Narragansett K-5 Math News

Question: How does Investigations approach the teaching and learning of mathematical vocabulary?

Answer:

“Students do not learn how to speak mathematics by memorizing definitions, but by hearing these words frequently and having many opportunities to use them in context.” (Mathematical Thinking at Grade 3, p. 21.)

Teaching and learning math vocabulary can be particularly tricky. The big question is; What does it mean to “know” a vocabulary word or term? For example, say square is a vocabulary word that students must know in Grade X. Does this mean that they can:

• identify or recognize a square?
• draw or construct a square?
• describe a square?
• categorize shapes into squares and not squares?
• explain what makes a square a square (four equal sides, four 90 degree angles)?
• explain why a square is also a rectangle but a rectangle is not a square?

From: http://investigations.terc.edu/library/curric-math/qa-1ed/math_vocab.com

For a detailed explanation of the geometry concepts in the Investigations program, see:  http://investigations.terc.edu/library/curric-math/geometry_2ed.pdf

Students in Grades 3- 5 can use a Frayer Model to deepen their understanding of geometry terms and concepts.  See a geometry example of a Frayer Model: http://wvde.state.wv.us/strategybank/FrayerModel.html 

Students in Grades K – 2 learn to describe and classify shapes through sorting activities, games such as Guess My Rule and building models.

http://www.carlscorner.us.com/Sorts/Shapes.pdf

These strategies are introduced in third grade and extended in grades four and five. Students use numerical relationships and visual models to solve single and multi-digit problems.

We had 19 parents attend either the Grade K-2 or the Grade 3-5 Parent Math Workshops in October and November!  The feedback was great.  Some comments were:

“Please post the posters on your blog.”  (Next month I’ll post the the strategies for multiplication and division.)

“I learned about encouraging mental recall using understanding and reasoning.”

“Math is not what is used to be.”

“Great presentation for me as a parent for my own understanding and new learning.  Now I can reinforce rather than confuse.”

“Great program.  I cannot wait for more!”

These posters reflect the strategies our students use to mentally add and subtract multi-digit numbers. These strategies are introduced in second grade and extended through grade 5 to numbers up to 10,000.  The subtaction strategies are posted above and the addition and compensation strategies are posted below.  Move your cursor over each picture for the title of each photo.  Ask your child to explain the difference between breaking numbers apart to add (“split strategy”) and counting backwards using tens and ones for subtraction (“jump strategy”).

Universal Screening:  All students in Kindergarten and Grade 1 are being screened using the K-1 Interview.  This interview is a shorter version of the AddVantage Math Recovery Diagnostic Tests we use for RTI.  Kindergarten teachers used this screening tool last year and revised it over the summer.  This is the first year it is being used in Grade 1. 

Students in Grades 2-5 recently completed their Universal Screening using NWEA MAP testing.  Unlike the K-1 Interview, this assessment is done on the computer.  For both screening tools, students’ responses are scored in terms of meeting the Fall benchmark for the grade level.  Any student who exceeds the benchmark will be challenged with enrichment.  Children whose scores fall below the Fall benchmark will receive intervention from the classroom teacher as part of RTI.

 K-5 Data Days:  Teachers will meet with their grade level and the mathematics coach to review the Universal Screening data and identify students who would benefit from intervention and extension.  The intervention focus is determined after the student has been given one or more of the AddVantage Math Recovery tests.  These tests are done as 1-1  interview and administered by either a classroom teacher, special educator or the mathematics coach.  This is the first year we will use a Personal Mathematics Plan (PMP) to identify and progress monitor the intervention goal for any student who falls below both the Universal Screening and AddVantage benchmarks for the grade level.

 Professional Development:  Teachers will spend a half day this Fall with the mathematics coach to learn more about the instructional strategies and visual representations used in both AddVantage (Teaching Number in the Classroom) and Contexts for Learning Minilessons Guides.   In addition to school based professional development,  a group of 18 teachers from Grades K-12 will meet to begin the work of curriculum writing using the newly adopted National Core Standards.  For more information about these standards, visit: www.nctm.org.

K-5 Parent Workshop Series:   Investigations and Narragansett core support materials provide students with multiple opportunities to deeply understand and apply mathematics concepts beyond rote learning.  It is essential for parents to know the rationale for why the teaching and learning of elementary mathematics has changed since they were in school.   Parents of students in Grades 3-5 are invited to attend the first session on October 13th from 5 to 6 pm at NES.  The second session will be for parents of students in Grades K-2 prior to the November PTO meeting at 6 pm.  Both sessions will teach parents the strategies and vocabulary being taught in school so that they can support their child’s learning at home.   Information about these workshops can be found in the NES and Pier PTO Newsletters, as well as in a note sent home via backpack express.  Please RSVP for these sessions with Sharon Martin or your child’s teacher.  Note babysitting will be provided.

Another year and we are ready!  NES classroom teachers and special educators met with me on August 30th to review “the familiar” and preview ”the new”.  

1.  RTI (Response to Intervention)

Teachers reviewed the Mathematics Protocol for Intervention and viewed video examples of intervention using Math Recovery/AddVantage.  This year teachers will meet monthly with the NES Core Team (and the Pier Core team) to review current assessment data (MAP, NECAP, AddVantage) to identify students who need support to meet grade level benchmarks.  Teachers will receive ongoing professional development to progress monitor and document mathematics interventions this year.

Core Curriculum:

The 2010-2011 Pacing Guide for Mathematics identifies core lessons for direction instruction, practice, games and problem solving.  Teachers met in teams to discuss this document, preview new pre-assessments and comment on the compacting of units for the first trimester.  This year classroom teachers will receive ongoing support to differentiate instruction using the workshop model (mini-lesson, small group instruction and centers, share-out).

Celebrations:

Students who return their completed “Summer Fun Mathematics Reflection” sheets will be honored with a NES certificate and sparkle pencil.  It is wonderful to know that our students enjoyed this summer practice!

As the school year comes to a close, it is time to think about relaxation, reading and of course math games!  Click on the Summer Fun link to download Investigations games and other problem solving opportunities.  In addition you may want to print the K-6 math calendars created by the Brookline Public Schools for Investigations: http://lawrence.brookline.k12.ma.us/math/index.html

These are suggestions for maintaining  students’  fluency over the summer in preparation for a new school year:

Grade K to 1: 

1.  Counting backward and forward from 1 to 30 and 30 to 1, counting by 10′s

2. Playing compare numeral card games (ask your child: “Which number is more?  How much more/less is it?”)

3.  Adding and subtraction combinations to 12 using counting on or counting back from strategies

Grade 1 to 2: 

1.  Counting forward and backward by 2′s, 5′s, 10′s and by 1′s to 100

2.  Flash card sets for both addition and subtraction combinations: doubles, near doubles, make ten, and ten plus

3.  Counting coins

Grade 2 to 3:

1.  Counting forward and backward by groups of 2′s, 3′s, 5′s, 10′s

2.  Flash card sets for both addition and subtraction combinations: doubles, near doubles, make ten, and ten plus (focus on subraction)

3.  Counting coins and making change from $2.00

4.  Solving 2-digit addition and subtraction problems mentally using the breaking numbers apart or jumps of tens and ones strategies

Grade 3 to 4:

1.  Counting forward and backward by multiples

2.  Flash card sets for both addition and subtraction combinations: doubles, near doubles, make ten, and ten plus (focus on subraction)

3.  Flash card sets for both multiplication and division combinations

3.  Counting coins and making change using the “counting up from” strategy (mentally and using open number line)

4.  Solving 3-digit addition and subtraction problems mentally using the breaking numbers apart,  jumps of tens and ones or compensation strategies

5.  Fraction Sharing Problems (for example “I have 6 brownies and 8 people.  What would each person’s “fair share” be?”)

Grade 4 to 5:

1.  Counting forward and backward by multiples

2.  Flash card sets for both addition and subtraction combinations: doubles, near doubles, make ten, and ten plus (focus on subraction)

3.  Flash card sets for both multiplication and division combinations

4.  Making change using the ” counting up from”  strategy (for example:  “It costs $13.59 and I give the cashier $20.  How much change do I get back?”)

5.  Sharing Problems (for example “I have 6 brownies and 8 people.  What would each person’s “fair share” be?”)

Grade 5 to 6:

1.  Counting forward and backward by multiples

2.  Flash card sets for both multiplication and division combinations

3.  Making change using the ” counting up from”  strategy (for example:  “It costs $13.59 and I give the cashier $20.  How much change do I get back?”)

4.  Sharing Problems (for example “I have 6 brownies and 8 people.  What would each person’s “fair share” be?”)

5.  Fraction and Percent Equivalents (“What is the percent equivalent of 1/4? 3/8? etc”)

6.  Addition and Subtraction of Fractions using equivalents (think: fraction track, clock model and percent equivalents)

The Investigations mathematics program encourages students to solve problems in a way that is meaningful to them.  The ultimate goals for our students when solving single and multi-digit problems are:  understanding, accuracy, flexibility and efficiency.  Narragansett students are expected to solve problems mentally at every grade level.  Mental computation requires a deep understanding of our number system.  In the early grades, students are given visual models (double ten frames, hundreds charts, arrays and number lines) to learn how to manipulate numbers using near-by landmarks. 

By the later elementary grades students are expected to internalize these concrete tools (representations) and use them as mental models for solving problems.  Students are given opportunities to explain how they solve problems so that the teacher can assess the depth of their understanding and plan the next steps for their learning.

Many parents ask me why students are not being taught to solve multi-digit problems using the U.S. Traditional Algorithm.  My response to them has been the same for the past 17 years as a teacher of Investigations.  We give our students explicit instruction and practice using strategies that make sense to them.  These strategies can be represented visually and can be used over and over again with the same result.  Once students have demonstrated proficiency with the alternative strategies in Investigations, they are able to solve problems using the U.S. Traditional Algorithm as long as they can prove their solution is correct in another way either orally or in writing when asked to do so.  We also expect these students to know why the procedures of “borrowing and carrying” work using place value concepts.

One of the best articles about this topic for parents is written for another conceptual based mathematics program called Every Day Mathematics.  You can download it from here:

http://everydaymath.uchicago.edu/about/research/algorithms.pdf

Celebrate and encourage your child to explain his/her thinking to you and try to solve a new problem using the same strategy.  Then look for oppotunities to solve problems mentally with your child when shopping or driving.  You will be amazed!

In January many Narragansett parents attended my K-5 Mathematics Presentation in the NES library.  After the presentation some parents asked me to write one of my blogs highlighting the new vocabulary of mathematics.  It is important to note that our K-5 mathematics program, Investigations in Data Number and Space does not have a vocabulary online resource, but the Investigations Student Math Handbook (sent home in September for Grades 1-5) offers examples for many of the strategies and terms listed below.

Addition Strategies

Doubles (examples of doubles include single digit equations such as 4+ 4 and multi-digit doubles such as 25 + 25)

Near Doubles (students use a double combinations to solve a near double equation, for example solving 4+5 using the double combination of 4+4 plus one more or 5+5 minus one)

Make Ten Combinations (combinations equal to 10 such as 3 + __ = 10 and 10 – 7)

Bridge to Ten Strategy (The goal for this strategy is to make a ten.   Students can add ten plus a number mentally rather than counting up by ones .  Using double ten frames students see how 9 + 5 is the same as 10 + 4 and 8 + 6 is the same as 10 + 4)

Double Ten Frames (a tool to support children’s development of number relationships based on five- and ten -referenced strategies and doubles strategies for numbers up to 2o)

Breaking Numbers Apart (breaking  or “splitting” multi-digit equations into tens and ones such as 34 + 27 = 30 + 20 + 4 + 7)

Jumps of 10 (solving multi-digit equations using the open number line as a mental model for making jumps of tens and ones:  34 + 27 is solved by starting at 34 and counting on two tens  “44, 54” and then counting on 7 ones)

Compensation (students solve 34 + 27 by rounding the 27 to 30 and mentally computing 34 + 30 minus 3)

Number Line (a line “marked” with numbers, “unmarked” or  ”open”  that students use to make jumps forward and backward of tens and ones)

U.S. Standard Algorithm

*Subtraction Strategies: (see an explanation for these strategies in my previous blog)

Multiplication Strategies

Skip Counting (counting groups of a number using the 300 chart, manipulatives or the open number line)

Repeated Addition (adding groups for example, 3 x 4 is the same as 4 + 4 + 4 or 3 groups of 4)

Array (a rectangular pattern arranged in rows and columns used for representing multiplication and division: see http://www.haelmedia.com/OnlineActivities_txh/mc_txh3_002.html)

Distributive Property (breaking numbers apart by place:  for example: 56 x 12 = (56 x 10) + (56 x 2) see http://www.themathpage.com/ARITH/mental-arithmetic-multiplication-2.htm)

Compensation  (rounding to the nearest multiple of ten for example:  solving 35 x 19 using  35 x 20 and subtracting a group of 35)

U.S. Standard Algorithm

Division Strategies

Successive Subtraction (subtracting groups of the divisor for example, 27 ÷ 3 = 27 – 3 + 24 – 3 + 21 – 3 etc)

Dealing Out Into Group  (dealing out tens and ones of the dividend to determine the quotient) see: ^{http://www4.uwm.edu/Org/mmp/PDFs/Unpacking%20Division%20article.pdf}

Dividing Groups of the Divisor (see #1 from http://investigations.terc.edu/library/curric-gl/sample_g5_u1_tn.pdf and also http://investigations.terc.edu/library/curric-gl/sample_g5_u1_smh_p38-39.pdf)

Breaking the Dividend into Parts (see #2 from http://investigations.terc.edu/library/curric-gl/sample_g5_u1_tn.pdf)

U.S. Standard Algorithm

 Additional K-5 Vocabulary Sites

http://www.amathsdictionaryforkids.com/

http://www.harcourtschool.com/glossary/math2/index_temp.html

This cold winter is a wonderful time to “talk and play math” at home.  The NES and Pier School websites offer Investigations game downloads and problem solving tasks for every grade level (K-5).  The games give students a playful way to practice the concepts learned in school and improve their recall of basic combinations.  The problem solving tasks offer families an occasion to work together to find the solution to a challenging task. When playing games or solving tasks, encourage familiy members to explain their thinking and note how different strategies can achieve the same solution!

Another great time to “talk and play math” is in the car or at the dinner table.  Counting games such as “Zap” give children practice counting forward and backward in various sequences and by groups (multiples, money, decimals, or fractional amounts).

ZAP Game: Count backward from 33 to 19 by ones.  Each player counts aloud.  The person who say’s 19 is “Zapped” and out of the game.  Counting begins again and ends with one remaining player.

Play sorting games such as “Guess My Rule” to sort people, shapes or objects by one of more attributes.  Encourage students to sort objects by one (red, not red) or more then one attribute (water animals, land animals and animals who live on BOTH land and in water). 

Finally use homework as a special occasion to talk about the big ideas students are learning in school.  Talk to your child’s teacher for ways to support their learning at home.  Be sure to have plenty of ten frames and number lines for Grade K-2 and graph paper and 300 charts for students in grades 2-5.  Students are expected to explain their thinking using these tools with the intention that tools become open and unmarked in the later grades.  For more information about open number lines or unmarked arrays, please see:

Open Number Line: www.dreambox.com (see teacher tools for open number line)

Unmarked Array: http://www.schultzcenter.org/mathvideos.shtml (see video for “generic rectangle array”)

Have fun talking and playing math at home!  Also, please join me for an hour presentation prior to the January NES PTO meeting. A note reminding parents to RSVP will  be sent home via backpacks.

Teachers are using some of the AddVantage Math Recovery tools and strategies to teach students to “structure”  or “subitize” numbers in order to improve recall of basic combinations in the range of 1 -20 without counting.  This is developed from the work of Treffers (1991); Gravemeijer, Cobb and colleagues (Gravemeijer, Cobb, Bowers, & Whitenack, 2000); and Wright and colleagues (Wright et al., 2007).

Subitizing is the ability to “see” numbers at a glance, without one-to-one counting.  Research shows that students who have strong subitizing skills can more easily manipulate and partition numbers, which aids in computation and calculation of numbers.  Most youngsters can “see” a group of three objects or less quickly.  They will break larger number patterns into smaller groups to find the value (see examples below).  The use of colorful visual models such as “pair wise” or ”five wise” ten and twenty frames illustrate numerical relationships for students.  Playing dominoes and dice games where children see a quantity quickly at a glance further develops subitizing skills, which is the underpinning of developing fluency in basic math facts. (from http://ruthrose.edublogs.org))

The following strategies for addition in the range 1 to 20 use non-count-by-one strategies:

• Doubles and Near Doubles combinations
• Five Plus (4 + 8 = “I know I can break 8 into 1 and 7.  4 + 1 is 5 and 5 + 7 =12”)
• Ten Plus Combinations (10 + 3 + 13)
• Make Ten Combinations (9+1,8+2, 7+3, 6+4, 5+5)
• Nine Plus or “Bridging to Ten” combinations (9+5= “I know I can take one from the 5 to make the 9 into a 10.  10 + 4 = 14” …or “I know 10 + 5 is 15 and one less is 14.”

Clustering like-combinations using numerical relationships for subtraction improves recall of basic facts.  The following are strategies for teaching and learning subtraction clusters:

“Count Back 1, 2, 3” Number of facts: 27

• 2-1,  5-2,  7-3,  10-1, 3-1,  5-3,  8-1,  10-2
• 3-2,  6-1,  8-2,  10-3, 4-1,  6-2,  8-3,  11-2
• 4-2,  6-3,  9-1,  11-3, 4-3,  7-1,  9-2,  12-3
•  5-1,  7-2 , 9-3,  12-3

“Count On 1, 2, 3” Number of Facts: 18

• 5-4,  7-6,  9-7,  10-9, 6-4,  8-5,  9-8,  11-8
• 6-5,  8-6,  10-7,  11-9, 7-4,  8-7,  10-8,  12-9
• 7-5,  9-6

“Subtracting 0” Number of Facts: 19

• 0-0, 1-0, 1-1, 4-0, 4-4, 7-0, 7-7, 2-0, 2-2, 5-0
• 5-5, 8-0, 8-8, 3-0, 3-3 , 6-0, 6-6, 9-0, 9-9

“Doubles Near Doubles” Number of Facts: 16

• 8-4, 9-4, 9-5, 10-5, 11-5, 11-6
• 12-6, 13-6, 13-7, 14-7,  15-7, 15-8
• 16-8,  17-8, 17-9

“Use Addition” Number of Facts: 20

• 10-4, 10-6 13-4, 13-9 15-6, 15-9, 11-4, 11-7 13-5, 13-8 16-7, 16-9
• 12-4, 12-8 14-5, 14-9, 12-5, 12-7 14-6, 14-8  

(Subtraction Clusters from: http://www.shorelineschools.org/departments/instruction/curriculum/math/Basic_Facts/Subtraction_Fact_Strategies.pdf )

Download Ten and Twenty Frames and other structuring numbers games from: http://www.solonschools.org/SolonNet/FIS/Files.aspx?ID=1176