by Sugi Sorensen
March 30, 2026
Hanover Research, the La Cañada Unified School District’s (LCUSD) hired consultant partner in its current elementary mathematics curriculum selection process, recently delivered its most anticipated and valued deliverable in its $25,000 contract — the High-Quality Instructional Material (HQIM) rubric – to be used to help the LCUSD Math Adoption Committee downselect the current four finalist curricula to two to pilot next year, and presumably to be used again next year to pick a winner among the two piloted curricula.
Hanover’s HQIM rubric itself and a four-page introductory document were made available to district K-8 families by LCUSD Associate Superintendent of Student Services Jim Cartnal in an email sent on Wednesday March 26, 2026.
Overview
The rubric itself, as with all documents Hanover has delivered during this textbook adoption, is a mixed bag – it contains genuine content in Priority #3 ( “teacher usability”) and several defensible attributes. But at its core it is a flawed instrument that is likely to misdirect district teachers towards attributes that the district itself has said it does not value in its next K-5 elementary math curriculum.
In addition, the rubric does not use differential weighting of criteria, meaning it values secondary indicators like P4-3 (school-to-home connections) the same as it does critically important attributes like P1-1 (“Develops conceptual understanding and procedural fluency at the same time. Provides sufficient activities for already-learned skills to support fluency.”) This leads to a rubric that treats curb appeal as co-equal with structural integrity. This is not a problem unique to Hanover’s rubric – every other instructional material evaluation rubric LCMP has seen LCUSD use in the past suffers from this same weakness. However, the implications of this design defect can be fatal and should be corrected.
The rubric is a politically calibrated instrument disguised as a research-informed one. Its most serious failure is structural: five of its nineteen indicators — controlling 26% of the total score — either have no grounding in the Science of Mathematics literature (P4-3 home connections, P4-4 positive disposition, P5-4 Learning Management System integration), are explicitly contraindicated by cognitive load theory for novice learners (P2-3 real-world application as an instructional vehicle), or address a procurement decision the district has already separated from the curriculum selection (P5-3 diagnostic screener ).
The research problems run deeper than omission. The rubric includes “learning styles” as a scored attribute without defining what it means — a construct so thoroughly discredited in the cognitive science literature that no education research firm with genuine expertise in the field would use the phrase unqualified in 2026. Hanover also introduces “written conceptual problems” as a ‘Look-For’ (i.e. attributes that Hanover says rubric users should explicitly look for in the materials being reviewed) under P4-2, an attribute that does not exist as a defined construct in any K–5 mathematics research framework. In practice, evaluators will pattern-match it to open-ended constructivist tasks, advantaging inquiry-oriented programs that the district’s own stakeholders explicitly deprioritized in the LCUSD Math Priorities Survey.
Meanwhile, the rubric omits as scored criteria content sequencing, instructional clarity on students’ first-exposure, practice volume, cumulative interleaved review, and assessment quality — the very elements that measure instructional efficacy. The explicit-instruction language in Priority 3 is real, but the criteria that would penalize discovery-first lesson designs are conspicuously absent. The presence of instructional elements is scored, sequencing is not. The net effect is a compressed score distribution across the four remaining finalists, deferring the actual selection decision to a qualitative judgment process the district controls — and that compression is the most plausible explanation for why the rubric was designed this way.
The Indicators That Should Not be in the Rubric
Here are the five indicators that should not have been included in an instructional materials rubric:
P2-3: Real-world application
Indicator 3 under priority #2 “Student Engagement and Support” reads “Opportunities for students to apply math learning to real-world concepts.” Although this sounds appealing and does have a role in math instruction, the main problem here is real-world applications must be used sparingly and with extreme care. Real-world application opportunities will favor scripted project-based learning (PBL) activities that favor engagement, but in actuality risk inducing cognitive overload in Sweller’s Cognitive Load Theory (CLT.) Students who are new to a topic – called novice learners in cognitive psychology – are distracted by the real-world example the curriculum or teacher centers, and that in-turn displaces instructional and practice time due to the inherent gross inefficiency of PBL tasks.1
Kirschner, Sweller and Clark (2006) distinguishes intrinsic cognitive load (inherent to the content) from extraneous load (generated by the task format).2 Real-world context problems, as implemented in inquiry-oriented curricula, systematically increase extraneous load on novice learners: the student spends cognitive resources navigating the scenario — figuring out what’s mathematically relevant, what to ignore, how the story maps onto operations — resources that would otherwise go toward building the mathematical schema itself.
There’s also Kalyuga’s Expertise Reversal Effect – worked examples are more effective than problem-solving for novice learners – only experienced learners benefit from open-ended application tasks.3 The real-world framing that makes mathematics feel relevant to curriculum designers actively increases the cognitive distance novice learners must cross before reaching the mathematics.
And then there’s the time-on-task argument from Haring and Eaton: fluency requires a high volume of correctly performed practice trials within the fluency stage of the Instructional Hierarchy.4 Every minute a student spends navigating a PBL project is a minute not spent on practice and application. PBL tasks are demonstrably time-inefficient relative to explicit instruction for building procedural fluency, which is precisely what LCUSD’s own stakeholder survey identified as its instructional priority.
Which finalist curriculum benefits? enVision+ (STEAM integration is a design pillar) and Eureka Math² (its Application Problems are rich contextual tasks). Math In Focus is relatively disadvantaged because it treats applications as a component of practice, not as an instructional vehicle.
P4-3: Home to School Connection
Priority #4 in the rubric is “User-Friendly Materials that Support Learning,” and indicator 4 is, “Materials that facilitate a connection between school and home.” The four ‘Look-Fors’ in the rubric for this indicator are: 1) “Home practice aligned to lessons” 2) “Family guides to math strategies” 3) “Family/caregiver support resources” & 4) “Suggested home math activities.”
The presence of these four ‘Look-Fors’ is almost entirely a publishing decision, not an instructional quality decision. A district that buys any of the four finalist curricula could produce a one-page family guide for every unit using AI in an afternoon. The inverse is also true: a curriculum with elaborate parent-facing digital portals or pre-written school-to-home letters as Everyday Mathematics had, does not thereby teach children mathematics more effectively in the classroom.
There’s also a subtle equity problem with this indicator that cuts against Hanover’s stated equity values: curricula with strong ‘Home Links’ assume parents have the time, digital literacy, and mathematical background to engage with the materials. Curricula that teach students exclusively at school — so that home support is optional — is arguably more equitable, not less.
Which finalist curriculum benefit? enVision+ (elaborate Family Engagement resources), Eureka Math² (Zearn companion platform, family math nights materials). Math in Focus’ parent materials are comparatively functional but not elaborate.
P4-4: Positive Math Dispositions
Indicator 4 of priority #4 is: “How students build on and use conceptual supports for a balanced inquiry path to mastery.” Positive math disposition is primarily a classroom/teacher issue and should not be tied to curriculum. After all, an easy curriculum that is unrigorous might make students feel great about their math abilities, but actually give them false confidence. Research shows that programs with high challenge consistently produce better long-term outcomes than programs optimized for enjoyment.5 Measuring disposition as a proxy for curriculum quality gets the causal direction backwards.
The positive math disposition construct is drawn directly from the growth mindset literature of Carol Dweck, and its derivative math mindset theory of Jo Boaler.6 Together these theories present the framing that curriculum should make students feel good about mathematics. The research base for this in mathematics curriculum design is thin and contested.7
Second, the specific ‘Look-Fors’ – “collaborative problem solving,” “multiple solution strategies,” “engaging or exploratory tasks” – are pedagogy descriptors, not curriculum quality criteria. More precisely, they describe a specific pedagogy — inquiry-based constructivism. Direct instruction curricula do not include these elements to produce students who feel competent and confident about mathematics. In fact, the research suggests the opposite – students who master material through explicit instruction and reach fluency report higher mathematical confidence than students in inquiry-based programs who never quite nail down the concepts. Genuine mastery produces genuine confidence. This indicator rewards the theater of engagement over the substance of learning.
Which finalist curriculum benefits? Heinemann’s Math Expressions (math talk communities, “Supports students in finding their mathematical voice”), Eureka Math² (rich discussion structures, multiple representations, debrief culture) and enVision+ (investigation-first, student-generated strategies). Math In Focus is disadvantaged because its lesson architecture is linear and teacher-directed, which the ‘Look-Fors’ systematically undervalue.
P5-3: Diagnostic Screener and Intervention Tools
Priority #5 in the rubric is “Implementation Success and Support” and its third indicator is, “Availability of math diagnostic screener, progress monitor, and intervention support tools.” Cartnal explicitly announced at the November 5th first parent informational meeting that they are evaluating a separate progress monitoring and MTSS intervention tool:
As such, a diagnostic screener and MTSS intervention components should be Nice-to-Haves, not Must-Haves in a core curriculum evaluation process. Absent a differentially weighted rubric, the presence or strength of a diagnostic screener and intervention components should not be co-equal with indicators for efficacy and usability.
In fact, Hanover itself produced the vendor scan on screeners and interventions as a separate deliverable. The district has already acknowledged that this is not a curriculum selection criterion — it’s a separate procurement decision. Including it as a co-equal curriculum indicator therefore double-counts the criterion for programs that happen to bundle these tools and disadvantages programs where the instructional design is excellent, but the publisher chose not to package diagnostic tools.
It also creates a perverse incentive: publishers learn to bundle mediocre screeners with their core products to pick up rubric points, regardless of whether the screener is actually the best available option or ever used. LCUSD would be better served selecting a diagnostic and MTSS intervention program like Spring Math or Acadience Math independently on their own merits, not choosing a core curriculum based on which one happens to bundle a screener.
Which finalist curriculum benefits? enVision+ (Savvas has Momentum Math online assessment tools), California Math Expressions (HMH offers MAP Growth and other diagnostics assessment tools as add-ons), Eureka Math² (Great Minds has its Affirm assessment platform). Math In Focus has limited bundled diagnostic tools.
P5-4: LMS/Tech Integration
Indicator five of priority #5 – Implementation Success and Support – is “Compatible with District Systems, like Classlink and Google Classroom.” The integration of core curriculum with the district’s Learning Management System (LMS) – Classlink – and the Google Classroom learning platform is unnecessary at best, and at most a nice-to-have. Again a good teacher can integrate any curriculum, including their own teacher-created assignments, with whatever student learning platform or LMS they have in place, whether a 3-ring binder, or Google Classroom.
In fact, several recent large studies have found mixed to negative long-term benefits from educational technology (ed-tech) that counsel caution. In light of these warnings, it would be better if the adopted curriculum did not require any integration with any online student learning system whether Google Classroom or Classlink.
Older trusted resources found the evidence base for ed-tech integration unconvincing. The IES What Works Clearinghouse has found limited evidence for most math-specific digital platforms, and the NMAP Final Report (2008) was already cautious about technology as a substitute for explicit instruction and systematic practice.
More pointedly: the Hanover survey itself found that 63% of LCUSD respondents ranked ‘paper and pencil math practice activities’ as a top-five priority for supporting effective instruction — the highest-ranked item on the list. Tech integration ranked 12th out of 16 priorities. Hanover included an indicator that directly contradicts its own survey findings about what LCUSD stakeholders want. That is not a rubric design error; it is a revealing inconsistency.
Google Classroom and Classlink integration is a school operations feature, not a mathematical instruction feature. A curriculum should be evaluated based upon what happens between teacher and student during a math lesson, not on whether its homework can be assigned through an LMS. These are genuinely different questions.
Which finalist curricula benefits? enVision+ and Math Expressions (large US publishers with full ed-tech departments and existing LMS partnerships). Math In Focus, as a Singapore-origin program distributed through US publishers, has historically weaker LMS infrastructure. This indicator is a structural tax on non-US-native curriculum designs.
Why This Matters
The rubric has 19 indicators at 3 points each — 57 points total. With equal weighting (remember I pointed out the Hanover rubric fails to use differential weighting), each indicator represents 5.26% of the total score. The five indicators that should not be in the rubric at all together control 26.3% of the rubric’s scoring weight — more than a quarter of the instrument.
When the denominator is inflated with low-validity or zero-validity items, you don’t just add error, you reduce the contribution from the high-validity items. A curriculum that scores 3/3 on every legitimate indicator but 1/3 on the five junk indicators finishes behind a curriculum that scores 2/3 on legitimate criteria but 3/3 on junk. This is how a mediocre program with a robust ed-tech sales team and engaging math games feature to be played at home beats a pedagogically superior program that simply didn’t build a parent portal.
The ‘Learning Styles’ Red Flag
Priority #2 “Student Engagement and Support” indicator one is, “Materials provide clear supports for all types of learners (e.g., English learners, advanced learners, and struggling learners) and different learning styles, and offer access to accessibility features like text to speech.” The inclusion of learning styles is a concerning red flag.
Learning Styles is a reference to a widely held educational myth that different students have different modes of learning, and their learning could be improved matching one’s teaching with that preferred learning mode.8
A 2020 systematic review published in Frontiers in Education analyzed 37 studies representing 15,045 educators from 18 countries, spanning 2009 to 2020, and found that 89.1% of educators self-reported a belief that individuals learn better when they receive information in their preferred learning style. Critically, the authors found no evidence that this belief has declined over time — and among pre-service teachers, the figure was even higher at 95.4%.9
While Hanover might have meant something other than this definition, they failed to define the term. More importantly, Hanover has a glossary in their introduction to the rubric. They defined “CPA,” “Math In Focus,” “Rigor,” “Scope and Sequence,” and “Differentiation.” They chose not to define “learning styles” — the one term in the rubric with a contested and largely discredited meaning in the research literature. That is not an oversight. Either they know the term is problematic and declined to define it to avoid scrutiny, or they don’t know it’s problematic, which is the more damning possibility for a firm claiming research expertise in education.
That numerous studies have all definitively demonstrated that matching instruction to preferred learning style produces no measurable learning benefit taints the rubric indicator and ‘Look-Fors.’10 The construct lacks both construct validity (there is no reliable way to classify learners into types) and predictive validity (classified types don’t respond differently to matched instruction).
Hanover’s use of this phrase is either an indicator of the firm’s research literacy level or a deliberate appeal to the teacher-audience that still widely believes the myth. Either way, it’s a red flag.
On ‘Written Conceptual Problems’
In Priority area #4: “User-Friendly Materials that Support Learning”, the second indicator reads, “Presence of paper and pencil tasks that promote student skill development, conceptual understanding, and automaticity.” And the ‘Look-Fors’ say: “Written conceptual problems”, “Procedural practice sets”, “Mixed practice tasks”, and “Repetition supports fluency.” This indicator suffers from multiple deficiencies. It is poorly written – the last look-for is a declarative statement about pedagogy. How “repetition supports fluency” relates to the presence of paper and pencil tasks that promote students’ skill development, conceptual understanding or automaticity, much less how it evinces evidence of those remains a mystery.
More concerning “written conceptual problems” does not exist as a defined category or construct in any mathematics education research framework we are currently aware of. So what does Hanover mean by “written conceptual problems”? Once again the Glossary does not help. It is anybody’s guess what they mean and is likely to place teachers on a wild goose chase looking for their own conception of “written conceptual problems” in curricula they are evaluating.
Hanover may have meant an application or generalization type problem in Haring & Eaton’s framework that is in the form of a word problem. If so, why didn’t they just say that? And if they meant an application/generalization word problem, would a fluency or an application-level word problem that merely asks a student to explain their answer or “explain their reasoning” suffice? Again, no guidance is offered.
They may have meant constructed response items or “low-floor, high-ceiling tasks” to borrow a term from the 2023 California Mathematics Framework.
If constructed response is what Hanover meant, it conflates a response format (written) with a cognitive level (conceptual) in a way that has no research basis. Adding “explain your work” to a computation problem doesn’t elevate it to a conceptual task — it adds a metacognitive reporting requirement on top of a procedural one. Worse, demanding verbal articulation at the wrong instructional stage can interfere with automated skill execution (called verbal overshadowing).
If they meant a ‘Big Ideas: Low-floor, High-ceiling’ task, that connotes constructivist teaching. Those are Jo Boaler-style open tasks. A teacher who encounters a “What do you notice? What do you wonder?” task in California Math Expressions or enVision+ will score it as a “written conceptual problem” because it looks like it addresses conceptual understanding. The problem is that these tasks are simultaneously the least efficient and the most cognitively overloading introduction to new content that a curriculum can offer — the opposite of what explicit instruction prescribes for novice learners.
The practical consequence is this ‘Look-For’ will produce a scoring advantage for curricula that include open-ended discussion tasks and explanation prompts — constructivist hallmarks — and a disadvantage for curricula that develop conceptual understanding through CPA progression, teacher modeling, and worked examples, which is how conceptual understanding is actually built at the K–5 level per the research. A curriculum reviewer holding a Math In Focus lesson will have a harder time locating a “written conceptual problem” than a reviewer holding a California Math Expressions or enVision+ textbook, even though MiF’s CPA architecture is more effective at building conceptual understanding than any “What do you notice?” prompt.
The Key Indicators That Were Left Out of the Rubric
Left out of the rubric entirely are five extremely important attributes of effective math curricula that experienced teachers recognize:
- Content coverage and sequencing – An instructional evaluation rubric that omits content coverage and sequencing is gross negligence in and of itself. It is particularly needed because it would force evaluators to ask whether K–5 content actually prepares students for MiF’s 6–8 sequence — not just whether the district claims alignment, but whether topics are introduced at the right grade level and in the right order. MiF and Eureka Math² have explicit, research-grounded scope-and-sequence documentation. enVision+’s sequence is SBAC-optimized, not MiF-preparation-optimized.
- Instructional clarity / what happens first – This is another extremely important missing criterion because it directly captures the explicit-vs.-inquiry distinction the rubric refuses to encode. A criterion that would have been more useful than almost all of those included is, “Key ideas made explicit through worked examples, structured investigation, and/or consolidation.”
- Practice opportunities – Again, Hanover omitted a key attribute that differentiate math curricula. Does the curriculum contain a sufficient number of scaffolded practice problems? This item would capture the volume and variety of practice — not just whether the Gradual Release structure exists (which Hanover measures via P3-2), but whether there is actually enough practice for students to achieve fluency. MiF’s practice volume is significantly higher than enVision+’s; its workbook structure provides the high-trial-density that Haring and Eaton identify as essential for the fluency stage.
- Cumulative/interleaved review – Two of the most efficacious instructional practices to emerge from the Science of Mathematics are interleaving and distributed practice. Both were mentioned as best practices in Robin Codding’s presentation to the LCUSD Elementary Math Selection Committee on December 13, 2025. Both would differentiate curricula sharply. Math In Focus and Eureka Math² both have systematic distributed practice that spans the year. enVision+’s review structure is primarily chapter-level, not interleaved. This element directly operationalizes the spacing effect from cognitive science — arguably the most robust finding in the learning science literature for long-term retention — and it is completely absent from the Hanover rubric.
- Assessment quality – This is perhaps the most staggering omission in a rubric ostensibly designed to support high-quality curriculum selection. The quality of embedded assessments — whether they measure conceptual understanding and procedural fluency at an appropriate depth, whether items are comparable to NAEP or TIMSS in cognitive demand, whether formative assessment guidance helps teachers actually adjust instruction — is entirely invisible to evaluators using Hanover’s rubric. This matters because MiF’s assessments are mathematically demanding in ways that directly reflect the content’s rigor, while programs optimized for SBAC performance may have assessments that look rigorous by state-test standards but don’t approach TIMSS-level depth.
What makes the omission of the above even more baffling is that I provided Associate Superintendent Cartnal with four examples of instructional material rubrics, including one from the Science of Mathematics organization,11 an Instructional Materials Evaluation Tool (IMET) developed by Student Achievement Partners for districts evaluating Common Core alignment, and a rubric I developed in my work with the Institute for Mathematics Instruction (IMI) specifically for LCUSD’s K-8 math adoption.
All four rubrics provided to Cartnal contain the five missing criteria.
Conclusion
The bottom line is that Hanover produced a rubric that:
- Awards roughly a quarter of its points to criteria with no grounding in the Science of Math literature (P4-3, P4-4, P5-3, P5-4) or with evidence suggesting they are counterproductive (P2-3 via CLT.)
- Omits the criteria that would most clearly differentiate programs on instructional design quality – content coverage & sequencing, instructional clarity, practice opportunities, cumulative/interleaved review, and assessment quality.
- Systematically advantages programs from large US publishers with robust ed-tech and parent-engagement features (e.g. Savvas enVision+) that should not even be in the rubric.
- Is silent on the diagnostic question that every Science of Math researcher would identify as most important: what happens when a student encounters a new concept for the first time?
If the district wants to ensure that it does not inadvertently lock students and teachers into a ten-year train wreck like it did in 2016, it needs to fix or replace the current HQIM rubric Hanover delivered. The IMI or the Science of Math rubrics already provided to the district could be used as correctives.
- Sweller, J. (1994) “Cognitive load theory, learning difficulty, and instructional design.” Learning and Instruction, 4(4), pp. 295–312. ↩︎
- Kirschner, P., Sweller J., & Clark, R. (2006) “Why Minimally Guided Instruction Doesn’t Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching.”Educational Psychologist, 41:2, pp.75-86. ↩︎
- Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003) “The expertise reversal effect,” Educational Psychologist, 38(1), pp.23–31. ↩︎
- Haring, N. G., & Eaton, M. D. (1978) “Systematic instructional procedures: An instructional hierarchy.” In N. G. Haring, T. C. Lovitt, M. D. Eaton, & C. L. Hansen (Eds.), The fourth R: Research in the classroom (pp. 23–40). Charles E. Merrill. ↩︎
- Stockard, J., Wood, T. W., Coughlin, C., & Rasplica Khoury, C. (2018) “The effectiveness of Direct Instruction curricula: A meta-analysis of a half century of research.” Review of Educational Research, 88(4), pp. 479–507. ↩︎
- LCUSD has long been a fan of Carol Dweck’s Growth Mindset work and if you are a parent in LCUSD you’ve undoubtedly heard it from your child and/or from back-to-school nights. For the basics on Growth Mindset theory see Dweck’s Wikipedia page on Mindset theory. For the full book-length treatment taught in most ed schools, see Dweck, C. S. (2006). Mindset: The new psychology of success. Random House. Stanford mathematics educational guru Jo Boaler extended growth mindset theory into the mathematics domain with what she called ‘mathematical mindset,’ primarily explained in Boaler’s book Mathematical Mindsets – Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Jossey-Bass. ↩︎
- The Institute for Educational Science (IES) Practice Guides don’t list disposition development as a primary instructional lever. Rosenshine’s 12 Principles for Instruction don’t mention it. And the National Mathematics Advisory Panel (NMAP) Final Report (2008) emphasizes fluency, procedural competence, and problem-solving — not affect. ↩︎
- The definition is quoted from Riener, C., & Willingham, D. (2008) “The Myth of Learning Styles.” Change: The Magazine of Higher Learning, 42(5), pp.32-35. Taylor & Francis Online. ↩︎
- Newton, P.M. and Salvi, A. (2008) “How Common Is Belief in the Learning Styles Neuromyth, and Does It Matter? A Pragmatic Systematic Review.” Frontiers in Education. 5:602451. ↩︎
- Pashler, H., McDaniel, M., Rohrer, D., & Bjork, R. (2008). “Learning styles: Concepts and evidence.” Psychological Science in the Public Interest, 9(3), pp.105–119. And Riener and Willingham, Ibid. ↩︎
- Kaye Andersen (2023), Science of Mathematics. ↩︎