FOSTERING AT-RISK PRIMARY-GRADE CHILDREN’S FLUENCY WITH BASIC ADDITION COMBINATIONS

Arthur J. Baroody, Michael Eiland, Veena Paliwal, Neet Priya Bajwa, and Sharon C. Baroody

Support for the research summarized in this review was provided by

U.S. Department of Education grant R305A080479 (“Developing an Intervention

to Foster Early Number Sense and Skill”)

ISSUE

All children need to learn the basic number combinations such as 7+1=8 and 9+5=14—to accurately and quickly generate the sums for 1-digit addition

expressions (e.g., Kilpatrick et al., 2001; NCTM, 2000, 2006; NMAP, 2008).

Many drill programs have been developed, and some have been evaluated.   Some previous research indicates that computer-based instruction can be more

effective than traditional instruction (Fuchs et al., 2006; see review by Kulik & Kulik, 1991; but cf. Fuson & Brinko, 1985; Hativa, 1988).

  Software designed to help children discover patterns and relations and

evaluations of these teaching tools are rare.

Can computer-based structured learning of patterns and relations

effectively promote learning of basic sums byK-grade 2 pupil at-risk for academic failure?

THEORETICAL FRAMEWORK

THEORETICAL FRAMEWORK

Number sense and fact fluency—to paraphrase Poincare (1905)— “is built up of facts as a house is of stones, but a collection of facts is no more

[fact fluency] than a pile of stones is a house” (p. 141). Patterns and relations are the mortar that ties facts together to form well-

structured knowledge.  In his 1892 “Talk to Teachers,” the eminent psychologist William James (1958)

underscored the need to build on existing knowledge to promote meaningful memorization of knowledge:

“…the art of remembering is the art of thinking; … when we wish to fix a new

thing in a pupil’s [mind], our conscious effort should not be so much to impress and retain it as to connect it with something already there. The connecting is the thinking; and, if we attend clearly to the connection, the connected thing will certainly be likely to remain within recall” (pp. 101-102).

EXAMPLE

INSIGHT: “adding one facts” are related to extant number-after knowledge

With PRACTICE

METHOD 8 training experiments with K to Grade 2 at-risk children

Training: Individually administered outside the classroom.

Structured discovery of zero pattern and one relation in Study 1

Unstructured discovery of zero pattern and one relation in Study 1

Doubles and structured discovery of the near doubles relation in Study 1

Subset 1:

Items 1 to 5

After 3 3+1 1+3 3+0 3+5

Items 6 to 10

After 4 4+1 1+4 0+4 After 6

Subset 2:

Items 1 to 5

After 7 7+1 1+7 0+7 5+7

Items 6 to 10

After 8 8+1 1+8 8+0 After 9Subset 1:

Items 1 to 5

After 6 1+4 3+0 7+1 5+7

Items 6 to 10

1+8 After 4 3+1 0+7 After 8

Subset 2:

Items 1 to 5

After 7 4+1 1+7 0+4 3+5

Items 6 to 10

After 9 8+1 1+3 8+0 After 3Subset 1:

Items 1 to 5

2+2 2+1+2

3+2 2+3 8+8

Items 6 to 10

5+5 5+1+5

6+5 5+6 5+7

Subset 2:

Items 1 to 5

3+3 3+1+3

4+3 3+4 3+5

Items 6 to 10

6+6 6+1+6

7+6 6+7 4+4

Structured Discovery of the Number-After Rule for n+1/1+n Items (Castle Wall Game)

In the case shown, the juxtaposition of the blocks underscores the relations between number-after and n+1 (After 78 and 7+18) and

between n+1 and 1+n (7+18 and 1+78).

RESULTS AND INSTRUCTIONAL IMPLICATIONS

Finding #1 (Study 1). ANCOVAs with the MA pretest scores as the covariate revealed—

Both the structured and unstructured adding 1 conditions were significantly more effective than business as usual in promoting fluency with practiced n+1/1+n items.

Structured, but not unstructured, instruction/practice was significantly more effective than business as usual in promoting transfer of fluency to unpracticed n+1/1+n items (i.e., in discovering a general “n+1/1+n=the number-after-n” rule).

  As suggested by both the NMAP (2008) and the number sense view, structured

discovery and practice is more effective than unstructured discovery and practice in promoting the learning of mathematical regularities and fact fluency, even with

relatively salient number-after/adding-one relation.

Instructional Implication of Finding #1: 

As suggested by both the NMAP (2008) and the number sense view, structured discovery/practice is more effective than unstructured

discovery/practice in promoting the learning of mathematical regularities and fact fluency, even with relatively salient number-

after/adding-one relation.

RESULTS AND INSTRUCTIONAL IMPLICATIONS

Finding #2 (Study 1 & 3). Learning varied appreciably with grade level, even among the easiest combination families.

Like first-graders in Study 1, Study 2 (kindergarten) participants who received structured instruction/practice exhibited significantly great gains in fluency with both practiced and unpracticed n+1/1+n items than those who received regular instruction/practice.

Whereas the first graders in Study 1 benefitted significantly or appreciably from the structured instruction/practice with the practiced doubles and near doubles, the kindergartners did not.

  Whereas the first graders in Study 1 did not benefit from the structured instruction/practice

with the unpracticed near doubles, at-risk second graders in Study 6 did.

 Instructional Implication of Finding #2:

 With at-risk children particularly, it is probably unrealistic to set as a goal “achieving fact

fluency with addition combinations by the end of first grade.”Developmentally appropriate goals should be guided by research.

Instructional Implication of Finding #2: With at-risk children particularly, it is probably unrealistic to set as a goal “achieving fact fluency with addition combinations by the

end of first grade.”Developmentally appropriate goals should be guided by research.

RESULTS AND INSTRUCTIONAL IMPLICATIONS

Finding #3 (Study 1 & 3). Improvement in fact fluency was dependent on mastering the developmental prerequisites for a particular fact family.

  In both Studies 1 and 3, only children who were proficient at the number-after

skill significantly their improved fluency with practiced and unpracticed (transfer) n+1/1+n items.

  The Study 3 kindergartners’ utter lack of success with the near doubles was

probably due to their failure to achieve fluency with (a) the doubles and (b) the adding-1 facts. 

  Educators need to consider carefully whether a ch combination family.

Providing diagnostic information on such prerequisites is essential for successful differentiated instruction.

Instructional Implications of Finding #3: 

Educators need to consider carefully whether a child has mastered the prerequisites for a particular combination family.

Providing diagnostic information on such prerequisites is essential for

successful differentiated instruction.

RESULTS AND INSTRUCTIONAL IMPLICATIONS

Instructional Implications of Finding #3 continued: Practice is an instructional tool that needs to be used judiciously.

Specifically, practice of even the simplest (adding 1 and doubles) combinations is likely to be ineffective before a child has mastered the developmental prerequisites for a combination family.  

Practice is not merely a vehicle for strengthening a factual association but an opportunity to actively create new connections. For example, recalling that the number after “seven” is “eight” while solving (calculating) 7+1=? may help children to construct and strengthen the successor rule—re-represent the count sequence as an integer sequence (n, n+1, [n+1]+1…).

This permits mathematical induction—making a logical generalization about a reoccurring pattern for an integer n and its successor (n+1).

Such a representation and reasoning allow children to use their (automatic) knowledge of the generative rules for counting to efficiently deduce the sum of any n+1/1+n item for any known part of the counting sequence.

Instructional Implications of Finding #3 continued:

Practice is an instructional tool that needs to be used judiciously.

Specifically, practice of even the simplest (adding 1 and doubles) combinations is likely to be ineffective before a child has mastered the developmental prerequisites for a combination family.  

Practice is not merely a vehicle for strengthening a factual association but an opportunity to actively create new connections. For example, recalling that the number after “seven” is “eight” while solving (calculating) 7+1=? may help children to construct/strengthen the successor rule—re-represent the count sequence as an integer sequence (n, n+1, [n+1]+1…). This permits mathematical induction—making a logical generalization about a reoccurring pattern for an integer n and its successor (n+1). Such a representation and reasoning allow children to use their (automatic) knowledge of the generative rules for counting to efficiently deduce the sum of any add 1 item for any known part of the counting sequence.

Finding #4 (Study 3). Targeted training on mental addition (MA) prerequisites, MA, and regular instruction combined to improve general number sense as measured by TEMA-3 achievement.

Instructional Implication of Finding

#4: 

Differentiated instruction can help at-risk children achieve at

norm or near normal levels.

Finding #5 (Study 3). Structured learning of the simplest rules (involving adding 0 or 1) appeared to promote general number sense.

An ANCOVA (using the TEMA-3 pretest as the covariate) revealed that the mathematics achievement scores of kindergarten participants trained with combinations involving 0 and 1 improved significantly more than the group that received the doubles training.  

In particular, the participants trained with combinations involving 0 and 1 improved significantly more on TEMA-3 item #22 (Number After: Numbers to 40) than the group that received the doubles training.

 

Instructional Implication of Finding #5:  Fluency with combinations involving 0 or 1 should be a key goal or

focal point for all kindergartners because it can have a significant impact on

various other aspects of mathematical thinking/knowledge. For example, Nancy Jordan and colleagues (Jordan et al., 2009; Jordan

& Levine, 2009) have found that mental addition is an important bridge between informal and formal mathematics and adding 0 or 1 is

the gateway to mental addition.

For a complete, updated report, e-mail baroody@illinois.edu