This document lives at www.mrfloresreads.info/resources/math/making_number_sense.html
Number sense is a foundational understanding of how numbers work. Researchers have demonstrated that every child can develop number sense. Unfortunately, it has also been found that many children don’t. This is because number sense does not develop by accident. It is not a “side effect” of doing puzzles, songs, or other informal activities that may be superficially related to math. Teachers are increasingly working to build number sense, especially in the early grades. However, you can give your children a jump-start by developing number sense before they start school, or if they are already in school, you can help them strengthen their number sense to be better prepared for deep mathematical understanding and success.
If you are interested in the reasoning behind the types of activities included, or want to know why number sense is so important, the “What is Number Sense?” explanation section is available in the full document at www.mrfloresreads.info/resources/math/making_number_sense.html, or contact your teacher for a printed version.
Encourage your child to explain his reasoning. This not only gives you insight into how he thinks, but it also helps the child evaluate his thinking and “cement” it in place.
Be sure to ask your child to explain his thinking all the time, not just when he makes a mistake. Constantly asking sends several important messages: your ideas are valued; math is about reasoning; and there are always other ways to look at a problem.
As much as possible, link math to real-world situations. When shopping, compare prices. When walking around the neighborhood or in the park, estimate distances using steps and/or standard measurement (feet, miles). Allow your children to pay at the checkout and count their change to make sure it is correct (matching the receipt, not necessarily doing a subtraction problem).
When teaching your child mathematics, make her verify estimates through doing. Point out that measuring is important, even though it is always an approximation since it can always be made more precise. For example, a pencil that looks 5 inches long might be 4 3/4 using a ruler with 1/4 inch marks or 4 11/16 on a ruler with 1/16 marks.
Numbers versus numerals
As you work with your child, remember that “number” is the amount of the things you are counting or measuring. The “numeral” is the shape we have decided to use to record it. For example, five beans on a plate can also be represented as five fingers on a hand, the oral or written word “five”, five tally marks on a page, a drawing of five beans, or the numeral 5. Connecting the glyph “5” to the amount of beans on the plate should be the last thing to focus on. It is far more useful for children to understand that five can be represented all those other ways.
These activities are divided into two types, prenumber activities and number activities. Prenumber activities are important for developing concepts that support and eventually lead to understanding the concept of numbers. They also help in developing understanding of problem situations (problem solving and critical thinking) and so should not be ignored.
You can do activities for as long or short a time as is comfortable (you will be there working along with your child). But you should be sure to do the activities multiple times over the course of several days. Later on, revisit activities that you haven’t done together in a while.
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– Include transformation patterns as the child gets more skilled at continuing patterns. Transformation patterns are those where the pattern changes by movement instead of color, shape, etc. For example, using toothpicks:
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Using everyday objects and situations, provide opportunities for your child to discover that change the size or organization of objects does not change the quantity of objects. (Don’t worry, you don’t have to teach your child the words “organization”, “conservation”, or “quantity”!)
After showing your child 5 pennies all grouped together, count them. Then spread the pennies apart and ask your child if there are more now. After responding, your child should count them again to see it is the same.
Make two groups of jellybeans, the same number in both groups. In one group, have the jellybeans all squished together, in the other have them spread apart. After guessing which has more, have your child count each group, or have him line up the jellybeans from one group next to a line of the jellybeans from the other group.
The activities used to develop this skill may take a little longer to prepare, but do not skip them! This skill is highly correlated to better understanding and scores in math.
– Practice counting out loud backwards, initially from 10 then later count backward from 20. After they are fluent with these, ask them to count backward from other numbers. Finally, ask them to tell you what number comes before a given number.
– Count larger numbers by rote starting from a multiple of ten, for example 40 or 70. Point out that there is a pattern when counting, that all they need to do is say the ten and then count by ones, for example “40, 1, 40, 2, 40, 3 …” for counting 41, 42, 43.
– Practice skip counting by 2s, 5s, and 10s. After reaching fluency with 10s, then count larger numbers starting from a ten, this time crossing the next ten. For example, start at 50 cross the next ten (60) and continue on to 70. Don’t forget to also count backwards!
– After reaching fluency with 10s, practice counting by 10s starting at any number other than a multiple of 10. Instead of just counting “10, 20, 30 …”, start at 27: “27, 37, 47, 57”.
Once your child is fluent with counting numbers to 20, you can begin doing these activities with her. Don’t forget to continue oral counting frequently!
Good estimation skills develop slowly. As you work on these activities, focus on having your child get “close enough” rather than exact answers. Have discussions when the estimates are not reasonable; explain that “reasonable” estimates have good reasons behind the number, for example we do not need to buy 50 Valentine cards because there are only four students at each table and 8 tables of four is a lot less than 50. Keep the number of objects used for estimating within a range of numbers familiar to your child. For many new first graders, numbers within the range of 10-30 are appropriate. When estimating and counting larger numbers, have your child put the items into groups of “about 10” to count by tens to find a number close to the actual amount.
Real life often requires mental computation. Our children need to be able to move numbers around in their heads and discuss their strategies. They also need to be able to do if fluently, though not necessarily “fast”. For these activities, you will be saying an extended sequence of operations to your child, one part at a time. Pause after each operation to give your child a chance to perform the operation in his head.
EXAMPLE: “Start with 4, (pause) add 3, (pause), add 2 more, (pause) then add 3 again. How much is that? (12)”
After hearing the answer, ask your child how he got the answer. He may repeat the sequence to you and explain his answers along the way, or he may explain how he manipulated the numbers. For example, “You said to add 4, 3, 2, and 3. I know that 3 and 3 are 6, and 4 plus 2 is 6. So I have two 6s. 6 plus 6 is 12!” The manner of arriving at the answer is not so important as being able to recall/explain how.
- Start with a sequence using 3 numbers. Gradually move up to longer sequences as your child gains proficiency.
- Use smaller numbers (1-5) first. Use single operations to begin with (only adding or only subtracting). Combine operations later as your child shows fluency and accuracy in single operations.
- Include adding and subtracting 10, but avoid numbers higher than 10 unless/until your child is very good at adding and subtracting 2-digit numbers without confusing the tens and ones.
- You can start simply by using “plus” and “minus” or “take away”. Later, use “add” and “subtract”. Don’t be afraid to use “increase it by” and “decrease it by” later on!
Be sure to start with composing and decomposing activities. Composing is just putting numbers together to make new number. Decomposing, then, is taking a number and breaking it into two or more smaller numbers. Doing these activities with counters for a long time before introducing written number sentences (equations like "4 + 5 = ") builds an understanding of how numbers work. Putting numbers together results in a larger number. A number can be broken into many different combinations of numbers that all make the same amount when recombined.
To do these activities, use counters like dry beans. (Beans work well because they are hard and cheap, but you can use beads, buttons, or any other small objects, though I would stay away from cereals for these activities.)
|Result unknown||Change unknown||Start unknown|
|Joining||Grandmother had 5 strawberries. Grandfather gave her 8 more strawberries. How many strawberries does Grandmother have now?||Grandmother had 5 strawberries. Grandfather gave her some more. Then Grandmother had 13 strawberries. How many strawberries did Grandfather give Grandmother?||Grandmother had some strawberries, Grandfather gave her 8 more. Then she had 13 strawberries. How many strawberries did Grandmother have before Grandfather gave her any?|
|Part-Part-Whole||Grandmother has 5 big strawberries and 8 small strawberries. How many strawberries does Grandmother have altogether?||Grandmother has 13 strawberries. Five are big and the rest are small. How many small strawberries does Grandmother have?|
|Separate||RGrandfather had 13 strawberries. He gave 5 strawberries to Grandmother. How many strawberries does Grandfather have left?||Grandfather had 13 strawberries. He gave some to Grandmother. Now he has 5 strawberries left. How many strawberries did Grandfather give Grandmother?||Grandfather had some strawberries. He gave 5 to Grandmother. Now he has 8 strawberries left. How many strawberries did Grandfather have before he gave any to Grandmother?|
|Compare||Grandfather has 8 strawberries. Grandmother has 5 strawberries. How many more berries does Grandfather have than Grandmother?||Grandmother has 5 strawberries. Grandfather has 3 more strawberries than Grandmother. How many strawberries does Grandfather have?||Grandfather has 8 strawberries. He has 3 more strawberries than Grandmother. How many strawberries does Grandmother have?|
|Multiplication & Division||Grandmother has 4 piles of strawberries. There are 3 strawberries in each pile. How many strawberries does Grandmother have?||Grandmother had 12 strawberries. She gave them to some children. She gave each child 3 strawberries. How many children were given strawberries?||Grandfather has 12 strawberries. He wants to give them to 3 children. If he gives the same number of strawberries to each child, how many strawberries will each child get?|
When students trade with base ten manipulatives, they can demonstrate that the number 14 may be represented as 14 ones or as 1 ten and 4 ones. Being able to understand this concept of place-value is essential for understanding higher math concepts.
- Encourage your child to count objects and empty cups from 5 instead of from 1 when amounts span a full row or more. For example, set A above should be counted, "Five, six", starting with the whole first row.
A * * * * * B * * * * * * *
These are just a few ways to develop number sense in your child. By working on these fun, simple activities with your child, you will be building important concepts and skills in number sense that are crucial for your child’s mathematics success. And remember to enjoy your time together!
If you are interested in reading more about number sense, contact your child’s teacher or find this document online at www.mrfloresreads.info/resources/math/ .
Number sense develops gradually in children. It comes about as a result of exploring numbers, playing with them, and paying attention to the ways numbers relate to other numbers. As number sense develops, children develop flexibility in thinking about numbers including the ability to represent numbers in multiple ways. They also develop an understanding of number magnitude (size) and are able to predict the effects of operations on numbers (e.g. adding positive numbers together results in a larger number).
Some children are able to develop number sense through careful observation and play beginning at a very young age. Many children need specifically guided instruction or participation in activities to develop number sense.
Number sense enables students to understand and express quantities in their world. For example, whole numbers describe the number of students in a class or the number of days until a special event. Decimal quantities relate to money or metric measures, fractional amounts describing ingredient measures or time increments, negative quantities conveying temperatures below zero or depths below sea level, or percent amounts describing test scores or sale prices.
Number sense is also the basis for understanding any mathematical operation and being able to estimate and make a meaningful interpretation of its result.
In a study by Jordan, Locuniak, Ramineni and Kaplan entitled Predicting First-Grade Math Achievement from Developmental Number Sense Trajectories, it was shown that “early sense of numbers (sic) is a reliable and powerful predictor of math achievement at the end of first grade.”
Knowing the why of how numbers work is of utmost importance, and children should not be shown the how until they understand the “why.” Techniques such as using ten frames and using concrete models to show place value concepts are daily necessities for young children. Inquiry-based approaches (such as math dice games) to teaching children mathematics should be utilized as primary teaching methods in the early grades. This is not to say that explicit teaching of sense of numbers skills is not essential, especially for those students from low socio-economic status. We absolutely need to do this. It is saying that teachers should provide multiple opportunities for students to experience numbers and make connections before putting the pencil to paper.
Number sense begins very early and must be a focus of primary math. This is the solid foundation in math that all kids need. A sense of numbers is critical for primary students to develop math problem solving skills. The National Council of Teachers of Mathematics increasingly calls for districts to give more attention to building this skill, and studies have found that number sense accounts for 66% of the variance in first grade math achievement. They council (USA 1989) have also addressed five critical areas that are characteristic of students who have good number sense:
The understanding that something stays the same in quantity even though its appearance changes. To be more technical (but you don’t have to be) conservation is the ability to understand that redistributing material does not affect its mass, number or volume.
represents an amount
being able to represent an amount with counters
and expressing a number different ways—5 is “4 + 1” as well as “7 - 2,” and 100 is 10 tens as well as 1 hundred
the last number counted in one-to-one counting is the amount
a student who has not yet grasped this concept will simply count the quantity over again.
first by rote then by applying rules and patterns
the ability to identify a number by an attribute—such as odd or even, prime or composite-or as a multiple or factor of another number.
how big a number is compared to another
sequencing and ordering numbers -Understand relative position and magnitude of whole numbers
Estimation - It’s helpful for kids to see that the way the objects are counted doesn’t change how many in all.
This should be embedded in problem solving. This is not referring to textbook rounding. Real life estimation is about making sense of a problem and using anchor numbers to base reasoning on.
Adding and Subtracting
Mental Math Real life requires mental computation. Students need to be able to move numbers around in their heads and discuss their strategies.
Teachers also encourage children to think in terms of the connections between numbers. That’s why you’ll probably see a lot about fact families in 2nd grade. These are groups of three numbers that work together in various combinations to create addition and subtraction facts — 4, 6, and 10, for example. When kids see these numbers together, they can create an addition sentence of 6 plus 4 equals 10. If asked for the result of 10 minus 6, they can easily find the correct answer of 4.
; knowing how to write and represent numbers in different ways
; writing and recognizing numbers in different forms such as expanded, word, and standard;
understanding place value in the context of our base 10 number system
always an approximation because it can always be made more precise
http://www.mrfloresreads.info/room19/ for parents
http://nlvm.usu.edu/en/nav/vlibrary.html Java based
Helping Your Child Learn Math (1999)
put a picture into odd even to show tower pairs
Daily Math in resources/math/dailymath add this to room19 symbaloo for parents
More activities for modeling cardinality Moomaw, Sally. “Cardinality in the Early Learning and Development Standards.” Early Childhood Building Blocks (n.d.): n. pag. Resources for Early Childhood. Web. 3 Nov. 2014. http://rec.ohiorc.org/orc_documents/orc/recv2/briefs/pdf/0020.pdf
“Activities to Help Your Child Develop Number Sense.” Fisherkindercorner, n.d. Web. 24 Oct. 2014. https://sites.google.com/site/fisherkindercorner/Home/ideas-to-help-your-child-develop-number-sense.
“Boosting Number Sense.” Scholastic.com. N.p., n.d. Web. 24 Oct. 2014. http://www.scholastic.com/parents/resources/article/what-to-expect-grade/boosting-number-sense.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. “Cognitively Guided Instruction: A Research-Based Teacher Professional Development Program for Elementary School Mathematics.” National Center For Improving Student Learning & Achievement In Mathematics & Science. Sep. 2000. Web. 29 Oct. 2017. http://ncisla.wceruw.org/publications/reports/RR00-3.PDF
Connell, Mike. “Number Sense: What It Is, Why It’s Important, and How It Develops.” Native Brain. N.p., 27 Nov. 2012. Web. 24 Oct. 2014. http://www.nativebrain.com/2012/11/number-sense-what-it-is-why-its-important-and-how-it-develops/
Glasgow, Brennan. “Ways to Work on Number Sense.” Regional School District #10, n.d. Web. 24 Oct. 2014. http://www.region10ct.org/math/region10mathsitefaq/whtwaysworknmbersense.html.
Glasgow, Brennan. “What is Number Sense?” Regional School District #10, n.d. Web. 24 Oct. 2014. http://www.region10ct.org/math/region10mathsitefaq/whatisnumbersense.html.
Kepner, Henry (Hank). “A Missed Opportunity: Mathematics in Early Childhood.” Editorial. NCTM Summing Up Feb. 2010: n. pag. NCTM.org. Web. 24 Oct. 2014. http://www.nctm.org/about/content.aspx?id=25054.
MacNee, Anne. “Activities Kids Can Do at Home to Develop Number Sense Handout2.” Scribd. N.p., n.d. Web. 24 Oct. 2014. https://www.scribd.com/doc/57759461/Activities-Kids-Can-Do-at-Home-to-Develop-Number-Sense-Handout2
McLeod, Saul. “Concrete Operational Stage.” Simply Psychology. N.p., n.d. Web. 29 Oct. 2014. http://www.simplypsychology.org/concrete-operational.html.
Moomaw, Sally. “Cardinality in the Early Learning and Development Standards.” Early Childhood Building Blocks (n.d.): n. pag. Resources for Early Childhood. Web. 3 Nov. 2014. http://rec.ohiorc.org/orc_documents/orc/recv2/briefs/pdf/0020.pdf
“Number Sense.” TeacherVision. N.p., n.d. Web. 24 Oct. 2014. https://www.teachervision.com/pro-dev/teaching-methods/48939.html.
Edited with StackEdit.