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The Coherence Map shows the connections between Common Core State Standards for Mathematics.

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English Learners (EL) Academic Language Function Toolkit | Missouri Department of Elementary and Secondary Education

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205 Jefferson St.
Jefferson City, MO 65101Map(link is external)

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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!

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Virtual Manipulatives | Mathematical Modelling | Toy Theater

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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!

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

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Common Core Instruction: Exploring Dividing Decimals to the Hundredths

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Solve problems and explain your thinking in a digital math workspace.

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To learn more about the standards, click here. To learn more about the NGSS Appendices,click here.

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2016 Science Framework - Science Framework (CA Dept of Education)

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The curriculum framework was adopted by the State Board of Education on November 3, 2016. This is the final published version.

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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."

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6 Ways to Merge Literacy With Mathematics

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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.

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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

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Curriculum frameworks provide guidance to educators, parents, and publishers, to support implementing California content standards.

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"Math in the World Around Us: Fibonacci Sequence" by Adventure Academy

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Imagine What's Possible.

Whether you're looking to try something new, wanting to better connect with your students or need to develop your curriculum for hybrid or online environments, AdvancED's team of skilled practitioners is ready to support your next step.

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Connect with our dynamic community at AdvancED! If you have a compelling story or upcoming event you'd like to highlight or need design expertise for, we're here to support you. Our dedicated team is committed to amplifying your message and championing groundbreaking, research-driven innovations to elevate training and development initiatives.

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Bloom’s Taxonomy - Center for Instructional Technology and Training - University of Florida

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The original Taxonomy of Educational Objectives, commonly referred to as Bloom’s Taxonomy, was created by Benjamin Bloom in 1956, and later revised in 2001. Bloom categorized and classified the cognitive domain of learning into varying levels according to complexity and richness. As you travel up th…

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Did you know that not all vocabulary words are created equally? This is especially true when it comes to classroom instruction. The type of words we focus on can make a big impact on students' vocabulary progress. Within this blog post, we'll chat about the three tiers of vocabulary. We'll also discuss what each vocabulary tier consists of, examples of words for each tier, and how we can use this information as teachers. Let's get started!

What are the Three Tiers of Vocabulary?

For instructional purposes, vocabulary words are sorted into three tiers: Tier 1, Tier 2, and Tier 3. The illustration below shows the three tiers of vocabulary. As you can see, we move from Tier 1 on the bottom of the triangle up to Tier 3 at the top of the triangle.

What is the difference between each vocabulary tier? The image below gives a quick, concise explanation of each tier. I will break down each tier for you and give examples of words for each tier.

As you can see in the image above, each tier consists of a very different type of word list. Let's break down each vocabulary tier and discuss what it means for us as classroom teachers and the focus for our instruction.

Tier 1 Vocabulary

Tier 1 vocabulary words consist of basic, familiar words. These words are commonly used by most students in everyday conversation. Students have the highest exposure with Tier 1 words. In fact, children are exposed to Tier 1 words from an early age, and as a result, they become easily familiar with this set of words. Perhaps the easiest way to think about Tier 1 words is to remember that the words are often used in oral conversations and familiar in speech. Examples of Tier 1 words are shown in the image below.

Tier 1 word examples in the image above include: sad, walk, elephant, baby, red, bus, clock, phone, and rain. Tier 1 vocabulary words are words students come to school knowing and using the most. In fact, Beck, McKeown, and Kucan explain in their book, Bringing Words to Life, that students are likely able to learn Tier 1 words naturally and independently!

Tier 2 Vocabulary

Tier 2 vocabulary words are robust, high-frequency words that students encounter across the content areas. They are not widely used in speech and daily conversation. Unlike Tier 1 words, Tier 2 words are not usually learned naturally or independently because students do not hear or use them in conversation. A key point to understand is that Tier 2 words are often used and found in books and written text. Tier 2 vocabulary words often have multiple meanings. This tier can also be referred to as academic vocabulary words. Examples of Tier 2 vocabulary words are listed in the image below.

Examples of Tier 2 vocabulary words are bolt, fine, amiable, rare, harmony, blush, astute, and valuable. As you can see, these words have multiple meanings and are not typically used in everyday, causal conversation. However, it is Tier 2 words that students will encounter frequently in written texts. Tier 2 words will have an important impact on your students' comprehension skills.
(Pssst! Guess what?! If you use any of my Vocabulary Curriculum Sets already in your kindergarten, first grade, or second grade classroom, you are already a ROCKSTAR at explicitly teaching Tier 2 words! Go you!)
Tier 3 Vocabulary

Tier 3 vocabulary words are low-frequency words. These words are content specific and have distinct meanings and purposes. Examples of Tier 3 words are listed in the image below.

Tier 3 words like pterodactyl, aorta, thesis, isosceles, osmosis, electrolyte, photosynthesis, and igneous are not words we often use in everyday conversations. These words have a distinct purpose and we use them only when they relate to a specific topic or domain. When we use a Tier 3 word, there is a specific reason to do so, such as when we are learning about plants in a science unit or triangles during math class.

Which Tier Should We Focus Instruction On?

Let's recap what we learned about the three tiers of vocabulary. We learned that Tier 1 vocabulary words are words students come to school knowing and using the most. Therefore, as teachers, we naturally do not (and should not) spend the bulk of our time explicitly teaching the meanings of Tier 1 words. Remember, most students can naturally and independently learn Tier 1 words. For example, if the word "dinosaur" is part of your district's reading curriculum, I think it's safe to say that most first graders have heard of the word dinosaur before. They know what a dinosaur is and what it looks like! A quick review or picture card would probably be all they need to remind them of what a dinosaur is. Spending a week on the word dinosaur is not an effective use of instructional time!

Students will need to know Tier 3 words, however, Tier 3 words will not make up the bulk of the words they encounter in written text. For example, if your students are reading a chapter series, the frequency of the word "mitosis" in each book is going to be really low! Furthermore, Tier 3 words typically do not have synonyms and antonyms, so learning the definition to a Tier 3 word is usually pretty "cut and dry". Students only need to learn Tier 3 words when a specific need is present.

So, did a lightbulb go off for you yet?! Do you know which vocabulary tier we want to spent the most time on each week? If you answered Tier 2, you are correct!

As teachers, we will get the most "bang for our buck" if we spend the bulk of our vocabulary instruction on Tier 2 words. These are words that are found and used across variety of topics, domains, areas, and content. Since MANY Tier 2 words have multiple meanings, they are more difficult to learn because students need to understand the word within multiple contexts, as well as understand the connection between the words' synonyms and antonyms.

Having a robust vocabulary and knowledge of Tier 2 words directly impacts our students' reading comprehension skills.

Ways To Focus on Tier 2 Vocabulary

Now that we know the importance of Tier 2 vocabulary words, let's chat about ways to incorporate these words into our daily routine! Several years ago, I created an engaging and easy-to-follow vocabulary routine for kindergarten, first-grade, and second-grade teachers! I often get asked, "Where did the words in each set come from?" The words I chose for my curriculum are Tier 2 words that often have multiple meanings and can be seen in written text, across multiple contents, topics, and domains! Sounds pretty spot-on to what we just learned about in this post, right?! Take a look at some of the word lists below. These Tier 2 words are words that students will not typically learn independently. It's important we have a routine and a plan for explicitly teaching Tier 2 vocabulary in our classrooms.

Synonyms and Shades of Meaning

We can expose students to multiple Tier 2 words by spending time on the synonyms and antonyms for Tier 2 words. When we teach our students even three to five Tier 2 words, we can actually expose them to up to TWENTY Tier 2 words throughout the week by discussing the synonyms associated with these words, and their shades of meaning. The image below shows you an easy routine for teaching and exposing your students to a variety of Tier 2 words:

Multiple Meanings and Multiple Contexts

As we learned in this post, many Tier 2 words have multiple meanings. It's important to explicitly point out these multiple meanings and help students understand how they are used in various contexts. Each week's vocabulary lessons in my curriculum includes a passage AND a read aloud! The read aloud (often presented on Day 2 each week) uses the Tier 2 words in a different context than the passage on Day 1. Exposing our students to various contexts and situations helps them to better understand the word and see it used in various ways! ( Bonus: Each read aloud comes in a printable version AND a digital version!)

Tier 2 Vocabulary Word Play

Pointing Tier 2 words out in the books you read and allowing students to "play" with words is also an important part of vocabulary instruction. We want our students to have a lot of opportunities for word play and practice using the words in their daily discussions. Examples of word play games include:

One of the best parts of my vocabulary sets are the included word play games for each week's Tier 2 words! You can see the table of contents in image above, and a page out of the word play section called Sing-Along Antonyms! Each word play activity has a teacher direction page and if required, the printable materials that are needed. You can read more about Vocabulary Word Play ideas in another blog post I wrote by clicking HERE.

Learn More About Vocabulary Instruction

That was ALOT of information on the three tiers of vocabulary, and some various activities and ideas for teaching Tier 2 words within your classroom! If you'd like to learn more about my simple, already-done-for-you vocabulary routine click HERE to check out my vocabulary curriculum blog post! In this post, I go into depth on even more vocabulary research and why it is so important to have an explicit plan in place for your vocabulary instruction.

You can check out my vocabulary curriculum bundles by clicking on your grade level below. The vocabulary bundles are also sold as individual sets, too!

If you'd like to refer back to this post, or share it with a colleague, feel free to pin the image below to your Pinterest page! Thank you so much for hanging out with me and learning about the three tiers of vocabulary and how they impact our classroom instruction. Don't forget to make sure you are following me on Instagram @missdecarbo for weekly tips, ideas, activities, resources, and freebies!

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The 3 Tiers of Vocabulary for Classroom Instruction - Miss DeCarbo

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Learn about the 3 tiers of vocabulary for the classroom and lesson planning. Discover how to choose your vocabulary words, and which ones matter!

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The DoK Levels added to questions will be reflected in reports. Teachers can easily track and show growth of students understanding in an individual assignment from start to end.

What do individual DoK Levels help you assess?

Level 1: Recall - Focus here is on specific facts, information, details, and procedures e.g specific definitions, details, applies simple steps.

Level 2: Skills and Concepts - Processing of conceptual knowledge, comparing and contrasting, extracting main points, interpreting facts etc. More content knowledge required here than Level 1. Actions require more than one step.

Level 3: Strategic Thinking - Focus is on reasoning based on deep knowledge: which include understanding and rationalizing thoughts, manifesting them into achievable steps. E.g Drawing conclusions from Examples, citing evidence to support a logical argument, concept based explanations.

Level 4: Extended Thinking - Focus is on applying complex reasoning, experimental design, and planning over an extended period of time. It requires extended investigation (research) and time to process information and problem solving. Students are required to make multiple connections, relate ideas to specific content area(s), and select an approach from among many as a solution.

Download the PDF Depth of Knowledge Question Stems.

To learn more about the DoK framework, you can refer to Brenda Matthews' presentation:
Brenda Mathew's DoK framework
Add Depth of Knowledge Levels:

STEP 1: Select "Options" then "Add DoK"

STEP 2: Select the DoK Level you want

View a student's proficiency by DoK Level in your Student Reports [Team Plan]

This will give you real-time metrics on how your students are engaging with texts and how they are performing by DoK Levels.

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The DoK Levels added to questions will be reflected in reports. Teachers can easily track and show growth of students understanding in an individual assignment from start to end. What do individual DoK Levels help you assess? Level 1: Recall - Focus here is on specific facts, information, details, and procedures e.g specific definitions, details, applies simple steps.

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Update 2022: For a recent article on how to talk about math, click here.
Think about the questions that you ask in your math classroom. Can they be answered with a simple "yes" or "no," or do they open a door for students to really share their knowledge in a way that highlights their true understanding and uncovers their misunderstandings? Asking better questions can open new doors for students, helping to promote mathematical thinking and encouraging classroom discourse. Such questions help students:

Work together to make sense of mathematics.

Rely more on themselves to determine whether something is mathematically correct.

Learn to reason mathematically.

Evaluate their own processes and engage in productive peer interaction.

Discover and seek help with problems in their comprehension.

Learn to conjecture, invent and solve problems.

Learn to connect mathematics, its ideas and its applications.

Focus on the mathematical skills embedded within activities.

Below are 100 questions from mathematics expert Dr. Gladis Kersaint to help you address these core areas and promote mathematical thinking and discourse in the classroom. Want these questions visible in your classroom? Curriculum Associates has released an infographic that you can print and have at your desk or in your class for quick access! Here's a preview:

For more, please download Dr. Kersaint's new whitepaper:Orchestrating Mathematical Discourse to Enhance Student Learning.
Dr. Gladis Kersaint
Help students work together to make sense of mathematics

1. What strategy did you use?

2. Do you agree?

3. Do you disagree?

4. Would you ask the rest of the class that question?

5. Could you share your method with the class?

6. What part of what he said do you understand?

7. Would someone like to share ___?

8. Can you convince the rest of us that that makes sense?

9. What do others think about what [student] said?

10. Can someone retell or restate [student]'s explanation?

11. Did you work together? In what way?

12. Would anyone like to add to this?

13. Have you discussed this with your group? With others?

14. Did anyone get a different answer?

15. Where would you go for help?

16. Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?

17. How could you help another student without telling the answer?

18. How would you explain ___ to someone who missed class today?

Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

1. Is this a reasonable answer?

2. Does that make sense?

3. Why do you think that? Why is that true?

4. Can you draw a picture or make a model to show that?

5. How did you reach that conclusion?

6. Does anyone want to revise his or her answer?

7. How were you sure your answer was right?

Help students learn to reason mathematically

1. How did you begin to think about this problem?

2. What is another way you could solve this problem?

3. How could you prove that?

4. Can you explain how your answer is different from or the same as [student]'s?

5. Let's see if we can break it down. What would the parts be?

6. Can you explain this part more specifically?

7. Does that always work?

8. Is that true for all cases?

9. How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

1. What do you need to do next?

2. What have you accomplished?

3. What are your strengths and weaknesses?

4. Was your group participation appropriate and helpful?

Help students with problem comprehension

1. What is this problem about? What can you tell me about it?

2. Do you need to define or set limits for the problem?

3. How would you interpret that?

4. Would you please reword that in simpler terms?

5. Is there something that can be eliminated or that is missing?

6. Would you please explain that in your own words?

7. What assumptions do you have to make?

8. What do you know about this part?

9. Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

1. What would happen if ___? What if not?

2. Do you see a pattern?

3. What are some possibilities here?

4. Where could you find the information you need?

5. How would you check your steps or your answer?

6. What did not work?

7. How is your solution method the same as or different from [student]'s?

8. Other than retracing your steps, how can you determine if your answers are appropriate?

9. What decision do you think he or she should make?

10. How did you organize the information? Do you have a record?

11. How could you solve this using (tables, trees, lists, diagrams, etc.)?

12. What have you tried? What steps did you take?

13. How would it look if you used these materials?

14. How would you draw a diagram or make a sketch to solve the problem?

15. Is there another possible answer? If so, explain.

16. How would you research that?

17. Is there anything you've overlooked?

18. How did you think about the problem?

19. What was your estimate or prediction?

20. How confident are you in your answer?

21. What else would you like to know?

22. What do you think comes next?

23. Is the solution reasonable, considering the context?

24. Did you have a system? Explain it.

25. Did you have a strategy? Explain it.

26. Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

1. What is the relationship of this to that?

2. Have we ever solved a problem like this before?

3. What uses of mathematics did you find in the newspaper last night?

4. What is the same?

5. What is different?

6. Did you use skills or build on concepts that were not necessarily mathematical?

7. Which skills or concepts did you use?

8. What ideas have we explored before that were useful in solving this problem?

9. Is there a pattern?

10. Where else would this strategy be useful?

11. How does this relate to ___?

12. Is there a general rule?

13. Is there a real-life situation where this could be used?

14. How would your method work with other problems?

15. What other problem does this seem to lead to?

Help students persevere

1. Have you tried making a guess?

2. What else have you tried?

3. Would another recording method work as well or better?

4. Is there another way to (draw, explain, say) that?

5. Give me another related problem. Is there an easier problem?

6. How would you explain what you know right now?

Help students focus on the mathematics from activities

1. What was one thing you learned (or two, or more)?

2. Where would this problem fit on our mathematics chart?

3. How many kinds of mathematics were used in this investigation?

4. What were the mathematical ideas in this problem?

5. What is the mathematically different about these two situations?

6. What are the variables in this problem? What stays constant?

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. In the next and final blog in this series, we will dive into the specific strategies that teachers can use to foster meaningful conversations about what students are thinking, doing and learning.

This blog is part of a three-post series on the importance of mathematical discourse from Curriculum Associates and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning. Download your free copy .
For more on mathematical discourse and Curriculum Associates, check out:
Dr. Gladis Kersaint is a Professor of Mathematics Education at the University of Connecticut. Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update. This post includes mentions of a Getting Smart partner. For a full list of partners, affiliate organizations and all other disclosures please see our Partner page

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Talking Math: 100 Questions That Help Promote Mathematical Discourse

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By Gladis Kersaint - 100 questions from a mathematics expert to help promote mathematical thinking and discourse in the classroom.

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Picture books about math are the perfect cross-curricular tool to introduce your kids to complicated concepts and mathematical thinking. We chose some of our favorite authors and stories, both new and old, to add to your classroom library and inspire your students' mathematical minds.

Our picks for the best books about math are listed by grade level and math concept so you can easily find the right selection for your class.

1.by Mac Barnett and Jon Klassen

A sassy story about sneaky Triangle trying to play a trick on his friend Square that turns into a shape showdown. Kids will be mesmerized by this whimsically illustrated tale.

Math concepts: Shapes, relative size

Best for: Grades K-4

2. Place Value by David Adler and Edward Miller

Love the story about monkeys jumping on the bed? You'll adore this version that tells the story of the monkeys at Banana Cafe making one very large banana cupcake.

Math concept: Place value

Best for: Grades 1-5

Also by this author: Fraction Fun, Mystery Math: A First Book of Algebra, How Tall ,How Short, How Far Away, Shape Up!: Fun with Triangles and Other Polygons, Perimeter, Area and Volume: A Monster Book of Dimensions, Millions, Billions and Trillions: Understanding Big Numbers, and more.

Mr. and Mrs. Comfort are having a family reunion and need to arrange the tables so that everyone in the family can enjoy their fabulous spaghetti and meatballs. A witty tale by math guru Marilyn Burns.

Math concepts: Area and perimeter

Best for: PreK-Grade 3

Also by this author: The Greedy Triangle, Math for Smarty Pants, The $1.00 Word Riddle Book, This Book is About Time, and more.

Brave knights, a strong ruler, and a dilemma that needs to be solved using math. This book about math is great for introducing geometry concepts in a fun and accessible way.

Math concepts: radius, diameter, circumference

Best for: Grades 2-7

Also by this author: The Sir Cumference series has more than 10 books that explore math concepts from algebra and geometry to measurement and fractions.

Delightful illustrations and a dreamy young girl pondering the vastness of the universe make a very difficult concept approachable and interesting in this book about math.

Math concept: Limitless numbers and infinity

Best for: Grades K-5

Watch as this busy construction crew creates impressive structures by grouping bricks by twos, fives, and tens. Kids will love the catchy repetitive verse and charming illustrations.

Math concept: Counting in quantities

Best for: PreK-Grade 2

Ethan wakes up one morning with a strange cat on his head and the only way it will agree to get it off is if Ethan agrees to play a game of probability with it. This kooky book about math will entertain your students as it teaches a difficult concept.

Math concept: Probability

Best for: Grades 2-5

Also by this author: Fractions in Disguise: A Math Adventure

Kids will love this fun story with delightful cartoon illustrations that teach the importance of zero as a placeholder in our number system. Without Zero the Hero, we wouldn't be able to count beyond the number nine!

Math concept: Concept of zero

Best for: Grades 1-5

A beautifully illustrated book about math that shows how flips, slides, and turns can create amazing symmetrical images.

Math concept: Geometry, symmetry

Best for: Grades 1-3

A playful book with silly characters and perfect rhyming that teaches not just the basics of telling time, but the tricky concept of relative time.

Math Concept: Time

Best for: Grades 1-4

Also by this author: Cleary has written a whole series of books called Math is CATegorical, including The Mission of Addition, The Action of Subtraction, A Dollar, a Penny, How Much and How Manym, A Fraction's Goal- Parts of a Whole, and more.

Mouse and her friends want to play a game of tug-of-war but have a hard time making the sides fair. This charming story teaches the concept of balance and equality.

Math concept: Equality

Best for: Kindergarten-Grade 3

Two more rhyming tales with colorful, busy illustrations that kids will love. In each of these books about math, a boy wakes up one day and finds that things are kookily out of order. Great as a read-aloud or for independent readers who will love hunting through the detailed pictures to find the odd or even items.

Math concepts: Odd and even

Best for: Kindergarten-Grade 3

Your students will love quirky Henry and his wild imagination as he uses his dog Frank as a unit of measure. Henry thinks about real life objects and how they can take up specific spaces. This story will get kids thinking creatively about numbers and many other math concepts.

Math concepts: Problem-solving, measurement, estimation, and calculatio n.

Best for: PreK-Grade 3

Colorful fish bob, glide, and swish through the ocean until POP, a shark appears! Kids not only love hearing and seeing this story, it will get them interested in learning and creating their own patterns.

Math concepts: Patterns and variations of pattern types

Best for: Kindergarten-Grade 2

16. A Remainder of One by Eleanor Pinczes and Bonnie MacKain

The 25th Beetle Army Corp is trying to solve the problem of how to divide their troops into equal groups for the parade march without leaving any stragglers. A fun, rhyming book about math that will get kids thinking about how numbers are put together and pulled apart.

Math concept: Grouping, division

Best for: Preschool-Grade 3
What picture books have you used in your math lessons? Come and share in our WeAreTeachers HELPLINE group on Facebook. Plus, Why It's Important to Honor All Math Strategies.

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16 Picture Books About Math to Inspire Curious Kids - We Are Teachers

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Picture books about math are the perfect cross-curricular tool to introduce your kids to complicated concepts and mathematical thinking. We chose some of our favorite authors and stories, both new and old, to add to your classroom library and inspire your students' mathematical minds.

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Erikson Institute Early Math Collaborative

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One sure fire way to warm up children's attitude towards mathematics is to build math activities and lessons out of the books they all love to listen to and read.

Children explore the Big Ideas about foundational mathematical concepts as they interact with math in everyday life. When reading and listening to books, mathematical situations come to life in new and fanciful ways. And, we're not just talking about books that are explicitly mathematical, since math is often central to the problem and resolution of our favorite stories. So many questions that drive children, such as "Is it fair?", "Who is bigger/faster/stronger?", "What will happen next?" are shared by the characters they read about.

But, don't forget to take the math out of the book! Play with the stories. Act them out and innovate. Ask the what-if questions. Create your own version. This is where mathematical thinking gets creative.

We have collected many lists and activity ideas related to some of the best children's books we know of, all infused with math concepts. We've outlined links below, organized under the early math topics of our website. Use books to bring math to life today!

Where's the Math? Books, Games, and Routines to Spark Children's Thinking

Make math learning both meaningful and fun by building on children's natural curiosity to help them grow into confident problem solvers and investigators of math concepts.
Learn more ›

Big Ideas of Early Mathematics:

What Teachers of Young Children Need to Know

The Big Ideas that convey the core concepts of mathematics are at the heart of this book that gives early childhood educators the skills they need to organize for mathematics teaching and learning during the early years.
Order a copy

Online MS in Early Childhood Ed - STEM Focus

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Elevate your practice and strengthen your knowledge of teaching and learning in the disciplines of science, technology, engineering, and mathematics (STEM), with an emphasis on children in preschool through third grade.
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Best Childrens Books for Early Math Learning | Early Math Collaborative

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One sure fire way to warm up children's attitude towards mathematics is to build math activities and lessons out of the books they all love to listen to and read. Children explore the Big Ideas about foundational mathematical concepts as they interact with math in everyday life.

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6th Grade Literature List

Related Links

See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more.

The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs.

McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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6th Grade Literature List - Everyday Mathematics

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See advice from Everyday Mathematics teachers on working with parents, using technology in the classroom, pacing, and more. The UChicago STEM Education offers strategic planning services for schools that want to strengthen their Pre-K-6 mathematics programs. McGraw-Hill Education's website features supplemental materials, games, assessment and planning tools, technical support, and more.

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Marilyn’s current thinking about math education and her ongoing classroom experiences and learning.

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Interview/Survey

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Tori Kelly - So Will I (100 Billion X) - Instrumental with Lyrics

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The following list of California initial teaching credential areas have been aligned with the California Commission on Teacher Credentialing (CTC) approved edTPA assessment areas. The related passing scores for each assessment area have been approved by the CTC.

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Welcome fellow Recovering Traditionalists to Episode 2 where we will explore the 3-Phase Lesson Format that is essential to having a student-centered math classroom.

In Episode 1 I shared about Cognitively Guided Instruction. A big part of CGI is teaching math through story problems. Well as I started making that shift away from my textbook and to learning math through problem solving I found the book Teaching Student Centered Mathematics and it was a HUGE help. It quickly became my Math Bible. This is another book that I will likely do multiple episodes about but today is all about the 3-phase lesson format.

The Three-Phase Lesson Format is on pages 24-26 of the edition I have. I know there is a newer edition. Here is the insight from the book:

"A 3-phase lesson format ( Before, During, After ) provides a structure for teaching mathematics through problem solving. refers to the time before the children start work on the problem. refers to the time during which the children work on the problem, and refers to the discussion that takes place after children work on the problem." p. 24

The next few pages go into detail about the phases but I'm going to highlight a big take-a-way that I had about each phase.

In the BEFORE phase it is very important to anticipate children's strategies & their misconceptions. I honestly had never done that before. Even as I made my transition away from the textbook I was just picking problems and giving them to the students without even solving the problem myself sometimes!!! But in order to have a productive DURING and AFTER phase you have to take time exploring the problem yourself and anticipating the way that your students will explore the problem.

The DURING phase is when the kiddos are working on the problem. Now this is hard to admit but during that time I was usually off doing something else. Maybe preparing for the next thing we were doing. Instead the DURING phase should be a time that while the students are exploring the problem, you are exploring their thinking. This is prime time to be looking over shoulders to see what strategies they have and asking questions like "Why do you think that's right?" "Can you tell me why you did _______?" and so on. These questions and you learning how they solved the problem leads into the AFTER phase.

The AFTER phase is honestly one of the most important. This is where you come together and share solutions and strategies. However, the one thing I didn't do correctly was the sharing of strategies. I would have kids share but I never helped them make connections between strategies and that is an essential piece to the AFTER phase.

As I started moving away from just doing the next thing in my textbook, this 3-phase lesson format really helped me design my lessons. If you want to learn more about the 3-phase lesson and get some great ideas on activities to do in those lessons then I highly recommend you go get this book.

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Episode 2: Three Phase Lesson Format to create a Student Centered Mathematics classroom

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Welcome fellow Recovering Traditionalists to Episode 2 where we will explore the 3-Phase Lesson Format that is essential to having a student-centered math classroom. In Episode 1 I shared about Cognitively Guided Instruction.

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The Next Generation Science Standards are now available. Twenty-six states and their broad-based teams worked together with a 40-member writing team and partners throughout the country to develop the standards.
The NGSS are composed of the three dimensions from the NRC Framework. Watch the videos below to learn more about the standards.

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The Next Generation Science Standards are now available. Twenty-six states and their broad-based teams worked together with a 40-member writing team and partners throughout the country to develop the standards. The NGSS are composed of the three dimensions from the NRC Framework. Watch the videos below to learn more about the standards.

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Engineering Integration for Science Article

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Engineering Integration for Science Article

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Engineering is a systematic and often iterative approach to designing objects, processes, and systems to meet human needs and wants (National Assessment Governing Board, 2010). It’s important for students to understand how science can be utilized to solve real-world problems through engineering. Students have ample opportunities to build their engineering design skills throughout Smithsonian Science for the Classroom.

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Place Value Pockets are amazing tools for developing strong number sense and place value understanding. This is a tool that will quickly become a daily routine in the elementary classroom. It is also an excellent tool for middle school students to practice divisibility rules and algebraic thinking. This product includes 30 sets of 0-9 digit cards in light pink cardstock and 30 place value pockets for ones, tens, hundreds and thousands in blue cardstock. Included are directions with games, questions and activities to facilitate the daily use of this tool. These place value pockets and digit cards are very powerful with Kim Sutton’s book, Place Value with Pizzazz.

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What is the CRA Math Model? - The Math Spot

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This is the power of the CRA Math Model.

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How To Use Manipulatives In Math for Elementary Students

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hand2mind Plastic Rainbow Fraction Tiles, Montessori Math Materials, Fraction Manipulatives, Unit Fraction, Fraction Bars Math Manipulatives, Fraction Games, Homeschool Supplies (Set of 51)

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Buy hand2mind Plastic Rainbow Fraction Tiles, Montessori Math Materials, Fraction Manipulatives, Unit Fraction, Fraction Bars Math Manipulatives, Fraction Games, Homeschool Supplies (Set of 51): Science Kits & Toys - Amazon.com ✓ FREE DELIVERY possible on eligible purchases

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Dry Erase Answer Paddle - White Board, Double Sided Paddles (25 Pack, 7.83 X 6.46 Inches)

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Amazon.com : Dry Erase Answer Paddle - White Board, Double Sided Paddles (25 Pack, 7.83 X 6.46 Inches) : Office Products

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Description

From measurement and 3-D shapes to fractions and decimals, our comprehensive manipulative library reinforces essential math concepts that support upper-grade curricula! Each 6 1/4" plastic tub includes enough hands-on supplies for the whole class to explore concepts at once, plus a guide filled with skill-building activities. Library includes all 8 tubs shown-organized in an easy-access, 26"w x 8 1/2"d x 13 1/2"h metal storage center. Each tub also available separately.

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From measurement and 3-D shapes to fractions and decimals, our comprehensive manipulative library reinforces essential math concepts that support upper-grade curricula! Each 6 1/4" plastic tub includes enough hands-on supplies for the whole class to explore concepts at once, plus a guide filled with skill-building activities.

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Description

Tons of manipulatives students need to explore essential math concepts-all in one place! Our handy 6 1/4" plastic tubs are filled with a huge variety of manipulatives designed to provide lots of hands-on experience with everything from counting and patterning to place value and money. Plus, the tubs come in a convenient storage center that gives kids easy access to the materials they need! Library includes all 8 tubs shown, each with an activity guide. Metal storage center measures 26"w x 8 1/2"d x 13 1/2"h. Each tub also available separately.

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Tons of manipulatives students need to explore essential math concepts-all in one place! Our handy 6 1/4" plastic tubs are filled with a huge variety of manipulatives designed to provide lots of hands-on experience with everything from counting and patterning to place value and money.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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In Ratio Rumble, players choose from among 18 different characters as their avatar in the tournament.

Ratio Rumble is all about the player re-creating each ratio recipe given to them by the game. In this case, the game asks for two blue potions and one red potion. The player used double the original ratio. This is possible because the ratio is still the same even if the quantities are doubled.

Players can use health potions found on the gameboard to heal their characters.

Potions with lightning symbols, when played as part of a ratio, will weaken the opponent. Potions with numbers are time bombs that tick down each turn. It's important to use these potions in your ratios. If they are not used before they finish their countdown, they explode and weaken the character's health.

In Ratio Rumble, players learn to create ratios following numerical recipes. Ratios get more complicated as the game progresses and as the player masters new levels of difficulty.

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Students engaged in Math Snacks in class learned more math than those with access to just classroom instruction.

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Science Manipulatives " Toy Theater | Learn * Create * Play

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Interactive STEM Simulations & Virtual Labs | Gizmos

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Unlock STEM potential with our 550+ virtual labs and interactive math and science simulations. Discover engaging activities and STEM lessons with Gizmos!

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Categories

Discover our collection of hands-on science materials to enhance your students' learning experience.

hand2mind offers a comprehensive assortment of science manipulatives to nurture and engage your curious STEM learners!

Shop our selection of interactive science and engineering resources for preschool through high school age students. Introduce core concepts through hands-on quality play with our science games and activity sets. Develop creativity and innovation through structured activities and laboratory experiences encompassing biology, botany, chemistry, earth science, physical science, coding, robotics and much more. Find the perfect lab kit, activity set, cart, game or supplies to keep your science classroom fully stocked for action!

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Hand2mind: Growing Minds with Hands-On Learning

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The store will not work correctly in the case when cookies are disabled. JavaScript seems to be disabled in your browser. For the best experience on our site, be sure to turn on Javascript in your browser. Shop our selection of interactive science and engineering resources for preschool through high school age students.

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Math Interest Survey - Diagnostic Example

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Math Interest Survey - Diagnostic Example

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Here’s how it works:

1. Post the lesson objectives in specific, kid-friendly language.

2. Give each student a small slip of paper or an index card.

3. Create a statement, based on the lesson objections, to which students can respond with “sometimes,” “always,” or “never.” Note: Responding in writing will boost all the students’ critical thinking skills, but creating a statement with a “sometimes” response gives you a more comprehensive picture of your students’ progress toward understanding the learning objective.

4. Ask students to respond to the statement with “sometimes true,” “always true,” or “never true” and give a justification in pictures, words, or numbers

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Here’s how it works:

1. Present a problem to the class along with four characters’ answers and reasoning.

2. Individually, have students determine who is correct and explain why.

3. After students have selected the response(s) they believe to be correct, have them provide a justification for why the other responses are incorrect

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Here’s how it works:

1. Post the lesson objectives in specific, kid-friendly language.

2. Give each student a small slip of paper or an index card.

3. Create a statement, based on the lesson objectives, to which students can respond “fact” or “fib.”

4. Ask students to respond to the statement with “fact” or “fib” and give a justification in pictures, words, or numbers

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Summative Assessment - Create your own board game

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Summative Assessment - Create your own board game

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Editable Board Game Templates - Board Game Project - Bundle of Blank Board Games

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Who doesn't love a good board game? With these editable blank board game templates you are able to make it meet your needs. You will be able to use it throughout the year for a variety or subjects and/or skills. Use the printable boards and write in your problems, or use the editable PowerPoint file...

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Summative Assessment - Create your own Jeopardy 

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Summative Assessment - Create your own Jeopardy 

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Create your own jeopardy template online, without PowerPoint. Or browse the pre-made templates created by other people!