How To Design An Elementary School Part 2: Spiral Math Curriculum

Is a spiral based math curriculum the best approach?

Is a spiral based math curriculum the best approach?

My children went to Flora Vista Elementary School within the Encinitas School District.  Flora Vista used a spiral approach to teach math. My children had trouble so my wife and I hired tutors to provide remedial math exercises so they could master primary school math and eventually Algebra in middle school.  My son has a daughter who attends a private school in Portland, Oregon.  She had trouble mastering arithmetic. He hired tutors so she would master primary school math and eventually Algebra in middle school.  Both the public and the private schools used math curricula based on a spiral approach.  I examined the textbooks. I wondered, whether a spiral based math curriculum was the best approach.  I was skeptical. I looked into it.

Math Curriculum

When selecting a math curriculum, it is essential to choose one that aligns with current research on effective math instruction. Researchers suggest that a Strand approach is superior over a Spiral approach.  An example of a Spiral approach is the Scott Foresman Addison Wesley Math (1990) used frequently in public schools. Examples of math programs that used the Strand approach include: McGraw Hill’s Direct Instruction Arithmetic, Connecting Math Concepts, and Essentials for Algebra. Singapore Math, another popular program, uses a combination of spiral and strand approaches.

The Spiral Approach. Bruner (1960) invented the spiral curriculum model.  Vickie Snider (2004), from the University of Washington, compared Spiral vs Strand approaches.  According to Snider, Spiral designs are organized as follows.  Various topics are revisited every year.  Each math concept is taught for a few lessons scattered over part of the school year, then removed from the curriculum only to return the next year.  Each topic is covered briefly. Snider writes, “Teachers devote less than 30 minutes of instructional time across an entire year to 70% of the topics they cover.”

Snider (2004) offers the Scott Foresman-Addison Wesley Math (SFAW) as an example of a spiral approach.  She compares how addition, subtraction and fractions are taught using SFAW versus Connecting Math Concepts (CMC). “SFAW introduces addition and subtraction concepts in chapters 3,4,6,12, and 13. The concepts are revisited every fall except second grade where they are both covered in Fall and Spring”. Addition and subtraction make their final appearance in 5th grade.  Fractions appear in 2nd grade in chapter 12, then in 3rd grade in chapter 10; 4th grade in chapters 9 & 10; 5th grade within chapters 7,8,& 9, and in 6th grade within chapters 6 & 7.  Since the instruction time on each concept is short, interspersed across the school year, Snider argues the students obtain a superficial understanding of math concepts -breadth over depth. Since the relatively short amount of instructional time is about equal regardless of the difficulty of the concept, the rate of introducing new concepts is at times too fast or too slow, she writes.

Direct Instruction vs Spiral Curricula. In the bar graph below is a comparison of Direct Instruction Math and three curricula that used a Spiral approach, namely: Southwest Labs, Bank Street and TEEM. They were published in the 70’s and 80’s and were a part of the Project Follow Through research project. Twelve School Districts participated. Other curricula were compared, also, however I only picked the data from those curricula that I could confirm a Spiral approach was used. By the way, the data from other curricula that I excluded (e.g., High Scope) was not any better than the data from the Spiral curricula.

Net Percentages of Statistically and Educational Significant Outcomes for Follow Through Models on MAT Math Subscales (Carnine & Gersten, 1982.)

As you can see, Direct Instruction produced better achievement in computation, problem solving, math concepts and in the overall total math score.

The Strand Approach. The curriculum that aligns itself most with the current research on Math instruction is Direct Instruction: Connecting Math Concepts, Corrective Math, and Essentials for Algebra.

The Direct Instruction curricula are organized into strands that focus on teaching prerequisite skills to mastery before more complex skills are taught. CMC provides consistent teaching time on addition and subtraction for grades 1,2 and half of grade 3. Fractions are introduced after addition and subtraction concepts are mastered.  Fractions are taught consistently throughout grades 3 and 4.

Another feature of Direct Instruction math curricula is the use of explicit instruction compared to spiral curricula. Explicit instruction involves modeling, guided practice, feedback, and practice. When signaled, students respond in unison during small group instruction.

Again, this can be contrasted with spiral approaches where teachers facilitate students, through math activities. Students explore and discover unique ways to solve problems on their own. Connecting Math Concepts downplays exploration and discovery.  According to Engelmann, it is much more effective to show students how to perform a skill and then give them practice opportunities to ensure success over discovery learning.  The metaphor, “discovery learning” gives a limited role to the teacher, who does not teach directly, but only helps the student learn.

A Hybrid Approach. Singapore Math uses a combination of a Spiral and Strand approach.  According to Wenxi Lee, author of “The Secrets To Singapore’s World Class Math Curriculum”, Singapore Math is a vertical spiral curriculum with a mastery approach.  Lee describes a typical spiral curriculum this way:   

“..children using a spiral curriculum may learn multiplication of groups of 1 for the first two lessons of the school year.  After that, they switch to a completely different topic, such as geometry, for the next two lessons.  A month or two later, multiplication is revisited, but this time they are taught multiplication in groups of 2. Then in the last few weeks of first grade they finally learn how to multiply in groups of 3, which concludes the learning goal for…multiplication in the school year.”  (P.46)

Singapore Math organizes it’s curriculum more like Connecting Math Concepts.

“Math topics are broken down into smaller chapters, but instead of alternating them with lessons from different topics throughout the year, these smaller chapters are taught sequentially, in single chunks at any grade level.  As children advance to the next grade, they spiral back to a similar sequence of topics, revisit some of the content taught previously, and build on their existing knowledge..” (p46)

In the United States schools use the Singapore Math curriculum “Math in Focus”(MIF). New Jersey’s Edison Township School District began a year long pilot of Math in Focus during the 2011‐2012 school year looking to improve students’ overall skill in math (Kuska 2014). Third graders showed an improvement from 17% to 76% on pre‐post tests. In fourth grade, 39% of students scored 70% or higher on the post test.  Only 5% scored 70% on the pre‐test. (Koska’s (2014) describes this study in her research paper, however, the authors of the study are not named and the link to the study was removed from the website).

Another group of researchers (Jaciw et al., 2012) looked at the effectiveness of Math in Focus on math achievement within the Clark County School District in Nevada. The dependent measure was scores on the SAT problem solving assessment.  Teachers from 3rd, 4th, and 5th grade classrooms participated in the research. The remaining grades were used as a control group.  The control group included teachers who used the following curricula: Envisions, Investigations, Scott Foresman, Pearson SuccessNet, Everyday Math, or No set curriculum. Compared to the control group, MIF was more effective in improving scores on the SAT problem solving skills assessment but was not better at improving procedural skills.  Procedural skills involve knowing the step by step procedures to solve mathematical problems. Scores on the Nevada Criterion Reference Test (CRT) were not significantly different between the experimental and control groups, possibly because MIF did not cover all the standards measured by the Nevada CRT.  As the percentage of standards were covered researchers saw an increase in CRT scores.

Fidelity. The majority of MIF teachers did not meet all the criteria for MIF fidelity. The teachers augmented the curriculum with other curricula and materials outside of MIF.  Many teachers did not use MIF completely the way the authors intended.  Some adjusted the curriculum to meet State standards.

Fidelity to a curriculum’s methods is important.  Connecting Math Concepts uses various methods to obtain the best performance from the student.  The program is scripted.  It tells the teacher what to say, how to correct errors, and how to introduce new lessons. When used in small group setting, the students respond in unison. It’s easy to see how the degree of fidelity to the these procedures improves student achievement.

What curriculum would I pick?

I’ve worked with teachers in many school districts. They do not like scripted curriculum. They like to use their own expertise, no matter how imperfect, to teach their students. They like to use teaching techniques they have honed over their years of training and experience.   All curricula have a format to follow.  Perhaps the procedures Direct Instruction requires that make it most effective is what teachers find most aversive.  There is little room for teacher improvisation.  But if I was picking a curriculum for my imaginary private school I would use Direct Instruction, and hire trained teachers who promote it’s use. When my wife and I operated a preschool in Tustin, CA we hired a woman from the University of Oregon to train the Kindergarten teacher in Reading Mastery and Language for Learning.  Once she learned the curriculum she became an advocate.  However, to be fair, the other kindergarten teacher in our preschool in Vista refused to use Reading Mastery and insisted on using her tried and true basal readers.

A list of curricula: Strand, Spiral and Hybrid

If you are not a fan of Direct Instruction I’ve listed other curricula you can review.

The following information on math curricula was obtained from: https://www.rainbowresource.com/pdfs/math.pdf  published in 2018.  I made some additions such as, Direct Instruction. The rainbow resource information was published in 2018.  Additional information about the curricula can be found on the rainbow resource website. Direct Instruction information can be found at: https://www.mheducation.com/prek-12/explore/direct-instruction.html . There is useful information on these websites.

I have not personally reviewed all of the curricula listed below.  In reading information online about the curricula it’s not always stated whether the curriculum is Strand, Spiral, or if it has a high mastery criterion for advancement. Other terms are used often, such as: sequenced, linear, network and computer based. Are the words synonymous? Also, publishers may have changed their curricula since this list was published.  I would take the following useful information with a grain of salt.

End of Part 2.  Part 3 on Classroom management systems will be posted soon.  I appreciate your comments positive or negative.

References

Adams, G., and Engelmann, S., (1996) Research on Direct Instruction: 25 Years Beyond DISTAR.

Carnine, D., & Gersten, R. (1982). Effective Mathematic Instruction for Low Income Students: Results of a Longitudinal Field Research in 12 Schools Districts. Journal of Research in Mathematics Education 13,145-152.

Bruner, J., (1960) The Process of Education 6th Ed., Cambridge Mass, Harvard University Press.

Ireland, J., and Mouthaan, M., (2020) Perspectives on Curriculum Design: Comparing the Spiral and the Network Models Research Matters Issue 30 Autumn

Kuska, C., (2014) Math in Focus: Singapore Math and Student Achievement of 4th Grade Students. Submitted to professional education faculty Northwest Missouri State University, Missouri, Department of Professional Education College of Education and Human Services, Maryville, MO 64468. Submitted in Fulfillment for the Requirements for 61‐683 Research Paper. (Kuska references: Study Finds Houghton Mifflin Harcourt’s Singapore Math Program Dramatically Raises Student Math Achievement in New Jersey School District, 2012).

http://ww.hmhco.com/media‐center/press‐releases/2012/june/SingaporeMath‐Raises‐NJ‐Achievement

Lee, W., (2020) The Secrets To Singapore’s World-Class Math Curriculum

Self published paperback.

Snider, V., (2004)  A Comparison of Spiral Versus Strand Curriculum  Journal of Direct Instruction Vol4, No 1 pp 29-39