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IXL | Math, Language Arts, Science, Social Studies, and Spanish

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IXL is the world’s most popular subscription-based learning site for K–12. Used by over 15 million students, IXL provides personalized learning in more than 17,000 topics, covering math, language arts, science, social studies, and Spanish. Interactive questions, awards, and certificates keep kids motivated as they master skills.

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Free worksheets for kindergarten to grade 5 kids. Over 10,000 math, reading, grammar and writing, vocabulary, spelling and cursive writing worksheets. Answer sheets included; no login required.

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Impact of Identity in K-8 Mathematics-Rethinking Equity-Based Practices

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Impact of Identity in K-8 Mathematics-Rethinking Equity-Based Practices

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Impact of Identity in K-8 Mathematics- Rethinking Equity-Based Practices.pdf

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From Stanford Online's "How To Learn Math for Teachers and Parents": Number Talks

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Arithmetic | Talking Math With Your Kids

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Problems are randomized, allowing students multiple opportunities to get the practice and feedback they need on the road to mastery.

Go beyond right and wrong. As soon as a student submits an answer, they get a detailed age-appropriate explanation.

DeltaMath allows teachers to mix and match problem-sets, control rigor, vary due dates, and, with PLUS or INTEGRAL, create tests and problems of their own.

Use DeltaMath's modules to create high-leverage assignments and track student learning. With DeltaMath PLUS or INTEGRAL, students also get access to help videos. Create and assign tests, assign specific problem-types, even create your own problem.

Learn More

Problems solved with DeltaMath

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Ensuring Equitable Access with Number Sense Routines

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Ensuring Equitable Access with Number Sense Routines

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Ensuring Equitable Access with Number Sense Routines

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Thanks to my bro for the music! He's at christopherhart2010@gmail.com Thanks to my other bro for hand dancin' and throwing money at me, and special thanks to my mamma and grandma for their guest appearances. My personal website, which you might like: http://vihart.com

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Using Children's Literature to Teach Math

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Using Children's Literature to Teach Math

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MARILYN BURNS MATH | Marilyn's current thinking about math education and her ongoing classroom experiences and learning.

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I've found that children's books are extremely effective tools for teaching mathematics. They can spark students' math imaginations in ways that textbooks or workbooks don't. Connecting math to literature can boost confidence for children who love books but are wary of math. And students who already love math can learn to appreciate stories in an entirely new way.

One book that I've read to many first and second graders, always to the delight of the children, is One Is a Snail, Ten Is a Crab by April Pulley Sayre and Jeff Sayre, illustrated by Randy Cecil (Candlewick Press, 2003). The book counts by groups of feet-1 is a snail (because a snail touches the sand on the beach at only one spot, with its belly), 2 is a person, 3 is a person and a snail, 4 is a dog, and so on, up to 10 is a crab. Then the book continues by multiples of 10 up to 100, which the illustration shows as 10 crabs and also as 100 snails. Read on for how I used the book in a lesson with a class of second graders.

I sometimes ask students to choose their own numbers for riddles, but with this class, I asked them all to write a riddle for the number 11. I made one of those decisions that we as teachers make all of the time because of our particular students. It was a good decision for this class.

The activity can spread over several days, with a few students presenting their riddles at a time. It can be extended so that students write riddles for other numbers. It's a wonderful lesson for differentiating so that students work with numbers that are appropriate for them. Some stick with numbers less than 20, others choose multiples of 10 up to 100, and some choose other two-digit numbers.

Using a Children's Book in a Math Lesson
There's a huge variety of books available that provide many opportunities for teaching math lessons. And while the books and lesson goals differ, I follow the same guidelines when planning these lessons:

1. Read the book aloud. (At times, reread the book if it's appropriate to do so.)

2. Allow time for class discussion of the book.

3. Introduce the math connection to the class and assign the math work to be done in class, typically an activity, a problem, an exploration, or a game.

4. After students have had time to complete the assignment, lead a class discussion for students to present their work and listen and respond to one another's ideas.

5. If appropriate, assign homework based on the lesson.

Choosing Books for Math Lessons
One important aspect of incorporating children's books into math instruction is making good choices about the books to use. I consider these three questions before selecting a book to use as a read-aloud springboard for a math lesson:
1. Is the book of high quality from a literary perspective?
2. Does the book present content that is mathematically sound and grade-level appropriate?
3. Is the book effective for helping students learn to think and reason mathematically?

A Final Tip
While the goal after reading a book aloud to a class is to launch a math lesson, I first allow time for students to enjoy the book. As with any read-aloud experience, I want students to savor the text and examine the illustrations. Sometimes I'll reread all or part of the book, and I always provide the opportunity for class discussion about it. Only then do I shift the students' attention to make the math connection. After the lesson, I make the children's book available for students to revisit on their own or to check out to take home and enjoy with their families.

Additional Suggestions
For more ideas, not only for young children but also for older students, see published by Scholastic and the Math, Literature, and Nonfiction series published by Math Solutions Publications. Most of my favorite ideas are in these resources. Also, you can download an At-a-Glance Chart of children's books featured in the Math, Literature, and Nonfiction series, listed with grade levels and topics.

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Using Children's Literature to Teach Math | MARILYN BURNS MATH

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I've found that children's books are extremely effective tools for teaching mathematics. They can spark students' math imaginations in ways that textbooks or workbooks don't. Connecting math to literature can boost confidence for children who love books but are wary of math. And students who already love math can learn to appreciate stories in an entirely new way.

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Dear Parents, Common Core Math" Isn't Out to Get You

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Dear Parents, Common Core Math" Isn't Out to Get You

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Kimmie Fink, M.Ed., Curriculum & Instruction

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Dear Parents, “Common Core Math” Isn’t Out to Get You, and Here’s Why

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“Carrying the one” isn’t all it’s cracked up to be.

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Van de Walle Ch 8 Developing Early Number Concepts & Number SenseLinks to an external site.

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Van de Walle Ch 8 Developing Early Number Concepts & Number SenseLinks to an external site.

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:The Progression of Early Number & CountingLinks to an external site.

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:The Progression of Early Number & CountingLinks to an external site.

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Week 3 SlidesLinks to an external site.

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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Number Sense Through the Elementary GradesLinks to an external site.

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Number Sense Through the Elementary GradesLinks to an external site.

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Jo Boaler-Number SenseLinks to an external site.

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Number Sense Through Elementary Grades - The Recovering Traditionalist

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A recent debate between my 4 and 6 year-olds about how to tell a digital time that was showing on my phone prompted me to want to share my tips on teaching kids to tell time digitally.

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Subitizing. It's our favorite new mathematical buzz word.

Subitizing is basically the ability to quickly conceptualize quantity. Subitizing helps students use their natural approximate number system (learn more about this natural math ability through the Natural Number course!) and connect that sense of quantity to the digits we have created for them.

Just because kids can count does not mean they know number any more than a kid who sings the alphabet knows phonics. A kid who can write numbers doesn't know number any more than a kid who writes letters knows phonics.

There has to be more tie to this sense of quantity. What these numbers truly are, how they relate, and how they are units.

Many of us are already on the subitizing train. Quick dots are a great number talk for young students. You show a dot card for 3-5 seconds and ask students to describe what they see and how they see it. This can lead to some great discussions related to additive thinking. For example, I can see 8 as 4 and 4 or 5 and 3 or 6 and 2, etc. This facilitates divergent mathematical thinking which will depend conceptual understanding when students need to use number to reason in problems.

If you teach a younger grade, I truly hope you use quick dots every day. If not, you should stop here, learn more, and get started!

If you teach an older grade, I know what you are thinking...
"Does not apply to me!" or "well, for my struggling math group...but nobody else"...my kids have early number sense"

Well, well, well, WELCOME. I'm going to show you how quick dot activities are actually extremely beneficial for any grade, any developmental level, any time.

So, let's break quick dot/subitizing tasks down a bit to walk through it. When I speak of quick dots, I mean showing students an image of dots for 3-5 seconds. They can get multiple looks, but the looks should be short so students are note resorting to rote counting. They should not be given time to count.

At this early stage, quick dots are to help students develop a sense of quantity. They can see that 2 is 2 no matter how it looks because 2 is a quantity of items. Young students need many experiences with dots 0-5 to build a strong foundation.

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Subitizing for Upper Elementary:Enhancing Mathematical Understanding with Subitizing Activities - Aubree Teaches

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Discover the power of subitizing activities and quick dot strategies to develop students' number sense and mathematical reasoning. Subitizing, the ability to quickly conceptualize quantity, is a vital skill in mathematics education. Explore how subitizing fosters a deep understanding of numbers, pro

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all.

Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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Build Math Minds | Learn to LOVE Teaching Math

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all. Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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 Analysis of CAS Standard One-Number Sense

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 Analysis of CAS Standard One-Number Sense

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Purpose of Mathematics

"Pure mathematics is, in its way, the poetry of logical ideas."
~Albert Einstein, Obituary for Emmy Noether (1935)

"Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics."
~Hans Freudenthal, Why to Teach Mathematics So as to Be Useful (1968)

Mathematics is the human activity of reasoning with number and shape, in concert with the logical and symbolic artifacts that people develop and apply in their mathematical activity. The National Council of Teachers of Mathematics (2018) outlines three primary purposes for learning mathematics:

1. To Expand Professional Opportunity. Just as the ability to read and write was critical for workers when the early 20th-century economy shifted from agriculture to manufacturing, the ability to do mathematics is critical for workers in the 21st century as the economy has shifted from manufacturing to information technology. Workers with a robust understanding of mathematics are in demand by employers, and job growth in STEM (science, technology, engineering, and mathematics) fields is forecast to accelerate over the next decade.

2. Understand and Critique the World. A consequence of living in a technological society is the need to interpret and understand the mathematics behind our social, scientific, commercial, and political systems. Much of this mathematics appears in the way of statistics, tables, and graphs, but this need to understand and critique the world extends to the application of mathematical models, attention given to precision, bias in data collection, and the soundness of mathematical claims and arguments. Learners of mathematics should feel empowered to make sense of the world around them and to better participate as an informed member of a democratic society.

3. Experience Wonder, Joy, and Beauty. Just as human forms and movement can be beautiful in dance, or sounds can make beautiful music, the patterns, shapes, and reasoning of mathematics can also be beautiful. On a personal level, mathematical problem solving can be an authentic act of individual creativity, while on a societal level, mathematics both informs and is informed by the culture of those who use and develop it, just as art or language is used and developed.

References
National Council of Teachers of Mathematics (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Reston, VA: National Council of Teachers of Mathematics.

Prepared Graduates in Mathematics

Prepared graduates in mathematics are described by the eight Standards for Mathematical Practice described in the Common Core State Standards (CCSSI, 2010). Across the curriculum at every grade, students are expected to consistently have opportunities to engage in each of the eight practices. The practices aligned with each Grade Level Expectation in the Colorado Academic Standards represent the strongest potential alignments between content and the practices, and are not meant to exclude students from engaging in the rest of the practices.

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).
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.
Math Practice MP2. Reason abstractly and quantitatively.
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.
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.
Math Practice MP4. Model with mathematics.
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.
Math Practice MP5. Use appropriate tools strategically.
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.
Math Practice MP6. Attend to precision.
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.
Math Practice MP7. Look for and make use of structure.
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 x 8 equals the well remembered 7 x 5 + 7 x 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 x 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.
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 + x^2 + 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.

References
Common Core State Standards Initiative. (2010). Standards for mathematical practice. http://www.thecorestandards.org/Math/Practice

Standards in Mathematics

The Colorado Academic Standards in mathematics are the topical organization of the concepts and skills every Colorado student should know and be able to do throughout their preschool through twelfth grade experience. The standards of mathematics are:
1. Number and Quantity
From preschool through high school, students are continually extending their concept of numbers as they build understanding of whole numbers, rational numbers, real numbers, and complex numbers. As they engage in real-world mathematical problems, they conceive of quantities, numbers with associated units. Students learn that numbers are governed by properties and understand these properties lead to fluency with operations.
2. Algebra and Functions
Algebraic thinking is about understanding and using numbers, and students' work in this area helps them extend the arithmetic of early grades to expressions, equations, and functions in later grades. This mathematics is applied to real-world problems as students use numbers, expressions, and equations to model the world. The mathematics of this standard is closely related to that of Number and Quantity.
3. Data Analysis, Statistics, and Probability
From the early grades, students gather, display, summarize, examine, and interpret data to discover patterns and deviations from patterns. Measurement is used to generate, represent and analyze data. Working with data and an understanding of the principles of probability lead to a formal study of statistics in middle in high school. Statistics provides tools for describing variability in data and for making informed decisions that take variability into account.
4. Geometry
Students' study of geometry allows them to comprehend space and shape. Students analyze the characteristics and relationships of shapes and structures, and engage in logical reasoning. Students learn that geometry is useful in representing, modeling, and solving problems in the real world as well as in mathematics.
Modeling Across the High School Standards
A star symbol (★) in the high school standards represents grade level expectations and evidence outcomes that make up a mathematical modeling standards category.

Modeling links classroom mathematics and statistics to everyday life, work, and decision making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. (For more on modeling, see Appendix: Modeling Cycle in the PDF download of the standards document.)

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2020 CAS - Mathematics Standards Introduction

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"Pure mathematics is, in its way, the poetry of logical ideas." ~Albert Einstein, Obituary for Emmy Noether (1935) "Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result.

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Achievethecore.org :: Instructional Content Nav - Mathematics: Focus by Grade Level

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The Common Core and other college- and career-ready (CCR) standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, CCR standards require us to significantly narrow and deepen the way time and energy are spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. The following documents illustrate these concepts using the Common Core State Standards, but Focus, Coherence, and Rigor are integral to all college- and career-ready standards.

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

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Site Index | CDE Offices | Staff Directory

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-Focusing on Fast Isn’t How to Build Fluency

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-Focusing on Fast Isn’t How to Build Fluency

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Focusing on FAST isn’t how you build elementary math fluency - The Recovering Traditionalist

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For far too long we have associated ‘fluent’ with ‘fast’, but being fast at math does NOT mean you are fluent with math. But, when we focus on just getting kids to be FAST at math, they aren’t going to be truly fluent.

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When we ask students to memorize their addition and subtraction facts, we are essentially asking them to memorize over 100 isolated equations. This memorization doesn't stick for many kids, and doesn't allow any students to explore the foundational relationships between numbers and the properties of the operations. After years of growing and learning alongside students and teachers, I have embarked on an amazing journey with my editor, colleague, and good friend, Tracy Zager, and I'm super excited to introduce Building Fact Fluency: A Toolkit for Addition & Subtraction . Throughout my teaching journey, I have tried to leverage videos and media to make math meaningful for all students. The toolkit focuses on a contextualized understanding of fluency which will help develop a deep understanding of number for your students, while simultaneously building fluency through purposeful practice.

The Building Fact Fluency toolkit explores seven different strategies...

Through 21 different contexts...

And each context contains a lesson string that includes the following resources...

If you'd like to know more, please check out this or head on over to the . At Stenhouse, you will also be able to download a sample context that you can test drive with your students. Building Fact Fluency page at Stenhouse Publishers

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When we ask students to memorize their addition and subtraction facts, we are essentially asking them to memorize over 100 isolated equations. This memorization doesn't stick for many kids, and doesn't allow any students to explore the foundational relationships between numbers and the properties of the operations.

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Elementary educators: watch this video series and get your FREE Building Math Fluency Starter Kit. Yes, even if you are a parent at home educating your child, a teacher doing online teaching, or a teacher back doing face-to-face learning, this series is for you.

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Download PDF Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler, Professor of Mathematics Education, co-founder youcubed With the help of Cathy Williams, […]

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The Relationship Between The Development Of Math Operation Proficiency And Student Confidence

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The Relationship Between The Development Of Math Operation Proficiency And Student Confidence

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The Relationship Between The Development Of Math Operation Proficiency And Student Confidence

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The research question of this study was, What is the relationship between math operation fluency and a student’s confidence and perception of math in general? A multi-faceted intervention was developed to address both students’ multiplication fact proficiency and student confidence, with middle school students in Special Education. Pre and post assessment data was analyzed to determine the effect of this intervention on student fact proficiency and confidence. Findings suggest that students that progressed throughout the intervention period by mastering multiplication fact families also generally had an increase in math confidence. Implications for future work include determining how to improve this intervention for Special Education students with more severe deficits in math, both in terms of improving fact proficiency and confidence.

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The Relationship Between Reading Fluency and Arithmetic Fact Fluency

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The Relationship Between Reading Fluency and Arithmetic Fact Fluency

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Frontiers | The Relationship Between Reading Fluency and Arithmetic Fact Fluency and Their Shared Cognitive Skills: A Developmental Perspective

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This study investigated the underlying cognitive abilities which are related to both fluency in reading and arithmetic across different developmental phases…

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Why Children Have Difficulty with Their Math Facts

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Why Children Have Difficulty with Their Math Facts

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Build Math Minds - Elementary Math Professional Development at your Fingertips

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Episode 74: Why Children Have Difficulties with their Math Facts - Build Math Minds

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Welcome fellow Recovering Traditionalists to Episode 74. Today we are talking about Why Children Have Difficulties with their Math Facts. Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them by Arthur Baroody is the article that opened my eyes to the true root of student difficulties with mastering their math facts: A Lack of Number Sense. The reason I fell in love with this article was because I saw myself in it. I saw myself as the learners he talks about and I even saw myself as some of the teachers he talks about. In the article he describes what ‘Conventional Wisdom’ says to help students who are struggling but then opposes that with the “Number Sense View.” He does this for 3 different scenarios related to helping kids learn their facts. Get the link to the article here: buildmathminds.com/74

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4 column lesson planning reading for week 8.pdf

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When I was in the classroom, I felt skilled using data. My students set daily personal goals, we had learning targets based on our data, and students had assessment goals. I often taught whole-group lessons, though, and thinking back, I wonder why. How did I know all the students I was delivering whole-group learning to needed that particular lesson? I didn't. Formative assessment could have helped me.

After I left teaching, I spent time learning more about formative assessment. I realize now that I struggled with my understanding of delivering and analyzing formative assessments. I let my curriculum map determine what I taught instead of my students' data. While there's nothing I can do about the past, I hope that by sharing what I've learned, I can help you in your classroom today. Knowing how to use formative assessment effectively can inform your whole-group, small-group, and in-the-moment instruction and help you meet students where they are.

Reminder: What formative assessment is (and isn't)

NWEA uses the following definition of "formative assessment," adapted with permission from the Council of Chief State School Officers (CCSSO): "Formative assessment is a planned, ongoing process used by all students and teachers during learning and teaching to elicit and use evidence of student learning to improve student understanding of intended disciplinary learning outcomes and support students to become self-directed learners."

Knowing how to use formative assessment effectively can inform your whole-group, small-group, and in-the-moment instruction and help you meet students where they are.

The words "planned, ongoing process" really stand out to me. There are moments where formative assessment is very intentional, or planned. You decide ahead of time to have an exit ticket at the end of every unit, for example. There are other opportunities where formative assessment is more spontaneous, where you realize there's a disconnect with your students and you pause to explore why. After multiple quizzical looks from your kids, for example, you stop, ask a specific question related to your lesson, and have them answer using a thumbs-up for "yes" or thumbs-down for "no."

While the definition doesn't spell this out explicitly, it's important to remember that formative assessment should not be graded. Its sole purpose is to increase your understanding so you can adjust your teaching. It's never about penalizing kids for what they don't know yet.

Tip #1: Use formative assessment to prepare for whole-group instruction

There is a time and a place for whole-group learning. Whole-group instruction is good for introducing new concepts and encouraging whole-class engagement. (For ideas on how to get kids excited to learn together, read "The best whole-group engagement strategies.") Typically, whole-group instruction is targeted toward the average student in your class.

Data can provide insight into what you can teach in a whole-group setting. Using formative assessment at the end of a previous day's lesson or at the start of a new lesson will allow you to see what your students already know about the standard you have targeted to teach. Consider using an exit or entrance ticket, for example.

The Teacher Toolkit walks you through how to use entrance tickets in your classroom. Start by having a question on the board related to the day's learning. Allow yourself time to review the data you gather before proceeding to the whole-group lesson. You may find you have misconceptions about what your students know or that there are things they've misunderstood that you can address before beginning. When reviewing your formative assessment data, you may also find that what you thought should have been a whole-group lesson may need to be taught in small groups instead. What would tip you off to this? A wide array of answers proving a big range of differing abilities.

Tip #2: Let formative assessment data guide small-group work

Like my colleague Tami Hunter mentioned in her post "How to use flexible grouping remotely," "Using flexible grouping strategies can be a great way to keep kids engaged in learning, empower students to own their learning, increase understanding through collaboration, and allow for social connections." To be flexible, your small groups should be fluid and change frequently based on the standard you are working on and on your formative assessment data.

Formative assessment [...] gives you the information you need to make instructional decisions based on exactly where students are, and it helps students have buy-in and ownership of their learning.

Small groups allow you to provide scaffolding based on where your students are. You can specifically target your lesson based on the needs of the students in that group. Your instruction will vary depending on which group you are working with and what level they are on. With some groups, you will need to help the students reach the grade-level expectation of a standard by filling gaps in previous skills. Other groups may be right on target and able to learn the new standard with ease. Some groups may have already mastered the standard you're on and are ready to explore it more deeply or even move on to the next standard.

By providing this targeted attention to student learning needs you can help students grow and be empowered in their learning paths. While working with small groups you can also differentiate seat work or centers that are not teacher directed. Work groups could have a designated tub or folder with assignments that are targeted to meet students' learning needs, for example. This does take more planning up front, but if you have planning or data conversations with your grade-level teams, you can plan these varied tasks together and share the load.

Tip #3: Wield formative assessment feedback to make in-the-moment adjustments

Whether you're teaching the whole class or have broken into groups, you'll have a multitude of opportunities to adjust your instruction-right before, during, or immediately after a lesson-based on the needs of your students. There are several simple strategies you can try right away to target your instruction and help students grow academically. Here are some examples of formative assessment techniques and when to use them.

Right before a lesson

Standards check-in. I have used a standards check-in in kindergarten through middle school and wrote more about it in a previous post. Start by posting the lesson's standard on a chart. Give each student a colored sticky note and tell them to write their name. Ask students to rate how they feel about the standard before they get started. This will help you create the learning target based on the standard as well as develop success criteria. It is also good to conduct another check at the end of the lesson to see if students have progressed in their understanding of the standard.

Pre-assessment. This type of assessment is given at the start of a unit or lesson to see what students know, so you can decide how to address learning needs before beginning the lesson. I encourage you to read this blog post to learn more.

During a lesson

Questioning. How you ask questions in your classroom can be a way of formatively assessing. Be intentional about planning your questions before lessons and make sure the questions you ask are higher order and train the brain to think deeply. Achieve the Core explains the purpose of developing questions to make meaning in their article "Creating and sequencing text-dependent questions to support English language learners" (note that their tips are useful for all students). Math teacher and trainer Craig Barton digs into this, too, in his article "What is a diagnostic question?"

All-student response systems. Having all students give you feedback at once can give you insight into how students are processing material as they're receiving it. I really like Emerging Ed Tech's article "Top 5 student response systems that work on multiple platforms." Want to try a response system that's lower tech? Ask students to use whiteboards and hold them up to answer questions during a lesson, or have them use hand signals.

Immediately after a lesson

Metacognition. When students are aware of their thinking throughout a lesson, they're practicing metacognition, something that helps them process the material and better understand how they think. End a lesson by having students reflect on what-and how-they learned. This practice can help them retain more of the material.

Exit ticket. While entrance tickets are given at the start of a lesson, exit tickets are given at the end. There are several ways to use exit tickets both virtually and on paper, as demonstrated by We Are Teachers.

Never stop learning

Formative assessment empowers both teachers and students. It gives you the information you need to make instructional decisions based on exactly where students are, and it helps students have buy-in and ownership of their learning. To learn more about how to make formative assessment a bigger part of your practice, download Making it work: How formative assessment can supercharge your practice.

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3 tips for understanding when-and how-to use formative assessment

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Use formative assessment to inform your whole-group, small-group, and in-the-moment instruction and meet your students where they are.

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Teaching Mathematics Equitably to All Children

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Teaching Mathematics Equitably to All Children

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Getting Started with Educational Equity.pdf

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Early number concepts are the building blocks of mathematical understanding. They provide the foundation for more complex mathematical skills, such as addition, subtraction, multiplication, and division. When students have a strong understanding of early number concepts, they are better equipped to:

Solve problems: They can break down problems into smaller, more manageable steps.

Reason mathematically: They can use logical thinking to analyze situations and find solutions.

Communicate mathematically: They can explain their thinking and justify their answers.

Key Early Number Concepts

Number sense: Understanding the meaning of numbers and how they relate to each other.

Counting: Being able to count objects accurately and fluently.

Cardinality: Understanding that the last number counted represents the total number of objects in a set.

One-to-one correspondence: Matching each object with one and only one number word.

Subitizing: Quickly recognizing small quantities of objects without counting.

Number operations: Understanding the concepts of addition, subtraction, multiplication, and division.

Place value: Understanding the value of digits based on their position in a number.

Strategies for Developing Early Number Concepts

Use manipulatives: Manipulatives, such as blocks, counters, and number lines, can help students visualize and understand number concepts.

Engage in hands-on activities: Activities like sorting, patterning, and measuring can help students develop a deeper understanding of numbers.

Use real-world contexts: Connect number concepts to real-world situations, such as cooking, shopping, and playing games.

Encourage mathematical language: Use precise mathematical language, such as "more than," "less than," and "equal to," to help students develop their vocabulary.

Provide opportunities for practice: Consistent practice is essential for developing strong number sense.

By focusing on early number concepts, we can help students build a solid foundation for future mathematical success.

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Basic fact fluency refers to the ability to quickly and accurately recall basic arithmetic facts. This includes addition, subtraction, multiplication, and division facts. When students are fluent in basic facts, they can:

Solve problems more efficiently: They can focus on problem-solving strategies rather than spending time calculating basic facts.

Develop a deeper understanding of mathematics: Fluency with basic facts can help students develop a better understanding of number relationships and mathematical concepts.

Increase their confidence in math: When students know their facts, they feel more confident in their abilities and are more likely to be engaged in math activities.

Strategies for Building Fact Fluency:

Here are some effective strategies to help students develop basic fact fluency:

1. Conceptual Understanding:

Use concrete manipulatives (like blocks or counters) to represent number operations.

Use visual models (like number lines or arrays) to illustrate strategies.

Connect facts to real-world situations.

2. Fact Strategies:

Teach students effective strategies, such as:

Doubles: 2 + 2, 3 + 3, etc.

Doubles Plus One: 2 + 3, 3 + 4, etc.

Near Doubles: 3 + 4, 4 + 5, etc.

Ten-Frame Strategies: Using ten-frames to visualize numbers and operations.

Subtraction as the Inverse of Addition: Using known addition facts to solve subtraction problems.

Multiplication and Division Relationships: Understanding the relationship between multiplication and division.

3. Practice and Games:

Timed Drills: Short, timed drills can help students build speed and accuracy.

Math Games: Engaging in math games can make practice fun and motivating.

Flashcards: Using flashcards is a classic way to practice facts.

Digital Tools: Many online and app-based tools can provide interactive practice.

4. Frequent Review:

Incorporate regular review into lessons to reinforce learning.

Use quick, daily practice sessions to keep skills sharp.

Additional Tips:

Positive Reinforcement: Praise and reward students for their efforts.

Individualized Instruction: Tailor instruction to meet the needs of each student.

Patience and Persistence: Building fact fluency takes time and practice.

Celebrate Success: Acknowledge and celebrate students' achievements.

By implementing these strategies and providing consistent practice, you can help your students develop strong basic fact fluency, setting them up for success in future math learning.

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The Concrete-Pictorial-Abstract (CPA) approach is a teaching method that helps students build a deep understanding of mathematical concepts by progressing through three stages of learning:

1. Concrete Stage

Hands-on Learning: Students use physical objects (manipulatives) to represent mathematical concepts.

Real-World Connections: Relate the concept to real-life situations.

Examples of Manipulatives:

Base-ten blocks

Counters

Fraction tiles

Pattern blocks

2. Pictorial Stage

Visual Representation: Students use drawings, diagrams, or pictures to represent the concept.

Transition from Concrete to Abstract: Bridge the gap between physical objects and abstract symbols.

Examples of Pictorial Representations:

Bar models

Number lines

Diagrams

Drawings

3. Abstract Stage

Symbolic Representation: Students use abstract symbols (numbers and mathematical operations) to represent the concept.

Problem-Solving: Apply the concept to solve word problems and equations.

Examples of Abstract Representations:

Number sentences

Algebraic equations

Formulas

Benefits of the CPA Approach:

Improved Understanding: By moving through these stages, students develop a strong foundation of understanding.

Enhanced Problem-Solving Skills: Students can apply their knowledge to solve complex problems.

Increased Confidence: A gradual progression from concrete to abstract helps students build confidence in their abilities.

Better Retention: Concrete experiences and visual representations help students remember concepts for longer periods.

Example: Teaching Addition

Concrete Stage: Use physical objects (e.g., blocks) to represent numbers and combine them to find the sum.

Pictorial Stage: Draw pictures of the blocks or use number lines to visualize the addition process.

Abstract Stage: Write number sentences (e.g., 2 + 3 = 5) to represent the addition problem.

By effectively implementing the CPA approach, teachers can create engaging and effective math lessons that cater to diverse learning styles and help students achieve mathematical success.

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Teaching through problem-solving is a pedagogical approach where students learn mathematics by actively engaging in problem-solving activities. Rather than simply memorizing procedures and formulas, students are encouraged to think critically, reason logically, and apply their knowledge to solve challenging problems.

Key Principles of Teaching Through Problem-Solving:

Problem-Based Learning: Students are presented with open-ended problems that require them to apply their knowledge and skills.

Active Engagement: Students are actively involved in the learning process, exploring, experimenting, and discussing ideas.

Collaborative Learning: Students work together in groups to share ideas, strategies, and solutions.

Metacognition: Students reflect on their thinking processes and learn from their mistakes.

Teacher as Facilitator: The teacher guides the learning process, asking probing questions and providing support as needed.

Benefits of Teaching Through Problem-Solving:

Deeper Understanding: Students develop a deeper understanding of mathematical concepts.

Enhanced Problem-Solving Skills: Students become skilled at breaking down problems, analyzing information, and developing creative solutions.

Increased Motivation: Engaging in challenging problems can motivate students and spark their curiosity.

Improved Communication Skills: Students learn to articulate their thinking and collaborate with others.

Positive Attitudes Toward Math: By experiencing the joy of discovery, students develop a positive attitude towards mathematics.

Strategies for Implementing Problem-Solving:

1. Pose Challenging Problems: Present problems that require students to think critically and apply their knowledge.

2. Encourage Multiple Approaches: Allow students to explore different strategies and solutions.

3. Foster a Growth Mindset: Emphasize the importance of effort, perseverance, and learning from mistakes.

4. Provide Timely Feedback: Offer constructive feedback to help students improve their problem-solving skills.

5. Create a Supportive Learning Environment: Encourage students to take risks, share ideas, and ask questions.

6. Use Real-World Connections: Relate problems to real-life situations to make them more meaningful.

Example Problem-Solving Activity:

Problem: A class of 25 students is going on a field trip. If each van can hold 6 students, how many vans are needed?

Strategies:

Use manipulatives (e.g., counters) to represent students and vans.

Draw a diagram or picture to visualize the problem.

Use repeated subtraction or division to find the answer.

By incorporating problem-solving into your math instruction, you can create a more engaging and effective learning experience for your students.

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Meeting the needs of all students in a mathematics classroom requires a multifaceted approach that considers individual differences, learning styles, and abilities. Here are some strategies to foster an inclusive and effective learning environment:

Differentiated Instruction

Content: Adjust the complexity of tasks and problems to match students' abilities.

Process: Provide various learning activities and strategies to accommodate different learning styles.

Product: Offer a range of assessment options to allow students to demonstrate their understanding in different ways.

Flexible Grouping

Small-Group Instruction: Target specific skills and concepts for small groups of students.

Partner Work: Encourage collaboration and peer learning.

Independent Work: Provide opportunities for self-paced learning.

Use of Technology

Educational Software: Utilize software to provide individualized practice and feedback.

Online Resources: Access a variety of online resources to supplement instruction.

Digital Tools: Use digital tools to enhance engagement and motivation.

Effective Communication

Clear and Concise Explanations: Use clear and concise language to explain concepts.

Active Listening: Pay attention to students' questions and concerns.

Encourage Student Questions: Foster a classroom culture where students feel comfortable asking questions.

Positive Classroom Culture

Create a Safe Space: Establish a positive and supportive learning environment.

Celebrate Diversity: Value and appreciate different perspectives and backgrounds.

Encourage Effort and Persistence: Promote a growth mindset and perseverance.

Assessment and Feedback

Formative Assessment: Use formative assessments to monitor student progress and adjust instruction.

Summative Assessment: Assess student learning through a variety of methods, such as tests, projects, and presentations.

Provide Specific and Timely Feedback: Offer constructive feedback to help students improve.

Additional Considerations:

Learning Disabilities: Provide accommodations and modifications as needed.

English Language Learners (ELLs): Use visual aids, real-world examples, and simplified language.

Gifted Students: Challenge them with advanced problems and projects.

Students with Behavioral Challenges: Use positive reinforcement and behavior management strategies.

By implementing these strategies, educators can create inclusive mathematics classrooms where all students can thrive and reach their full potential.

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Facilitating Discourse: Fostering Mathematical Conversations

Facilitating discourse in a mathematics classroom involves creating opportunities for students to engage in meaningful conversations about mathematical ideas. By encouraging students to share their thinking, justify their reasoning, and listen to the perspectives of others, we can foster a deeper understanding of mathematical concepts.

Strategies for Facilitating Discourse:

1. Ask Open-Ended Questions:

Pose questions that have multiple correct answers or require students to explain their thinking.

For example, instead of asking "What is the answer to 5 x 7?", ask "How can you use known facts to find the product of 5 and 7?"

2. Create a Supportive Environment:

Establish a classroom culture where students feel safe to share their ideas, even if they are incorrect.

Encourage respectful dialogue and active listening.

3. Use Think-Pair-Share:

Give students time to think individually about a problem or question.

Then, have them discuss their ideas with a partner.

Finally, facilitate a whole-class discussion to share different perspectives.

4. Use Math Talk Moves:

Revoicing: Restate a student's idea to clarify and extend their thinking.

Asking for Elaboration: Prompt students to provide more details or examples.

Waiting Time: Allow sufficient wait time for students to formulate their thoughts.

Cueing: Use verbal or nonverbal cues to encourage participation.

5. Use Mathematical Representations:

Encourage students to use diagrams, models, or other visual representations to explain their thinking.

Use these representations as a starting point for class discussions.

6. Use Collaborative Problem-Solving:

Assign group tasks that require students to work together to solve problems.

Encourage students to share their strategies and justify their solutions.

By implementing these strategies, you can create a classroom where students are actively engaged in mathematical discourse, leading to deeper understanding and improved problem-solving skills.

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Flexible Methods for Addition & Subtraction

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Flexible Methods for Addition & Subtraction

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Flexible methods for addition and subtraction involve using a variety of strategies to solve problems, rather than relying on a single algorithm. This approach helps students develop a deeper understanding of number sense and problem-solving skills.

Key Strategies for Flexible Methods:

1. Number Bonds:

Break numbers into smaller, more manageable parts.

Use number bonds to decompose and recompose numbers.

Example: To add 8 + 7, break 7 into 2 and 5. So, 8 + 7 = 8 + 2 + 5 = 10 + 5 = 15.

2. Mental Math Strategies:

Compensation: Adjust numbers to make calculations easier.

Example: To add 48 + 27, add 2 to 48 to make 50, then subtract 2 from 27 to make 25. So, 50 + 25 = 75.

Breaking Apart Numbers: Decompose numbers into place values to add or subtract.

Example: To subtract 34 - 17, break 34 into 30 + 4 and 17 into 10 + 7. Subtract the tens and ones separately: 30 - 10 = 20, 4 - 7 = -3. So, 20 + (-3) = 17.

3. Using Number Lines:

Visualize addition and subtraction as movement on a number line.

Use jumps of different sizes to represent different strategies.

4. Using Base-Ten Blocks:

Manipulate physical objects to represent numbers and operations.

Model regrouping and borrowing using base-ten blocks.

5. Using Algorithms:

Teach standard algorithms, but also encourage students to use mental math and estimation to check their answers.

Benefits of Flexible Methods:

Deeper Understanding: Students develop a deeper understanding of number relationships and operations.

Problem-Solving Skills: Students become more flexible problem-solvers and can choose the most efficient strategy.

Increased Confidence: Students gain confidence in their ability to solve problems in different ways.

Reduced Math Anxiety: Flexible methods can make math less intimidating and more enjoyable.

By encouraging students to use a variety of strategies, we can help them become more confident and skilled mathematicians.

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Flexible Methods for Multiplication & Division

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Flexible Methods for Multiplication & Division

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Flexible methods for multiplication and division involve using a variety of strategies to solve problems, rather than relying solely on traditional algorithms. This approach helps students develop a deeper understanding of number relationships and problem-solving skills.

Key Strategies for Flexible Methods:

1. Number Sense and Properties of Operations:

Commutative Property: The order of factors does not affect the product (e.g., 3 × 4 = 4 × 3).

Associative Property: The grouping of factors does not affect the product (e.g., (2 × 3) × 4 = 2 × (3 × 4)).

Distributive Property: Break down numbers into smaller parts to make multiplication easier (e.g., 7 × 6 = (7 × 5) + (7 × 1)).

2. Mental Math Strategies:

Breaking Apart Numbers: Break down numbers into smaller, more manageable parts (e.g., 6 × 8 = (6 × 4) × 2).

Using Doubling and Halving: Double one factor and halve the other to simplify the calculation (e.g., 4 × 16 = 8 × 8).

3. Using Arrays and Area Models:

Visualize multiplication as the area of a rectangle.

Break down arrays into smaller, more manageable parts.

4. Using the Lattice Method:

A visual method for multiplying multi-digit numbers that involves breaking down numbers into place values and multiplying digit by digit.

5. Using Division Strategies:

Partial Quotients: Break down the divisor into smaller parts and divide in stages.

Traditional Algorithm: Teach the standard algorithm, but also encourage students to use mental math and estimation to check their answers.

Benefits of Flexible Methods:

Deeper Understanding: Students develop a deeper understanding of number relationships and operations.

Problem-Solving Skills: Students become more flexible problem-solvers and can choose the most efficient strategy.

Increased Confidence: Students gain confidence in their ability to solve problems in different ways.

Reduced Math Anxiety: Flexible methods can make math less intimidating and more enjoyable.

By encouraging students to use a variety of strategies, we can help them become more confident and skilled mathematicians.

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Early number concepts are the foundational building blocks of mathematical understanding. They include:

Number sense: Understanding the meaning of numbers and how they relate to each other.

Counting: Being able to count objects accurately and fluently.

Cardinality: Understanding that the last number counted represents the total number of objects in a set.

One-to-one correspondence: Matching each object with one and only one number word.

Subitizing: Quickly recognizing small quantities of objects without counting.

Number operations: Understanding the concepts of addition, subtraction, multiplication, and division.

Place value: Understanding the value of digits based on their position in a number.

Strategies for Developing Early Number Concepts:

1. Hands-on Activities:

Use manipulatives like blocks, counters, and number lines to represent numbers and operations.

Engage in activities like sorting, patterning, and measuring.

2. Real-world Connections:

Connect number concepts to real-life situations, such as cooking, shopping, and playing games.

3. Language and Vocabulary:

Use precise mathematical language to describe quantities and relationships.

Encourage students to use mathematical language in their discussions.

4. Games and Puzzles:

Play games that involve number recognition, counting, and problem-solving.

Use puzzles to develop spatial reasoning and number sense.

5. Number Sense Activities:

Practice counting objects, recognizing number patterns, and comparing quantities.

Use ten-frames to visualize numbers and operations.

6. Problem-Solving:

Present word problems that require students to apply their number sense and problem-solving skills.

Encourage students to use different strategies to solve problems.

By providing a variety of learning experiences, we can help young children develop a strong foundation in early number concepts, which will benefit them in their future mathematical learning.