Showing posts with label common core. Show all posts
Showing posts with label common core. Show all posts

What is Number Sense?

A person's ability to use and understand numbers:
• knowing their relative values,
• how to use them to make judgments,
• how to use them in flexible ways when adding, subtracting, multiplying or dividing
• how to develop useful strategies when counting, measuring or estimating.

What is number sense?

The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well-organized conceptual framework of number information that enables
a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

  • mental calculation
  • computational estimation 
  • judging the relative magnitude of numbers 
  • recognizing part-whole relationships and place value concepts and;
  • problem solving 

How does number sense begin?

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding. Piaget called this ability to instantaneously recognize the number of objects in a small group 'subitizing'. As mental powers develop, usually by about the age of four, groups of four can be recognized without counting. It is thought that the maximum number for subitizing, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognize more than five objects if they are arranged in a particular way or practice and memorization takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognized when presented this way.

Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

What teaching strategies promote early number sense?

Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts the combination of 'four and two makes six'. If the four is not subitised, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.

If mental strategies such as these are to be encouraged (and just counting discouraged) then an element of speed is necessary. Seeing the objects for only a few seconds challenges the mind to find strategies other than counting. It is also important to have children reflect on and share their strategies. This is helpful in three ways:

  • verbalizing a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
  • it provides other children with the opportunity to pick up new strategies;
  • the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.
Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.
Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

What games can assist development of early number sense?

Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.

Deal and Copy (4-5 years) 3-4 players

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

Memory Match (5-7 years) 2 players

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

What's the Difference? (7-8 years) 2-4 players

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

Common Core Resources

So, over the last two weeks, I've given the state testing (which is changing next year) and worked on replacing my now died laptop. With that now behind me, I can look forward to the second have of state testing for 4th and 5th grader in science and social studies on the computers. I still have no clue what the accommodations look like. With that all over for now, I left for Spring Break.

Something that keeps coming up in my school, is where to look for resources that help teachers and parents to understand that the depth looks like across the grades. Colorado is a CMAS and PARCC state--fewer standards at a deeper level but we have not told what that deeper level looks like. As I have gone searching to resources that break down math skills, I have come across a couple of sites that might help explain what Common Core is parents and help teachers continue to content with resources.

For Parents
Parents' Guide to Student Success
The Parents’ Guide to Student Success (listed below in English and Spanish) was developed in response to the Common Core State Standards in English language arts and mathematics that more than 45 states have adopted. To find out if your state has adopted the standards, visit Created by teachers, parents, education experts, and others from across the country, the standards provide clear, consistent expectations for what students should be learning at each grade in order to be prepared for college and career.

Common Core State Standards Initiative
The Common Core State Standards (CCSSI) is a joint effort led by the National Governors Association Center for Best Practices and the Council of Chief State School Officers to develop a common core of K-12 standards in English language arts and Mathematics.

Achieve The Core
As educators, as researchers, and as citizens, we view the changes brought by the college and career readiness focus of the Common Core State Standards as a once-in-a-generation opportunity for kids of all backgrounds and ability levels to better fulfill their potential. Like the standards themselves, we are evidence-based in our approach. Our work is aimed at ensuring that teachers across the country are able to put the standards to work, quickly and effectively, to help their students and colleagues aspire to a higher standard and reach it. Accordingly, the content available on this site is assembled by and for educators and is freely available to everyone to use, modify and share.

We invite educators and people curious about the Common Core State Standards to explore what the site has to offer, including hundreds of math and literacy resources for teachers, resources for leaders who are putting college and career readiness standards into action in their own schools, and opportunities to become an advocate for the Common Core.

Parent Roadmaps to the Common Core Standards
The Council of the Great City Schools' parent roadmaps in mathematics provide guidance to parents about what their children will be learning and how they can support that learning in grades K-8. These parent roadmaps for each grade level also provide  three-year snapshots showing how selected standards progress from year to year so that students will be college and career ready upon their graduation from high school.

Common Core for Parents
Our Core-ready programs and materials help students become college and career ready while keeping the joy in learning. A great place to get questions answered from what the DRA (Developmental Reading Assessment) is to figuring out if your child has a learning disability.

For Teachers

Common Core Explorer
Graphite is a free service from nonprofit Common Sense Media designed to help preK-12 educators discover, use, and share the best apps, games, websites, and digital curricula for their students by providing unbiased, rigorous ratings and practical insights from our active community of teachers.

At your fingertips is a wealth of information and resources about the Common Core State Standards (CCSS) for teachers, educational leaders, schools, and school districts. It has a great collection of evidence-based tools, strategies, videos, and supporting documents to learn about the implementation and transition to CCSS. A great place to build your capacity for understanding the shifts of the CCSS.

Defining the Core
The Common Core State Standards (CCSS) are the culmination of an extended, broad-based effort to fulfill the charge issued by the states to create the next generation of K-12 standards in order to help ensure that all students are college and career ready no later than the end of high school. This is one of the most important changes in education in the United States in the last fifty years and stands to positively affect students, parents, teachers, communities, and the workforce as we take a firm grasp on what 21st century learning truly means.

ELA/Literacy Learning Progressions
The online tool organizes the Common Core ELA strands of Writing, Speaking & Listening, and Language into visible student skill progressions in an effort to keep the learning of all students moving forward. These progressions continually guide students and teachers to "what's next" and are a great foundation for assessing reading comprehension within the standards. The following descriptors outline how the various parts of the tool are constructed. This is great for backwards planning, so you can meet students where they are.

Using Common Core Standards to Enhance Classroom Instruction & Assessment

Not everything Robert Marzano publishes is worth buying but I picked this one up earlier and love the rubrics (0-4 point scale) that have been created for all the standards. This is the closest I have come to finding something that gets at the depth of the standards. His website has freebies that are tied to each of the books.

I hope that this list of resources helps you better understand Common Core and where its headed. If you have a favorite resource, please share. Have a great week.

Reading Fluency and Common Core

I have this love hate relationship with reading fluency. My building uses DIBELS for all our primary students but only use the DRA decoding rubric for the intermediate students. Both of these assessments have speed ranges are the bare minimum of how fast students need to be reading.

Now factor in Common Core. Fluency is addressed starting in kindergarten with standard RF 1.4 RF.K.4 with students reading emergent-reader texts with purpose and understanding. And the standard changes in first grade with RF.1.4 where students have to read with sufficient accuracy and fluency to support comprehension. It remains the same through fifth grade. Common Core doesn't address in detail what that looks like. The key in reading fluency is finding that perfect balance of accuracy and understand what is being read. Common Core wants students to understand what they are reading and then have meaning deep conversation about what they are being taught. This is a hard balance and a tough shift this year for me.

It's the readability and the depth in which students have to understand material that really counts. This has more to do with Lexile Measures than DRA or Fountas and Pinnell levels than anything else. This makes for some very fun shifts in what students have to read and what they need to be able to read. The focus needs to stay on text just slightly harder than their independent reading level (at 96% to 98% accuracy and 90 to 125 wcpm) to have a conversation with the depth needed for them to demonstrate that they really understand what they are reading. The shift is in the text complexity. Check Common Core Appendix A for more information on what the Lexile expectations look like.

I'm not sure this is a huge problem for my student's who struggle with reading fluency but I do know that if they can't read fast enough, they won't finish the test. Students should not see difficult material for the first time on these tests. They have to be prepared to closely read, examine, decode, and digest material that is not within their “fluency” or comfort range.

I'm not sure if the best way to accomplish this is by giving them more challenging texts or by guiding students through have a more in-depth conversation about what they are reading.  If students can talk about what they are reading, then they can write about it.

Close Reading where students markup the text is a great way to support students. I have had student do this but I'm having problems with them transferring this idea outside of their time with me. Plus, our new assessment will be on the computer. No marking the text. Markup the text has helped them find the evidence to support their thinking which is I've been told is way more important. They need to work on telling why they picked that specific chunk of text though.

Why Fluency is important:

As great as close reading of complex text maybe for instruction, we should not measure independent reading. Also from Appendix A (p.4):

Students need opportunities to stretch their reading abilities but also to experience the satisfaction and pleasure of easy, fluent reading within them, both of which the Standards allow for…. Students deeply interested in a given topic, for example, may engage with texts on that subject across a range of complexity.

It's the reading easy material that a student enjoys a book and builds fluency. For independent reading recommendations, students need to read and enjoy whatever they choose, at whatever level for independent reading. That is how to build lifelong readers.

Common Core and Shifting the Cognitive Load

Since, coming back from Thanksgiving Break, I have worked to shift the cognitive load in my small groups. This has not been as easy as you may think. Why?? Well, mostly because of the rubric I'm evaluated and I do most of the work. My students will never be able to tackle more complex text, if I can't find ways for them to take that load on.

The Common Core State Standards have changed the way I look at teaching reading. I have had to shift my focus to increase the rigor and cognitive load on my students to gathering evidence, knowledge, and insights from what they read. In fact, 80-90% of the reading standards in every grade require text-dependent analysis — being able to answer questions only by referring back to the assigned text, not by drawing upon and referencing prior knowledge and experiences. This is the first year where 90% of everything my students has read has been informational text.

The hard part getting students to talk more about the text without my direct support--ie; me needing to talk way, way less. But this means that they have to take it having the conversation about the book at a depth that is meaningful and with a high level of rigor.

Gretchen Owocki's book has some many strategies to support reader and show the progression of rigor from kindergarten to fifth grade for the Common Core Standards. My students have been working on Reading Anchor 1: Read closely to determine what the text says explicitly and to make logical inferences from it; cite textural evidence when writing or speaking to support conclusions drawn from the text. (Key ideas and Details)

Students have to read closely, to determine what the text says to find the evidence to support their thinking. Easier said than done. This can be seen in what students need to be able to do on their own at the end of each reading level. One way that I have been able to build my student's ability to talk about books has been to use Accountability stems. These sentence frames have scaffolded my student's as they move towards these benchmarks.

I have many factors to keep in mind how I introduce students to the advanced language of informational text analysis because they don't have the skill set necessary to access more complex text or the relevant terminology to be successful without direct support.. Informational and narrative text features, organization, genres, comprehension questions, and constructed response tasks differ strikingly.

Accountability talk is one way that I'm able to increase the cognitive load on my students because they get to do all the talking. I have included a sample of the sentence frames I use with my students.

The ultimate goal is to ensure that students are more familiar with the text structure and content. This also gives all students daily opportunities to communicate using more sophisticated social and academic English. The more they talk the better. Have a great week.

Subtracting the Common Core Way

The beginning of the week started by teaching my 3rd grade math group subtraction. We worked on addition before leaving for break--I was amazed that they remembered how to do addition after getting back. With addition using unifix cubes and drawing the pictures out had them adding within minutes.

Common Core Standards highlight building number sense with third graders.  To build number sense, we have taught several strategies for addition and subtraction: using a hundreds chart, an open number line, and place value blocks.  

This anchor chart shows how to subtract using an open number line.  The hardest part for the students is using the skill of counting backwards, especially when going back over a ten-mark. We struggle with remembering friendly numbers.

Here is my anchor chart for using place value blocks to subtract when there is no need to regroup. We started with real place value blocks.  Then we moved to the picture model.  After a bit of modeling, my kiddos really caught on to this strategy well. When subtracting without regrouping, this method tended to be especially easy for them. 

Ultimately, the boys decided on the standard algorithm for problems needing regrouping. This poem helped them remember to regroup.

A great online manipulatives resource is from the National Library of Virtual Manipulatives. They have base ten blocks for addition and subtraction. This can be used from a SMART board--my students would do the problems from the board. They also have Number Pieces. This app is free and is easy for the students to build the problems and solve them.  Both work great to provide students with practice. 


I hope this gives you some inspiration on how to help students master subtraction. Common Core has many twists and turns. It's important that whatever strategy students find and have success with that they are able to explain their thinking and tell is the strategy they choose is effective and efficient.  

Parent Common Core Website

We have all familiar with Common Core but our parents well that's a whole different thing. If your parents are like mine, understanding Common Core is more than a challenge.  Explaining what it is and how it will change things is hard if not close to impossible. Early this week, I came across a wonderful website that breaks it down for parents. The Parent Toolkit was designed to help parents track and support their child’s progress from preschool to graduation from High School.

NBC News and education company Pearson have come together to form Parent Toolkit, a plain-talk, grade-by-grade digital guide to Common Core and benchmarks for each grade. Parents can drill down by grade to see what the math and English language arts requirements are, and get sample problems that illustrate the concept.

Since parents are also their children’s teachers, the site provides ideas for everyday ways to support what your child’s learning at school. For instance, if he’s working on fractions, give him a real-world lesson by dividing a sandwich. The site is also planning to add social and health and wellness milestones for each grade in 2014.

A heads up I'll be throwing a sale at my Teachers Pay Teachers store the beginning of next month.

What's mastery?

In the world of Common Core, we have to shift our thinking to mastery and what it looks like as we move through core. Everyone has there own definition of mastery. Which makes it hard to figure out how mastery is defined.

Determining what's acceptable evidence of mastery is key.  It's not enough to simply identify what knowledge and skills are essential. You have to determine what evidence will show that students have  mastered the essential knowledge and skills.  If not, how will you know if they have mastered the information???

Robyn Jackson (Never Work Harder than Your Students), points out that to figure out what mastery is to ask two questions:
1) What will students be able to do? 
Meaning you have to look at your core curriculum and determine what is the essential content and  processes that students need to know. 

2) What criteria you judge this demonstration of mastery?

Example might be: students correctly multiply fractions 80% of the time; correctly identify 45 of the 50 states; or correctly answer 75% of the reading questions on a novel.

There is no answer to this question. A lot of this boils down to your end of the year testing. It also depends on your stated learning goals.  Once you have determined the criteria for mastery, you can determine what summative assessment will best reveal this mastery.

The key elements in mastery learning are:
  • Clearly specifying what is to be learned and how it will evaluated
  • Allowing students to learn at their own pace
  • Assessing student progress and providing appropriate feedback or remediation
  • Testing that final learning criterion has been achieved
In fact, the end of the unit or summative assessment should be planned first. That's right before you even plan your lessons. If you use Backwards Planning, you know that it's the easy way to make sure students will master your objectives. The summative assessment should only test the need to know things that you have to cover. I give mine as a pre and post test. It helps me know if they have mastered the material. On last note, all students are held to the same standards. Differentiation is not about having different standards for different students. One set of standards and the how you present, teach, and support your students is differentiated. How do you define mastery?

Counting to 10 and 20

A change that Common Core has brought to teaching math is helping students understand the relationship between numbers and quatities and connect counting to cardinality. I created Match Me a Turtle for my students to count a set (dice dots) and see sets and then match the digits to the turtle. Then they can match the digit and turtle with the number word. By using dice students learn that no matter how you arrange a number of dots, they will digit doesn't change.

These connections are higher-level skills that require students to analyze, reason about, and explain relationships between numbers and sets of objects. Once they have mastered numbers and sets to 10, they will be comfortable with working numbers to 20. Common Core states that students should have this skill mastered by the end of Kindergarten.
Students implement correct counting procedures by pointing to one object at a time (one-to-one correspondence), using one counting word for every object (synchrony/ one-to-one tagging), while keeping track of objects that have and have not been counted. This is the foundation of counting.
Students answer the question “How many are there?” by counting objects in a set and understanding that the last number stated when counting a set (…8, 9, 10) represents the total amount of objects: “There are 10 bears in this pile.” (cardinality). Since an important goal for children is to count with meaning, it is important to have children answer the question, “How many do you have?” after they count. Often times, children who have not developed cardinality will count the amount again, not realizing that the 10 they stated means 10 objects in all.
Young children believe what they see. Therefore, they may believe that a pile of cubes that they counted may be more if spread apart in a line. As children move towards the developmental milestone of conservation of number, they develop the understanding that the number of objects does not change when the objects are moved, rearranged, or hidden. Children need many different experiences with counting objects, as well as maturation, before they can reach this developmental milestone.

The first math game is to 10 and the second to 20 also includes base 10 blocks along with the number word.
 Match Me a Turtle

Fall Couting to 20 With Base 10 and Words

Common Core State Standards

The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

In August 2010, Colorado rewrote the State Standards from preschool to twelfth grade to reflect that adoption of Common Core. The state standards and Common Core define mastery and help students and teachers achieve clearer results in order to guide schools to greater outcomes.

State academic standards are the expectations of what students need to know and be able to do. They also stand as the values and content organizers of what Colorado sees as the future skills and essential knowledge for our next generation to be more successful. State standards are the basis of the annual state assessment. Standards are not the same as lesson plans or curriculum. They are the content understandings and abilities that lead a student to success beyond school.

The content areas include Mathematics, Science, Reading and Writing, Social Studies, Music, Visual Arts, Theatre, Dance, Comprehensive Health and Physical Education, and World Languages. In addition, the state had developed standards for Expanded Benchmarks and English Language Learners.

Colorado Academic Standards are created to support an aligned P-20 system which provides an inclusion of early school readiness expectations and postsecondary competencies. Historically, these standards have been organized by grade spans but have evolved to be articulated by grade. Additionally, state standards reflect workforce readiness and 21st century skills such as critical thinking and reasoning, information literacy and invention. The ability to take responsibility for additional learning, self-direction and interaction with others to learn new information quickly and more naturally is the new emerging direction of our work.

What is P-20 Education?
P-20 is short for an integrated education system that extends from pre-school through higher education.

What is the Goal of P-20?
The goal of P-20 is to help create a more seamless and integrated education experience for all students through Academic and Career Pathways.
P-20 Academic and Career Pathways in the following areas:
  • Arts & Communication
  • Business
  • Health Sciences
  • Science, Technology, Engineering & Math
What are P-20 Academic and Career Pathways?
Academic and Career Pathways are an integrated collection of learning experiences intended to develop students’ core academic skills; and provide them with continuous education and employability credentials; in order to place them in high-demand, high-opportunity jobs.

Common Core Standards-Reading Foundational Skills

K-2 Math Common Core

About Me

Welcome to my all thing special education blog. I empower busy elementary special education teachers to use best practice strategies to achieve a data and evidence driven classroom community by sharing easy to use, engaging, unique approaches to small group reading and math. Thanks for Hopping By.
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