods

Physical or digital manipulatives like snap cubes, pattern blocks, and relational rods are all tools that can help students bring mathematics into being-a process called representation. Teachers can use decks of cards, dice, or counting objects to help students practice their basic skills.

For example, younger students can practice multiplication facts up to 6 times 6 by rolling two dice and multiplying the results. Older students can use decks of cards to practice integer operations, where red suits are negative and black suits are positive. For young students learning basic number skills, designate one day a week for purposeful practice using games.

Visual presentation of mathematical ideas can be playful. For example, give students a can lid or other circular object, some string, and a measuring tape to try and find the relationship between circumference and diameter.

Using physical or digital elements creates space for students to play with more abstract concepts and ideas. The freedom of a playful environment creates opportunities for deeper engagement. As I said in my talk, "Let them play. Let them talk and think and conjecture and wonder. Let them play."

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Math doesn't have to be serious. One teacher believes play is the key to math comprehension for all ages.

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4 Tips for Engaging Reluctant Math Learners

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4 Tips for Engaging Reluctant Math Learners

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Showing students real-world applications of math helps boost their willingness to struggle with difficult concepts.

Brendan Fitterer / © St. Petersburg Times / ZUMA Press Inc / Alamy Stock Photo

Think about these three quotes:

"I'm not a reading person."

"Reading? I haven't done any reading since ninth grade!"

"I don't do reading."

It's hard to imagine anyone talking about reading like this in a professional setting. So why is it socially acceptable to say, "I don't do math"? It's true that students-and adults-have very real fears about their mathematical abilities, a mindset that often becomes a self-fulfilling prophecy leading people to handle even simple math problems poorly. I saw this firsthand in law school.

When I taught middle school math while going to law school at night, I took a class on wills, trusts, and estates. One night we discussed a scenario in which a father had three daughters. One of the daughters had three sons. When the father died, each one of his three daughters received one-third of the estate. Simple, right? But here's what seemed complicated to my classmates: The daughter with three sons died next. What fraction of the estate would each son inherit?

Students learn fraction division in fifth grade. Law students have undergraduate degrees, and many have master's degrees. But in this room of highly educated adults, there was serious confusion when the professor explained that each son would receive one-ninth of the estate. Some of my peers wondered: "The sons should have less their mother-how does that work when 9 is higher than 3?" What bothered me more than the error was the complete lack of embarrassment of students who laughed off their mistake by saying, "That's why I'm in law school, not business school."

It will not be easy to fix math education in the United States, but educators who have "I don't do math" students in their classrooms can try these four strategies to get them excited about math.

1. Exploring Intersections of Math and Social Justice

Math and social justice add up: Although some have ridiculed the idea of weaving social and political issues into math content, real-world applications that tap into students' strong sense of fairness and justice are powerful motivators for diving into rigorous math concepts.

"I don't do math" students who love being tucked away with a book at their neighborhood libraries may perk up when they understand the numerous mathematical implications of completing the census, for example. And a circle is the set of all points equidistant from a specific point, but when that specific point is an elementary school in a low-income community and students need to figure out how far they would need to walk to access a grocery store for healthy food options, the equity issue of food deserts may be compelling enough that they grapple with the geometry.

2. Leveraging Healthy Competition as Motivation

Competition only works if competitors are motivated to compete, and if your math competitions are based on who finishes the fastest or who gets the most right answers, you have created a perfect setup for disengaged students to give up before they even start.

For this reason, leveraging competition as a motivation strategy works best when students compete against themselves. How many consecutive problems can I answer correctly? Can I break my own record once a week?

Competition can also be motivating for disengaged students when they're part of a team effort. The stakes are somewhat lower when they aren't on their own, for one thing. When their successes, however few or small, count toward a class total in a competition against different classes, these students may feel more motivated because they're helping their team succeed.

3. Emphasizing "Math and..."

"I don't do math" students may hate fractions. But if they love playing music, that's a perfect opportunity for "math and": Fractions connect seamlessly with rhythmic patterns, time signatures, and the length of notes in musical pieces. Budding hip-hop artists might find graphs of trigonometric functions far more compelling when "math and" leads student rappers to understand beats per minute as the frequency and class DJs to recognize the period as a value that shifts based on slowing down or speeding up a beat loop.

Math is everywhere. Take advantage of interdisciplinary applications because you may be just one "math and" connection away from giving an "I don't do math" student a compelling reason to fall in love with math.

4. Explaining That "I Don't Do Math" People Are Crucial in STEM Fields

As a computer science graduate-turned-math teacher, I had many conversations early in my career that went like this:

Student: I don't like math.

Me: But math is so important for careers in science, technology, engineering, and math.

Student: Didn't I just tell you I don't like math? Why would I want to spend my life doing something I don't like doing?

Fortunately, there's a very good response to that question: The future workforce requires flexible, adaptable thinkers, and learning how to learn is crucial. Lessons learned from math struggles give students a huge leg up because they know how to struggle, are willing to struggle, and know how to leverage 21st-century collaboration skills to work with others to push through the struggle.

Thomas Edison was an "I don't do math" person who exemplified the value of learning how to learn. He considered himself deficient in math, lacking an understanding of scientific formulas like Ohm's law even as he invented the incandescent lamp. How did this owner of over 1,000 patents become such a prolific inventor? Edison was great at assembling teams and pushing them to think creatively. He celebrated mistakes and surprising results, and his story is proof that "I don't do math" people can play a crucial role in STEM fields.

It's OK for educators to dream of a pi-in-the-sky world where everyone cares deeply about math. But we have to be rational and understand that for reasons out of our control, "I don't do math" students will almost always be in our classrooms. These four powerful but practical tools can help educators negate students' negative relationship with math and turn it into a positive.

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4 Tips for Engaging Reluctant Math Learners

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Showing students real-world applications of math helps boost their willingness to struggle with difficult concepts. Brendan Fitterer / © St. Petersburg Times / ZUMA Press Inc / Alamy Stock Photo Think about these three quotes: "I'm not a reading person." "Reading? I haven't done any reading since ninth grade!"

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Game-Based Learning in Middle School Math

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Game-Based Learning in Middle School Math

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Working with teachers, researchers created free online math games that encourage students to persist in grappling with difficult concepts.

In the Math Snacks game Creature Caverns, teachers and their classrooms can build creatures with different attributes, such as eyes and antennae, and then see the ratio of those mapped in a ratio table and a graph.

Teachers all over the country are coming up with different solutions to the problem of motivating students to persist with math learning. One potential solution that excites a lot of teachers-and their students-is game-based learning, and I recently had a chance to get an inside look at how games can be used in math education by talking to the creators of a free online platform called Math Snacks.

Funded by the National Science Foundation, Math Snacks is a set of games and animations for middle school students that has brought math educators and learning theory researchers together with multimedia developers in a fruitful collaboration.

I sat down with Barbara Chamberlin, director of the Learning Games Lab at New Mexico State University, and Ted Stanford, associate professor in the Department of Mathematical Sciences at New Mexico State, to discuss Math Snacks and effective strategies for integrating multimedia in the classroom. Our conversation has been edited for length and clarity.

AMANDA ARMSTRONG: Let's jump right in: How did Math Snacks come into being?

BARBARA CHAMBERLIN: It happened many years ago when we were talking about some of the gaps we kept seeing with middle school math learners. In the Learning Games Lab, with our background in game development and animations, we were starting to say we could narrow in on those gaps and counter some of those in exciting and interesting ways.

We asked, "What are the things kids often struggle with?" As we drilled down on that, we found that these concepts usually were concepts they didn't fully understand in grades 2 and 3 but were emerging now that those kids were in middle school and were dealing with fractions and decimals.

TED STANFORD: There are several big ideas that sort of hit all at once in middle school, like ratio, rate, and scale factor, in particular. The big issue is, there are these key ideas and you can put an image to them. We used to talk a lot about concept image, your idea of what a rate is. It isn't just a calculation, it's an idea, it fits into a story, it has a picture associated with it, and maybe a need-why do you need to do this?

ARMSTRONG: You wanted the games to relate to a real-world application?

CHAMBERLIN: Yes. With games like Agrinautica and Curse Reverse, for example, we hope students become very comfortable with variables. Our goal was not to get students to learn how to solve expressions-our goal was to help them think through solving problems in logical ways. Our goal with these games was to help students look at algebra as a really logical way to address those problems. We're not teaching students algebra-we're teaching the concepts that help them deal with problem-solving, and notation, and written algebra.
NMSU Learning Games
STANFORD: I hope what Curse Reverse does is, when you're done with that game, you'll feel like using algebraic notation, using variables, and it makes sense in this context. Then when you get in my algebra class, you can take a deep breath and say, "OK, I know that this tool is a logical tool that makes sense and is useful for some things." You might feel more comfortable that this is a sensible way to approach the world.

ARMSTRONG: So the games are engaging for students?

STANFORD: A game can draw somebody in-the same ideas may be laid out on a piece of paper or in a textbook, but they don't draw you in in the same way. You get instant feedback from a game. The game responds very quickly to what you do, and it's private. You don't get judged by it. There is this idea that you can try things. You can fail, and you instantly find out what happens. Then you try again, and it's not discouraging. It can challenge kids.

ARMSTRONG: What was the role of teachers in your design process? How were they involved in the creation of Math Snacks?

CHAMBERLIN: A lot of people on our team have been classroom teachers, which was critically important and very helpful. Even those who haven't been classroom teachers do a lot of classroom instruction in this age group.

We would take prototypes to teachers and have them play with the prototypes. We would watch how they would deal with it and talk with it. We brought teachers and kids in over the summer and asked teachers to come up with a workshop around this kind of game prototype and teach it to the kids. Teachers would work together to come up with something and teach the session. The kids would go home, and the teachers would talk to each other about what they observed, what should be changed, what worked, and what didn't. The teachers would reorient their lesson plans and then the next day would bring in new kids, and then teach the lesson again with their changes.

We got to watch the teachers iterate on lesson plans. We got to see them teach with the game, change their lesson plan and activity again, do it again, change it a little bit, do it again.

STANFORD: The social interaction, the social learning, that was a big surprise because one point of view about game development is you want a kid to be able to go off in their room and play this game for two hours and learn something. And that's good. That happens, and that's not a bad thing to shoot for.

But there is so much more that a game can be. Two kids playing together is so cool-there are all these things that happen when kids play a game together that don't happen when they play alone. The technology isn't just a thing that one person does with a computer-it drives all this other learning that's based on interactions between people.

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Game-Based Learning in Middle School Math

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Working with teachers, researchers created free online math games that encourage students to persist in grappling with difficult concepts.

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Bringing a Culturally Responsive Lens to Math Class

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Bringing a Culturally Responsive Lens to Math Class

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Culturally Responsive Teaching

A middle school math project gives students a chance to apply the skills they learn in class to an issue they care about.

Richard Levine / Alamy Stock Photo

Contrary to popular belief, math is more than just numbers, algebraic formulas, and ancient algorithms. It can serve as a vehicle to help our students make sense of the world in which we live, but current events and real-world issues have generally been integrated only into subjects such as English, science, and social studies-math has been considered its own little island.

After years of searching for lesson plans and resources that would help me bridge the gap between math and real-world issues, I decided to create my own unit project to do that. I polled my seventh-grade students to find the right topic and created a three-week project that focused on the intersection between law enforcement and communities of color in Boston. Other topics of interest to them included immigration, poverty, homelessness, pollution, and creating more youth jobs in our economy-there are many possibilities for this kind of work.

A 4-Step Process for Connecting Math to Students' Lives

1. Selecting the topic or issue: I started with the idea that the topic would have to be relevant to the racial and cultural composition of my students, and to the communities in which they lived. The best way to select the topic was to ask them what would really matter to them. My students, who were predominantly African American and Latinx, chose the topic of police brutality because it had personally impacted them.

If you try this, choosing the right topic will guarantee a high level of student engagement throughout the project. You can create an interest survey or questionnaire to give your students so they can share in great detail the real-world topics they would like to explore-this might work better than initially asking them to discuss ideas as a whole class, as it takes some pressure off.

Once you've determined the topic, write it up as an open-ended focus question that your students will work toward answering throughout the project. For the topic of police brutality, I worked with my students to come up with the focus question: "Can more diverse police forces prevent instances of racial profiling?"

2. Gathering background information: The next big question for a math teacher is, "What specific data points are needed to effectively answer the focus question?" Since students are going to explore the topic through a mathematical lens, you'll need to consider what quantitative data points they can study in order to draw their own conclusions about the topic.

These data points-statistics, graphical representations, geometric diagrams, or functional relationships, for example-must not only be available for contextualization within the scope of the topic but also accessible to your students so that they can apply the appropriate math skills. My project highlighted national stop and frisk statistics, population data, and demographic data on police officers as data points that would help answer the focus question.

In addition to the quantitative data your students will use in their math work, they need qualitative data-news reports, books, etc.-to more deeply understand the issue they're exploring. In my opinion, gathering this information is the most difficult step of the process because you have to compartmentalize your personal biases, which can influence how your students think about the topic. You need to present the multiple perspectives that people have about the issue so that your students have the information necessary to build background knowledge and develop their own thinking around the issue.

Primary sources are the most reliable information sources to use because they're original and were written or created during the time of the issue. Examples of primary sources include manuscripts, books, newspaper articles, historical documents, videos, photographs, and interviews. In other words-and they'll probably ask-students should not use Wikipedia.

3. Identifying math skills and connecting them to standards: Once you've identified the appropriate data points, you'll want to determine the specific math skills your students will use to analyze the data points and construct mathematical arguments.

The math skills should be developmentally appropriate for the grade level you teach: Recognizing that our racial profiling unit was heavily focused on statistical concepts, I determined that students could analyze the data points using such skills as conversions between rational numbers (decimals, fractions, and percentages), measures of central tendency (mean, median, and mode), and two-way frequency tables.

Next, you'll need to align the math skills you've identified to the appropriate academic standards for your state, whether that's the Common Core State Standards or the academic standards for your state. Identify the standards that best align with the math skills that you expect your students to master for this lesson.

4. Determining the final work product: Finally, in thinking about the product your students will produce, you need to take into account their diverse academic needs. If you teach in an inclusive classroom setting, I highly recommend that you collaborate with the special education teacher on this step. They can assist you in creating a scaffolded or modified version of the final work product that is developmentally appropriate for each individual student receiving instructional support.

For this project, I gave students three options for the final product. They could participate in a whole-class Socratic seminar and engage in dialogue with each other by providing textual and statistical evidence that would support their response to the focus question. The second option was to write a letter to the police commissioner to express their concerns around the issue of racial profiling by law enforcement in Boston, including relevant data to support their argument. The final option was to write a paper stating their argument, with statistical evidence to support it and at least two solutions they thought would address the issue. My students opted for a whole-class Socratic seminar.

Ultimately, the final product should provide multiple access points for the diverse learners in your class and allow them to best demonstrate mastery of math skills and their knowledge of the topic.

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Bringing a Culturally Responsive Lens to Math Class

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Culturally Responsive Teaching A middle school math project gives students a chance to apply the skills they learn in class to an issue they care about.

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An English teacher co-teaching trigonometry asked students to explain an equation to a child and to turn discrete problems into a story.

©Shutterstock/Evgeny Atamanenko

I had always hated math. Now I suddenly found myself teaching trigonometry. I was an English teacher in Chicago Public Schools with certification in special education, and when my school was facing a shortage of certified special education teachers, I was pulled in mid-year to co-teach a junior-level trigonometry class with the math teacher.

My students struggled with the calculations, thinking they just weren't good at math. Like me, they hated it. What was the point in working and reworking these calculations? What were we trying to figure out anyway? And I originally agreed with them.

Yet trig slowly became my favorite class of the day. After spending years teaching English and reading, I was being challenged to move beyond what I had always been doing. When you're new to something, you have a fresh perspective. You're willing to take risks. You're willing to try anything because you don't know how something should be done.

I worked with my co-teacher to create a series of supplementary lessons through a different lens to let students experience personal meaning and creativity in their math.

Explaining It to a Baby

I found that many students felt frustrated with math because they needed to come to one single correct answer. This was especially hard with my diverse learners, who struggled with multistep equations. Instead of focusing on coming to the correct answer, my students and I focused on the process of getting there.

I brought in some books from Chris Ferrie's Baby University series-books like General Relativity for Babies and Optical Physics for Babies. The idea is that you don't fully know something unless you can break it down so simply that you can explain it to a young child.

That's the task I gave my students. We started by reading Ferrie's board books to see how simple language and illustrations could be used to explain complex subjects. Next, students chose a multistep equation they had initially struggled with. Working in pairs or small groups, they talked through their thinking and the steps needed to solve the equation. Their partners were encouraged to ask questions and get clarification so the ideas were explained at the simplest level.

Using the books as models, students revised and wrote down their explanations to make them so simple that they could be explained to a young child. After they wrote out their explanations, my co-teacher and I challenged them to create short books using card stock and colored pencils. Students worked with their small groups to talk through ideas and illustrate their books. If they struggled, they were able to pair with another student to create a book together.

Sharing with other students helped them explain ideas in new ways, which helped them develop a deeper understanding. Students were pushed to think metacognitively in order to explain their thinking and their process to others, and the class as a whole gained access to varying perspectives in math by hearing their peers' thought processes. And they were all excited to see how they could use writing and art skills in an authentic way in math class.

Putting the 'Story' in 'Story Problem'

The interesting thing often overlooked in math class is that it already includes stories and real-life connections, in the form of story problems. But the story problems are generally discrete-each is an individual unit, and they don't tell a larger story.

Another issue is that the real-life elements usually don't relate to things that are real issues in students' lives. They might include calculating area so that someone can buy new carpet for their home. Or a story problem might be about landscapers planting a new tree, and needing to calculate the length of wire required to support the tree. These might be things the students will do later as adults, but they're not current issues in the teens' experience.

I used story problems as an opportunity to connect math to students' lives by creating fictional math-based stories. First, students would work in small groups to go through the chapter in their math textbook and collect the story problems, writing them on index cards. Next, students would lay out the cards to see the questions as a whole: Out of 10 or more story problems in the chapter, were there five similar ones they could group together? What problem-solving skills were called for to work on these problems?

Looking at these five unconnected stories, students thought why they needed to solve them, and used their reasons to come up with some type of connected ideas. They created backstories for the names in the problems, in the process turning them into more developed characters. They identified challenges or reasons why the characters needed to solve the problem.

Finally, they combined the story problems they had created and developed a longer narrative to connect these scenarios, an overarching story rooted in authentic math story problems. Survival was a common theme: One group wrote about a zombie apocalypse and another imagined an alien invasion, situations in which characters needed to solve the problems and employ skills that would help them survive. It's true that these stories were not rooted in students' actual lives, but they were more engaging than rug purchases or landscaping.

When they used creative writing skills to develop math story problems about things they were interested in, students became more engaged. They wanted to read the other groups' stories and work on the math in them because they had a real investment in the outcome. The stories helped students find motivation because they created an answer to the question "Why do we need to learn this?"

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An English teacher co-teaching trigonometry asked students to explain an equation to a child and to turn discrete problems into a story.

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A Strategy for Boosting Student Engagement in Math

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A Strategy for Boosting Student Engagement in Math

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A Strategy for Boosting Student Engagement in Math

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A four-step approach to group work can get students talking and boost their mathematical and metacognitive thinking.

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A national experiment reveals where a growth mindset improves achievement

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A national experiment reveals where a growth mindset improves achievement

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A national experiment reveals where a growth mindset improves achievement - Nature

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A US national experiment showed that a short, online, self-administered growth mindset intervention can increase adolescents’ grades and advanced course-taking, and identified the types of school that were poised to benefit the most.

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Gender similarities in the brain during mathematics development

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Gender similarities in the brain during mathematics development

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Introduction

Limited evidence for intrinsic, biological gender differences in mathematics ability has fueled debate about the underrepresentation of girls and women in STEM fields (science, technology, engineering, and mathematics). Some have suggested that girls and women are underrepresented in careers in STEM owing to biological differences.1 Biological sex differences manifest in aspects of brain function, particularly those related to neuroendocrinology,2,3,4 but many measures indicate that neural variability is a continuum wherein the brains of males and females reflect one heterogenous population rather than two distinct groups.4 In the domain of mathematics, the evidence for biologically based gender differences is weak because when gender differences are observed, the studies fail to differentiate intrinsic biological factors from sociocultural ones.5,6,7,8,9 Moreover, behavioral studies often find no gender differences in mathematical cognition in early childhood, and there are no prior functional neuroimaging studies of biological gender differences in mathematical cognition during early childhood.5,8,9,10,11,12,13 In order to understand the origins of mathematics ability, and whether there are any gender differences, it is important to ask whether boys and girls begin development with biological differences in mathematical processing. Here, we combine frequentist and Bayesian statistical approaches to test for gender similarities and differences in the neural processing of mathematics during early childhood. We use "gender" instead of "sex" throughout this manuscript, which accords with the relevant literature,12,13,14 and because we collected parental report of children's gender and did not measure their chromosomes.

Although evidence for behavioral gender differences in mathematics is weak in older children, adolescents, and adults, it is important to consider when and how any differences might emerge. One possibility is that despite established gender similarities on behavioral tasks in early childhood,5,10,12,13 the underlying biological or neural processes could differ between boys and girls. For example, boys' and girls' incorrect responses could result from different neural processes (e.g., inefficient recruitment of math processing regions vs inefficient response selection mechanisms). Boys' and girls' error rates could yield the same levels on behavioral tests of mathematics but the biology that underlies the errors in each gender group could differ. Alternatively, boys and girls may show significant, widespread biological similarities in the neural processes of mathematics during early childhood. This outcome would be consistent with yet untested claims that boys and girls share a core biology for mathematical cognition.

To compare the neural processes underlying mathematics development, we used functional magnetic resonance imaging (fMRI) to measure neural activity in 3-10-year-old children while they watched video clips that targeted early childhood mathematics skills (e.g., counting, addition; see Methods for more details). In total, 104 children (55 girls) participated in one of three natural viewing tasks (2 published studies15,16 + 1 unpublished study, under review). Data were combined across natural viewing tasks by normalizing each subject to a within-task adult baseline using intersubject correlations.17 This approach yields 80% power to achieve a medium-effect size of d = 0.55, p < 0.05 for independent samples t tests.

Intersubject correlations were conducted by comparing each child with every other child and comparing each child with every adult within a comparison group (63 total adults, 25 women, who watched one set of video clips). This resulted in an index of 'neural maturity' (child-to-adult intersubject correlations) and an index of 'neural similarity' (child-to-child intersubject correlations). The majority of analyses focus on similarities and differences in 'neural maturity', which indicates how well-developed and adult-like each child's brain is during mathematical processing.15,16 Neural maturity was calculated by conducting intersubject correlations17 of the neural timecourses across the entire video between children and adults in every voxel of the brain (see Methods for details). Thus, it assesses the degree to which children's neural activity resembled that of adults who watched the same video and allows the data to be combined across studies in a meaningful way. Within-child comparisons of neural maturity calculated to women vs men revealed that children's neural maturity did not statistically differ based on the gender of the adult comparison group (see Methods for details). Therefore, for each child, their measure of neural maturity is averaged across all adults who watched the same video.

Girls' and boys' neural maturity were statistically compared across five whole-brain analyses which test for differences in mean neural maturity, similarities in mean neural maturity, differences in variance of neural maturity, and differences in the rate of mathematics development. First, we conducted frequentist statistical tests of differences (two-sided, independent samples t tests) and similarities in neural maturity. Similarities in mean neural maturity were assessed using statistical equivalence statistics. Testing for statistical equivalence is critical for evaluating gender similarities because a null result from a t test only suggests that there is not enough evidence to conclude that a difference exists-to address this we conducted a statistical test of similarity, Schuirmann's Test of Equivalence.18 This test uses two one-sided t tests to determine the likelihood that the mean difference between two groups falls within a specified similarity range (consistent with previous work that tested for similarities and differences in SAT scores, we used a similarity range of 2/3 of a standard deviation19). Complementary to this approach, we conducted a Bayes Factor analysis, which also allows for the interpretation of both significant differences and significant similarities. The Bayes Factor analysis weighs the evidence for an alternative hypothesis against the evidence for the null hypothesis by taking the ratio of the posterior probabilities for the two hypotheses (the Bayes Factor). Bayes Factors > 3 indicate substantial, interpretable evidence for the corresponding hypothesis, and Bayes Factors < 3 suggest that the evidence is only anecdotal. Following previous work,13 the prior for the alternative hypothesis of gender differences was the default Cauchy distribution centered on the prior for the null hypothesis with a width of 0.707. The prior for the null hypothesis was 0. To test for differences in neural variance between boys and girls across the whole brain, we used Levene's Test of Variance. Some have claimed that differences in the upper and lower tails of the distributions drive gender differences.20 Variance thus is an important measure because previous work shows that gender differences in variance can exist even when mean performance is the same.10,21 Following the whole-brain analyses, we present more-detailed region-of-interest analyses on regions of the number processing network (bilateral intraparietal sulcus, bilateral inferior frontal gyrus, and anterior cingulate cortex), which were defined from an independent functional localizer scan. In a final analysis, we calculated intersubject correlations across children to obtain measures of within-gender and between-gender neural similarity. We then directly compared children's neural similarity as calculated with children of the same and of the different gender. If gender differences in neural activity have a biological categorical origin rooted in childhood, these categorical differences should be evident in the brain. In contrast, if gender differences in neural activity do not originate from categorical differences in early childhood, there should instead be widespread similarities.

Results

Frequentist and Bayesian comparisons of neural maturity

We used intersubject correlations to compare girls' and boys' temporal patterns of neural activity across the whole brain during the educational videos. First, we compared girls' and boys' overall neural maturity using frequentist statistics. A whole-brain t test revealed no differences in neural maturity between groups (threshold =  t(102) ≥ 2.36, voxel-wise p ≤ 0.01, cluster corrected to p < 0.05, cluster threshold = 29 voxels). Next, gender similarities were assessed using Schuirmann's Test of Equivalence.18 Findings of statistical equivalence would suggest that children's neural processing of mathematics comprises one heterogenous group rather than two distinct gender groups. In fact, girls and boys showed statistically equivalent levels of neural maturity throughout the brain (Fig. 1a, light purple; minimum t(102) ≥ 2.36, maximum one-sided p ≤ 0.01), suggesting that the neural processing of mathematics develops at similar rates in boys and girls. In terms of differences in neural variance, Levene's Test of Variance revealed one small region of right posterior parietal cortex where groups differed in variance but not in mean activation, with girls showing greater variance than boys (TAL peak: 18, −64, 43; F(1,102) ≥ 6.89, voxel-wise p ≤ 0.01, cluster corrected to p < 0.05, cluster threshold = 11 voxels, Supplementary Fig. 1). This variance cluster was small (15 voxels) and girls exhibited more variance than boys at equal neural amplitudes to boys, which did not result in a mean difference between groups in this region.

Whole-brain analyses. a Frequentist statistical tests of gender differences and similarities (light purple) of neural maturity. Using frequentist analyses ( t tests), there are no regions showing significant gender differences at the standard threshold. b Whole-brain Bayes Factor analysis showing substantial evidence of gender differences (orange) and gender similarities (dark purple) of neural maturity. The plot to the right shows the percent of voxels across the brain that show substantial support for gender similarities and differences. The regions that show evidence of gender similarities are consistent across frequentist and Bayesian approaches

The pattern of large-scale statistical similarities between boys and girls from the frequentist analyses was replicated in the Bayesian analysis. In each voxel of the brain, the weight of evidence for the null and alternative hypotheses were indexed by the Bayes factor (B ₀₁ for the null hypothesis of gender similarities, B ₁₀ for the alternative hypothesis of gender differences). Bayes factors suggesting that the data provide substantial support of the hypotheses are displayed in Fig. 1b (B ₀₁ > 3, in purple indicating gender similarities, B ₁₀ > 3 in orange indicating greater neural maturity in girls; no cortical regions showed substantial support for greater neural maturity in boys). Less than 1% of voxels (0.8%) showed substantial or strong (B ₁₀ > 10) evidence of gender differences (Fig. 1b; 23.6% voxels did not substantially support either similarities or differences in the Bayesian analysis).

Similarities in math processing networks

Importantly, across all three natural viewing tasks, children engaged numerical processes in the brain. Children showed number-selective neural activation in the intraparietal sulci (IPS) during the mathematics content in the educational videos (Fig. 2) consistent with previous fMRI research on numerical cognition in children and adults.15,16 Boys and girls showed equivalent mathematics-related neural responses (see Fig. 2). This is evidence that children engaged mathematical neural processes during the educational videos, and that boys and girls did so equally.

Number and math selectivity in the intraparietal sulci (IPS). Individual data for boys are shown in blue and for girls are shown in red. Error bars represent ±1 standard error of the mean. a Percent signal change for number clips vs non-number clips in Study 1-redrawn data15 (RIPS: t(25) = 4.01, p = 0.0005, 95% CI = 0.19-0.60, Cohen's d = 0.79; LIPS: t(25) = 1.51, p = 0.26, 95% CI = −0.09-0.33, Cohen's d = 0.23). b Preference for math clips vs non-math clips in Study 2-redrawn data16 (RIPS: t(34) = 6.86, p < 0.0001, 95% CI = 0.12-0.23, Cohen's d = 1.16; LIPS: t(34) = 4.12, p = 0.0002, 95% CI = 0.05-0.15, Cohen's d = 0.70). c Slope of % signal change across increasing counting vs alphabet sequences in Study 3 (higher slope = greater sensitivity to sequence; RIPS: t(42) = 3.21, p = 0.003, 95% CI = 0.35-1.56, Cohen's d = 0.49; LIPS: t(42) = 2.55, p = 0.014, 95% CI = 0.14-1.17, Cohen's d = 0.39). Independent t tests suggest no differences between girls and boys (max t value = 0.84, min p value = 0.41) and Bayes Factor analyses suggest anecdotal to substantial evidence for gender similarities (BF ₁₀ for gender differences: 0.30-0.47; BF ₀₁ for gender similarities: 2.14-3.25). Although, samples for each individual study are small, effects are consistent with the larger patterns reported in the following analyses

Next, we compared the rate of mathematics development in boys and girls. Ninety-seven children completed the Test of Early Mathematics Ability to evaluate their mathematics skills (TEMA-322; n = 50 girls, 3.12-8.96 years; 47 boys, 3.33-9.08 years). Math ability was statistically equivalent across children and did not show gender differences in mean ability or variance (Fig. 3a; t test: t(95) = 0.57, p = 0.57, girls' mean = 33.62, boys' mean = 35.96, 95% CI = −10.42-5.74; Tests of Equivalence: t ₁(95) = 3.84, p < 0.001, t ₂(95) = −2.70, p = 0.004; Test of Variance: F(1,95) = 0.29, p = 0.59, girls' sd = 20.19, boys' sd = 19.87; descriptively there were more girls than boys in the upper tail of the distribution: 14 girls and 10 boys). Nor did the relation between gender and math ability change across age (Fig. 3a; Regression of TEMA-3 on gender and age: R ² = 0.79, F(3,93) = 118.5, p < 0.0001, Gender: b = 1.22, t(93) = 0.163, p = 0.871; Age: b = 11.74, t = 13.97, p < 0.0001; Gender × Age Interaction: b = 0.26, t = 0.21, p = 0.84). These behavioral data were previously included as part of a larger behavioral study that showed no differences and statistical equivalence in math ability in this age group.12

Gender similarities in math ability. a Left: distributions of TEMA-3 scores for girls (red) and boys (blue). Right: TEMA-3 scores increase with age for girls (red) and boys (blue), shaded regions around the line represent ±1 standard error of the mean. b Regions where boys and girls showed a relation between neural maturity and math ability. Importantly, no cortical regions showed an interaction between math ability and gender. Abbreviations: IPS = intraparietal sulcus, PFC = prefrontal cortex, MTG = middle temporal gyrus, R = right, L = left

We then identified mathematical processing networks by testing for regions that showed higher neural maturity in children with stronger math skills. Math ability, gender, and the interaction between math ability and gender were entered as predictors of neural maturity in a whole-brain regression. This regression revealed that math ability predicted neural maturity in both gender groups in the IPS, prefrontal cortex, and middle temporal gyrus (Fig. 3b; t(93) ≥ 2.63 voxel-wise p ≤ 0.01, cluster corrected to p < 0.05, girls: r(48) ≥ 0.36 with cluster threshold of 109 voxels, boys: r(45) ≥ 0.37 with cluster threshold of 69 voxels, see Supplementary Fig. 2 for separate maps). These regions are consistent with those that show a correlation between neural maturity and math ability when collapsed across gender (r(95) ≥ 0.26, p ≤ 0.01, corrected, Supplementary Fig. 2). No cortical regions showed significant interactions between math ability and gender (t(93) ≤ 2.63, threshold: voxel-wise p ≤ 0.01, cluster corrected to p  < 0.05, threshold = 110 voxels), indicating that the relation between math ability and neural maturity does not depend on gender. In other words, mathematical processing networks develop at the same rate for girls and boys.

To visualize patterns of gender similarities, neural maturity was extracted from an independently-defined number processing network15 consisting of bilateral IPS, bilateral inferior frontal gyrus, and anterior cingulate cortex (number > face, shape, and word matching; Fig. 4a; t(17) ≥ 4.04, FDR corrected to p < 0.05). In accord with the whole-brain analyses, these regions showed statistical equivalence, not statistical differences, and no differences in variance (Fig. 4b; t tests: max t(102) = 1.08, p = 0.28; equivalence-tests: min t ₁(102) = 3.82, p = 0.0001; min absolute value of t ₂(102) = 2.33, p = 0.01; see Supplementary Table 1 for full statistics). Regression analyses revealed that math ability predicted neural maturity throughout the number processing network, particularly in the IPS, but did not interact with gender (interaction predictors: max t(93) = 1.42, p = 0.16, Supplementary Table 1). This shows that within key number processing regions of the brain, girls and boys show the same degree of development in mathematical processing.

Region-of-interest analyses. a Number processing network identified from an independent localizer (number matching > face, shape, & word matching). b Mean neural maturity in the number network (top) and relation between neural maturity and math ability in the number network (bottom). Boxplot center line identifies the median, the upper whiskers extend from the 75th percentile to the 75th percentile + 1.5 interquartile range, the lower whiskers extend from the 25th percentile to the 25th-1.5 interquartile range. Outliers are those data points beyond the whisker ranges. Shaded regions around the lines in the scatterplots indicate ±1 standard error. Abbreviations: IPS = intraparietal sulcus, ACC = anterior cingulate cortex, IFG = inferior frontal gyrus, R = right, L = left

Whole-brain child-to-child similarity

Finally, we examined neural similarity in children of the same versus different genders. Intersubject correlations were calculated between children, resulting in maps of same-gender neural similarity (comparing girls with girls and boys with boys) and different-gender neural similarity (comparing girls with boys and boys with girls). To determine whether there were differences between same-gender versus different-gender similarity, each child's whole-brain different-gender similarity map was subtracted from their same-gender similarity map. These difference maps were then subjected to a one-sample t test vs 0. This whole-brain t test revealed no regions that showed a difference in neural activity (threshold: t(103) ≥ 2.62, voxel-wise p ≤ 0.01, corrected to p < 0.05 with a cluster threshold of 19 voxels). Figure 5 shows average neural similarity calculated to children of the same gender (yellow), a different gender (dark green), and the overlap of those maps (light green; r(500) ≥ 0.115, p  ≤ 0.01). The regions that showed strong neural similarity between children were identical for statistical comparisons of the same gender and different genders. This again indicates that children's patterns of neural activity reflect one heterogenous group, rather than two distinct groups based on gender.

Child-to-child neural saimilarity. Average neural similarity (r(500) ≥ 0.115, p  ≤ 0.01) when calculated across children of the same gender (yellow, column 1) and children of different genders (green, column 2). The third column shows the overlap of the first two columns in light green. Importantly, regions are identical across neural similarity calculated to children of the same gender and to children of different genders

Discussion

Across multiple neural analyses, we show that girls' and boys' brains function similarly during mathematical processing. We saw no evidence of gender differences in neural responses to mathematics content, neural responses during educational video viewing, or rates of neural development for mathematical processing in early childhood, and in fact we found statistical equivalence between boys and girls throughout the brain. Tests of statistical equivalence and a Bayes Factor analysis show gender similarities throughout the number processing network. Furthermore, boys' and girls' math abilities related to the rate of neural mathematics development in the same brain regions, and neural similarity was consistent across children of the same and of different genders.

Our results are consistent with the 'Gender Similarities Hypothesis', which argues that boys and girls function similarly in most areas of cognition.10,23 In particular, gender similarities in early childhood mathematics show, as proposed in the domain of spatial cognition,14 that gender differences in STEM fields in adults are not derived from intrinsic differences in children's brains but likely from a complex environmental origin.

Any test of cognitive ability that shows gender differences faces the difficulty of disentangling biological factors from social ones. For instance, 4- to 7-year-old boys show an advantage over girls in tests of spatial skills, but parents also report more-spatial play with their boys compared with their girls,14 suggesting a possible sociocultural influence on gender differences in spatial cognition. Similarly, in math and science, teachers tend to show differential distributions of time spent encouraging students, praising students, and explaining concepts to students, with boys receiving more time than girls.24,25,26,27 This is important because teachers' perceptions of children's math ability predicts later math achievement scores.28 Parents' expectations about their children's success also correlate with children's own self-concepts of their abilities and their performance on math tasks.29,30 A strong sociocultural influence on early childhood math achievement makes it difficult to tease apart intrinsic gender differences from sociocultural factors in older children and adults.8,9

Given the broad similarities between boys and girls, gender differences observed in STEM performance during adolescence or adulthood are unlikely to originate from early childhood differences in the brain or cognition. Although gender differences in STEM may emerge later in development or from interactions between STEM training and sexually dimorphic behaviors (e.g., differences in hormone levels following puberty),6,31,32 the findings of widespread gender similarities in boys' and girls' brains do not support claims of biological gender differences in childhood. Instead, the data show that the neural functions underlying mathematical cognition are similar between genders and represent one heterogeneous population rather than two categorical groups.

Methods

Participants

In total, 104 typically-developing 3- to 10-year-old children (55 girls) and 63 adults (35 women) participated in one of three studies. Age was statistically equivalent between girls and boys (t ₁(102) = 3.32, p = 0.0006, t ₂(102) = 3.43, p = 0.0004), and there were no differences in age variability, reflecting an even distribution of age across-gender groups (F(1,102) = 0.07, p  = 0.79, girls' sd = 1.65, boys' sd = 1.63). Informed written consent was obtained from adult participants and parents of children, and informed written assent was obtained for children 7 years and older. All protocols were approved by the University of Rochester Research Subjects Review Board.

fMRI paradigms

Study 1

Twenty-six 4- to 10-year-old children (15 girls; girls' mean age = 6.93 years, boys' mean age = 7.13 years; range = 4.32-10.80 years) and 20 adults (13 women; women's mean age = 20.52 years, men's mean age = 20.98 years; range = 18.9-25.4 years) successfully participated in Study 1. This paradigm consisted of a 20.3-min video containing clips from children's educational television shows. Clips ranged from 12 to 176 s in length and were edited into a continuous movie. These data have been previously reported15 as the "natural viewing" task.

Study 2

Thirty-five 4- to 8-year-old children (17 girls; girls' mean age = 6.61 years, boys' mean age = 6.35 years; range = 4.08-8.67 years) and 23 adults (12 women; women's mean age = 22.13 years, men's mean age = 22.65 years; range = 18.44-28.09 years) successfully participated in Study 2. The 11.6-min video contained clips from children's educational television shows. Clips ranged from 12.5 to 32.4 s in length and were edited into a continuous movie. These data have been previously reported16 as the "natural viewing" task.

Study 3

Forty-three 3- to 5-year-old children (23 girls; girls' mean age = 4.54 years, boys' mean age = 4.71 years; range = 3.12-5.96 years) and 20 adults (10 women; women's mean age = 23.43 years, men's mean age = 24.17 years; range = 20.15-31.55 years) successfully participated in Study 3. In this study, participants listened to pre-recorded audio tracks of someone counting or saying the alphabet. Short clips from child-friendly cartoons were presented on the screen during the sequences. Audio tracks were removed from the cartoons and were replaced with quieter, child-friendly instrumental music. Cartoon tracks were matched across sequences. Sequences were presented in 70 s blocks of 60 items presented at a rate of one item every 1.1-1.2 s. Fifteen blocks were presented throughout the experimental paradigm and were separated by 4-s of blackscreen. The scan began and ended with 12 s of blackscreen resulting in a total scan time of 19.2 min.

Number Localizer

Eighteen children from Study 1 (11 girls, 7 boys; girls' mean age = 6.78 years, boys' mean age = 7.47 years; range = 4.32-10.8 years) completed a traditional fMRI paradigm in which they compared pairs of stimuli (isolated images of faces, numbers, words or shapes) that were presented on the left and right sides of the screen. Participants reported whether they were the same or different by pressing a button only if the two stimuli matched. Half of the pairs presented were "matches", whereas the other half were "non-matches". Number comparisons were made across notation (i.e., dot array compared with Arabic numeral), face comparisons were made across a frontal shot and an oblique view, word comparisons were made across a word in all capital letters and the other in all lowercase letters, and shape comparisons were made across two shape images. Stimuli were presented in a blocked design. Each block consisted of three 2-s trials from the same condition, separated by a 2-s intertrial interval. Three blocks of each condition were semi-randomly presented throughout a run with 8 s of fixation between blocks. These data have been previously reported15 as the "traditional functional task".

fMRI session

Prior to the scanning sessions, children participated in a 30-minute training session in a mock scanner to familiarize them with the scanning environment and to practice remaining motionless. Children who completed the Number Localizer paradigm practiced the task prior to the MRI session. Adults received verbal instructions prior to scanning and did not participate in a training session. During the scan, children's heads were secured with medical tape, headphones, and foam padding, and adults' heads were secured with headphones and foam padding.

MR parameters

Whole-brain BOLD imaging was conducted on a 3-Tesla Siemens MAGNETOM Trio scanner with a 12-channel head coil at the Rochester Center for Brain Imaging. High-resolution structural T1 contrast images were acquired using a magnetization prepared rapid gradient echo pulse sequence at the start of each session (repetition time (TR) = 2530 ms, echo time (TE) = 3.44 ms, flip angle = 7, field of view (FOV) = 256 mm, matrix = 256 × 256, 192, 176, or 160 slices depending on head size, 1 × 1 × 1 mm sagittal left-to-right slices). An echo-planar imaging pulse sequence with online motion correction was used for T2*contrast (TR = 2000 ms, TE = 30 ms, flip angle = 90 degrees, FOV = 256 mm, matrix 64 × 64, 30 axial oblique slices, parallel to the AC-PC plane, voxel size = 4 × 4 × 4 mm). The primary paradigms from Studies 1, 2, and 3 were 610 volumes, 348 volumes, and 567 volumes, respectively, and the Number Localizer paradigm was distributed over two to four functional runs of 132 volumes each.

Preprocessing

fMRI data were analyzed in BrainVoyager33 using in-house scripts drawing on the BVQX toolbox. Data from previously published studies were analyzed as originally reported for consistency.15,16 For the Number Localizer and Study 1, which were collected during the same scanning session, the first six TRs of each run were discarded prior to analysis to allow for signal equilibration. For Studies 2 and 3, the first two TRs of each run were discarded. Functional data were registered to high-resolution anatomy images for each participant in native space. Preprocessing consisted of slice scan time corrected (cubic spline interpolation), motion correction with respect to the first volume in the run, and linear trend removal in the temporal domain (cutoff: two cycles within the run). A Gaussian spatial filter with an 8 mm full-width at half-maximum was applied to each volume for Study 1,15 and a 6 mm full-width at half-maximum was applied to each volume for Studies 216 and 3. The functional data from the Number Localizer were not smoothed. Adult and child echo-planar and anatomical volumes were then normalized into Talairach space34 using piecewise affine transformation based on manual identification of anatomical landmarks. Analyses were performed on processed data in Talairach space. Average framewise displacement35,36 was regressed across the brain for each child to control for sudden changes in volume-to-volume head motion.

fMRI data analyses

Neural maturity: neural data were analyzed using an intersubject correlation approach.15,16,17 Between-group intersubject correlations were performed by using the full timecourse of each voxel for each child as a predictor for activation of the corresponding voxel in each adult brain. Functional data from each child were then correlated with that of each adult from the same study to produce paired r-maps. Paired r-maps represent the neural similarity of each child compared with each adult in every voxel of the brain. A single, average brain map was then calculated to represent the neural similarity of each child to all adults. This map is referred to as a map of "neural maturity" because it shows how "adult-like" each child's neural timecourse appears. To ensure that these similarity maps were not confounded by any adult gender differences prior to averaging, we compared neural maturity calculated to women vs men. For each child, we conducted an Independent Samples t-test to compare neural maturity when calculated to adults of the same vs different gender as the child. This resulted in a map of between-group t values for each child. To determine whether there were any differences that were consistent at the group level, the absolute value of these individual-level between-group t value maps were then subjected to a one-sample t test vs a critical t value of 2.08. The critical value represents the t value at which a difference could be considered significant given the smallest sample of adults ( n = 20 for studies 1 and 3; t(18) = 2.08, p = 0.05). If there were a significant difference between neural maturity calculated to adults of the same vs different gender as the child, the one-sample t test should reveal a positive and significant effect, which would indicate that the absolute values of the individual-level between-group t values are significantly above 2.08. Instead, the whole-brain one-sample t test revealed that at the group level, the absolute values of the individual-level between-group t values were significantly below the critical t value of 2.08 (one-sample t test of absolute value of individual-level between-group t values vs critical t value of 2.08: t(102) ≤ −2.62, p ≤ 0.01). In other words, the majority of individual-level between-group t values were less than the critical t value of 2.08 (range of average absolute value of individual-level between-group t values across voxels: 0-1.90). This indicates that overall, the differences between neural maturity when calculated to adults of the same vs different gender were not different, so neural maturity was not biased by the gender of the adult.

Neural similarity

To compare how similar children were to each other, intersubject correlations were calculated across children following the same procedures as for calculating neural maturity. This resulted in two maps for each child: one representing the average neural similarity of a child to children of the same gender and one representing the average neural similarity of a child to children of the different gender. To compare neural similarity within and across gender, these maps were then subtracted from each other (within-gender similarity−across-gender similarity) and these difference maps were subjected to a one-sample t test vs 0.

Number Localizer

Functional data collected during the traditional fMRI paradigm were analyzed using a general linear model (random effects analysis). Experimental events (duration = 10 s) were convolved with a standard dual gamma hemodynamic response function. There were four regressors of interest (corresponding to the four stimulus categories), one regressor for button press, and six regressors of no interest (corresponding to the motion parameters obtained during preprocessing).

Region-of-interest (ROI) analyses

Data were extracted from number network ROIs using MATLAB. Analyses were then conducted using R (version 3.3.1) and R-Studio (version 0.99.902). Independent samples t tests were conducted using the " t test" function assuming equal variance. Tests of equivalence were conducted using the "TOSTtwo.raw" function from the "TOSTER" package (upper and lower bounds set to ±0.667 × standard deviation of the entire group; alpha = 0.05). This function returns two t values ( t ₁ and t ₂). For statistical equivalence, both t values must be statistically significant. Statistical equivalence is rejected if either t ₁ or t ₂ does not reach significance. Levene's Test of Variance was carried out using the "leveneTest" function in the "car" package. Regression analyses were conducted using the built-in "lm" function.

Assessment of math ability

Math skills were evaluated by administering the TEMA-322 to participants aged 8 and younger. The TEMA-3 tests a variety of math concepts and is standardized for 3-8-year-old children.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Data availability

These data sets are available from the corresponding author on reasonable request.

Code availability

All code for R and MATLAB are available from the corresponding author on reasonable request.

References

1. 1.
Levy, J. & Kimura, D. Women, men and the sciences. in The Science on Women and Science (ed. Sommers, C. H.) 202-284 (American Enterprise Institute Washington, 2009).

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Spelke, E. S. & Ellison, K. Gender, math and science. in The Science on Women and Science (ed. Sommers, C. H.) 24-53 (American Enterprise Institute Washington, 2009).

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Hutchison, J. E., Lyons, I. M. & Ansari, D. More similar than different: gender differences in children's basic numerical skills are the exception not the rule. Child Dev. 00, 1-14 (2018).

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Ginsburg, H. & Baroody, A. J. Test of Early Mathematics Ability, 3rd Edn (TEMA-3). (Pro-Ed, 2003).

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American Association of University Women Educational Foundation. How Schools Shortchange Girls. AAUW Rep. 1-13 (1992).

6. 34.
Talairach, J. & Tournoux, P. Co-Planar Stereotaxic Atlas of the Human Brain. (Thieme, 1988).

Download references

Acknowledgements

We thank Rosa Li, Courtney Lussier, and Pat Weber for help with data collection and Rosa Li and Kamy Wakim for help with data processing. This work was supported by the National Science Foundation (DRL-1459625 to J.F.C. and DGE- 1419118 to A.J.K.) and the National Institutes of Health (R01 HD064636 to J.F.C).

Author information

Affiliations

1. Department of Brain and Cognitive Sciences, University of Rochester, Rochester, NY, 14627, USA

Alyssa J. Kersey, Kelsey D. Csumitta & Jessica F. Cantlon

2. Department of Psychology, University of Chicago, Chicago, IL, 60637, USA

Alyssa J. Kersey

3. Department of Psychology, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

Jessica F. Cantlon

Contributions

All authors designed the research, analyzed the data, and revised and approved the manuscript.

Corresponding author

Correspondence to Alyssa J. Kersey.

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# data[].attributes.attachment_link.title
Gender similarities in the brain during mathematics development

# data[].attributes.attachment_link.description
Some scientists and public figures have hypothesized that women and men differ in their pursuit of careers in science, technology, engineering, and mathematics (STEM) owing to biological differences in mathematics aptitude. However, little evidence supports such claims. Some studies of children and adults show gender differences in mathematics performance but in those studies it is impossible to disentangle intrinsic, biological differences from sociocultural influences.

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The 3-read protocol and math tasks that CVES uses were developed by Exemplars, a company that provides a library of rich performance tasks and professional development for K-12 schools.

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Demystify math word problems with this simple technique that helps kids see the story beyond the numbers.

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A Simple Math Activity That Engages All Students

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A Simple Math Activity That Engages All Students

# data[].attributes.attachment_link.readable_text
Differentiated Instruction

This engaging low floor, high ceiling activity boosts students' confidence and helps them think creatively about mathematical relationships.

As a rule, teachers seek economy in teaching. The more bang we can get for the buck when we're with students, the better. Two years ago, I began looking for math activities that promote creative thinking, require persistence, necessitate the practice that many believe is required for learning, and could be given to the entire class. On top of that, I wanted to improve motivation and inspire a love of learning.

Impossible? As it turns out, it's not as difficult as one might think.

I believe that all students can learn and, if not love, at least appreciate math-and researchers agree with me. Every day in my fifth- and sixth-grade combination class (in which more than a third of students have IEPs), we spend about 10-15 minutes focused on a math challenge. I've found that the key to deep, engaged learning is to provide a challenge everyone can participate in (meaning it has a low floor) and that has many different ways of being solved (meaning it has a high ceiling).

For instance, last week, the challenge was to use all the numbers in a given set (in this case, 1, 2, 3, and 4) to come up with the answer of 5. The students could use any operation or math symbol that they wanted.

Some of the results students came up with were: (3 + 4 - 2) ÷ 1; 12 - 4 - 3; and 2 ³ - 4 + 1.

The first time someone came up with an out-of-the-box response (making 12 by combining 1 and 2, for example), there was an uproar of "That's not fair! That's cheating!" I had to remind them, "No rules, remember?" And they madly went back to their calculations.

Of course, while during this short time students work primarily at their desks, I make it a point to have students come up to the board to write down their solutions as soon as they discover them. This encourages discussion of how the solution works and reveals new possibilities to those who might have hit a wall. It becomes collaborative. Often, it allows me to reteach incorrect thinking about mathematical concepts or introduce new ones, making the learning both relevant and timely.

Let's look at the ways in which this 15-minute activity can meet all the conditions I hoped for above, including requiring deep thinking and cognitive flexibility.

Creative Thinking and Cognitive Flexibility

Students know that there's no set procedure, and they can't just parrot something they're supposed to have learned. As a result, they're free to approach the problem in any way that catches their attention.

Those who may struggle with remembering procedural rules are able to use alternative methods to show what they know.

Thinking outside the box is encouraged. Students who come up with original ideas are celebrated, which opens opportunities for recognition for those who may not typically do so well with memorizing and repeating.

Practice and Perseverance

I find that when questions are posed that have only one answer, the class can get in the habit of allowing the fastest student to answer-without bothering to put in any effort. And why should they? Once the answer has been found, there's no longer any need. With more than one way to solve the challenge, students know that they can still discover a solution, and I find that my students continue working longer than I even have scheduled time for.

Motivation

There are disparities between students in every classroom. Providing opportunities for creative thinking for students who don't typically stand out can be a means for them to garner respect from their peers through open-ended challenges-which, in turn, increases their motivation to learn. These challenges, which empower students to access the tasks in ways that suit them, enable students to expand their own conceptions of what they're capable of.

So, it seems deceptively simple-15 minutes a day of mathematical play. Give it a try for a week. Even in my virtual classroom, I have been able to engage students by posting a challenge as an assignment and asking them to respond in the comments section. I have gotten four to five responses in a row from students who couldn't stop playing with the numbers, along with shouts of congratulations from their peers. By turning numbers from a "problem" into objects to play with, we can bring the joy and inquisitiveness into mathematics for all students.

# data[].attributes.attachment_link.title
A Simple Math Activity That Engages All Students

# data[].attributes.attachment_link.description
Differentiated Instruction This engaging low floor, high ceiling activity boosts students' confidence and helps them think creatively about mathematical relationships. As a rule, teachers seek economy in teaching. The more bang we can get for the buck when we're with students, the better.

# data[].attributes.attachment_link.readable_text
Whether math class is happening online or in person, when kids are encouraged to talk about their math thinking, it can help teachers assess progress, improve students' math proficiency, and be a "powerful equity strategy" that allows kids to see themselves as "valued math thinkers," writes Catherine Gewertz for Education Week.

"A well-designed math conversation can make it easier for all students-even those who rarely talk in class-to participate," writes Gewertz. There are also academic benefits. At Robbins Elementary School in Robbins, California, for example, district superintendent Dawn Carl says that when teachers began focusing on math conversations, the proficiency rates of the school's English language learners improved by 5 percentage points and proficiency rates for native English speakers grew by 7 points.

The benefits of math talks are well-known-Gewertz points to research and publications from the National Council of Teachers of Mathematics dating back 40 years-but the new Common Core State Standards brought the practice back into focus. "Mastery of the common core's standards for mathematical practice means students must know how to do things like construct viable arguments and critique others' reasoning."

Math talk also translates well to online learning. The author highlights four ways teachers can help students improve their math-talk skills.

Encourage rough-draft math thinking

When students share their initial efforts at solving a math problem with their peers, it helps "deepen students' understanding of math ideas and practices," says Amanda Jansen, a mathematics education professor at the University of Delaware.

In the classroom, have your students share and comment on each other's rough-draft solutions to a problem, revise their work based on peer feedback, and then explain how and why their thinking evolved.

Highlight the value of each student's work

When students have the opportunity to think about how and why their thinking changed, it allows teachers "to call attention to the value each student's draft brought to the progression of the class' thinking. ... This can really elevate the thinking of their peers. They realize their ideas can help their classmates grow their thinking," Jansen explains.

Ask your students to examine a math problem together, work on solutions, then agree upon one solution to share with the entire class. Then have a collective discussion about "what's powerful and what could be improved" about each draft, says Jansen.

It's a technique that also functions as an equalizer for students-especially those who might not feel comfortable sharing their thinking-once they realize how their individual math thinking contributes to the whole group's problem-solving abilities.

Set up small-group conversations

Especially for English language learners who might shy away from speaking in front of the entire class, small-group settings can be helpful for tackling word problems. Teachers at Robbins Elementary first model problem-solving to the class, says Carl, the superintendent, then break kids up to work in pairs or small groups.

Teachers observe and ask open-ended questions like "tell me why you believe that, or, let's think about this," says Carl. When teachers gently guide the discussion like this without supplying correct answers, kids can start deepening their math thinking.

Model math talk

Kids need help learning how to engage in productive math conversations so teachers at Robbins Elementary use sentence starters-for example, writes Gewertz: "When Robert uses this strategy, it makes me think of..." or, "This makes sense to me because..."-to help get math talks started.

Teachers can also use cartoon-inspired thought bubbles, says Robert Q. Berry, past president of the National Council of Teachers of Mathematics, to help spur math conversations-he suggests modeling a solution, then circling a specific section of it and drawing in a thought bubble. "What do you think I'm thinking here?" a teacher might ask students, indicating the thought bubble. On paper, students can fill in the thought bubble; online this can be done using a text box.

# data[].attributes.attachment_link.description
Whether math class is happening online or in person, when kids are encouraged to talk about their math thinking, it can help teachers assess progress, improve students' math proficiency, and be a "powerful equity strategy" that allows kids to see themselves as "valued math thinkers," writes Catherine Gewertz for Education Week.

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Bringing Number Talks to the Online Classroom

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Bringing Number Talks to the Online Classroom

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Elementary school teachers use number talks to guide students to engage deeply with concepts, and they can work online, with a little creativity.

Before the coronavirus forced schools to close, if you walked into any elementary classroom during math instruction, you would notice students interacting with concepts through by using math manipulatives, playing games, engaging in discussion while problem-solving, and evaluating their own work as well as the work of their peers.

The sudden and swift transition to remote learning due to the Covid-19 pandemic required educators to adapt the idea of school to work in a digital environment. Among the many challenges of remote learning is providing elementary mathematics instruction that is engaging, rigorous, and equitable.

While there is no universal solution, one strategy is to use virtual number talks as a way to provide students an opportunity to critically think about math concepts. Based on the work of Sherry Parish, a number talk is a classroom conversation about mental math problems. In a number talk, teachers pose a problem to students and use lean prompting and questioning to elicit student thinking and dialogue. As students present their solutions, the class as a whole evaluates their work and considers the variety of methods one can use to generate a solution.

I'm an instructional coach working with classroom teachers in grades K-5, and when we first considered adapting number talks to be done virtually, we thought of the barriers that could impede student success and ways to address them.

Establish Norms

Just like in your classroom, students need to be explicitly taught how a productive number talk works. Working with our students, we developed and explained our norms, modeled them, and practiced. The three norms that worked for us are as follows. First, we are "all in," meaning that we are all in a good spot to focus on math and that we will remain focused on our number talk.

Our second norm, "one speaker," means there is only one speaker at a time and we take turns so that everyone has an equal chance to participate. Finally, our last norm, "step in, step out, and hold your thought," means that when it is a student's turn to talk, they make sure to say what they need to about the problem and then pass to the next speaker. If a student has a new thought or question that they want to share, they wait until it's their turn.

Think Beyond Mental Math Problems

While a number talk is traditionally done using mental math problems, physical classroom environments generally have scaffolds readily available, like number lines and anchor charts-tools that can assist students with their thinking. In order to successfully adapt number talks to a remote environment, encourage students to model their thinking by using pencil and paper or by creating their own at-home math tool kits of everyday items that can be used as math manipulatives, such as counters made of dried beans or pasta.

Furthermore, present your problems digitally with a presentation slide using easy-to-read fonts and, if possible, provide visual anchors like pictures to help support all students, including English learners, with comprehension.

Facilitate Discussion and Elicit Thinking

As educators, we often default to wanting to help our students by doing the heavy lifting. In challenging circumstances like remote learning, we may find ourselves doing this even more. However, during a digital number talk, we need to remember that our role as educators is to facilitate and guide discussion. Students may present an incorrect solution, and that is OK. The goal is to get students to critically evaluate their own work as well as their peers'. You can use lean prompts, like "Tell me more," "Is there anything you might change?" or "Can you explain it in a different way?"

At the conclusion of the digital number talk, have the class reflect on the work and articulate the different strategies you can use to solve a problem. You may find yourself at the end modeling a different strategy that students did not use or highlighting the work of students.

Practice With Digital Tools

You will want to find a teleconferencing application that meets your needs and your district's technology requirements. Once you've found that platform, practice and get to know the features. One of our teachers found it useful to practice number talks with her colleagues during meeting times. Just as in the classroom, technological issues may happen; however, practicing it made our digital number talks much more seamless for all participants.

We found it very helpful to learn how to utilize the "mute all" feature to eliminate background noise that can make it hard to hear students, as well as the hand-raising and chat box features. Students were made aware that they would be muted and when called on would need to unmute themselves. For our much younger students in kindergarten and first grade, our teachers found themselves manually doing this for students.

You may also want to utilize the digital whiteboard features to record student responses as the number talk progresses so that students have a visual reminder of the discussion.

# data[].attributes.attachment_link.title
Bringing Number Talks to the Online Classroom

# data[].attributes.attachment_link.description
Elementary school teachers use number talks to guide students to engage deeply with concepts, and they can work online, with a little creativity.

# data[].attributes.attachment_link.keywords[]
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Mathematics is one of the fundamental 3Rs of early education. Known as "arithmetic" in those levels, it involves counting and the basic operations of addition, subtraction, multiplication, and division. As kids advance through the grade levels, they progressively deal with more complex mathematical problems. Also, more new concepts get introduced as they progress.

For the most part, math is a lot about problem-solving. Some students may view this as tedious, others would say it's such a bother. However, the point of making students solve math problems is not to annoy them or make them hate math. On the contrary, it's for them to exercise their minds. Each math problem draws on a specific concept, and repeatedly solving similar problems allows students to master the concepts. A good grasp of various principles is very important in math because each higher topic builds upon an earlier topic.

Math also builds logical thinking skills in kids. Since math problems are mostly solved through step-by-step processes, there is always a sense of order. Proving theorems in geometry is one example of such a logical process. That and other forms of mathematical problem solving equips students to think in a highly ordered manner.

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Math is also crucial in careers like a business. Accounting, budgeting and projecting sales all require some level of competence in math.

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Math, Math education videos for students, teachers and parents on Teachertube.com

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Mathematics is one of the fundamental 3Rs of early education. Known as "arithmetic" in those levels, it involves counting and the basic operations of addition, subtraction, multiplication, and division. As kids advance through the grade levels, they progressively deal with more complex mathematical problems. Also, more new concepts get introduced as they progress.

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Where do math symbols come from? - John David Walters

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Overview

Learning from the practices of others is a powerful way to improve teaching. Explore real mathematics teaching and learning by exploring everyday classrooms where educators are working to refine their mathematics teaching practices.

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Learning from the practices of others is a powerful way to improve teaching. Explore real mathematics teaching and learning by exploring everyday classrooms where educators are working to refine their mathematics teaching practices.

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National Library of Virtual Manipulatives

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National Library of Virtual Manipulatives

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National Library of Virtual Manipulatives

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A digital library containing Java applets and activities for K-12 mathematics

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A required part of this site couldn’t load. This may be due to a browser extension, network issues, or browser settings. Please check your connection, disable any ad blockers, or try using a different browser.

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PBS Math Club helps you with your homework... and makes you laugh. They cover the 6th-9th grade Common Core Standard for math; topics like adding and subtracting integers, equations, ratios and proportions, and statistics. They use uncommon examples like Mean Girls to explain adding negative numbers in our show.

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Would You Rather Math – ASKING STUDENTS TO CHOOSE A PATH AND JUSTIFY IT WITH MATH

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Teachers, sign up for my mini-workshop to learn about how to implement Would You Rather Math into your classroom routines. We will explore some sentence frames, what happens when things don’t go as planned, how to use the prompts in a remote learning setting, and much more. See you there!

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The IM Community Hub creates effective strategies for teaching IM K–12 Math, especially in distance and hybrid learning environments.

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Community Hub

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Represent numbers by placing counters inside of five-, ten-, double ten-, and custom frames.

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Visualize and work with numbers in sequence on a virtual number line with or without tick marks.

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Represent fractions with denominators from 1 to 100 with circle and bar models.

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Slide beads to show and work with numbers and think in groups of 5s and 10s.

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Those you share this with will see the current array and settings.

Make a new rectangle

Change the size of the rectangle by entering whole numbers (up to 50) in the boxes. Then press the MAKE button.

Add and move the sliders

Drag the slider handles from the left or top to divide the rectange into smaller parts. Include up to 4 sliders on each axis.

View partial products

Watch how the equation changes depending on the position of the sliders. Point at any region to see the corresponding piece of the equation.

Show or hide answers

Toggle these buttons to show or hide the results of the equation.

Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination. The sides of the rectangle can then be divided into parts to reveal partial products that make up the total product. This helps students develop the flexibility and conceptual understanding required to build fluency with basic facts and strategies for multiplication of larger numbers.

We Welcome Your Feedback

Help us guide the ongoing development of this app by sharing your reactions or suggestions.

Partial Product Finder is brought to you by The Math Learning Center, a nonprofit organization serving the education community since 1976. Visit us at www.mathlearningcenter.org, where you can learn more about our organization, the Bridges in Mathematics PK-5 curriculum we publish, and the many free math education resources we provide, including a robust suite of free virtual manipulatives for use online or as downloadable apps for a variety of operating systems, including Apple iOS and Google Chrome.

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Partial Product, by The Math Learning Center

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Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination.

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Use virtual base ten pieces to represent and work with multi-digit numbers.

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GeoGebra - the world's favorite, free math tools used by over 100 million students and teachers

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Free digital tools for class activities, graphing, geometry, collaborative whiteboard and more

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It’s an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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Parents know that we need to read 20 minutes a day with our kids.

But teaching math to your kids doesn't have to be intimidating. It can be fun. Just as children say goodnight to the moon, they can say hello to numbers, shapes and measurement.

Talking Math with Your Kids provides ideas and products that get children and caregivers talking about math with creativity and joy.

Whether you're reading books on the couch, cooking dinner, running errands, or scheduling your day, opportunities for math talk abound.

This website collects research about, and examples of, math talk with children. Each post offers suggestions for adapting the examples for your own children, interests, and lives.

Enjoy and let me know how those conversations are going in your home. Use the contact form to ask questions, send feedback or to send along a conversation for highlighting on this site.
Thanks for reading. Read more about Christopher

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Parents know that we need to read 20 minutes a day with our kids. But teaching math to your kids doesn't have to be intimidating. It can be fun. Just as children say goodnight to the moon, they can say hello to numbers, shapes and measurement.

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Home

CHALLENGING MATH PROBLEMS WORTH SOLVING

DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL

WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS?

HERE'S OUR GROWING COLLECTION

WANT TO SHARE OPEN MIDDLE WITH OTHERS?

CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK

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CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE’S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS? CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK Elementary Version

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Home Improvement Tips & Advice from HomeAdvisor

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Are you a home schooling parent struggling to keep your kids focused on math? Many kids would much rather be playing their favorite sport, talking with their friends, or playing a video game than learning their multiplication tables. These distractions are that much harder to avoid when they're within reach.

One of the ways you can help get your child interested in math at home is to introduce them to some educational games. These may be played online or on your living room floor. Take some time to check out this list of math games and resources. Some are online games, while others are instructions for creating math activities at home.

On This Page
Tip: Before putting your child in front of a screen for educational activities, install browser plugins or adjust browsing settings to prevent them from visiting sites that might distract them from the lesson.
Addition and Subtraction

Multiplication and Division

Fractions and Decimals

Geometry and Graphing

Algebra

Math Games

Math Resource Websites

You can be the first to comment!

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Home-Schooling Help: Math Games and Resources - HomeAdvisor

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Are you a home schooling parent struggling to keep your kids focused on math? Many kids would much rather be playing their favorite sport, talking with their friends, or playing a video game than learning their multiplication tables. These distractions are that much harder to avoid when they're within reach.

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Count on us for math riddles for kids and adults! Add to your brain cells and multiply your fun with funny, easy and hard math riddles and answers. Get ready to give your undivided attention, because you'll need it to solve these fun math riddles. There are few things better for exercising your mind than riddles and math, and here we combine the two to give your brain a stellar workout. Test yourself, your friends, kids and family with these math brain teasers and you'll not only be working your mind, but you'll also have some fun! We have divided these into math riddles for kids (for those looking for more simple math riddles) and math riddles for adults (more challenging and difficult math riddles and more age appropriate).

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23 Math Riddles - Math Brain Teasers and Answers | Get Riddles

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Count on us for math riddles for kids and adults! Add to your brain cells and multiply your fun with funny, easy and hard math riddles and answers. Get ready to give your undivided attention, because you'll need it to solve these fun math riddles.

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Math and Business Activities for Young Entrepreneurs

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Math and Business Activities for Young Entrepreneurs

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You might already have an idea of what you want to be when you grow up. Maybe you want to be a scientist, an astronaut, a doctor, or a farmer. Or maybe you want to be a business owner, a fashion designer, an engineer, or a pilot. Did you know that you need to learn math to do all of those jobs? Most jobs use math in some way each day. You'll also need to use math to count up and spend the money you earn. Understanding how to use math in your daily life can help you to do well in life, no matter what type of job you want to have!

Math is everywhere in our world, not just in school. When you think of math, you might just think of things like homework, but the truth is that math can be found in fun places, too, like when you're shopping with your parents, when you're following a recipe to cook something, or even when you're playing games. After all, you can't even have a baseball game without math: How would you know who won without adding up the score?

The Internet also has lots of games that use math in fun ways. You can play these games whenever you're allowed to go online to have fun while working on your math skills. You might even start getting better grades in school thanks to the extra math practice you'll be getting!

Math Games

Dunk Tank: In this game, you can learn about ratios, proportions, and variables. This fun interactive activity lets you dunk Dolores the duck, Rocky the dog, and Iggy the cat when you get the answers right.

Monkey Jump: Help Curious George the monkey to count the balls!

Ratio Rumble: Pretend you have the power to create magic potions in this game. You'll work with ratios to see if you're the best potion master out there. Are you ready to test your abilities?

Chances: This probability and statistics game shows us how the results of a random test like rolling dice can be distributed on a chart. Try rolling a lot of times or a few to see what happens.

Penguin Party Addition: Use your addition skills to feed the penguins! Each penguin has a number on its belly. The penguin with the right answer to the question gets the fish.

Chinese Dragon Game: Work on your ordering and sequencing skills knowledge while playing this game. There are many different levels, so you can work your way up and learn while you play!

Mint Pals' Counting With Coins: Learn about each coin's worth and add them up to answer the questions. Can you complete all seven levels?

Island Chase Subtraction: Subtraction is fun when it helps you win a boat race! See how fast you can answer the questions to win the game while getting better at math.

Gasha Go: Solve different types of math puzzles to fill up the toy machine.

That's a Pattern: Sharpen your pattern knowledge with this fun game. After you choose your theme (space, polar animals, insects, or flowers), you can help fill in the patterns by picking the missing picture.

Math Racing Mania: Do you like to race cars? This game quizzes us while we drive down the road! Pick your favorite car to drive and get started.

Multiplication Games: Catch the Fish: Look at the fish swimming by, then pick the right fish to catch to solve the multiplication problem.

Fraction Matcher: Match the fractions shown in shapes and numbers to level up in this brain game!

Comparing Fractions: Two classes are having a bubble-blowing contest, and you are the judge. The classes need your help to figure out the winner, and using number lines can make it easier for you to see which class did better.

Business Games

Dollar-a-Glass Game: Have you been thinking about starting a lemonade stand? How about practicing first! With this game, you can learn about running a business, which includes a lot of different things to keep track of. Can you keep your customers happy and make a good amount of money, too?

Garage Sale Pricing: Sell your garage sale items to whoever will pay the highest price. You will need to count up the amount of money each person has offered to make the right choice!

Let's Shop: You can't always go to the store and buy new toys when you want to, but with this game, you can learn how to count your dollars and use them to pay for things.

Burger Restaurant Business Game: Run your own fast-food place in this game to learn more about how to run a business. Can you make the most money possible while keeping your customers happy?

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Math and Business Activities for Young Entrepreneurs

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You might already have an idea of what you want to be when you grow up. Maybe you want to be a scientist, an astronaut, a doctor, or a farmer. Or maybe you want to be a business owner, a fashion designer, an engineer, or a pilot.

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In the voiceover, Cologuard claims it finds 92% of colon cancers. The ad also provides a generously large and easy legible banner stating that false positives and false negative results can occur.

The next banner gives more detail.

If your grandparent gets a "Yes" response, what is the probability that they do have colon cancer? If they get a "No", what are the chances they don't have it?

https://www.ispot.tv/ad/op0r/cologuard-finding-things

If you've never paid much attention to advertisements before, this is pretty amazing. Usually the fine print is illegible even on a 62" ultra hi-definition tv paused at just the right moment.

So ... thank you Cologuard for being honest and up-front.

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180 Days of Ideas for Discussion in Math Class. (as of 9July2014, we're in overtime!)

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How do I get started? 1 USE THE FREE MATH LESSONS Estimation 180 Tasks 3-Act Tasks 2 SUBSCRIBE Email Newsletter Podcast

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Hosted by Chris Shore of The Math Projects Journal

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A garden snail's average speed is about ​0.029 mi/h. The North American X-15's top speed is about 4,520 mi/h. How much time would it take for each to travel across the United States, a...

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From Performance to Learning:Assessing to Encourage Growth Mindsets

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From Performance to Learning:Assessing to Encourage Growth Mindsets

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Introduction

The complex ways that children understand mathematics is fascinating to me. Students ask questions, see ideas, draw representations, connect methods, justify, and reason in all sorts of different ways. But recent years have seen all of these different nuanced complexities of student understanding reduced to single numbers and letters that are used to judge students' worth. Teachers are encouraged to test and grade students, to a ridiculous and damaging degree; students start to define themselves - and mathematics - in terms of letters and numbers. Such crude representations of understanding vastly under-describe and in many cases misrepresent children's knowledge.

In the United States students are over-tested to a degree that is nothing short of remarkable, particularly in mathematics. For many years students have been judged by narrow, procedural mathematics questions presented with multiple-choice answers. The knowledge needed for success on such tests is so far from the adaptable, critical and analytical thinking that students need in the modern world that leading employers such as Google have declared that they are no longer interested in students' test performance as it in no way predicts success in the workplace (Bryant, 2013).

One critical principal of good testing is that it assesses what is important.

For many decades in the United States, tests have assessed what is easy to test instead of important and valuable mathematics. This has meant that mathematics teachers have had to focus their teaching on narrow procedural mathematics, not the broad, creative and growth mathematics that is so important. The new common core assessments promise something different, with few multiple-choice questions and assessments of problem solving, but they are being met with considerable opposition from parents.

The damage does not only end with standardized testing, for math teachers are led to believe they should use tests in classrooms that mimic low quality standardized tests, even when they know the tests assess narrow mathematics.

They do this to help prepare students for later success. Some teachers, particularly at the high school level, test weekly or even more frequently. Mathematics teachers feel the need to test regularly, more than any other subject, because they have come to believe that mathematics is about performance, and they usually don't consider the role that tests play in shaping students' views of mathematics and themselves. Many mathematics teachers I know introduce a class with a test, which gives a huge performance message to students on the first day of class, the time when it is so important to be giving growth messages about mathematics and learning.

Finland is one of the highest-scoring countries in the world on international mathematics tests yet students do not take any tests in school. Instead teachers use their rich understanding of their students' knowledge gained through teaching to report to parents and make judgments about student work. In a longitudinal study I conducted in England students worked on open-ended projects for three years (ages 13 through 16) leading to national standardized examinations. They did not take tests nor was their work graded. Students encountered short questions assessing procedures in the last few weeks before the examination, as the teachers gave them examination papers to work through. Despite the students' lack of familiarity with examination questions or working under timed conditions of any kind, they scored at significantly higher levels than a matched cohort of students who spent three years working through questions similar to the national exam questions and taking frequent tests (Boaler, 1998, 2015). The students from the problem solving school did so well in the standardized national exam because they had been taught to believe in their own capabilities; they had been given helpful, diagnostic information on their learning; and they had learned that they could solve any question, as they were mathematical problem solvers.

Students with no experience of examinations and tests can score at the highest levels because the most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problem-solving mathematical tools to equip them for any mathematical situation.

The testing regime of the last decade has had a large, negative impact on students but it does not end with testing; the communication of grades to students is similarly negative. When students are given a percentage or grade, they can do little else besides compare it to others around them, with half or more deciding that they are not as good as others. This is known as "ego feedback," a form of feedback that has been found to damage learning. Sadly when students are given frequent test scores and grades they start to see themselves as those scores and grades. They do not regard the scores as an indicator of their learning or of what they need to do to achieve, but as indicators of who they are as people. The fact that US students commonly describe themselves saying "I'm an A student" or "I'm a D student" illustrates the ways students define themselves by grades. Ray McDermott wrote a compelling paper about the capturing of a child by a learning disability, describing the ways a student who thought and worked differently was given a label and was then defined by that label (McDermott, 1993). I could give a similar argument about the capturing of students by grades and test scores.

Students describe themselves as A or D students because they have grown up in a performance culture that valued frequent testing and grading, rather than persistence, courage, or problem solving. The traditional methods of assessing students that have been used across the United States for decades were designed in a less enlightened age (Kohn, 2011) when it was believed that grades and test scores would motivate students, and that the information they provided on students' achievement would be useful.

Now we know that grades and test scores demotivate rather than motivate students and that they communicate fixed and damaging messages to students that result in lower achievement in classrooms.

In studies of grading and alternatives to grading researchers have produced consistent results. Study after study shows that grading reduces the achievement of students. Elawar and Corno, for example, contrasted the ways teachers responded to math homework in sixth grade, with half of the students receiving grades and the other half receiving diagnostic comments without a grade (Elawar & Corno, 1985). The students receiving comments learned twice as fast as the control group, the achievement gap between male and female students disappeared, and student attitudes improved.

A study by Ruth Butler added a third condition, which gave students grades and comments -as this could be thought of as the best of both worlds (Butler, 1987, 1988). In this study, students who received grades only and those who received grades and comments scored equally badly, and the group that achieved at significantly higher levels was the comment only group. This showed that when students received a grade and a comment, they focused on only the grade. Butler found that both high-achieving (the top 25% GPA) and low-achieving (the bottom 25% GPA) fifth and sixth graders suffered deficits in performance and motivation in both graded conditions, compared with the students who received only diagnostic comments. Further research showed that that students only needed to think they were working for a grade to lose motivation, resulting in lower levels of achievement (Pulfrey, Butera, and Buchs, 2011).

The move from grades to diagnostic comments is a crucial one, and is a move that allows teachers to give students an amazing gift - the gift of their knowledge and insights about ways to improve.

Teachers, quite rightly, worry about the extra time this can take, as good teachers already work well beyond the hours they are paid for. My recommended solution is to assess less; if teachers replaced grading weekly with diagnostic comments given occasionally, they could spend the same amount of time, remove the fixed mindset messages of a grade, and provide students with insights that would propel them onto paths of higher achievement. Teachers who have made these changes see increases or no changes in test performance and significant increases in motivation and confidence.

When we give assessments to students we create an important opportunity. Well-crafted tasks and questions accompanied by clear feedback offer students a growth mindset pathway that helps them to know that they can learn to high levels, and, critically, how they can get there. Unfortunately most systems of assessment in U.S. classrooms do the opposite of this, communicating information to students that causes many of them to think they are a failure and they can never learn math. I have worked with teachers in recent years who have shifted their methods of assessment from standard tests with grades and scores to assessments that are focused upon giving students the information they need in order to learn well accompanied with growth mindset messages. This resulted in dramatic changes in their classroom environments.

Math anxiety, formerly commonplace among students, disappeared and was replaced by student self-confidence, which led to higher levels of motivation, engagement and achievement.

I am a strong supporter of teachers and know that the No Child Left Behind era stripped the professionalism and enthusiasm of many teachers as they were forced (and I choose that word carefully) to use teaching methods that they knew to be unhelpful. An important part of my work with teachers now is to help them regain their sense of professionalism. My aim in working with teachers is to help them see themselves as creators again, people who can design teaching environments infused with their own ideas for creative, engaging math. I have watched teachers come alive when they are encouraged in these ways.

In a new film by Vicki Abeles, director of Race to Nowhere (watch the trailer to learn more! ), her team interviewed the middle school students in a district I was working in, helping the teachers shift their teaching and assessment. In the film one girl, Delia, talks about getting an F for her homework in the previous year, and how it had caused her to stop trying in math and - shockingly - all of her classes across the school. In the interview she poignantly said:

"When I saw the F on my paper I felt like a nothing. I was failing in that class so I thought I may as well fail in all my other classes too. I didn't even try." Later in the film she talks about the change in her math class and how she now feels encouraged to do well. "I hated math," she says "I absolutely hated it, but now I have a connection with math, I'm open, I feel like I'm alive, I'm more energetic."

Beyond Measure Source: Image courtesy of Reel Link Films

Delia's use of the word "open" in describing how she felt about math is a sentiment I hear frequently from students when they are taught mathematics without the impending fear of low test scores and grades. But it goes further than assessment - when we teach creative, inquiry math students feel an intellectual freedom that is powerful. In interviews with 3rd graders who experienced number talks in class, I ask the students how they feel about number talks. The first thing young Dylan said in the interview was "I feel free." He went on to describe how the valuing of different mathematics strategies allowed him to feel he could work with mathematics in any way he wanted, to explore ideas and learn about numbers. The students' use of words such as 'free', and 'open' demonstrate the difference that is made when students work on growth mindset mathematics; this goes well beyond math achievement to an intellectual empowerment that will affect students throughout their lives (Boaler, 2015).

The perceptions students develop about their own potential affect their learning, their achievement, and, of equal importance, their motivation and effort-as Delia describes in the film. When she got an F in math, she gave up not only in math but also in all of her other classes; she felt like a failure. This is not an unusual response to grading.

When students are given scores that tell them they rank below other students, they often give up on school, deciding that they will never be able to learn and they take on the identity of an underperforming student.

The grades and scores given to students who are high achieving are just as damaging. Students develop the idea that they are an "A student" and begin a precarious fixed mindset learning path that makes them avoid harder work or challenges for fear that they will lose their A label. Such students often are devastated if they get a B or lower, for any of their work.

In another research study on grading Deevers found that students who were not given scores but instead given positive constructive feedback were more successful in their future work. He also, sadly, found that as students got older teachers gave less constructive feedback and more fixed grading. He found a clear and unsurprising relationship between teachers' assessment practices and student attitudes as students' beliefs about their own potential and the possibility of improving their learning declined steadily from 5th to 12th grade (Deevers, 2006).

We want students to be excited about and interested in their learning. When students develop interest in the ideas they are learning, they increase their motivation and their achievement. There is a large body of research that has studied two types of motivation. Intrinsic motivation comes from interest in the subject and ideas you are learning; extrinsic motivation is the motivation provided by the thought of getting better scores and grades. Because mathematics has been taught for decades as a performance subject, the students who are most motivated in math classrooms are usually those who are extrinsically motivated. One result of this is that students who feel positive about math class are usually only those students who are getting high scores and grades. Most of the teachers who believe in grades, use them because they think they motivate students to achieve. They do motivate some students - those who would probably achieve at high levels anyway, but they de-motivate the rest. Unfortunately the extrinsic motivation that the high achieving students develop is not helpful in the long term. Study after study shows that students who develop intrinsic motivation achieve at higher levels than those who develop extrinsic motivation (Pulfrey, Buchs, & Butera 2011; Lemos & Verissimo, 2014), and that intrinsic motivation to learn ideas motivates students to pursue subjects to higher levels and to stay in subjects rather than drop out (Stipek, 1993).

Assessment for Learning

A few years ago two professors from England - Paul Black and Dylan Wiliam -conducted a meta-analysis of hundreds of research studies on assessment. They found something amazing: a form of assessment so powerful, that if teachers used it the impact would be so great that it would raise the achievement of a country in international studies from the middle of the pack to a place in the top five. (Sir Paul Black and Professor Dylan Wiliam were both good colleagues of mine at London University; Paul Black was also my dissertation advisor and mentor.) Black and Wiliam found that if teachers were to use what is now called "assessment for learning" the positive impact would be far greater than other educational initiatives such as reductions in class size (Black, Harrison, Lee, Marshall, & Wiliam, 2002; Black & Wiliam, 1998a, 1998b). They published their findings in a small booklet that sold over 20,000 copies in the first few weeks in England. Assessment for Learning is now a national initiative in many countries; it has a huge research evidence base and it communicates growth mindset messages to students.

A little background will be helpful. There are two types of assessment-formative and summative. Formative assessment informs learning and is the essence of assessment for learning or A4L. Formative assessments are used to find out where students are in their learning so that teachers and students can determine what they need to know next. The purpose of summative assessment, in contrast, is to summarize a student's learning-to give a final account of how far a student has gotten, as an end point. One problem in the United States is that many teachers use summative assessment formatively; that is, they give students an end score or grade when they are still learning the materials. In mathematics classrooms teachers often use summative tests weekly and then move on to the next subject without waiting to see what the tests reveal. In A4L, students become knowledgeable about what they know, what they need to know, and ways to close the gap between the two. Students are given information about their flexible and growing learning pathways that contributes to their development of a growth mathematics mindset. In the weeks and months that students are learning in a course, it is very important to assess formatively, not summatively. Further, the A4L approach, which can also be thought of as assessing for a growth mindset, offers a range of strategies and methods.

One important principle of A4L is that it teaches students responsibility for their own learning.

At its core A4L is about empowering students to become autonomous learners who can self-regulate and determine what they most need to learn and who know ways to improve their learning. Assessment for learning can be thought of as having three parts: (1) clearly communicating to students what they have learned, (2) helping students become aware of where they are in their learning journey and where they need to reach, and (3) giving students information on ways to close the gap between where they are now and where they need to be.

Developing Student Self-Awareness and Responsibility

The most powerful learners are those who are reflective, who engage in metacognition - thinking about what they know - and who take control of their own learning (White & Frederiksen, 1998). A major failing of traditional mathematics classes is that students rarely have much idea of what they are learning or where they are in the broader learning landscape. They focus upon methods to remember but often do not even know what area of mathematics they are working on. I have visited math classes many times and stopped at students' desks to ask them what they are working on. In many cases students answer with the question they are working on. Many of my interactions have gone something like this:

JB: What are you working on?

Student: Exercise 2

JB: So what are you actually doing, what math are you working on?

Student: Oh, I'm sorry - question 4

Students are often not thinking about the area of mathematics they are learning, they do not have an idea of the mathematical goals for their learning, and they expect to be passively led through work with teachers telling them whether they are "getting it" or not. Alice White, an assessment expert, likens this situation to workers on a ship who are given jobs to do each day but don't have any idea where the ship is travelling to.

One research study, conducted by Barbara White and John Frederikson (1998), powerfully illustrated the importance of reflection. The researchers studied twelve classes of seventh grade students learning physics. The researchers divided the students into experimental and control groups. All groups were taught a unit on force and motion. The control groups then spent some of each lesson discussing the work whereas the experimental group spent some of each lesson engaging in self- and peer assessment, considering criteria for the science they were learning. The results of the study were dramatic. The experimental groups outperformed the control groups on three different assessments. The previously low-achieving students made the greatest gains. After they spent time considering the science criteria and assessing themselves against them, they began to achieve at the same levels as the highest achievers. The middle school students even scored at higher levels than AP physics students on tests of high school physics. The researchers concluded that a large part of the students' previous low achievement came not from the fact that they lacked ability but that they had not previously known what they should really be focusing upon.

This is true for many students and this is why it is so important to communicate to students what they should be learning. This both helps the students know what success is, and starts a self-reflection process that is an invaluable tool for learning.

There are many strategies for encouraging students to become more aware of the mathematics they are learning and their place in the learning process, several of which are in Jo's upcoming book, Mathematical Mindsets.

Conclusion

Tests and grading can lead students to disengage from mathematics and even school itself. Assessment for Learning, by contrast, represent an incredible opportunity for teachers to provide students with information on their learning that accelerates pathways to success and gives students powerful growth mindset messages about mathematics and learning. Research shows that a change from grading and testing to Assessment for Learning has a powerful impact on students' achievement, self-beliefs, motivation and future learning pathways.

By using assessments to empower students to learn and grow, we can help our students develop positive attitudes towards mathematics and themselves.

This article contains excerpts from Jo Boaler's new book, Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching
Why are we doing this??

"Many students have felt a sense of anxiety/stress when it comes to learning math. Many fear that math is something that you are either good at or not. There is also this sense of pressure that students feel because they view math as something that you are either right or wrong. What many students struggle to understand is that the beauty of math is not so much on the solution but the thinking and creativity that goes into trying to solve the problem. In addition to this, students constantly feel pressure to perform due to grades. The anxiety students feel with regards to grades is like a dark cloud that hangs over them. No matter what is going on in class, or how much they are enjoying the topic they are learning, there is still this looming pressure to perform in order to receive the highest grade possible. Whether it's higher achieving students or lower achieving students, many students' focus is on performing in order to achieve a certain grade. What this has caused is a focus on performing (memorizing) rather than a focus on learning. We want students to enjoy learning. Isn't that the purpose of going to school? With this being, last year we tried to make a shift by encouraging students to take risks and not be afraid to make mistakes. However, when it came time to grading tests, students were still penalized for having wrong answers. How can we encourage students to not be afraid to make mistakes and then penalize them for doing it on a test? This sent mixed messages and has been on our minds over the course of the summer. After careful thought, we're curious to try this. What would happen if everyone got an A at the beginning of the year?! Would students no longer feel pressure or worry about their performance? Would this free students up to be more creative and to take more risks? Would this lead to students being more curious and wanting to go deeper and to understand why things work the way they do? Would this lead to more of an intrinsic motivation of students wanting to do well and to learn? Let's Find Out..." -An extract from letter to parents, from High Tech High Math Department, Chula Vista, San Diego.

References

Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2002). Working inside the black box: Assessment for learning in the classroom. London: Department of Education & Professional Studies, King's College.

Black, P. J., & Wiliam, D. (1998a, October). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 139-148.

Black, P. J., & Wiliam, D. (1998b). Assessment and classroom learning. Assessment in Education, 5(1), 7-74.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 41-62.

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest and performance. Journal of Educational Psychology, 79, 474-482.

Butler, R. (1988). Enhancing and undermining intrinsic motivation: The effects of task-involving and ego-involving evaluation on interest and performance. British Journal of Educational Psychology, 58, 1-14.

Deevers, M. (2006). Linking classroom assessment practices with student motivation in mathematics. Paper presented at the American Educational Research Association, San Francisco.

Elawar, M. C., & Corno, L. (1985). A factorial experiment in teachers' written feedback on student homework: Changing teacher behavior a little rather than a lot. Journal of Educational Psychology, 77(2), 162-173.

Kohn, A. (2011, November). The case against grades. Retrieved from https://www.alfiekohn.org/ article/case-grades/

Lemos, M. S., & Veríssimo, L. (2014). The relationships between intrinsic motivation, extrinsic motivation, and achievement, along elementary school. Procedia - Social and Behavioral Sciences, 112, 930-938.

Pulfrey, C., Buchs, C., & Butera, F. (2011). Why grades engender performance-avoidance goals: The mediating role of autonomous motivation. Journal of Educational Psychology, 103(3), 683-700. Retrieved from https://www.researchgate.net/profile/Fabrizio_Butera/publication/232450947_Why_grades_engender_performanceavoidance_goals_The_mediating_role_of_autonomous_ motivation/links/02bfe50ed4ebfd0670000000.pdf

Stipek, D. J. (1993). Motivation to learn: Integrating theory and practice. New York: Pearson.

White, B. Y., & Frederiksen, J. R. (1998). Inquiry, modeling, and metacognition: Making science accessible to all students. Cognition and Instruction, 16(1), 3-118.

Introduction

The complex ways that children understand mathematics is fascinating to me. Students ask questions, see ideas, draw representations, connect methods, justify, and reason in all sorts of different ways. But recent years have seen all of these different nuanced complexities of student understanding reduced to single numbers and letters that are used to judge students' worth. Teachers are encouraged to test and grade students, to a ridiculous and damaging degree; students start to define themselves - and mathematics - in terms of letters and numbers. Such crude representations of understanding vastly under-describe and in many cases misrepresent children's knowledge.

In the United States students are over-tested to a degree that is nothing short of remarkable, particularly in mathematics. For many years students have been judged by narrow, procedural mathematics questions presented with multiple-choice answers. The knowledge needed for success on such tests is so far from the adaptable, critical and analytical thinking that students need in the modern world that leading employers such as Google have declared that they are no longer interested in students' test performance as it in no way predicts success in the workplace (Bryant, 2013).

One critical principal of good testing is that it assesses what is important.

For many decades in the United States, tests have assessed what is easy to test instead of important and valuable mathematics. This has meant that mathematics teachers have had to focus their teaching on narrow procedural mathematics, not the broad, creative and growth mathematics that is so important. The new common core assessments promise something different, with few multiple-choice questions and assessments of problem solving, but they are being met with considerable opposition from parents.

The damage does not only end with standardized testing, for math teachers are led to believe they should use tests in classrooms that mimic low quality standardized tests, even when they know the tests assess narrow mathematics.

They do this to help prepare students for later success. Some teachers, particularly at the high school level, test weekly or even more frequently. Mathematics teachers feel the need to test regularly, more than any other subject, because they have come to believe that mathematics is about performance, and they usually don't consider the role that tests play in shaping students' views of mathematics and themselves. Many mathematics teachers I know introduce a class with a test, which gives a huge performance message to students on the first day of class, the time when it is so important to be giving growth messages about mathematics and learning.

Finland is one of the highest-scoring countries in the world on international mathematics tests yet students do not take any tests in school. Instead teachers use their rich understanding of their students' knowledge gained through teaching to report to parents and make judgments about student work. In a longitudinal study I conducted in England students worked on open-ended projects for three years (ages 13 through 16) leading to national standardized examinations. They did not take tests nor was their work graded. Students encountered short questions assessing procedures in the last few weeks before the examination, as the teachers gave them examination papers to work through. Despite the students' lack of familiarity with examination questions or working under timed conditions of any kind, they scored at significantly higher levels than a matched cohort of students who spent three years working through questions similar to the national exam questions and taking frequent tests (Boaler, 1998, 2015). The students from the problem solving school did so well in the standardized national exam because they had been taught to believe in their own capabilities; they had been given helpful, diagnostic information on their learning; and they had learned that they could solve any question, as they were mathematical problem solvers.

Students with no experience of examinations and tests can score at the highest levels because the most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problem-solving mathematical tools to equip them for any mathematical situation.

The testing regime of the last decade has had a large, negative impact on students but it does not end with testing; the communication of grades to students is similarly negative. When students are given a percentage or grade, they can do little else besides compare it to others around them, with half or more deciding that they are not as good as others. This is known as "ego feedback," a form of feedback that has been found to damage learning. Sadly when students are given frequent test scores and grades they start to see themselves as those scores and grades. They do not regard the scores as an indicator of their learning or of what they need to do to achieve, but as indicators of who they are as people. The fact that US students commonly describe themselves saying "I'm an A student" or "I'm a D student" illustrates the ways students define themselves by grades. Ray McDermott wrote a compelling paper about the capturing of a child by a learning disability, describing the ways a student who thought and worked differently was given a label and was then defined by that label (McDermott, 1993). I could give a similar argument about the capturing of students by grades and test scores.

Students describe themselves as A or D students because they have grown up in a performance culture that valued frequent testing and grading, rather than persistence, courage, or problem solving. The traditional methods of assessing students that have been used across the United States for decades were designed in a less enlightened age (Kohn, 2011) when it was believed that grades and test scores would motivate students, and that the information they provided on students' achievement would be useful.

Now we know that grades and test scores demotivate rather than motivate students and that they communicate fixed and damaging messages to students that result in lower achievement in classrooms.

In studies of grading and alternatives to grading researchers have produced consistent results. Study after study shows that grading reduces the achievement of students. Elawar and Corno, for example, contrasted the ways teachers responded to math homework in sixth grade, with half of the students receiving grades and the other half receiving diagnostic comments without a grade (Elawar & Corno, 1985). The students receiving comments learned twice as fast as the control group, the achievement gap between male and female students disappeared, and student attitudes improved.

A study by Ruth Butler added a third condition, which gave students grades and comments -as this could be thought of as the best of both worlds (Butler, 1987, 1988). In this study, students who received grades only and those who received grades and comments scored equally badly, and the group that achieved at significantly higher levels was the comment only group. This showed that when students received a grade and a comment, they focused on only the grade. Butler found that both high-achieving (the top 25% GPA) and low-achieving (the bottom 25% GPA) fifth and sixth graders suffered deficits in performance and motivation in both graded conditions, compared with the students who received only diagnostic comments. Further research showed that that students only needed to think they were working for a grade to lose motivation, resulting in lower levels of achievement (Pulfrey, Butera, and Buchs, 2011).

The move from grades to diagnostic comments is a crucial one, and is a move that allows teachers to give students an amazing gift - the gift of their knowledge and insights about ways to improve.

Teachers, quite rightly, worry about the extra time this can take, as good teachers already work well beyond the hours they are paid for. My recommended solution is to assess less; if teachers replaced grading weekly with diagnostic comments given occasionally, they could spend the same amount of time, remove the fixed mindset messages of a grade, and provide students with insights that would propel them onto paths of higher achievement. Teachers who have made these changes see increases or no changes in test performance and significant increases in motivation and confidence.

When we give assessments to students we create an important opportunity. Well-crafted tasks and questions accompanied by clear feedback offer students a growth mindset pathway that helps them to know that they can learn to high levels, and, critically, how they can get there. Unfortunately most systems of assessment in U.S. classrooms do the opposite of this, communicating information to students that causes many of them to think they are a failure and they can never learn math. I have worked with teachers in recent years who have shifted their methods of assessment from standard tests with grades and scores to assessments that are focused upon giving students the information they need in order to learn well accompanied with growth mindset messages. This resulted in dramatic changes in their classroom environments.

Math anxiety, formerly commonplace among students, disappeared and was replaced by student self-confidence, which led to higher levels of motivation, engagement and achievement.

I am a strong supporter of teachers and know that the No Child Left Behind era stripped the professionalism and enthusiasm of many teachers as they were forced (and I choose that word carefully) to use teaching methods that they knew to be unhelpful. An important part of my work with teachers now is to help them regain their sense of professionalism. My aim in working with teachers is to help them see themselves as creators again, people who can design teaching environments infused with their own ideas for creative, engaging math. I have watched teachers come alive when they are encouraged in these ways.

In a new film by Vicki Abeles, director of Race to Nowhere (watch the trailer to learn more! ), her team interviewed the middle school students in a district I was working in, helping the teachers shift their teaching and assessment. In the film one girl, Delia, talks about getting an F for her homework in the previous year, and how it had caused her to stop trying in math and - shockingly - all of her classes across the school. In the interview she poignantly said:

"When I saw the F on my paper I felt like a nothing. I was failing in that class so I thought I may as well fail in all my other classes too. I didn't even try." Later in the film she talks about the change in her math class and how she now feels encouraged to do well. "I hated math," she says "I absolutely hated it, but now I have a connection with math, I'm open, I feel like I'm alive, I'm more energetic."

Beyond Measure Source: Image courtesy of Reel Link Films

Delia's use of the word "open" in describing how she felt about math is a sentiment I hear frequently from students when they are taught mathematics without the impending fear of low test scores and grades. But it goes further than assessment - when we teach creative, inquiry math students feel an intellectual freedom that is powerful. In interviews with 3rd graders who experienced number talks in class, I ask the students how they feel about number talks. The first thing young Dylan said in the interview was "I feel free." He went on to describe how the valuing of different mathematics strategies allowed him to feel he could work with mathematics in any way he wanted, to explore ideas and learn about numbers. The students' use of words such as 'free', and 'open' demonstrate the difference that is made when students work on growth mindset mathematics; this goes well beyond math achievement to an intellectual empowerment that will affect students throughout their lives (Boaler, 2015).

The perceptions students develop about their own potential affect their learning, their achievement, and, of equal importance, their motivation and effort-as Delia describes in the film. When she got an F in math, she gave up not only in math but also in all of her other classes; she felt like a failure. This is not an unusual response to grading.

When students are given scores that tell them they rank below other students, they often give up on school, deciding that they will never be able to learn and they take on the identity of an underperforming student.

The grades and scores given to students who are high achieving are just as damaging. Students develop the idea that they are an "A student" and begin a precarious fixed mindset learning path that makes them avoid harder work or challenges for fear that they will lose their A label. Such students often are devastated if they get a B or lower, for any of their work.

In another research study on grading Deevers found that students who were not given scores but instead given positive constructive feedback were more successful in their future work. He also, sadly, found that as students got older teachers gave less constructive feedback and more fixed grading. He found a clear and unsurprising relationship between teachers' assessment practices and student attitudes as students' beliefs about their own potential and the possibility of improving their learning declined steadily from 5th to 12th grade (Deevers, 2006).

We want students to be excited about and interested in their learning. When students develop interest in the ideas they are learning, they increase their motivation and their achievement. There is a large body of research that has studied two types of motivation. Intrinsic motivation comes from interest in the subject and ideas you are learning; extrinsic motivation is the motivation provided by the thought of getting better scores and grades. Because mathematics has been taught for decades as a performance subject, the students who are most motivated in math classrooms are usually those who are extrinsically motivated. One result of this is that students who feel positive about math class are usually only those students who are getting high scores and grades. Most of the teachers who believe in grades, use them because they think they motivate students to achieve. They do motivate some students - those who would probably achieve at high levels anyway, but they de-motivate the rest. Unfortunately the extrinsic motivation that the high achieving students develop is not helpful in the long term. Study after study shows that students who develop intrinsic motivation achieve at higher levels than those who develop extrinsic motivation (Pulfrey, Buchs, & Butera 2011; Lemos & Verissimo, 2014), and that intrinsic motivation to learn ideas motivates students to pursue subjects to higher levels and to stay in subjects rather than drop out (Stipek, 1993).

Assessment for Learning

A few years ago two professors from England - Paul Black and Dylan Wiliam -conducted a meta-analysis of hundreds of research studies on assessment. They found something amazing: a form of assessment so powerful, that if teachers used it the impact would be so great that it would raise the achievement of a country in international studies from the middle of the pack to a place in the top five. (Sir Paul Black and Professor Dylan Wiliam were both good colleagues of mine at London University; Paul Black was also my dissertation advisor and mentor.) Black and Wiliam found that if teachers were to use what is now called "assessment for learning" the positive impact would be far greater than other educational initiatives such as reductions in class size (Black, Harrison, Lee, Marshall, & Wiliam, 2002; Black & Wiliam, 1998a, 1998b). They published their findings in a small booklet that sold over 20,000 copies in the first few weeks in England. Assessment for Learning is now a national initiative in many countries; it has a huge research evidence base and it communicates growth mindset messages to students.

A little background will be helpful. There are two types of assessment-formative and summative. Formative assessment informs learning and is the essence of assessment for learning or A4L. Formative assessments are used to find out where students are in their learning so that teachers and students can determine what they need to know next. The purpose of summative assessment, in contrast, is to summarize a student's learning-to give a final account of how far a student has gotten, as an end point. One problem in the United States is that many teachers use summative assessment formatively; that is, they give students an end score or grade when they are still learning the materials. In mathematics classrooms teachers often use summative tests weekly and then move on to the next subject without waiting to see what the tests reveal. In A4L, students become knowledgeable about what they know, what they need to know, and ways to close the gap between the two. Students are given information about their flexible and growing learning pathways that contributes to their development of a growth mathematics mindset. In the weeks and months that students are learning in a course, it is very important to assess formatively, not summatively. Further, the A4L approach, which can also be thought of as assessing for a growth mindset, offers a range of strategies and methods.

One important principle of A4L is that it teaches students responsibility for their own learning.

At its core A4L is about empowering students to become autonomous learners who can self-regulate and determine what they most need to learn and who know ways to improve their learning. Assessment for learning can be thought of as having three parts: (1) clearly communicating to students what they have learned, (2) helping students become aware of where they are in their learning journey and where they need to reach, and (3) giving students information on ways to close the gap between where they are now and where they need to be.

Developing Student Self-Awareness and Responsibility

The most powerful learners are those who are reflective, who engage in metacognition - thinking about what they know - and who take control of their own learning (White & Frederiksen, 1998). A major failing of traditional mathematics classes is that students rarely have much idea of what they are learning or where they are in the broader learning landscape. They focus upon methods to remember but often do not even know what area of mathematics they are working on. I have visited math classes many times and stopped at students' desks to ask them what they are working on. In many cases students answer with the question they are working on. Many of my interactions have gone something like this:

JB: What are you working on?

Student: Exercise 2

JB: So what are you actually doing, what math are you working on?

Student: Oh, I'm sorry - question 4

Students are often not thinking about the area of mathematics they are learning, they do not have an idea of the mathematical goals for their learning, and they expect to be passively led through work with teachers telling them whether they are "getting it" or not. Alice White, an assessment expert, likens this situation to workers on a ship who are given jobs to do each day but don't have any idea where the ship is travelling to.

One research study, conducted by Barbara White and John Frederikson (1998), powerfully illustrated the importance of reflection. The researchers studied twelve classes of seventh grade students learning physics. The researchers divided the students into experimental and control groups. All groups were taught a unit on force and motion. The control groups then spent some of each lesson discussing the work whereas the experimental group spent some of each lesson engaging in self- and peer assessment, considering criteria for the science they were learning. The results of the study were dramatic. The experimental groups outperformed the control groups on three different assessments. The previously low-achieving students made the greatest gains. After they spent time considering the science criteria and assessing themselves against them, they began to achieve at the same levels as the highest achievers. The middle school students even scored at higher levels than AP physics students on tests of high school physics. The researchers concluded that a large part of the students' previous low achievement came not from the fact that they lacked ability but that they had not previously known what they should really be focusing upon.

This is true for many students and this is why it is so important to communicate to students what they should be learning. This both helps the students know what success is, and starts a self-reflection process that is an invaluable tool for learning.

There are many strategies for encouraging students to become more aware of the mathematics they are learning and their place in the learning process, several of which are in Jo's upcoming book, Mathematical Mindsets.

Conclusion

Tests and grading can lead students to disengage from mathematics and even school itself. Assessment for Learning, by contrast, represent an incredible opportunity for teachers to provide students with information on their learning that accelerates pathways to success and gives students powerful growth mindset messages about mathematics and learning. Research shows that a change from grading and testing to Assessment for Learning has a powerful impact on students' achievement, self-beliefs, motivation and future learning pathways.

By using assessments to empower students to learn and grow, we can help our students develop positive attitudes towards mathematics and themselves.

This article contains excerpts from Jo Boaler's new book, Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching
Why are we doing this??

"Many students have felt a sense of anxiety/stress when it comes to learning math. Many fear that math is something that you are either good at or not. There is also this sense of pressure that students feel because they view math as something that you are either right or wrong. What many students struggle to understand is that the beauty of math is not so much on the solution but the thinking and creativity that goes into trying to solve the problem. In addition to this, students constantly feel pressure to perform due to grades. The anxiety students feel with regards to grades is like a dark cloud that hangs over them. No matter what is going on in class, or how much they are enjoying the topic they are learning, there is still this looming pressure to perform in order to receive the highest grade possible. Whether it's higher achieving students or lower achieving students, many students' focus is on performing in order to achieve a certain grade. What this has caused is a focus on performing (memorizing) rather than a focus on learning. We want students to enjoy learning. Isn't that the purpose of going to school? With this being, last year we tried to make a shift by encouraging students to take risks and not be afraid to make mistakes. However, when it came time to grading tests, students were still penalized for having wrong answers. How can we encourage students to not be afraid to make mistakes and then penalize them for doing it on a test? This sent mixed messages and has been on our minds over the course of the summer. After careful thought, we're curious to try this. What would happen if everyone got an A at the beginning of the year?! Would students no longer feel pressure or worry about their performance? Would this free students up to be more creative and to take more risks? Would this lead to students being more curious and wanting to go deeper and to understand why things work the way they do? Would this lead to more of an intrinsic motivation of students wanting to do well and to learn? Let's Find Out..." -An extract from letter to parents, from High Tech High Math Department, Chula Vista, San Diego.

References

Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2002). Working inside the black box: Assessment for learning in the classroom. London: Department of Education & Professional Studies, King's College.

Black, P. J., & Wiliam, D. (1998a, October). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 139-148.

Black, P. J., & Wiliam, D. (1998b). Assessment and classroom learning. Assessment in Education, 5(1), 7-74.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 41-62.

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest and performance. Journal of Educational Psychology, 79, 474-482.

Butler, R. (1988). Enhancing and undermining intrinsic motivation: The effects of task-involving and ego-involving evaluation on interest and performance. British Journal of Educational Psychology, 58, 1-14.

Deevers, M. (2006). Linking classroom assessment practices with student motivation in mathematics. Paper presented at the American Educational Research Association, San Francisco.

Elawar, M. C., & Corno, L. (1985). A factorial experiment in teachers' written feedback on student homework: Changing teacher behavior a little rather than a lot. Journal of Educational Psychology, 77(2), 162-173.

Kohn, A. (2011, November). The case against grades. Retrieved from https://www.alfiekohn.org/ article/case-grades/

Lemos, M. S., & Veríssimo, L. (2014). The relationships between intrinsic motivation, extrinsic motivation, and achievement, along elementary school. Procedia - Social and Behavioral Sciences, 112, 930-938.

Pulfrey, C., Buchs, C., & Butera, F. (2011). Why grades engender performance-avoidance goals: The mediating role of autonomous motivation. Journal of Educational Psychology, 103(3), 683-700. Retrieved from https://www.researchgate.net/profile/Fabrizio_Butera/publication/232450947_Why_grades_engender_performanceavoidance_goals_The_mediating_role_of_autonomous_ motivation/links/02bfe50ed4ebfd0670000000.pdf

Stipek, D. J. (1993). Motivation to learn: Integrating theory and practice. New York: Pearson.

White, B. Y., & Frederiksen, J. R. (1998). Inquiry, modeling, and metacognition: Making science accessible to all students. Cognition and Instruction, 16(1), 3-118.

# data[].attributes.attachment_link.title
Aligning Assessment to Brain Science - YouCubed

# data[].attributes.attachment_link.description
Introduction The complex ways that children understand mathematics is fascinating to me. Students ask questions, see ideas, draw representations, connect methods, justify, and reason in all sorts of different ways. [...]

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This page gives advice and resources to help teachers move to a method of assessment that can offer students a changed relationship with their learning. It is called "Assessment for Learning" and it is both described in the "aligning assessment to brain science" article and outlined in ten strategies from David Wees. Our performance and testing culture harms students and the continual assessment of students through tests, grades and homework is part of why so many students have a negative relationship with mathematics. There is a much better way to collect evidence on students learning, that also helps them learn, that we recommend. My book: 'Mathematical Mindsets' also includes a chapter on assessment that shares the work of innovative teachers who changed their assessment methods with impressive results.

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Assessment & Grading Archives - YouCubed

# data[].attributes.attachment_link.description
This page gives advice and resources to help teachers move to a method of assessment that can offer students a changed relationship with their learning. It is called "Assessment for Learning" and it is both described in the "aligning assessment to brain science" article and outlined in ten strategies from David Wees.

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Get pumped for learning! Stay in shape! SEt new goals!

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HESD Mission Control for Distance Learning

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HESD Mission Control for Distance Learning