Response to Intervention Research in Mathematics – Series #1

Written By jmosier

On September 10, 2010

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Recommendation 6 – Building Fluent Retrieval of Basic Arithmetic Facts

Connected to the CCSSO Standards and Instructional Recommendations

The Institute of Education Sciences (IES) practice guide, Assisting Students Struggling with Mathematics: Response to Intervention (RTI) for Elementary and Middle School Students identifies eight recommendations that are designed to help teachers, principals, and administrators use Response to Intervention for early detection, prevention, and support of students struggling with mathematics. This guide is a synthesis of research that provides instructional recommendations in support of engaging those struggling mathematics students.

In viewing the eight IES research recommendations; I was thrilled to see the common threads between the research recommendations and newly adopted CCSSO Mathematics Standards for Grades K-8. Additionally, given my classroom experiences and work as a mathematics consultant, I spent some time conceptualizing what these recommendations would look like instructionally.

For the next series of blogs I will take five of those eight recommendations, one at a time, and connect each to the CCSSO Standards indicators, and then provide an instructional look alike for that recommendation. I invite others to join in the discussion and add other instructional prototypes.

RTI Recommendation 6: Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
CCSSO Standards:

  • Compose and decompose numbers; grouping in tens and whole numbers; Think of whole numbers between 10 & 100 in terms of tens and ones.
  • Develop strategies for operating with numbers based on prior experiences with small numbers.
  • Use properties of whole numbers; look for and make use of structures.
  • Build number sense through activities.
  • Compare a variety of strategies for understanding the relationship between: addition & subtraction, multiplication & division, addition & multiplication, subtraction & division.
  • Balance procedure and understanding.

What might build fluent retrieval of arithmetic facts look like instructionally?

I.  On a regular basis, students visualize the number line model as a way to obtain a visual of basic arithmetic facts. Take ten minutes to have the students interact with the number line to demonstrate proficiency.

number line 1-20

Such as:

  • 1/2 of 4 is the same as 1/2 the distance from 0 to 4. Counting spaces from zero.
  • Adding 10 + 15 is 15 moves to the right from 10.
  • Multiplying 3 × 5 is the same as moving 3 groups of 5 from zero on the number line.
  • Visualizing that 1/2 has the same position on the number line as .50 as 50 %.

II.  Hold the number in your head:

State a series of short problems for students to compute in their head – no pencil, calculators, or fingers allowed; keep a running tab in your head of answers obtained; record only a final answer. For immediate feedback for correct computations, use something as simple as an index card for students to record their final answer. Student can write the final answer on the provided card, hold them up, and there is instant feedback as to students who have obtained the correct answer. Choose a student to go through each of the series of computations, with answers. With practice students improve in their arithmetic fluency.

So that students can easily compute in their head and retrieve those basic arithmetic facts, use rational numbers that will support number sense.

Here is an example:
99 + 1 (students hold 100 in their heads)
Take one-tenth of that answer (hold 10)
Multiply by 6 (hold 60)
Subtract 35 (hold 25)
Divide by 5 (hold 5)
Take 50% of that.
Students then record their final answer of 2.5 or 2 1/2

Upon proficiency with this process, have students create their own mental arithmetic drills and let them present to classmates.

III.  Having a mental image of number families for multiple ways of viewing numbers and fluent retrieval.

Multiplication & Division

Addition & Subtraction
7 · 5 = 35 7 + 8 = 15
5 ·   7 = 35 8 + 7 = 15
35 ÷ 5  = 7 15 – 7 = 8
35 ÷ 7 =  5

15 – 8 = 7

III.  Use of structures of numbers

  • 6 x 7 because of the distributive property is the same as 6 x 5 + 6 x 2 or 6 (5 + 2)

distributive property

  • 1 + 99 because of the commutative property is the same as 99 + 1
  • 12 + 14 + 8 can be added using a combination of the commutative and associative properties (12+8)+14 to take advantage of the tens groups

What other examples could we include for building fluent retrieval of arithmetic acts?