# data[].attributes.attachment_link.readable_text
Improving Science Education Through Three-Dimensional Learning

Quick Search Next Generation Science Standards

THE THREE DIMENSIONS OF SCIENCE LEARNING

Within the Next Generation Science Standards (NGSS), there are three distinct and equally important dimensions to learning science. These dimensions are combined to form each standard-or performance expectation-and each dimension works with the other two to help students build a cohesive understanding of science over time.

Crosscutting Concepts help students explore connections across the four domains of science, including Physical Science, Life Science, Earth and Space Science, and Engineering Design.

When these concepts, such as "cause and effect", are made explicit for students, they can help students develop a coherent and scientifically-based view of the world around them.

Science and Engineering Practices describe what scientists do to investigate the natural world and what engineers do to design and build systems. The practices better explain and extend what is meant by "inquiry" in science and the range of cognitive, social, and physical practices that it requires. Students engage in practices to build, deepen, and apply their knowledge of core ideas and crosscutting concepts.

Disciplinary Core Ideas (DCIs) are the key ideas in science that have broad importance within or across multiple science or engineering disciplines. These core ideas build on each other as students progress through grade levels and are grouped into the following four domains: Physical Science, Life Science, Earth and Space Science, and Engineering.

More about 3D Learning

GET TO KNOW

The Next Generation Science Standards (NGSS) are K-12 science content standards. Standards set the expectations for what students should know and be able to do. The NGSS were developed by states to improve science education for all students.

A goal for developing the NGSS was to create a set of research-based, up-to-date K-12 science standards. These standards give local educators the flexibility to design classroom learning experiences that stimulate students' interests in science and prepares them for college, careers, and citizenship.

Get to Know the Standards

IMPLEMENT

The NGSS call for a three-dimensional approach to K-12 science instruction. This represents a significant transition from previous state standards. That's why effective implementation demands a great deal of collaboration and patience among states, districts, schools, teachers, and students.

Thoughtful and coordinated approaches to implementation will enable educators to inspire future generations of scientifically literate students. That is the vision of the NGSS. This website provides a range of high-quality resources that empower educators, administrators, parents, and the general public to help bring this vision to life.

Find Tools and Resources

WHY

Science-and therefore science education-is central to the lives of all Americans.

A high-quality science education means that students will develop an in-depth understanding of content and develop key skills-communication, collaboration, inquiry, problem solving, and flexibility-that will serve them throughout their educational and professional lives.

# data[].attributes.attachment_link.title
Home Page | Next Generation Science Standards

# data[].attributes.attachment_link.description
Improving Science Education Through Three-Dimensional Learning Quick Search Next Generation Science Standards Within the Next Generation Science Standards (NGSS), there are three distinct and equally important dimensions to learning science. These dimensions are combined to form each standard-or performance expectation-and each dimension works with the other two to help students build a cohesive understanding of science over time.

# data[].attributes.attachment_link.description
The Coherence Map shows the connections between Common Core State Standards for Mathematics.

# data[].attributes.attachment_link.readable_text
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council's report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy).

Standards in this domain:

CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x ² + 9 x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3( x - y) ² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y - 2)/( x - 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x - 1)( x + 1), ( x - 1)( x ² + x + 1), and ( x - 1)( x ³ + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

# data[].attributes.attachment_link.title
Standards for Mathematical Practice | Common Core State Standards Initiative

# data[].attributes.attachment_link.description
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

# data[].attributes.attachment_link.readable_text
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

The progressions can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. They would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Progressions documents also provide a transmission mechanism between mathematics education research and standards.

For more information on the project, please visit the project website.

# data[].attributes.attachment_link.title
Progressions Documents for the Common Core State Standards for Mathematics

# data[].attributes.attachment_link.description
The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics.

# data[].attributes.attachment_link.title
Common Core Instruction: Exploring Dividing Decimals to the Hundredths

# data[].attributes.attachment_link.title
Time for Class - Mr. Adler Science Lesson

# data[].attributes.attachment_link.title
Before During After Lesson Plan Template

# data[].attributes.attachment_link.title
Math Lesson Plan Picture Graphs to Bar Graphs

# data[].attributes.attachment_link.description
(1) See Subject-Specific Pedagogical Skills in Section 2 for reference.

# data[].attributes.body
Various videos of multiple ways to use CRA in math.

# data[].attributes.attachment_link.provider_name
iExploreScience | Because science is meant to be discovered.

# data[].attributes.attachment_link.title
Integrate Engineering: A How To Guide For Science Teachers | iExploreScience

# data[].attributes.attachment_link.description
Because science is meant to be discovered.

# data[].attributes.attachment_link.readable_text
There are a lot of misconceptions out there about Guided Math. I've seen many things labeled "Guided Math" that truly have me scratching my head. With so much (mis)information out there, I thought I'd take some time to help break down what Guided Math *IS* and what Guided Math *IS NOT*. Let's dive in!

Guided Math IS NOT:

Guided Math is NOT teaching the same lesson to students in small groups.

While teaching content in small groups is important and allows you more face time with students, Guided Math is NOT teaching the same lesson multiple times to different groups.

Why not? Guided Math focuses on what Vygotsky called a student's Zone of Proximal Development, which is individualized to each student. The model of teaching the same content to different groups doesn't allow the teachers to meet students at their instructional level. Instead, the teacher is still "teaching to the middle," but in a smaller group setting.

Guided Math is NOT students rotating through the same centers

Centers are a great way to give students time to practice skills independently! Especially while you are also trying to pull those small math groups. However, Guided Math is not setting up weekly centers (or something similar) where all students complete the same centers throughout the week.

Why not? Guided Math focuses on differentiating student work, and ensuring that students are working on skills at their instructional level. Just like with reading, the most meaningful gains come when students are pushed *just slightly* above their independent level. This can't happen when we aren't strategic about the centers we choose, and instead have everyone rotate through the same activities.

Guided Math is NOT Teaching a Mini Lesson and checking in with students individually

Providing common math experiences is awesome, so mini-lessons and Number Talks are very much a part of Guided Math. However, Guided Math is not teaching mini-lessons and then following up with individual students as they engage in independent work/centers.

Why not? The mini-lesson is a way to establish a community of mathematicians, and allow students to hear from one another, practice mathematical discourse, and engage in common experiences around math topics. But, the mini-lesson is not the only direct instruction that takes place in a Guided Math model.

Guided Math *has* to include targeted small group instruction beyond the mini-lesson to ensure that we are meeting students where they are in their understanding of a concept.

SO... What IS Guided Math?

Chances are, if you are familiar with Guided Reading, you have a great foundation for understanding Guided Math. Why? Both models follow very similar blueprints. They are both based in meeting students at their instructional level via targeted small group instruction, followed up with practicing those same skills at independent/partner/small group centers.

You might be asking, okay... then what does Guided Math look like? Here are some key components:

Use Assessments to Find Instructional Levels:

Just like with reading, Guided Math utilizes assessments (formal and informal) to guide instruction (hence "GUIDED" math). Those assessments guide targeted small group instruction as well as the selection of centers and independent work students engage with while not working directly with a teacher.

Create Flexible Learning Groups

Flexible grouping is also a key component of Guided Math. It uses assessments to regularly switch groupings in order to work within a student's zone of proximal development. We know student growth is not linear. This is why groupings are constantly changing as students gain and make meaning of new knowledge that pushes them forward.

Teach Targeted Lessons to Small Groups

Within the flexible groups mentioned above, teachers are working to push students at their instructional level within a concept. Each group should be engaging in a targeted lesson that is meant to bring students forward from where they are in their understanding.

Students Work Independently at Targeted Centers

While the teacher is pulling small groups, the rest of the class is engaged in targeted center work. That means that each group will be working through different centers that extend the learning they worked on within their small group time.

I know that sounds like a lot of extra work, but it is actually much less! You will most often be using the same materials for each group, just at different times to reflect their current understanding.

Built on Strong Structures and Routines

Just like the Guided Reading model, Guided Math is only made possible by creating strong structures and routines. This means that before any instruction truly begins, students need to have internalized the expectations or how to engage in learning within a whole group, small group, and independent center, AND how to transition between those settings.

It takes front loading and consistency, but it is SO WORTH the effort! Why? Because when you are able to work with a small group of learners and provide powerful lessons that push their understanding ALL WHILE the rest of your group is engaged in meaningful, independent learning, there is NOTHING BETTER!

Are you looking to learn EVEN more about making Guided Math work in your classroom?

Be sure to get on the waitlist for my online professional development course called Guided Math Academy.

Guided Math Academy is an interactive six-week course for teachers who want to learn how to teach ALL their students WITHOUT adding hours of prep! It includes a step-by-step blueprint, Q&A sessions, and incredible bonuses to help you AND your students.

This course opens every June. Be sure to get on the waitlist so you don't miss out!

# data[].attributes.attachment_link.description
There are a lot of misconceptions about Guided Math. I'm taking some time to break down what Guided Math *IS* and what Guided Math *IS NOT*.

# data[].attributes.attachment_link.title
Integrating science and engineering practices: outcomes from a collaborative professional development - International Journal of STEM Education

# data[].attributes.attachment_link.description
Background The Next Generation Science Standards accentuate engineering design along with scientific inquiry, emphasizing the relationship between scientific investigations and engineering design in solving problems and devising new ideas and technologies. The goal is for students to realize the importance of science and engineering in innovation and in solving many of today’s challenges. The Next Generation Science Standards contends that a working knowledge and practicality of engineering design prepares students for embracing the challenges of the future. To support students in developing these capabilities, teachers are tasked with the responsibility of facilitating science instruction that integrates science and engineering practices. This is a challenge since a majority of them have little to no understanding of engineering applications. Results An interdisciplinary team, consisting of science education and mechanical engineering faculty and doctoral students from each discipline, and science, mathematics, and career and technical curriculum supervisors, collaborated with middle school science, mathematics, and career and technical education teachers to develop a framework for integrating engineering practices into their curricula. The exploratory nature of the project, and instructional outcomes with their students, supported teachers in developing an understanding and value for science and engineering practices. As a result, they were motivated to critique and revise their practices, aiming to develop and implement instruction that they perceived as beneficial to their students. Conclusion With the surge in emphasis on preparing K-12 students for the STEM workforce, initiatives devoted to exposing teachers and students to STEM applications have also increased. The findings from this study could be useful for informing these initiatives, since they reveal the learning experiences of the teachers while processing instructional strategies for integrating science and engineering practices into their curriculum. The findings highlight factors that motivated these teachers to reform their instructional practices, as well as their deliberations while endeavoring to assimilate the strategies into their curricular activities.

# data[].attributes.attachment_link.title
Guided Math Small Groups in First Grade - Happy Hearts in 1st

# data[].attributes.attachment_link.description
Guided Math Small Groups have been such a game changer in my 1st grade classroom! I LOVE the opportunity for differentiation and all of the light bulb moments! 💡💗 Small…

# data[].attributes.attachment_link.title
Guided Math with Dr. Nicki - What is Guided Math?

# data[].attributes.attachment_link.readable_text
Online manipulatives bring all of the learning and fun without the mess. Our math virtual manipulatives help students grasp abstract concepts through modeling. Literacy virtual manipulatives create a fun space for practicing spelling, rhyming, and more!

# data[].attributes.attachment_link.title
Virtual Manipulatives | Mathematical Modelling | Toy Theater

# data[].attributes.attachment_link.description
Original teaching tools aimed at supporting teachers in the elementary school classroom. Two color counters, 3D dice, probability spinners, graph builders, fraction bars, base ten blocks, and more!

# data[].attributes.attachment_link.readable_text
Download Math Exit Ticket (Grades 1-2, addition)

Download Math Exit Ticket (Grades 3-4, finding quotients)

Download Math Exit Ticket (Grades 3-4, distributive property)

Download Math Exit Ticket (Grades 5-6, evaluate an expression)

Download Math Exit Ticket (Algebra 1)

Thumbs Up/Down

Ideally, formative assessment checks in on student progress quickly and adaptively. One technique to try is to have students use a thumbs-up/down technique. It commonly looks like this. The teacher presents a problem or makes a statement, and then students react using their thumbs. Thumbs up means "I've got this." Thumbs down means "I'm confused." This can be a fast method to divide students into groups or to determine whether you are free to continue with the lesson or need to stop and slow down. It is also a way to add physical movement to the classroom, a potential aid to learning. In Judy Willis' 2007 book Brain-Friendly Strategies for the Inclusion Classroom, she clarifies that the simple act of having students physically gesture with their hands "has the added benefit of linking the material to be learned with sensory input, thereby increasing access to the brain's memory banks."

There are many ways to extend this idea. The thumbs expressions can have different meanings, for example one problem's thumbs-down reaction could mean "the solution is incorrect." Many teachers include a thumb to the middle (that is, pointing to the side) for students who feel between thumbs-up and thumbs-down. Students could also use other hand gestures, for example holding up three fingers to mean, "I have something to say." Ultimately you get to decide what the hand gestures mean and when you want to use them.

In-Class Discussions

Discussions in and of themselves are not necessarily assessments. But they can be! Discussions can reveal misconceptions and help guide teachers in what to teach next. Facilitating classroom discourse with and among students can be a form of formative assessment. Teachers must constantly make moment-by-moment classroom decisions like "should I ask a follow-up question?" or "what's another way of showing this concept?" In-class discussions are a way for teachers to come up with quick answers, check in on student knowledge, and figure out how to proceed.

Just because in-class discussions aren't formal, that doesn't mean they can't be prepared for. Teachers can build formative assessment opportunities into classroom conversations by being strategic about what questions to use. Being cognizant of your students' most common errors and referencing program teacher guides can be invaluable.

Homework

The structure and frequency of homework will no doubt vary depending on the student, teacher, and school. However, when a class environment permits it, homework does not need to be graded for performance and can instead be leveraged as a tool for students to think through problems on their own pace and for you to check in on their strengths and challenges. Homework becomes in effect a guide for your teaching. Education professor Cathy Vatterott offers her perspective in Rethinking Homework: Best Practices That Support Diverse Needs: "The current consensus is that homework should be formative assessment that helps prepare students for summative assessment. Therefore, in a truly standards-based system, homework is not graded [but] is reviewed and feedback is given."

In this vein, homework is not usually rote practice of what was learned that class. Instead it becomes a way for students to independently explore new ideas. In fact, homework doesn't always need to be directly related to the skill or concept being taught! Homework can be an opportunity for students to learn deeply about topics they're already interested in that already have rich mathematical connections, for example sports, homebuilding, fashion design, or entrepreneurship.

Polls

Trying to figure out what your students should do next? Ask them! Polls can be administered during or after any classroom activity. They also work well whether the teaching is in person or online. They can range from a variation of an exit ticket ("What do you think the solution is?") to metacognitive reflections ("How do you feel about this math topic?").

Polls are an especially flexible tool. Teachers can exploit polls to cater to many specific instructional purposes in mind. As examples, polls can do the following:

Give the teacher a status check by polling student confidence

Stimulate discussion with questions having multiple reasonable answers

Quickly assess prior knowledge

Elicit a misconception

These are powerful tools that grab students' attention and, depending on how the polls are used, can foster the activation of other formative assessment strategies.

KWL Chart

A KWL chart is a tool that students can use to organize their knowledge. It can broadly be used across any subject, as it helps students to synthesize what they know (K), what they want to know (W), and what they have learned (L) about a topic. In math class, a KWL chart can be a valuable opportunity for students to take a step back and reflect on their learning avoid thinking that mathematics is just a "series of steps" but rather a world of connected ideas.

The KWL chart is useful as formative assessment in the classroom. It allows the teacher to find out students' prior knowledge on a particular topic and then gear upcoming lessons based upon this information. The KWL chart can be completed when starting a new topic and be added to throughout the unit. Further, when planning across a year, the teacher is able to find out what the students have learned by the end of their lessons.

# data[].attributes.attachment_link.description
This article discusses different math formative assessment examples to inform both your teaching and your students' learning.

# data[].attributes.attachment_link.readable_text
When teachers find creative ways to integrate reading and writing with math curricula, it humanizes the subject matter and allows kids to process unfamiliar material linguistically.

Research has long linked strong reading skills with improved math achievement -which may seem obvious when it comes to typical math tasks like understanding directions on a test, or parsing word problems.

But literacy skills also support and deepen math understanding in unique and powerful ways.

"Reading and writing are complex, fundamental, integrative learning skills that should be used to their full potential in math class," writes Alessandra King, a middle school math teacher in Bethesda, Maryland. For all students, and especially those who "enjoy reading and writing more than the computational side of math," King says that finding ways to fold literacy into math class gives them "a deeper understanding and stronger appreciation of the usefulness and effectiveness of mathematics."

Blending the disciplines, when the lesson allows for it, also sets students up to "engage with math in a new-and often far more inviting-way," explains educator and tutor Chelsea Miro in an article about math read-alouds. "Kids who have struggled with geometry may love the opportunity to write a shape story, for instance-and it might even help them to remember that a square is a type of rectangle but a rectangle is not a type of square."

Here are six ideas, sourced from teachers and recent research, for merging math and literacy in ways that challenge students' abstract reasoning and build critical thinking and collaborative learning skills.

Explore Picture Books

Early-grade math problems tend to be straightforward and linear-much like the picture books that introduce them to literacy concepts. These shared properties make mathematical reasoning fertile ground for early-grade book authors.

But not all children's books that feature math are created equal. Just like in language arts classes, text selection is important, say educators Kathleen Crawford-McKinney and Asli Özgün-Koca. The trick is to find options that present a rich mathematical problem that students can work through alone or in groups, and then compare their answers to the solution in the book.

In one of their favorite examples, Molly Bang's When Sophie Thinks She Can't, main character Sophie is challenged to form as many different rectangles as possible using 12 tiles. It's a low-threshold, high-ceiling problem, offering a challenge for students across a range of skill levels. For some kids, a related exercise might involve creating their own rectangles using manipulatives and comparing solutions as a class; a more advanced exercise could involve asking them to create squares instead and discussing the differences between shapes.

When designing lessons around picture books, Crawford-McKinney and Özgün-Koca suggest creating content specific questions for students to answer, which will vary by book. But they also favor asking general questions about how math concepts shape the story and how the author establishes this understanding-which can teach students to think more like mathematicians.

For more book ideas, see former elementary school teacher Chelsea Miro's list of favorite books for math read-alouds, including more abstract titles like Kate Hosford's Infinity and Me.

Connect Thinking Through Math Journals

Teachers in subjects like English and science often use journals or logbooks to give students space to record their learning progress, make observations, and brainstorm solutions in a low-stakes format.

In math class, however, the journals might look a little different, explains former high school math teacher Nell McAnelly, who used them often with her students. "It is a record of personal experience showing what a student tried, what worked and what didn't, what practices should be continued, and what improvements a student should focus on going forward," she says. "Consequently, journals can promote initiative and confidence, which we know are gateways to learning."

There's still space for written reflections, but also for illustrations, diagrams and charts-making math journals more formal than scrap paper but less polished than an exam paper.

In her experience, students use them to make connections across mathematical ideas and delve into the reasoning behind their answers-much like they might use a journal to analyze a poem or text in language arts. Entries can also be used as discussion prompts for group work, where students can learn directly from one another as they explore new strategies.

Design Math Magazines That Relate to Real-World Topics

In Alessandra King's math classroom, while writing prompt activities are helpful, she prefers assignments that create strong connections between what students are learning in math class and the real world. "Reading and writing are particularly effective in developing a quantitative understanding of the world around us because they can be used to lead students to reflect on everyday experiences," King writes.

To harness the benefits of reading and writing, King has students create math magazines, collaborative publications that they compile with summaries they've written of math-related articles and images from publications like Scientific American, The Economist, the New York Times' Upshot section, and the Atlantic. It's one of her most popular assignments of the year.

King keeps a thick file of articles she's clipped or printed out that touch on math-related topics-from political gerrymandering to fractals in modern art and using machine learning to predict March Madness upsets-and lets students choose ones that pique their interest. They compose a page-long summary of the piece with special emphasis on the math concepts discussed and package their work with related charts or graphics.

At the end of the assignment, everyone's contributions are compiled into a magazine, and King uses a simple rubric to assess their content understanding, clarity of communication, editing, critical thinking, initiative, and creativity.

Identify and Teach Math Vocabulary

When fifth-grade math teacher Kathleen Palmieri discovered that only about 40 percent of her students could accurately define key math vocabulary (of various levels of complexity), she realized she needed to make learning math terms a regular part of her instruction, Palmieri writes in an article for MiddleWeb.

But rather than simply memorizing a list of words, she found that students needed to understand how the mathematical concept worked before the term itself would stick. In other words, if her students didn't know how fractions work, learning the definition probably wouldn't mean much. Additionally, math concepts typically make the most sense when learned in sequential order.

Drawing on ELA vocabulary building skills, Palmieri began asking students to pull out key terms they don't understand at the beginning of a unit for a math glossary that they add to as they learn each concept. Students also used colorful Post-its to create a math word wall that they add to as they learn new words-and can refer to at any time. Lastly, the class regularly decides on a word of the day or even a word of the week, with students responsible for tallying every time they see the word and describing how it's used.

Identify Patterns and Connections Via Word Sorts

Another strategy inspired by ELA instruction, word sorts can easily be adapted to math class to help students recognize patterns and relationships between words. Working alone or in groups, students identify each term in a given vocabulary list and sort terms into categories-either ones created by the teacher (known as closed sorts) or that they come up with themselves (open sorts).

Word sorts can be implemented using mathematical terms, symbols, or even expressions. Students can sort these based on various attributes-say, operation type (addition, subtraction, multiplication, division), geometric shapes (circle, triangle, square), or algebraic properties (commutative, associative, distributive).

To sort a list of mathematical expressions, kids might separate them based on whether they involve addition, multiplication, or both (e.g., the expression 5𝑥+ 3).

Younger students might perform a similar sort by placing the word "rhombus" into multiple categories, including parallelogram, quadrilateral, and two-dimensional, write Miranda L. Sigmon, Kavin Ming, and Daniel Herring-the authors of a 2022 piece on word sorts for the journal Mathematical Teacher.

To drive meaning home, the authors suggest using sentence stems, such as "I think (insert term) goes under the (category name) category because . . ." -which gets students thinking more about the choices they're making. Model a few examples yourself and split students into small groups to encourage genuine discussion and comprehension.

Encourage Collaboration With Math Circles

Inspired by literature circles, a common practice in ELA, history, and occasionally even science classes, math circles involve assigning roles to students that they perform as part of collaborative work in small groups. The goal is to foster math-related group learning, discussion, and advanced collaboration skills like leadership and flexible thinking.

In Patricia Kridler's middle school math classroom, student roles include Situation Summarizer, Vocabulary Master, Idea Guy/Gal, Model Maker, and Computation Kid, Kridler writes in the journal Mathematics Teaching in the Middle School. Since math topics can vary greatly, she often creates new roles depending on the assignment, and asks groups to complete a role worksheet that includes space for every student to share their contributions. After every cycle, she asks students to swap roles.

At first, she assigns simple word problems and lets students take turns in their roles before moving to more complicated or abstract projects that could take several class periods to complete. Typically, students use the role sheets to explain their thinking and their mathematical reasoning in depth, and the Situation Summarizer presents the findings to the entire class.

Math circles allow Kridler to structure a math task "so that all students get to focus on different aspects of how to solve a problem," she writes. "Students also develop the skills necessary to become better independent thinkers. In the process, I am learning much about how students think through problems."

# data[].attributes.attachment_link.title
6 Ways to Merge Literacy With Mathematics

# data[].attributes.attachment_link.description
When teachers find creative ways to integrate reading and writing with math curricula, it humanizes the subject matter and allows kids to process unfamiliar material linguistically.

# data[].attributes.attachment_link.readable_text
2023 Mathematics Framework

At its meeting on July 12, 2023, the State Board of Education (SBE) adopted the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve ( Mathematics Framework). The framework is important guidance designed to help educators align classroom teaching with California's rigorous math learning standards.

The Mathematics Framework reflects input from California parents, educators, students, and others who commented during the two 60-day field review periods that were held in 2021 and 2022.

California is committed to achieving excellence in math teaching and learning through curriculum and instructional approaches grounded in research and reflective of best practices across the globe. The Mathematics Framework provides guidance for mathematics learning for all students at all levels of math-including calculus-and ensures students have a wide variety of options including pursuing Science, Technology, Engineering, and Mathematics (STEM) in college and career.

Frequently Asked Questions
Information about the draftMathematics Framework.

Schedule of Significant Events for the Revision of the Mathematics Framework
At its meeting on July 13, 2022, SBE revised the Schedule of Significant Events (Timeline) for the Revision ofMathematics Framework. The Timeline was originally approved by the SBE on May 8, 2019, and was previously revised on September 11, 2019, May 7, 2020, July 14, 2021, and November 3, 2021.

A professionally-edited and designed PDF version of the framework will be posted at a later date. The content will not substantively change from the version posted below.

Curriculum Framework and Evaluation Criteria Committee

At their January 08, 2020, meeting the State Board of Education appointed 20 members to the Mathematics Curriculum Framework and Evaluation Criteria Committee (Mathematics CFCC) to assist in the development of the Mathematics Framework.

The Mathematics CFCC Meetings and Notices

Focus Groups

Pursuant to the California Code of Regulations, Title 5, Section 9511(c), the California Department of Education (CDE) convened four focus groups across California.

Listserv

Additional Resources

2013 Mathematics Framework

Assembly Bill 250 (Chapter 608, Statutes of 2011) called for the revision of the Mathematics Framework to align with the California Common Core State Standards for Mathematics.

Video: Overview of the Mathematics Framework (Dated 26-Feb-2015)
This video for teachers, administrators, and other educators provides an overview of the Mathematics Framework and how it supports implementation of the California Common Core State Standards for Mathematics.

Common Core State Standards for Mathematics (PDF; 3MB)
Links to the standards adopted by the California State Board of Education in August 2010 (Modified January 2013 Publication Version).

Common Core State Standards (CCSS) Initiative
The multi-state initiative released the CCSS on June 2, 2010. The initiative site includes information about the development and research behind the standards, appendices, the states that have adopted the CCSS, and frequently asked questions.

Appendix A
Designing High School Mathematics Courses Based on the CCSS.

2005 Mathematics Framework (Available on request)
The Framework was adopted by the State Board of Education on March 9, 2005.
*To order a copy of this framework please contact CDE Press.

Content Standards
Links to downloads of all Content Standards.

All Curriculum Frameworks
Links to downloads of all Curriculum Frameworks.

Questions: Curriculum Frameworks and Instructional Resources Division | CFIRD@cde.ca.gov | 916-319-0881

Last Reviewed: Friday, October 20, 2023

# data[].attributes.attachment_link.description
Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

# data[].attributes.attachment_link.title
What Is the Common Core? The Controversial Standards Explained

# data[].attributes.attachment_link.description
Printable graph paper mystery pictures. Plot the points on the graph paper and connect to reveal a special picture.

# data[].attributes.attachment_link.title
How to Help Students Meet the CCSS for Math Infographic - e-Learning Infographics

# data[].attributes.attachment_link.description
The How to Help Students Meet the CCSS for Math Infographic outlines what students should be doing to meet the Common Core State Standards for Mathematics.

# data[].attributes.attachment_link.readable_text
The State Board of Education adopted the Science Framework on November 3, 2016. Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

Science

2016 Revision of the Science Framework

SB300 (Chapter 480, Statutes of 2013) calls for the revision of the Science Framework to align with the Next Generation Science Standards (NGSS).

California Science Framework

Science Framework For California Public Schools, Kindergarten Through Grade Twelve (2003).
*To order a copy of this framework please contactCDE Press

# data[].attributes.attachment_link.description
The State Board of Education adopted the Science Framework on November 3, 2016. Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

# data[].attributes.attachment_link.title
What Is the NGSS? and Why Is It Important?

# data[].attributes.attachment_link.title
NGSS - Next Generation Science Standards

# data[].attributes.attachment_link.title
Science Manipulatives " Toy Theater | Learn * Create * Play

# data[].attributes.attachment_link.title
Interactive STEM Simulations & Virtual Labs | Gizmos

# data[].attributes.attachment_link.description
Unlock STEM potential with our 550+ virtual labs and interactive math and science simulations. Discover engaging activities and STEM lessons with Gizmos!

# data[].attributes.attachment_link.title
5 Math Formative Assessment Examples That Work Wonders! | ClassPoint

# data[].attributes.attachment_link.description
Explore these 5 innovative Math Formative Assessment Examples and Techniques to transform the way you teach and conduct assessments in your math classes!

# data[].attributes.attachment_link.title
Formative and Summative Assessment Tools for Middle School Math Teachers

# data[].attributes.attachment_link.authors[].name
eLearning & Instructional Design for Beginners

# data[].attributes.attachment_link.title
Assessing for Success: Strategies for Effective Summative Evaluation | eLearning and ID for Beginner

# data[].attributes.attachment_link.provider_name
SplashLearn Blog – Educational Resources for Parents, Teachers & Kids

# data[].attributes.attachment_link.title
14 Best Assessment Tools for Teachers in 2025

# data[].attributes.attachment_link.description
Learn about the top assessment tools for teachers to identify learning gaps, tailor instruction, and offer personalized feedback for every student.

# data[].attributes.attachment_link.readable_text
1. Write the descriptors. As a student is judged to move up the performance continuum, previous level descriptions are considered achieved in subsequent description levels. Therefore, it is not necessary to include "beginning level" descriptors in the same box where new skills are introduced.

2. Evaluate the rubric. As with any instructional tool, evaluate the rubric each time it is used to ensure it matches instructional goals and objectives. Be sure students understand each criterion and how they can use the rubric to their advantage. Consider providing more details about each of the rubric's areas to further clarify these sections to students. Pilot test new rubrics if possible, review the rubric with a colleague, and solicit students' feedback for further refinements.

Types of Rubrics

Determining which type of rubric to use depends on what and how you plan to evaluate. There are several types of rubrics including holistic, analytical, general, and task-specific. Each of these will be described below.

Holistic

All criteria are assessed as a single score. Holistic rubrics are good for evaluating overall performance on a task. Because only one score is given, holistic rubrics tend to be easier to score. However, holistic rubrics do not provide detailed information on student performance for each criterion; the levels of performance are treated as a whole.

"Use for simple tasks and performances such as reading fluency or response to an essay question . . .

Getting a quick snapshot of overall quality or achievement

Judging the impact of a product or performance" (Arter & McTighe, 2001, p 21)

Analytical

Each criterion is assessed separately, using different descriptive ratings. Each criterion receives a separate score. Analytical rubrics take more time to score but provide more detailed feedback.

"Judging complex performances . . . involving several significant [criteria] . . .

Providing more specific information or feedback to students . . ." (Arter & McTighe, 2001, p 22)

Generic

A generic rubric contains criteria that are general across tasks and can be used for similar tasks or performances. Criteria are assessed separately, as in an analytical rubric.

"[Use] when students will not all be doing exactly the same task; when students have a choice as to what evidence will be chosen to show competence on a particular skill or product.

[Use] when instructors are trying to judge consistently in different course sections" (Arter & McTighe, 2001, p 30)

Task-specific

Assesses a specific task. Unique criteria are assessed separately. However, it may not be possible to account for each and every criterion involved in a particular task which could overlook a student's unique solution (Arter & McTighe, 2001).

"It's easier and faster to get consistent scoring

[Use] in large-scale and "high-stakes" contexts, such as state-level accountability assessments

[Use when] you want to know whether students know particular facts, equations, methods, or procedures" (Arter & McTighe, 2001, p 28)

Summary

Grading rubrics are effective and efficient tools which allow for objective and consistent assessment of a range of performances, assignments, and activities. Rubrics can help clarify your expectations and will show students how to meet them, making students accountable for their performance in an easy-to-follow format. The feedback that students receive through a grading rubric can help them improve their performance on revised or subsequent work. Rubrics can help to rationalize grades when students ask about your method of assessment. Rubrics also allow for consistency in grading for those who team teach the same course, for TAs assigned to the task of grading, and serve as good documentation for accreditation purposes. Several online sources exist which can be used in the creation of customized grading rubrics; a few of these are listed below.

# data[].attributes.attachment_link.title
Rubrics for Assessment | Center for Innovative Teaching and Learning | Northern Illinois University

# data[].attributes.attachment_link.description
A rubric is an explicit set of criteria used for assessing a particular type of work or performance (TLT Group, n.d.) and provides more details than a single grade or mark. Rubrics, therefore, will help you grade more objectively.