Structuring Numbers to Improve the Recall of Math Facts
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
Mathematics: Teaching All Children
K-5 Core Instruction:
Students receive daily instruction in mathematics using our core instructional program called Investigations in Data, Number and Space (revised edition) in grades K-5. A typical class lesson format:
- Mini-lesson: Students learn about the big idea for the day and participate in an interactive and short lesson. They also receive directions for their centers or for the task of the day.
- Workshops: Students rotate through centers such as: independent practice, teacher guided group, games, computer or problem solving. Sometimes students work on the same task in needs based or flexible groups. During this time some students might also receive additional support from the teacher or be assigned extension tasks for additional challenge.
- Share Out: Teachers reconvene the group to share new learning, ask a question or share strategies. Sometimes teachers use this time to do a quick check called an “exit assessment” to monitor and document students’ understanding of the lesson.
Assessment:
Prior to beginning a new Investigations unit, teachers pre-assess students’ knowledge of the concepts in the unit. During the unit, all students are assessed using informal (observation) and formal (pencil-paper or interview) formative assessments as part of the Investigations program. These assessments help the teacher plan for further instruction.
At the end of each unit, all students are assessed using a common unit assessment that teachers score using a common scoring rubric. These scores are analyzed for the grade level and trends are noted compared to the data from previous year assessment scores.
In addition to classroom assessments, students are assessed at least two times as year using the NWEA MAP assessment. This universal screening tool identifies students who are not meeting the grade level benchmark. Further assessment is done in a one-to-one format using the AddVantage Math Recovery assessment. This assessment reveals students’ misunderstanding of concepts.
Intervention:
Students who are not meeting the standard on classroom assessments and who have not met the benchmark for their grade level on MAP and /or AddVantage receive additional support from the classroom teacher during intervention blocks. This instruction is focused and targets a small group of students. Students are assessed often using progress monitoring probes. Teachers meet with administrators and other members of the Response to Intervention (RTI) team to share evidence of student progress.
Extension:
NES students who do not receive intervention in mathematics have the opportunity during intervention block to solve challenging and open ended problem solving tasks, play games to improve fluency, or explore unit concepts using technology.. Students at Pier School participate in the school enrichment program during the intervention block.
A New School Year!
September is a great time for our teachers to learn how their new students learn best. This month they will be observing how students interact with eachother while they play familiar Investigations games and listen carefully when students share their thinking to problems. Teachers are using formal and informal pre-assessments to learn what students already know about the concepts from their first Investigations units so they can plan instruction to meet their needs.
In addition to collecting pre-assessment information, teachers are assessing students’ ability to use numerical relationships to recall basic number combinations in Grades 2-5. They are mindful of the three phases of “fact aquisition” discussed in depth at our August 31 professional day. These three stages were presented by Nelson Palmer to math educators at the March 2009 NCTM National Conference in Washington D.C. For your information, the three Phases of Learning are:
1)Constructing meaning and counting strategies: (counting strategies and making meaning of operations)
2)Reasoning strategies (using known combinations to solve unknown combinations)
3)Working toward quick recall (using highly organized and planned practice for the purpose of devoting facts to memory
Finally, students will begin their new Investigations units this month and families can learn more about these units form the Investigations Student Mathematics Handbook (Grades 1-5) and the Investigations parent information newsletters (one or two letters per unit).
All families are welcome to email me any questions they have so that I can support their involvement in their child’s mathematics learning at home (smartin@narragansett.k12.ri.us). Have a wonderful school year!
Narragansett K-5 Math News
This blog has been created to inform Narragansett Elementary and Pier School families about the mathematics students are learning in school and the professional development being offered to teachers of Grades K-5.
