A Mathematical Mystery

My colleagues and I wrote a review of research on elementary mathematics (Pellegrini, Lake, Inns, & Slavin, 2018). I’ve written about it before, but I wanted to hone in on one extraordinary set of findings.

In the review, there were 12 studies that evaluated programs that focused on providing professional development for elementary teachers of mathematics content and mathematics –-specific pedagogy. I was sure that this category would find positive effects on student achievement, but it did not. The most remarkable (and depressing) finding involved the huge year-long Intel study in which 80 teachers received 90 hours of very high-quality in-service during the summer, followed by an additional 13 hours of group discussions of videos of the participants’ class lessons. Teachers using this program were compared to 85 control teachers. After all this, students in the Intel classes scored slightly worse than controls on standardized measures (Garet et al., 2016).

If the Intel study were the only disappointment, one might look for flaws in their approach or their evaluation design or other things specific to that study. But as I noted earlier, all 12 of the studies of this kind failed to find positive effects, and the mean effect size was only +0.04 (n.s.).

Lest anyone jump to the conclusion that nothing works in elementary mathematics, I would point out that this is not the case. The most impactful category was tutoring programs, so that’s a special case. But the second most impactful category had many features in common with professional development focused on mathematics content and pedagogy, but had an average effect size of +0.25. This category consisted of programs focused on classroom management and motivation: Cooperative learning, classroom management strategies using group contingencies, and programs focusing on social emotional learning.

So there are successful strategies in elementary mathematics, and they all provided a lot of professional development. Yet programs for mathematics content and pedagogy, all of which also provided a lot of professional development, did not show positive effects in high-quality evaluations.

I have some ideas about what may be going on here, but I advance them cautiously, as I am not certain about them.

The theory of action behind professional development focused on mathematics content and pedagogy assumes that elementary teachers have gaps in their understanding of mathematics content and mathematics-specific pedagogy. But perhaps whatever gaps they have are not so important. Here is one example. Leading mathematics educators today take a very strong view that fractions should never be taught using pizza slices, but only using number lines. The idea is that pizza slices are limited to certain fractional concepts, while number lines are more inclusive of all uses of fractions. I can understand and, in concept, support this distinction. But how much difference does it make? Students who are learning fractions can probably be divided into three pizza slices. One slice represents students who understand fractions very well, however they are presented, and another slice consists of students who have no earthly idea about fractions. The third slice consists of students who could have learned fractions if it were taught with number lines but not pizzas. The relative sizes of these slices vary, but I’d guess the third slice is the smallest. Whatever it is, the number of students whose success depends on fractions vs. number lines is unlikely to be large enough to shift the whole group mean very much, and that is what is reported in evaluations of mathematics approaches. For example, if the “already got it” slice is one third of all students, and the “probably won’t get it” slice is also one third, the slice consisting of students who might get the concept one way but not the other is also one third. If the effect size for the middle slice were as high as an improbable +0.20, the average for all students would be less than +0.07, averaging across the whole pizza.


A related possibility relates to teachers’ knowledge. Assume that one slice of teachers already knows a lot of the content before the training. Another slice is not going to learn or use it. The third slice, those who did not know the content before but will use it effectively after training, is the only slice likely to show a benefit, but this benefit will be swamped by the zero effects for the teachers who already knew the content and those who will not learn or use it.

If teachers are standing at the front of the class explaining mathematical concepts, such as proportions, a certain proportion of students are learning the content very well and a certain proportion are bored, terrified, or just not getting it. It’s hard to imagine that the successful students are gaining much from a change of content or pedagogy, and only a small proportion of the unsuccessful students will all of a sudden understand what they did not understand before, just because it is explained better. But imagine that instead of only changing content, the teacher adopts cooperative learning. Now the students are having a lot of fun working with peers. Struggling students have an opportunity to ask for explanations and help in a less threatening environment, and they get a chance to see and ultimately absorb how their more capable teammates approach and solve difficult problems. The already high-achieving students may become even higher achieving, because as every teacher knows, explanation helps the explainer as much as the student receiving the explanation.

The point I am making is that the findings of our mathematics review may reinforce a general lesson we take away from all of our reviews: Subtle treatments produce subtle (i.e., small) impacts. Students quickly establish themselves as high or average or low achievers, after which time it is difficult to fundamentally change their motivations and approaches to learning. Making modest changes in content or pedagogy may not be enough to make much difference for most students. Instead, dramatically changing motivation, providing peer assistance, and making mathematics more fun and rewarding, seems more likely to make a significant change in learning than making subtle changes in content or pedagogy. That is certainly what we have found in systematic reviews of elementary mathematics and elementary and secondary reading.

Whatever the student outcomes are compared to controls, there may be good reason to improve mathematics content and pedagogy. But if we are trying to improve achievement for all students, the whole pizza, we need to use methods that make a more profound impact on all students. And that is true any way you slice it.


Garet, M. S., Heppen, J. B., Walters, K., Parkinson, J., Smith, T. M., Song, M., & Borman, G. D. (2016). Focusing on mathematical knowledge: The impact of content-intensive teacher professional development (NCEE 2016-4010). Washington, DC: U.S. Department of Education.

Pellegrini, M., Inns, A., Lake, C., & Slavin, R. E. (2018). Effective programs in elementary mathematics: A best-evidence synthesis. Paper presented at the Society for Research on Effective Education, Washington, DC.

This blog was developed with support from the Laura and John Arnold Foundation. The views expressed here do not necessarily reflect those of the Foundation.



What Works in Elementary Math?

Euclid, the ancient Greek mathematician, is considered the inventor of geometry. His king heard about it, and wanted to learn geometry, but being a king, he was kind of busy. He called in Euclid, and asked him if there was a faster way. “I’m sorry sire,” said Euclid, “but there is no royal road to geometry.”

Skipping forward a couple thousand years, Marta Pellegrini, of the University of Florence in Italy, spent nine months with our group at Johns Hopkins University and led a review of research on effective programs for elementary mathematics  (Pellegrini, Lake, Inns & Slavin, 2018), which was recently released on our Best Evidence Encyclopedia (BEE). What we found was not so different from Euclid’s conclusion, but broader: There’s no royal road to anything in mathematics. Improving mathematics achievement isn’t easy. But it is not impossible.

Our review focused on 78 very high-quality studies (65 used random assignment). 61 programs were divided into eight categories: tutoring, technology, professional development for math content and pedagogy, instructional process programs, whole-school reform, social-emotional approaches, textbooks, and benchmark assessments.

Tutoring had the largest and most reliably positive impacts on math learning. Tutoring included one-to-one and one-to-small group services, and some tutors were certified teachers and some were paraprofessionals (teacher assistants). The successful tutoring models were all well-structured, and tutors received high-quality materials and professional development. Across 13 studies involving face-to-face tutoring, average outcomes were very positive. Surprisingly, tutors who were certified teachers (ES=+0.34) and paraprofessionals (ES=+0.32) obtained very similar student outcomes. Even more surprising, one-to-small group tutoring (ES=+0.32) was as effective as one-to-one (ES=+0.26).

Beyond tutoring, the category with the largest average impacts was instructional programs, classroom organization and management approaches, such as cooperative learning and the Good Behavior Game. The mean effect size was +0.25.


After these two categories, there were only isolated studies with positive outcomes. 14 studies of technology approaches had an average effect size of only +0.07. 12 studies of professional development to improve teachers’ knowledge of math content and pedagogy found an average of only +0.04. One study of a social-emotional program called Positive Action found positive effects but seven other SEL studies did not, and the mean for this category was +0.03. One study of a whole-school reform model called the Center for Data-Driven Reform in Education (CDDRE), which helps schools do needs assessments, and then find, select, and implement proven programs, showed positive outcomes (ES=+0.24), but three other whole-school models found no positive effects. Among 16 studies of math curricula and software, only two, Math in Focus (ES=+0.25) and Math Expressions (ES=+0.11), found significant positive outcomes. On average, benchmark assessment approaches made no difference (ES=0.00).

Taken together, the findings of the 78 studies support a surprising conclusion. Few of the successful approaches had much to do with improving math pedagogy. Most were one-to-one or one-to-small group tutoring approaches that closely resemble tutoring models long used with great success in reading. A classroom management approach, PAX Good Behavior Game, and a social-emotional model, Positive Action, had no particular focus on math, yet both had positive effects on math (and reading). A whole-school reform approach, the Center for Data-Driven Reform in Education (CDDRE), helped schools do needs assessments and select proven programs appropriate to their needs, but CDDRE focused equally on reading and math, and had significantly positive outcomes in both subjects. In contrast, math curricula and professional development specifically designed for mathematics had only two positive examples among 28 programs.

The substantial difference in outcomes of tutoring and outcomes of technology applications is also interesting. The well-established positive impacts of one-to-one and one-to-small group tutoring, in reading as well as math, are often ascribed to the tutor’s ability to personalize instruction for each student. Computer-assisted instruction is also personalized, and has been expected, largely on this basis, to improve student achievement, especially in math (see Cheung & Slavin, 2013). Yet in math, and also reading, one-to-one and one-to-small group tutoring, by certified teachers and paraprofessionals, is far more effective than the average for technology approaches. The comparison of outcomes of personalized CAI and (personalized) tutoring make it unlikely that personalization is a key explanation for the effectiveness of tutoring. Tutors must contribute something powerful beyond personalization.

I have argued previously that what tutors contribute, in addition to personalization, is a human connection, encouragement, and praise. A tutored child wants to please his or her tutor, not by completing a set of computerized exercises, but by seeing a tutor’s eyes light up and voice respond when the tutee makes progress.

If this is the secret of the effect of tutoring (beyond personalization), perhaps a similar explanation extends to other approaches that happen to improve mathematics performance without using especially innovative approaches to mathematics content or pedagogy. Approaches such as PAX Good Behavior Game and Positive Action, targeted on behavior and social-emotional skills, respectively, focus on children’s motivations, emotions, and behaviors. In the secondary grades, a program called Building Assets, Reducing Risk (BARR) (Corsello & Sharma, 2015) has an equal focus on social-emotional development, not math, but it also has significant positive effects on math (as well as reading). A study in Chile of a program called Conecta Ideas found substantial positive effects in fourth grade math by having students practice together in preparation for bimonthly math “tournaments” in competition with other schools. Both content and pedagogy were the same in experimental and control classes, but the excitement engendered by the tournaments led to substantial impacts (ES=+0.30 on national tests).

We need breakthroughs in mathematics teaching. Perhaps we have been looking in the wrong places, expecting that improved content and pedagogy will be the key to better learning. They will surely be involved, but perhaps it will turn out that math does not live only in students’ heads, but must also live in their hearts.

There may be no royal road to mathematics, but perhaps there is an emotional road. Wouldn’t it be astonishing if math, the most cerebral of subjects, turns out more than anything else to depend as much on heart as brain?


Cheung, A., & Slavin, R. E. (2013). The effectiveness of educational technology applications for enhancing mathematics achievement in K-12 classrooms: A meta-analysis. Educational Research Review, 9, 88-113.

Corsello, M., & Sharma, A. (2015). The Building Assets-Reducing Risks Program: Replication and expansion of an effective strategy to turn around low-achieving schools: i3 development grant final report. Biddeford, ME, Consello Consulting.

Inns, A., Lake, C., Pellegrini, M., & Slavin, R. (2018, March 3). Effective programs for struggling readers: A best-evidence synthesis. Paper presented at the annual meeting of the Society for Research on Educational Effectiveness, Washington, DC.

Pellegrini, M., Inns, A., & Slavin, R. (2018, March 3). Effective programs in elementary mathematics: A best-evidence synthesis. Paper presented at the annual meeting of the Society for Research on Educational Effectiveness, Washington, DC.

Photo credit: By Los Angeles Times Photographic Archive, no photographer stated. [CC BY 4.0  (https://creativecommons.org/licenses/by/4.0)], via Wikimedia Commons

This blog was developed with support from the Laura and John Arnold Foundation. The views expressed here do not necessarily reflect those of the Foundation.

More Chinese Dragons: How the WWC Could Accelerate Its Pace


A few months ago, I wrote a blog entitled “The Mystery of the Chinese Dragon: Why Isn’t the WWC Up to Date?” It really had nothing to do with dragons, but compared the timeliness of the What Works Clearinghouse review of research on secondary reading programs and a Baye et al. (2017) review on the same topic. The graph depicting the difference looked a bit like a Chinese dragon with a long tail near the ground and huge jaws. The horizontal axis was the dates accepted studies had appeared, and the vertical axis was the number of studies. Here is the secondary reading graph.


What the graph showed is that the WWC and the U.S. studies from the Baye et al. (2017) review were similar in coverage of studies appearing from 1987 to 2009, but after that diverged sharply, because the WWC is very slow to add new studies, in comparison to reviews using similar methods.

In the time since the Chinese Dragon for secondary reading studies appeared on my blog, my colleagues and I have completed two more reviews, one on programs for struggling readers by Inns et al. (2018) and one on programs for elementary math by Pellegrini et al. (2018). We made new Chinese Dragon graphs for each, which appear below.*



*Note: In the reading graph, the line for “Inns et al.” added numbers of studies from the Inns et al. (2018) review of programs for struggling readers to additional studies of programs for all elementary students in an unfinished report.

The new dragons look remarkably like the first. Again, what matters is the similar pattern of accepted studies before 2009, (the “tail”), and the sharply diverging rates in more recent years (the “jaws”).

There are two phenomena that cause the dragons’ “jaws” to be so wide open. The upper jaw, especially in secondary reading and elementary math, indicate that many high-quality rigorous evaluations are appearing in recent years. Both the WWC inclusion standards and those of the Best Evidence Encyclopedia (BEE; www.bestevidence.org) require control groups, clustered analysis for clustered designs, samples that are well-matched at pretest and have similar attrition by posttest, and other features indicating methodological rigor, of the kind expected by the ESSA evidence standards, for example.

The upper jaw of each dragon is increasing so rapidly because rigorous research is increasing rapidly in the U.S. (it is also increasing rapidly in the U.K., but the WWC does not include non-U.S. studies, and non-U.S. studies are removed from the graph for comparability). This increase is due to U. S. Department of Education funding of many rigorous studies in each topic area, through its Institute for Education Sciences (IES) and Investing in Innovation (i3) programs, and special purpose funding such as Striving Readers and Preschool Curriculum Education Research. These recent studies are not only uniformly rigorous, they are also of great importance to educators, as they evaluate current programs being actively disseminated today. Many of the older programs whose evaluations appear on the dragons’ tails no longer exist, as a practical matter. If educators wanted to adopt them, the programs would have to be revised or reinvented. For example, Daisy Quest, still in the WWC, was evaluated on TRS-80 computers not manufactured since the 1980s. Yet exciting new programs with rigorous evaluations, highlighted in the BEE reviews, do not appear at all in the WWC.

I do not understand why the WWC is so slow to add new evaluations, but I suspect that the answer lies in the painstaking procedures any government has to follow to do . . ., well, anything. Perhaps there are very good reasons for this stately pace of progress. However, the result is clear. The graph below shows the publication dates of every study in every subject and grade level accepted by the WWC and entered on its database. This “half-dragon” graph shows that only 26 studies published or made available after 2013 appear on the entire WWC database. Of these, only two have appeared after 2015.


The slow pace of the WWC is of particular concern in light of the appearance of the ESSA evidence standards. More educators than ever before must be consulting the WWC, and many must be wondering why programs they know to exist are not listed there, or why recent studies do not appear.

Assuming that there are good reasons for the slow pace of the WWC, or that for whatever reason the pace cannot be greatly accelerated, what can be done to bring the WWC up to date? I have a suggestion.

Imagine that the WWC commissioned someone to do rapid updating of all topics reviewed on the WWC website. The reviews would follow WWC guidelines, but would appear very soon after studies were published or issued. It’s clear that this is possible, because we do it for Evidence for ESSA (www.evidenceforessa.org). Also, the WWC has a number of “quick reviews,” “single study reports,” and so on, scattered around on its site, but not integrated with its main “Find What Works” reviews of various programs. These could be readily integrated with “Find What Works.”

The recent studies identified in this accelerated process might be identified as “provisionally reviewed,” much as the U. S. Patent Office has “patent pending” before inventions are fully patented. Users would have an option to look only at program reports containing fully reviewed studies, or could decide to look at reviews containing both fully and provisionally reviewed studies. If a more time consuming full review of a study found results different from those of the provisional review, the study report and the program report in which it was contained would be revised, of course.

A process of this kind could bring the WWC up to date and keep it up to date, providing useful, actionable evidence in a timely fashion, while maintaining the current slower process, if there is a rationale for it.

The Chinese dragons we are finding in every subject we have examined indicate the rapid growth and improving quality of evidence on programs for schools and students. The U. S. Department of Education and our whole field should be proud of this, and should make it a beacon on a hill, not hide our light under a bushel. The WWC has the capacity and the responsibility to highlight current, high-quality studies as soon as they appear. When this happens, the Chinese dragons will retire to their caves, and all of us, government, researchers, educators, and students, will benefit.


Baye, A., Lake, C., Inns, A., & Slavin, R. (2017). Effective reading programs for secondary students. Manuscript submitted for publication. Also see Baye, A., Lake, C., Inns, A. & Slavin, R. E. (2017, August). Effective reading programs for secondary students. Baltimore, MD: Johns Hopkins University, Center for Research and Reform in Education.

Inns, A., Lake, C., Pellegrini, M., & Slavin, R. (2018). Effective programs for struggling readers: A best-evidence synthesis. Paper presented at the annual meeting of the Society for Research on Educational Effectiveness, Washington, DC.

Pellegrini, M., Inns, A., & Slavin, R. (2018). Effective programs in elementary mathematics: A best-evidence synthesis. Paper presented at the annual meeting of the Society for Research on Educational Effectiveness, Washington, DC.

Photo credit: J Bar [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/), GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons

This blog was developed with support from the Laura and John Arnold Foundation. The views expressed here do not necessarily reflect those of the Foundation.