The proposed California Mathematics Framework generated a storm of controversy when the first draft was released in early 2021. Critics objected to the document’s condemnation of tracking and negative portrayal of acceleration for high achieving students. Indignation focused on the recommendation that schools stop offering Algebra I to mathematically precocious eighth graders. A revised draft was released in 2022, softening the harsh language of the original text while leaving intact the framework’s dim view of course acceleration or other forms of tracking.
Those are important issues; however, this post is concerned with students on the opposite end of the distribution of achievement: students who struggle with math. Over the past decade, math scores on the National Assessment of Educational Progress (NAEP) have been declining at the 25th percentile, indicating that struggling students are falling even further behind their peers. Moreover, as schools recover from the pandemic, the percentage of students with disappointing math achievement is sure to go up. What does the framework portend for them? What evidence does the framework rely upon to build its recommendations for these vulnerable youngsters?
The IES Practice Guide on Struggling Students
About the same time that the first draft of the framework came out, the What Works Clearinghouse of the Institute of Education Sciences (IES) published “Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades” (hereafter referred to as “Struggling Math Students”) as part of its ongoing series of practice guides for educators.[i] The practice guides distill the latest high-quality research on a specific, practical topic into a few clear recommendations that are useful to practitioners, in particular, classroom teachers.[ii]
The practice guides are developed following set protocols. First, a literature search is conducted of terms related to the guides. In the case of “Struggling Math Students,” 2,653 records were identified. Then the studies are screened for topic relevance and eligibility, the latter including criteria for sample size, clear learning outcomes, and research design. These criteria are meant to single out research producing findings with a strong causal warrant. The design criteria are especially important because they include longstanding hallmarks of good policy evaluations, including either randomized assignment of subjects to treatment and control groups or an acceptable quasi-experimental approach, standardized measures of outcomes, verification of group equivalence at baseline, and low sample attrition. Fifty-six studies survived these requirements.
After a final screen that focused on quality and relevance to the topic at hand, forty-three studies met criteria, encompassing 6,990 students and 490 schools. The studies are described in Appendix C of the practice guide, including explanations for how they support six recommendations:
1. Provide systematic instruction
2. Teach clear and concise mathematical language
3. Use well-chosen representations
4. Use number lines
5. Provide deliberate instruction on word problems
6. Regularly include timed activities to build fluency
The guide offers examples of how teachers can implement recommendations in classrooms, along with suggestions for addressing obstacles that may arise. By focusing on students who struggle with mathematics, the studies are limited in what they can say about whole-class instruction (Tier 1), the exception being middle schools, in which struggling students may be grouped together in the same class. The findings mostly pertain to instruction of students in small groups, often referred to as Tier 2 interventions, or one on one instruction (Tier 3). The latter two forms of intervention could involve working with a math specialist, special education teacher as part of an individualized education plan (IEP), or a tutor.
The practice guides are not infallible. Their value is in following a transparent, replicable process to summarize the best scholarship on practical problems facing educational practitioners. The practice guide on students who struggle to learn mathematics represents what we currently know about addressing those students’ needs.
Are the Studies Cited in the Practice Guide Cited in the 2022 California Math Framework?
I searched to see how many of the practice guide’s forty-three studies are cited in the California Math Framework.[iii] None. Zero. When I couldn’t find a single citation in the list of references, I searched the framework text to make sure a cited study hadn’t been inadvertently omitted. Nope. None are mentioned in the document’s 1,100+ pages.
California publishes frameworks to provide guidance to local educators in implementing standards, in the case of mathematics, the Common Core Math Standards. Some students have struggled with that learning. The 2022 California Math Framework overlooks the body of scientific evidence addressing how educators can best serve the needs of students for whom math is difficult.
How could that possibly happen?
How the Framework Addresses Struggling Students
The framework‘s treatment of how struggling students’ needs should be met seems to be a mixture of wishful thinking—every student will succeed if teachers simply follow the framework’s commands--and the attitude that the topic is outside the framework’s mandate.
Students develop at different times and at different rates; what educators perceive as an apparent lack of understanding may not indicate a real lack of understanding. The implementation of mathematics routines that encourage students to use language and discuss their mathematics work are of benefit to all students, particularly those who are learning English or who are challenged by the demands of academic language for mathematics. Such supports allow educators to help students strengthen understandings that may have been weak or incomplete in their previous learning without formal intervention program. When more support is warranted, teachers can access California’s Multi-Tiered System of Support (MTSS) (California Department of Education, n.d.), which is designed to provide the means to quickly identify and meet the needs of all students (Chapter 6).
This is a strange passage. The first sentence almost denies that many students struggle with math. It then asserts that adopting classroom “routines” involving mathematical discussions, as urged by the framework, can be counted on to take care of misunderstandings by English learners or those “challenged by the demands of academic language for mathematics.” No evidence for either claim is provided. Multi-tiered support is not mentioned until the end of the passage—and only then by pointing to another state document.
The Role of Ideology
It appears that the framework’s ideological commitment to the principle that all students should be treated the same—same curriculum, same instruction-- is the primary reason why the extensive literature on struggling students is ignored. Effective interventions require identifying students who are falling behind and creating supplemental instructional settings for them, either in small groups or individually. In contrast, the framework places all its bets on instruction that attends to mindset theory, lessons using math to explore social justice topics, and Universal Design for Learning (UDL) to reduce the number of students who need extra help. The framework doesn’t say it out loud, but the idea that students could fall behind once this instructional regime is established is treated as unlikely.
The framework’s second ideological commitment is to inquiry. Topics are organized around “big ideas” and “drivers of investigation.” Inquiry methods have a century-long checkered history, particularly for struggling students in the primary grades.[iv] As a long time reader of California’s frameworks, I can say that the 2022 Math Framework is the most inquiry-oriented that I’ve seen since the 1992 California Math Framework. This statement from the 1992 framework could have easily come from the 2022 version: “Children often misinterpret and misapply arithmetic and algebraic procedures taught the traditional way. This program, in contrast, values developing number and symbol sense over mastering specific computational procedures and manipulations.”[v] The 1992 framework flew under the radar until a coalition of concerned parents and mathematicians, in what became known as “The Math Wars,” rallied against the textbooks and instructional methods that the framework spawned and drove them all out of state policy.
The “traditional way” that inquiry advocates would like to upend goes by several terms, the most common being “direct” or “explicit” instruction. Take another look at the six recommendations supported by evidence. Three of them--provide systematic instruction, teach clear and concise mathematical language, and provide deliberate instruction on word problems—are predicated on explicit instruction. Teachers intentionally explain the how and why of mathematics, ask students to practice new knowledge, and check to see if students have learned the material. Exploration or discovery is not the governing activity.
The recommendation to use timed activities to build fluency is opposed by the 2022 framework. The Common Core math standards define fluency as “speed and accuracy,” identifying two components also common to instruction in reading and foreign languages.[vi] The framework believes that speed should be dropped in favor of flexibility, arguing that an emphasis on speed creates math anxiety.[vii]
This is a mistake. Students who struggle with math often have not mastered basic facts of whole number addition, subtraction, multiplication, or division (e.g., 4 + 7 = 11, 16 – 8 = 8, 4 x 9 = 36, 72 / 8 = 9). Asked to apply these facts quickly in multidigit calculations, the students flounder. So much working memory is devoted to simple calculations, the new procedures students encounter with multidigit arithmetic and fractions—and later algebra--cannot command sufficient cognitive resources. Fluency with basic facts often goes by the term “automaticity,” the ability to retrieve math facts effortlessly from long term memory. The word does not appear in the California Math Framework. Nor do the terms “retrieval practice” or “interleaved practice” appear, instructional strategies for enhancing students’ automaticity and long term memory.
The 2022 California Math Framework does not reflect current scholarship on how to serve students who struggle when learning mathematics. A search of studies cited in a recent What Works Clearinghouse publication, “Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades,” reveals absolutely no overlap. None of the studies cited in “Struggling Math Students” are cited in the framework. This is particularly troubling because of the transparent, rigorous process followed in producing the practice guide, ensuring that recommendations are based on scientifically sound research. In sharp contrast, the process employed to search literature and select evidence for the framework’s recommendations is unknown. It is not described in the document or on the framework’s website.
The California State School Board will consider the framework for adoption in July, 2022. All students will be poorly served if the state endorses inquiry over explicit instruction. Students who dream of pursuing a STEM major will arrive at college unprepared.[viii] Students who have difficulty learning math will see their frustrations increase and challenges multiply as they fall further behind their peers.
The Board should reject this framework.
[i]All references to Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades can be found in the document here: https://ies.ed.gov/ncee/wwc/PracticeGuide/26
[ii]Full disclosure. I served as a elementary math content expert for the What Works Clearinghouse from 2013-2018. I did not work on any of the practice guides.
[iii]All references to the 2022 California Mathematics Framework can be found in the document here: https://www.cde.ca.gov/ci/ma/cf/
[iv] Paul A. Kirschner, John Sweller, and Richard E. Clark (2006). “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential and Inquiry-Based Teaching.” Educational Psychologist, 41 (2), 75-86.
[v]Mathematics Framework for California Public Schools, 1992, p. 54.
[vi]In the introduction to the Common Core State Standards for Mathematics: “Procedural skills and fluency: The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.” (http://www.corestandards.org/other-resources/key-shifts-in-mathematics/)
[vii] From the framework’s glossary: “Fluency. The ability to select and flexibly use appropriate strategies to explore and solve problems in mathematics.”
[viii]Brian Conrad, Professor of Mathematics and Director of Undergraduate Studies in Math at Stanford University, offers a detailed analysis of the framework’s failures in preparing STEM majors: https://sites.google.com/view/publiccommentsonthecmf/