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THE DAILY 3 MATH STRUCTURE

The Daily 3 Math Structure

Kari Lovett

University of Alaska Southeast


Introduction

The standard math instruction in elementary classrooms today consists of a

period of direct instruction with guided practice followed by a period of

independent practice. These often take place in 30 to 45 minute blocks. This

format requires students to focus for long periods and allows for only limited

social learning opportunities. The Daily 3 math structure is a framework for

providing math instruction. It splits math into three mini lessons, no longer than

ten minutes each. Between the mini lessons, students rotate between math by

myself (independent practice), math with a partner, and writing in math. During

these stations, the teacher provides support to one of the small groups.

My research question will be, “Does the Daily 3 math structure increase

student engagement and depth of knowledge in math?”

Literature Review

In looking at research literature regarding the Daily 3 math structure themes

emerged. Interwoven into each of the three components present in the Daily 3 math

structure, increased student engagement and increased depth of knowledge were present.

Student engagement is vital to ensuring students learn to their fullest potential.

The Daily 3 structure encourages student engagement in a variety of ways. In

“Effectiveness of using iPads to build math fluency” by O’Malley it was reported that

student engagement increased with the use of iPads. “Teachers reported that the students

appeared to be eager to participate with the iPad activities.” (p. 11 O’Malley) The

students were excited about using the iPads and disappointed when having to return to the


traditional means of instruction during the various portions of the study. This study

focused on increasing fluency in math skills. This is often one of the most disengaging

aspects of math. Technology tools, such as iPads, can be incorporated into the Daily 3

math structure. The structure has students working on different tasks at the same time, so

an entire class set of iPads is not required.

The study “The effects of the math daily 3 structure on student achievement and

growth in an elementary school setting” by Ktytor covered the effects of the structure of

the Daily 3. It stated, “Student choice was given throughout the math structure; students

were interested and engaged in their learning and kept a positive attitude towards math

content.” (p. 15 Ktytor) It also discussed how the students developed a greater sense of

responsibility in their learning when under the Daily 3 structure. This sense of

responsibility is a direct result of the increase in engagement.

The Daily 3 structure is able to offer up more real world, complex problems to

students. The students have support whether via partners or small group interventions

with the teacher. This increase in support allows the teacher to challenge the students

with more complex problems. As stated in “The effects of blended instructional models

on math performance”, “First, the hands-on problems immersed students directly in

content-rich, relatively complex, and stimulating learning activities.” (p. 434 Bottge)

This stimulating environment allows the student to understand that math is important and

present in their daily lives. The authors of this study go on further to state, “An attitude

that develops when instruction conveys to students that math is worthwhile. To stimulate

this feeling, instruction should help them understand the connection between the math


skills they are learning and the reasons for learning them.” (p. 435 Bottge) This sense of

importance is key in encouraging student engagement.

Students are engaged by the Daily 3 structure because of the social opportunities.

The study “Mere belonging: The power of social connectedness” researched how a sense

of connectedness can affect a person’s attitude towards a mathematical task. The

findings indicated that just having a feeling of connecting with someone in a social aspect

increases engagement enough that the student will persist longer at mathematical

problems. The Daily 3 structure allows students to work cooperatively together on math

problems everyday. This social aspect will increase engagement and perseverance

towards difficult mathematical problems.

The Daily 3 structure also supports an increase in depth of knowledge through a

variety of routes. A study on writing in math reported an increase in depth of knowledge

as “Writing in math class supports learning because it requires students to organize,

clarify, and reflect on their ideas”. (p.30 Burns) This goes on further to describe how

writing can challenge student’s thinking of math in a variety of ways. For example, some

writing can be a journal of the steps to a procedure, whereas other writing can ask a

student to apply a concept to a new problem. These writing activities can challenge the

students to solve difficult problems as well as to explain their thinking process that led to

their solution. It forces students to defend their answers and thus increases their

understanding.

The study titled, “The effects of the math daily 3 structure on student achievement

and growth in an elementary school setting” by Ktytor also discovered that the students’

depth of knowledge grew under the Daily 3 structure. It commented on how the students


initially were using rudimentary strategies to arrive at the answers, but as the structure

became in place, they began to use more complex means. The study suggested that the

students’ individual needs were often met better with the Daily 3 structure, “By using

flexible grouping, based on our observational records and mini-conferences with

students, we were able to give explicit instructions individually to meet each student’s

needs in small groups.” (p. 15) The structure allows for more small group and individual

conferencing with students, thus a greater chance of meeting the needs of all students.

A study titled, “The effects of blended instructional models on math performance”

noticed a difference in pacing between traditional teaching practices and those of a more

blended approach. The blended approach teachers spent more time on each concept,

making sure that the students fully understood the concept before moving on to the next

topic. Traditional structures tend to go through the curriculum quickly, developing more

of a basic procedural knowledge of the topic. The blended approach teachers moved at a

slower pace and did not change topics until the student could display a further depth of

knowledge. The Daily 3 structure is a blended approach because it uses three different

modes of learning math everyday.

The study, “Interpersonal relations and group processes” researched the effects of

social connectedness to the perseverance of challenging math problems. The continued

perseverance of working through difficult problems will lead to an increase in depth of

knowledge. Learning is tough work. Students need to be able to push through these

difficulties to fully understand the concepts.


The research studies have shown that the Daily 3 structure increases student

engagement and depth of knowledge. Both of these are desirable when teaching math to

students.

Theoretical Framework

The literature has shown that the Daily 3 format allows for an increase in

student engagement as well as an increase in depth of knowledge.

Student engagement increased because there was a greater opportunity to

integrate technology into the classroom. In addition, the problems presented in this

instructional format can be more real world and complex. Finally, the engagement

increases because students worked with a partner. Taking part in social learning is

more engaging than traditional instructional strategies.

The depth of knowledge increases when students are more reflective. This

occurs through writing prompts. In addition, students allowed more time to process

the knowledge before learning a new concept. Finally, students tend to preserve for

longer when in a social setting. This continued perseverance leads to a greater

depth of knowledge.

Methods

In this study on the Daily 3 math structure, I take a qualitative inquiry approach

examining a real world situation, without manipulating it. I take a naturalistic approach

in this research (Merriam, 2009) through a case study design that is both historical and

observational (Merriam). I believe that through multiple sources of data I can determine

the effect of the Daily 3 math instructional technique on student engagement and depth of

knowledge.


The participants are a third grade class from the village in Northwest Alaska.

Participants who are absent from more than 25% of the lessons were dropped from the

report. There are 14 students in the class, four of whom receive special education

support. The participants are all of Alaska Native heritage and the entire school qualifies

for free breakfast/lunch. On the last Measures of Achievement Progress test (MAP test),

taken in early January, one of the students scored above grade level, the rest scored

below. Five students scored only one grade level below. Most are showing growth in

their test scores.

One document analyzed is the pre/post test. (See Appendix B.) I gave this to my

students at the beginning and end of the study. This is based off the pre test provided in

the curriculum that the district has adopted. It was designed so that there would be

several easier and several harder questions. The daily learning tasks received

assessments for understanding as well as effort and were assessed with a rubric. (See

Appendix C.) The third set of data was through a student effort survey given out once a

week. (See Appendix D.) Internal validity in this study was through triangulation, the

analysis of multiple sources of qualitative data (Merriam, 2009).

Analysis of Data Collected

After collecting data for three weeks, several themes emerged. The first

theme was that students preferred various aspects of the Daily 3 structure. One

response was, “The computer games were my favorite.” Another was, “I liked

working with Annie.” Not many liked the writing. Writing is an area where many of

my students struggle. By looking at the student work produced each week it became

clear that students often were more engaged with one modality. I believe that this


distribution is also a reflection of the many diverse learning styles present in

classrooms. I also believe that the Daily 3 structure forces me to teach in these

different learning styles everyday, thus serving my students better.

Students were engaged when operating under the Daily 3 math

structure. The math block was broken up into shorter sections and involved more

movement than the typical direct instruction/guided practice/independent practice

model. My students especially liked the fact that the worksheets they worked on

had fewer problems than before. Instead of doing one worksheet with lots of

problems, they worked through several shorter worksheets. The Daily 3 structure

also allowed me to work with small groups more. It also allowed more technology

to be utilized as a teaching tool. Many students loved the ability to practice math

concepts through technology. These computer/iPad games often have fun sound

effects and graphics. They were also more engaged during the posttest. They all

attempted all of the problems with reasonable guesses. They answered the

questions with increased confidence. It was evident that this model increases

student engagement for a variety of reasons.

Students did display and increase in depth of knowledge of the subject over the

period. The posttest results showed a greater depth of knowledge than the pretest

results. (See Table A1) Students made big gains in being able to measure to the

nearest half inch and beginning to understand converting between feet, yards, as

well as inches. They made gains in reading a thermometer, but we did not have time

to fully explore this concept. They also made gains in their depth of knowledge

through their weekly work. I was able to see an improvement as the week went on


in their understanding of the concepts. I would be curious to see how these results

would have compared to if I had taught them using a more traditional approach.

Under this structure, they were given more modes in constructing and displaying

their knowledge.

Discussion

The Daily 3 structure is an effective form of delivering math instruction. It

increases student engagement and depth of knowledge. This structure has many

elements of good teaching practices built into it, thus making student learning

better.

The Daily 3 breaks the math block up into several smaller segments. This

shorter time engaged in a particular task is very good at ensuring student

engagement. These short time periods add a sense of urgency to the student’s

learning.

The Daily 3 structure asks students to work collaboratively as well as

independently. Students utilize numbers as well as writing responses for math

problems. These different settings and different tasks allow for many different

learning styles. Diverse student groups are able to construct knowledge and show

growth through the diverse learning opportunities provided with this structure.

The Daily 3 structure allows for small group instruction. In most classrooms

today, the skill sets our students possess ranges widely. This means that whole

group direct instruction is often inadequate for a portion of our students. Whole

group direct instruction only benefits the students who are at the level of the


instruction. Small group allows the teacher to give appropriate instruction to all of

the students.

The Daily 3 structure allows for the implementation of technology because of

the small group rotations. As noted in the literature review, technology is a very

effective method of increasing student engagement, especially when it comes to rote

memorization. In small group rotations, technology is easily implemented, thus

increases student engagement.

The Daily 3 structure asks students to write everyday. This increases depth

of knowledge forcing students to process information into a new format.

As is discussed above, there are many lessons learned from the Daily 3 math

structure. This structure also has elements of good teaching practices. These

practices allow teachers to be more effective when delivering instruction to their

students.

Conclusion

This study has shown that the Daily 3 structure helps to increase student

engagement and depth of knowledge. It is a structure, which combines many of the

best instructional strategies used in education today.

The Daily 3 allows for a large amount of differentiation. Students have a

wide variety of methods to construct and display knowledge on a daily basis. They

learn in social environments and independently. The teacher is able to tailor their

learning opportunities through small group interventions and the tailoring of

expectations for the products produced during the math block. This differentiation

allows the student to work at their level. When students are performing at their


level, they are often happier and more engaged.

The Daily 3 allows a format for introducing the use of technology in small

groups. Technology can also provide students with new methods of acquiring

information. Some learning tasks are more meaningful on a computer/iPad than on

the traditional worksheet. The Daily 3 asks participants to be changing tasks

quicker than the traditional math instructional structure. Engagement increases

because the learning tasks are so varied. The continually moving forward pace gives

the learning a sense of urgency that is critical in keeping students engaged. It also

allows for short breaks, which can help refocus a student’s attention back to the task

of learning.

The research has shown that the Daily 3 structure is beneficial for students’

depth of knowledge and engagement in the task of learning math.


Appendix A

Table A1.

PostTest PreTest Change

AT 6 6 0

AC 6 6 0

AD 8 6 2

DB 8 6 2

DJ 6 6 0

DN 7 6 1

KS 7 6 1

TJ 6 5 1

*This shows the depth of knowledge scores on the pre/post test. The same test was

given at the beginning and the end of the study. Below is the rubric used to assign

point values.

0-3 4-6 7-10

Depth of

Knowledge

Displays limited

knowledge of

measurement skills.

(Cannot measure to

the nearest inch or

degree.)

Displays limited

knowledge of

measurement skills.

(Can measure to the

nearest inch and nearest

degree.)

Displays wide knowledge

of measurement skills.

(Can measure to the

nearest ½ inch and

nearest degree. Can

convert between units.)

Appendix B


Pre/Post Test

The screwdriver is ____________ inches long.

The hammer is __________ inches long.

The temperature is _________________.

Challenge Questions

1. 24 inches = _________ feet

2. 72 inches = _________ yards

3. __________ yards = 9 feet

4. Outside it is ________________ than 50° F?

a. warmer b. cooler c. same


Rubric for Pre/Post Test

0-3 4-6 7-10

Depth of Knowledge

Displays limited knowledge of measurement skills.

(Cannot measure to the nearest inch or degree.)

Displays limited knowledge of measurement skills.

(Can measure to the nearest inch and nearest degree.)

Displays wide knowledge of measurement skills. (Can measure to the nearest ¼ inch and nearest degree. Can convert between units.)

Engagement Attempts no problems.

Attempts to answer all the questions. If an answer is unknown, writes, “I don’t know” or “?”.

Attempts to answer all the questions with reasonable guesses.


Appendix CStudent Weekly Work Analysis

Name Effort Understanding Comments

Rubric0-3 4-6 7-10

Effort Does not attempt any problems.

Attempts to complete 50% of the work.

Attempts to complete 100% of the work.

Understanding Displays no or minimal understanding of the concept.

Displays a partial understanding of the concept. (Answers at least 50% of the questions correctly.)

Displays a full understanding of the concept. (Answers at least 85% of the questions correctly.)


Appendix DStudent Self Survey

Name:___________________________________

My favorite part of math today was:

____________________________________________________________________________________.

I tried my best during math when I:

____________________________________________________________________________________.

One thing that I learned today in math was:

____________________________________________________________________________________.


References:

Bottge, B. Ma, X., Gassaway, L., Toland, M., Butler, M., & Cho, S. (2014). Effects of

blended instructional models on math performance. Exceptional Children, 80

(4), 423-437 Retrieved from:

http://ecx.sagepub.com/content/80/4/423.full.pdf+html

Burns, M. (2004, October) Writing in math: Innovative teachers can make writing

an invaluable part of math instruction. Educational Leadership. 30-33.

Retrieved from:

http://www-tc.pbs.org/teacherline/courses/rdla230/docs/session_1_burns.

pdf

Ktytor, L. & Waechter, K. (2014, December) The effects of the math daily 3 structure

on student achievement and growth in an elementary school setting.

Masters of Arts in Education Action Research Papers. Paper 82. Retrieved

from: http://sophia.stkate.edu/cgi/viewcontent.cgi?

article=1077&context=maed

O’Malley, P., Jenkins, S., Wesley, B., Donehower, C., Rabuck, D.,& Lewis, M. (April

2013). Effectiveness of using iPads to build math fluency. Retrieved from

http://eric.ed.gov/?id=ED541158

Walton, G. ,Cohen, G., Cwir, D., & Spencer, S. (2012) Interpersonal relations and

group processes. Journal of Personality and Social Psychology, 102 (3), 513-

532 Retrieved from:

http://eric.ed.gov/?id=ED541158

http://sophia.stkate.edu/cgi/viewcontent.cgi?article=1077&context=maed

http://sophia.stkate.edu/cgi/viewcontent.cgi?article=1077&context=maed

http://www-tc.pbs.org/teacherline/courses/rdla230/docs/session_1_burns.pdf

http://www-tc.pbs.org/teacherline/courses/rdla230/docs/session_1_burns.pdf

http://ecx.sagepub.com/content/80/4/423.full.pdf+html


https://web.stanford.edu/~gwalton/home/Research_files/WaltonCohenCwi

rSpencer2012.pdf

https://web.stanford.edu/~gwalton/home/Research_files/WaltonCohenCwirSpencer2012.pdf

https://web.stanford.edu/~gwalton/home/Research_files/WaltonCohenCwirSpencer2012.pdf

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