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Academic Journals that are my references

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Academic Journals that are my references

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This study explored whether and how teachers' mathematical knowledge for teaching contributes to gains in students' mathematics achievement. The authors used a linear mixed-model methodology in which first and third graders' mathematical achievement gains over a year were nested within teachers, who in turn were nested within schools. They found that teachers' mathematical knowledge was significantly related to student achievement gains in both first and third grades after controlling for key student- and teacher-level covariates. This result, while consonant with findings from the educational production function literature, was obtained via a measure focusing on the specialized mathematical knowledge and skills used in teaching mathematics. This finding provides support for policy initiatives designed to improve students' mathematics achievement by improving teachers' mathematical knowledge.

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Effects of Teachers' Mathematical Knowledge for Teaching on Student Achievement - Heather C. Hill, Brian Rowan, Deborah Loewenberg Ball, 2005

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This study explored whether and how teachers' mathematical knowledge for teaching contributes to gains in students' mathematics achievement. The authors used a linear mixed-model methodology in which first and third graders' mathematical achievement gains over a year were nested within teachers, who in turn were nested within schools.

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Interactive whiteboards produce small gains in elementary students' self-reported motivation in mathematics

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A treatment/control study (N=773) was conducted to determine the extent to which use of interactive whiteboard technology (IWB) was associated with up...

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This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.

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Early Identification and Interventions for Students With Mathematics Difficulties - Russell Gersten, Nancy C. Jordan, Jonathan R. Flojo, 2005

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This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3.

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This article reviews research on the achievement outcomes of three types of approaches to improving elementary mathematics: mathematics curricula, computer-assisted instruction (CAI), and instructional process programs. Study inclusion requirements included use of a randomized or matched control group, a study duration of at least 12 weeks, and achievement measures not inherent to the experimental treatment. Eighty-seven studies met these criteria, of which 36 used random assignment to treatments. There was limited evidence supporting differential effects of various mathematics textbooks. Effects of CAI were moderate. The strongest positive effects were found for instructional process approaches such as forms of cooperative learning, classroom management and motivation programs, and supplemental tutoring programs. The review concludes that programs designed to change daily teaching practices appear to have more promise than those that deal primarily with curriculum or technology alone.

References

Carroll, WM, Isaacs, A Senk, SL, Thompson, DR Achievement of students using the University of Chicago School Mathematics Project's Everyday MathematicsStandards-based school mathematics curricula: What are they? What do students learn?2003 Mahwah, NJ Lawrence Erlbaum 79 108
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Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis - Robert E. Slavin, Cynthia Lake, 2008

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This article reviews research on the achievement outcomes of three types of approaches to improving elementary mathematics: mathematics curricula, computer-assisted instruction (CAI), and instructional process programs. Study inclusion requirements included use of a randomized or matched control group, a study duration of at least 12 weeks, and achievement measures not inherent to the experimental treatment.

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Abstract

People's fear and anxiety about doing math-over and above actual math ability-can be an impediment to their math achievement. We show that when the math-anxious individuals are female elementary school teachers, their math anxiety carries negative consequences for the math achievement of their female students. Early elementary school teachers in the United States are almost exclusively female (>90%), and we provide evidence that these female teachers' anxieties relate to girls' math achievement via girls' beliefs about who is good at math. First- and second-grade female teachers completed measures of math anxiety. The math achievement of the students in these teachers' classrooms was also assessed. There was no relation between a teacher's math anxiety and her students' math achievement at the beginning of the school year. By the school year's end, however, the more anxious teachers were about math, the more likely girls (but not boys) were to endorse the commonly held stereotype that "boys are good at math, and girls are good at reading" and the lower these girls' math achievement. Indeed, by the end of the school year, girls who endorsed this stereotype had significantly worse math achievement than girls who did not and than boys overall. In early elementary school, where the teachers are almost all female, teachers' math anxiety carries consequences for girls' math achievement by influencing girls' beliefs about who is good at math.

Results

As expected, at the beginning of the school year, there was no significant relation between teachers' math anxiety and students' math achievement (girls: r = −0.13, P = 0.31; boys: r = 0.12, P = 0.40). However, by the end of the school year, the higher a teacher's math anxiety, the lower was the girls' ( r = −0.28, P = 0.022) but not the boys' ( r = −0.04, P = 0.81) math achievement. ‡

Why might female teachers' math anxiety relate to girls' math achievement? Gender is an individuating feature that early elementary school-aged children notice ( 11, 13). Moreover, children at this age are aware of commonly held beliefs about gender and ability ( 12, 14) and are most likely to embrace behaviors and attitudes that they think are gender-appropriate ( 9). One possibility is that female teachers' math anxiety helps to confirm stereotypes about which gender is good at math and this, in turn, has an impact on girls' math achievement. If so, teachers' math anxiety should relate to the beliefs that girls hold about which gender is good at math.

As with math achievement, there was no significant relation between teachers' math anxiety and the gender ability beliefs of boys ( r = −0.04, P = 0.80) or girls ( r = 0.20, P = 0.10) § at the beginning of the school year, before teachers had spent a significant amount of time with their students. This was true for boys at the end of the school year as well ( r = 0.09, P = 0.52). In contrast, for girls, by the end of the school year, the higher a teacher's math anxiety, the more likely girls' ability beliefs were to fall along traditional gender lines ( r = 0.28, P = 0.022). ‖ In addition, the more girls at the end of the year endorsed the notion that boys are good at math and girls are good at reading, the lower was their math achievement ( r = −0.28, P = 0.025).

Female teachers' math anxiety negatively relates to girls' math achievement and also to girls' gender ability beliefs at the end of the school year. If this link between teacher anxiety and student math achievement occurs because teachers influence girls' gender ability beliefs and this, in turn, has an impact on girls' math performance, the relation between teacher and student should be mediated (or accounted for) by girls' gender ability beliefs.

As seen in Fig. 1, the relation between teacher math anxiety and girls' math achievement at the end of the school year became nonsignificant when girls' gender ability beliefs at the end of the year were also used to predict student math achievement. Only a strong negative relation between girls' gender ability beliefs and their math achievement remained. This relation did not hold for boys.

By the school year's end, female teachers' math anxiety negatively relates to girls' math achievement, and this relation is mediated by girls' gender ability beliefs. We speculate that having a highly math-anxious female teacher pushes girls to confirm the stereotype that they are not as good as boys at math, which, in turn, affects girls' math achievement. If so, it follows that girls who confirm traditional gender ability beliefs at the end of the school year (i.e., draw boys as good at math and girls as good at reading) should have lower math achievement than girls who do not and than boys more generally. This is exactly what we found. As seen in Fig. 2, students' math achievement at the end of the year depended on their gender and whether they confirmed common gender ability beliefs [a student gender × gender ability belief interaction; F(1,113) = 3.79, P = 0.05].

Girls who confirmed traditional gender ability beliefs had significantly lower end-of-year math achievement [mean (M) = 102.5, SE = 2.41] than girls who did not [M = 107.84, SE = 1.61; 95% confidence interval (CI): 104.66-111.03; Cohen's d = 0.66]. Moreover, girls who confirmed traditional gender ability beliefs had significantly lower math achievement than boys overall at the end of the year as well (M = 107.69, SE = 1.62; 95% CI: 104.49-110.90; d = 0.37). Boys' end-of-year math achievement did not differ as a function of gender ability beliefs (Don't Confirm: M = 106.14, SE = 1.80; Confirm: M = 109.25, SE = 2.69; 95% CI: 103.92-114.58).

Importantly, these differences were not seen at the beginning of the school year; at that point, teachers presumably had not had ample time to influence gender ability beliefs or relations between gender ability beliefs and math achievement. Indeed, in terms of the four groups displayed in Fig. 2, there were no significant differences in math achievement at the beginning of the school year [no gender × gender ability belief interaction; F(1,113) = 2.11, P = 0.15; girls: Don't Confirm: M = 101.44, SE = 1.86; Confirm: M = 99.0, SE = 2.74; boys: Don't Confirm: M = 98.83, SE = 2.04; Confirm: M = 103.56, SE = 3.06]. In concert with the mediation analysis above, these data suggest that girls' math achievement is, at least in part, related to their confirmation of traditional academic gender beliefs-beliefs that are affected by the math anxiety levels of their female teachers.

Discussion

Using a mediation analysis that depicts a model of a causal chain of events, we showed that female students' math achievement at the end of the school year is negatively affected by the way in which their teachers' math anxieties alter these girls' gender ability beliefs.

Similar to previous work ( 15), we did not find gender differences in math achievement at either the beginning [ t(115) = 0.18, P = 0.86] or end [ t(115) = 0.44, P = 0.66] of the school year. However, as Fig. 2 clearly shows, by the school year's end, girls who confirmed traditional gender ability roles performed worse than girls who did not and worse than boys more generally. We show that these differences are related to the anxiety these girls' teachers have about math.

If it is simply the case that highly math-anxious teachers are worse math teachers, one would expect to see a relation between teacher anxiety and the math achievement of both boys and girls. Instead, teachers with high math anxiety seem to be specifically affecting girls' math achievement-and doing so by influencing girls' gender-related beliefs about who is good at math.

This study explores the relation between female teachers' math anxieties and their students' math achievement. Thus, it is an open question as to whether there would be a relation between teacher math anxiety and student math achievement if we had focused on male instead of female teachers. In one sense, the lack of male elementary school teachers in the United States makes this a hard question to answer. Yet, it is an important question, given research suggesting that girls are more socially sensitive than boys in early elementary school ( 16). Thus, it is possible that even with male teachers, a relation between teacher anxiety and female student achievement might occur. Nevertheless, the literature on math anxiety, gender modeling, and the impact of negative stereotypes on achievement lead us to speculate that any relation between male teacher anxiety and girls' math achievement would be obtained through a different route than the one proposed here. Moreover, in the current work, the relation between female teachers' math anxieties and girls' math achievement was mediated (or accounted for) by girls' beliefs that boys are better at math. Hence, it seems unlikely that a male teacher's math anxiety would affect girls' math achievement by pushing girls to confirm that boys are good at math.

In addition, children do not blindly imitate adults of the same gender. Instead, they model behaviors they believe to be gender-typical and appropriate ( 9). Thus, it may be that first- and second-grade girls are more likely to be influenced by their teachers' anxieties than their male classmates, because most early elementary school teachers are female and the high levels of math anxiety in this teacher population confirm a societal stereotype about girls' math ability ( 2). This match between teacher math anxiety and societal norms would not hold for male teachers exhibiting math anxiety. However, if such a correspondence is important in influencing student achievement, we would expect that for school subjects for which girls are stereotyped to excel (e.g., language arts), male teachers' anxieties would have an impact on male more than female students' achievement.

It is important to note that the effects reported in the current work, although significant, are small. There are likely many influences on girls' math achievement and gender ability beliefs over and above their current teachers' anxieties. For instance, previous teachers, parents, peers, and siblings who either do or do not model traditional academic gender roles may play an important part in shaping girls' gender ability beliefs and their math achievement more generally. Exploring these relationships-in addition to the influence of both male and female teachers-will help to elucidate the full range of social influences on student achievement.

In conclusion, we show that female teachers' math anxiety has consequences for the math achievement of girls in early elementary school grades. Given that this relation is mediated by girls' gender ability beliefs, we speculate that female teachers model commonly held gender stereotypes to their female students through their math anxieties. These findings open a window into gender differences in math achievement and attitudes that emerge over the course of schooling.

Interestingly, math anxiety can be reduced through math training and education ( 17 - 19). This suggests that the minimal mathematics requirements for obtaining an elementary education degree at most US universities need to be rethought. If the next generation of teachers-especially elementary school teachers-is going to teach their students effectively, more care needs to be taken to develop both strong math skills and positive math attitudes in these educators.

Methods

Acknowledgments

This research was supported by a National Science Foundation Spatial Intelligence Learning Center grant (to S.L.B. and S.C.L.) and by National Science Foundation Grants BCS-0751336 and CAREER DRL-0746970 (to S.L.B.).

Footnotes

¹To whom correspondence should be addressed at: Department of Psychology, University of Chicago, 5848 South University Avenue, Chicago, IL 60607. E-mail: beilock{at}uchicago.edu .

Author contributions: S.L.B., E.A.G., G.R., and S.C.L. designed research; E.A.G. and G.R. performed research; S.L.B. and E.A.G. analyzed data; and S.L.B., E.A.G., G.R., and S.C.L. wrote the paper.

↵*This Direct Submission article had a prearranged editor.

↵† Indeed, a 2000 survey of kindergarten through second-grade teachers in the United States revealed that while almost all teachers (93%) had taken college courses in mathematics education, far fewer had completed more advanced math courses such as probability and statistics (33%), geometry for elementary/middle school teachers (19%), and calculus (13%) (1).

↵‡Controlling for teachers' math knowledge did not alter these results (see SI Methods for more detail).

↵§One might notice that this relation approached significance. Because it is unlikely that students are deliberately assigned to classrooms based on the teacher's math anxiety, this relation likely occurred by chance. Nonetheless, to ensure that it was not influencing our end-of-year findings, we ran an alternate version of our mediation model controlling for girls' beginning-of-year gender ability beliefs. Girls' end-of-year beliefs significantly continued to mediate (or account for) the relation between teachers' math anxiety and girls' end-of-year math achievement (95% CI: −2.84 to −0.061; P < 0.05). In contrast, girls' beginning-of-year gender ability beliefs were not a significant predictor (β = −0.02, t = −0.17, P = 0.862), suggesting that this factor was not driving our end-of-year mediation effects.

↵‖When controlling for beginning-of-year gender ability beliefs, the correlation between teachers' math anxiety and girls' end-of-year gender ability beliefs remained significant ( r = 0.31, P = 0.012).

↵‖ When teachers' math anxiety was in the equation, their math knowledge was controlled, and when girls' end-of-year math achievement was in the equation, their beginning-of-year math achievement was controlled.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0910967107/DCSupplemental.

Freely available online through the PNAS open access option.

View Abstract

# data[].attributes.attachment_link.title
Female teachers' math anxiety affects girls' math achievement

# data[].attributes.attachment_link.description
People's fear and anxiety about doing math-over and above actual math ability-can be an impediment to their math achievement. We show that when the math-anxious individuals are female elementary school teachers, their math anxiety carries negative consequences for the math achievement of their female students.

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Abstract

Students and teachers affiliated with Spring 2009 online mathematics courses offered by the North Carolina Virtual Public School (NCVPS) completed end-of-course surveys detailing the effectiveness of their courses. Several significant differences were noted when comparing results from math participants to results from participants in five other subject areas, suggesting math students were learning less online, were less likely to recommend online learning to peers, were more satisfied with content breadth and depth but could benefit from more detailed and authentic projects, were less likely to associate their courses with certain 21st century skills and tools, and were in need of increased communication from the teacher and increased collaboration with peers. A follow-up survey with math participants as well as related literature suggest these significant differences can most likely be attributed to needed improvements in online content and teaching, as well as characteristics of math students that are independent of but potentially improved by quality online instruction.

Citation

Oliver, K., Kellogg, S. & Patel, R. (2010). An Investigation into Reported Differences Between Online Math Instruction and Other Subject Areas in a Virtual School. Retrieved June 22, 2019 from Journal of Computers in Mathematics and Science Teaching, 29(4), 417-453. Waynesville, NC USA: Association for the Advancement of Computing in Education (AACE). https://www.learntechlib.org/primary/p/33079/.

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Cited By

View References & Citations Map

Sandra Fital-Akelbek & Mahmud Akelbek, Weber State University, United States

Journal of Computers in Mathematics and Science Teaching Vol. 37, No. 2 pp. 93-109

Sandra Fital-Akelbek, Weber State University, United States

EdMedia + Innovate Learning 2016 pp. 1177-1182

These links are based on references which have been extracted automatically and may have some errors. If you see a mistake, please contact info@learntechlib.org.

# data[].attributes.attachment_link.title
An Investigation into Reported Differences Between Online Math Instruction and Other Subject Areas in a Virtual School

# data[].attributes.attachment_link.description
Students and teachers affiliated with Spring 2009 online mathematics courses offered by the North Carolina Virtual Public School (NCVPS) completed end-of-course surveys detailing the effectiveness of their courses.

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Abstract

Computer Education and Instructional Technology (CEIT) Departments train computer teachers to fill gap of computer instructor in all grades of schools in Turkey. Additionally graduates can also work as instructional technologist or software developer. The curriculum of CEIT departments includes mathematics courses. The aim of this study is to identify attitudes of undergraduate students at CEIT departments towards math. In order to investigate the research question quantitative methods was used. Specifically survey research was preferred. Mathematics Attitude Questionnaire (MAQ) that was developed by Duatepe and Cilesiz (1999) was used and the questionnaire includes 38 items. The instrument was conducted with 122 undergraduate students from CEIT departments of four different universities in Turkey in the spring semester of 2010-2011. Data were analyzed through independent samples t-test and one-way ANOVA by means of PASW Statistics-18. In conclusion, some differences were found in terms of math attitudes of CEIT's students. (Contains 9 tables.)

Citation

Tekerek, M., Yeniterzi, B. & Ercan, O. (2011). Math Attitudes of Computer Education and Instructional Technology Students. Retrieved June 23, 2019 from Turkish Online Journal of Educational Technology, 10(3), 168-174. https://www.learntechlib.org/p/53303/.

Keywords

# data[].attributes.attachment_link.title
Math Attitudes of Computer Education and Instructional Technology Students

# data[].attributes.attachment_link.description
Computer Education and Instructional Technology (CEIT) Departments train computer teachers to fill gap of computer instructor in all grades of schools in Turkey. Additionally graduates can also work as instructional technologist or software developer. The curriculum of CEIT departments includes mathematics courses.

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More than 40 states have now signed onto the Common Core standards in English language arts and math, which have been both celebrated as a tremendous advance and criticized as misguided and for bearing the heavy thumbprint of the federal government. Assessing the merits of the Common Core math standards are Ze'ev Wurman and W. Stephen Wilson. Wurman, who was a U.S. Department of Education official under George W. Bush, is coauthor with Sandra Stotsky of "Common Core's Standards Still Don't Make the Grade" (Pioneer Institute, 2010). Wilson is a professor of mathematics at Johns Hopkins University, served on the National Governors Association-Council of Chief State School Officers "feedback group" for the Common Core standards, and was mathematics author of Stars by which to Navigate? Scanning National and International Education Standards in 2009: An Interim Report on Common Core, NAEP, TIMSS, and PISA.
Education Next: Are the Common Core math standards "fewer, higher, and clearer" than most state standards today? Can you provide some specific examples where you think the Common Core marks a step forward or backward?
Ze'ev Wurman: Common Core standards may in fact be clearer and more demanding than many, though not all, of the state standards they replaced. The Fordham Institute reviewed them last year and found them so. While I have no reason to doubt the technical quality of that review, there is good cause to note what it does not say.

It does not say that Common Core standards are fewer. Indeed, if one compares them to the better state mathematics standards like those of Minnesota or California, they are more numerous. Minnesota's standards fill 42 pages and California's 59 pages, while the Common Core takes 73 pages even without the advanced statistics or calculus sections that are included in California's standards. Counting the standards rather than pages, in grades 1 to 4 California has, on average, a few more standards than Common Core, but in grades 5‒8 the Common Core standards are more numerous than California's.

Fordham's review does not unequivocally say the standards are higher, either. They may be higher than some state standards but they are certainly lower than the best of them. For example, the 2008 report of the National Mathematics Advisory Panel, Foundations for Success, called for fluency in addition and subtraction of whole numbers by the end of grade 3, and fluency in multiplication and division by the end of grade 5. This is also what California calls for, along with high achievers like Singapore and Korea. (Japan and Hong Kong finish with multiplication and division of whole numbers even earlier, by grade 4.) Yet the Common Core defers fluency in division to grade 6. Fractions are touted as the Common Core's greatest strength, yet the Common Core pushes teaching division of fractions to grade 6 without ever expecting students to master working with a mix of fractions and decimals. Students in Singapore, Japan, Korea, and Hong Kong achieve fluency in fractions and decimals in grade 5.

Nor are the Common Core standards necessarily clearer. They may be clearer than many state mathematics standards, but they still tend to be wordy and hard to read. Table 1 compares a few grade 4 California standards with their Common Core counterparts.

Andrew Porter, dean of the University of Pennsylvania's Graduate School of Education, recently evaluated the Common Core standards with his colleagues, and their conclusion was stark:

Those who hope that the Common Core standards represent greater focus for U.S. education will be disappointed by our answers. Only one of our criteria for measuring focus found that the Common Core standards are more focused than current state standards...Some state standards are much more focused and some much less focused than is the Common Core, and this is true for both subjects.

We also used international benchmarking to judge the quality of the Common Core standards, and the results are surprising both for mathematics and for [ELA].... High-performing countries' emphasis on "perform procedures" runs counter to the widespread call in the United States for a greater emphasis on higher-order cognitive demand.

Another recent analysis, by University of Southern California professor Morgan Polikoff, found the Common Core mathematics standards similarly repetitive, and hence as unfocused across elementary grades as the state content standards they attempt to replace, with only somewhat less redundancy in the middle grades.

In summary, analyses of the Common Core standards find them to be mediocre and not obviously better than many sets of state standards.

W. Stephen Wilson: It turns out that nearly everyone was in favor of Common Core standards in mathematics if, and this is a big if, they got to write them. As it turns out, no one got to write the standards. A committee wrote them. Worse, the committee was hired by the very states whose standards would be replaced, so states got first crack at suggesting "corrections" to the standards. The pressures on the writing committee must have been enormous. The only reasonable expectation was that the result would resemble some sort of middle way between the states' various standards. What is surprising is that the standards don't rank in terms of quality in the middle 20 percent of state standards, but, instead, fall in the top 20 percent.

There is much to criticize about them, and there are several sets of standards, including those in California, the District of Columbia, Florida, Indiana, and Washington, that are clearly better. Yet Common Core is vastly superior-not just a little bit better, but vastly superior-to the standards in more than 30 states.

Where this gap is most obvious, and most important, is in laying the foundation for college readiness in mathematics early, by grade 6 or 7. Judging by state standards, few people see a connection between elementary school mathematics and college math, let alone really understand how the foundation is built.

Arithmetic is the foundation. Arithmetic has to be a priority, and it has to be done right. A number of things can and do go wrong with state standards for arithmetic in elementary school.

With the introduction of calculators, many states have downplayed the importance of arithmetic, apparently not realizing its true educational value. Instead, they spend time on statistics and probability, both of which Common Core has tossed out of early elementary school. Another thing that states love is geometric slides, turns, and flips, sometimes presented every year in grades K‒11, perhaps under the mistaken belief that they are really doing mathematics.

Fewer than 15 states are explicit about the need for students to know the single-digit number facts (think multiplication tables) to the point of instant recall. States love to have kids figure out many ways to add, subtract, multiply, and divide, but often leave off the capstone standard of fluency with the standard algorithms (traditional step-by-step procedures for the addition, subtraction, multiplication, and division of whole numbers). For example, only seven states expect students to know explicitly the standard algorithm for whole number multiplication. Fractions are even harder to find done well. Standards for fractions are generally so vague that nearly everything is left to the reader. Often states expect students to develop their own strategies or a variety of strategies for dealing with fractions. For example, only 15 states mention common denominators. Common Core does a pretty good job with arithmetic, even a very good job with fractions.
EN: Will the Common Core put an end to what has sometimes been termed the "math wars"? In your view, do the math standards resemble those recommended by the National Council of Teachers of Mathematics (NCTM), and what do you make of that similarity (or lack thereof)?
WSW: The end of the math wars! You must be joking.

There will always be people who think that calculators work just fine and there is no need to teach much arithmetic, thus making career decisions for 4th graders that the students should make for themselves in college. Downplaying the development of pencil and paper number sense might work for future shoppers, but doesn't work for students headed for Science, Technology, Engineering, and Mathematics (STEM) fields.

There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is bad for a student, perhaps believing that it means students can no longer understand them. Of course this permanently slows students down, plus it requires students to think about 3rd-grade mathematics when they are trying to solve a college-level problem.

There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach "high-level critical thinking" than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.

There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics.

There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.

There will always be people who think that statistics and probability are more important than arithmetic and algebra, despite the fact that you can't do statistics and probability without arithmetic and algebra and that you will never see a question about statistics or probability on a college placement exam, thus making statistics and probability irrelevant for college preparation.

There will always be people who think that teaching kids to "think like a mathematician," whether they have met a mathematician or not, can be done independently of content. At present, it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core, which they sometimes think is the "real" mathematics, are more important than the 75 pages of content standards, which they sometimes refer to as the "rote" mathematics. They are wrong. You learn Mathematical Practices just like the name implies; you practice mathematics with content.

There will always be people who think that teaching kids about geometric slides, flips, and turns is just as important as teaching them arithmetic. It isn't. Ask any college math teacher.

The end of the math wars! You must be joking.

ZW: Math wars erupted as a result of the unfocused and mostly math-less 1989 NCTM standards. NCTM rewrote those terrible standards in 2000, yet much of what mathematicians found objectionable remained in place. Only in 2005, with the publication in Notices of the AMS [American Mathematical Society] of "Reaching for Common Ground in K-12 Mathematics Education," did the two sides make a serious attempt to bridge the chasm. NCTM followed shortly with its 2006 Curriculum Focal Points, a document that finally focused on what mathematics is all about: mathematics. Since then, NCTM seems to have regressed, as evidenced by its 2009 publication Focus in High School Mathematics, a document that is full of high-minded prose yet contains little rigor or specificity.

The Common Core mathematics standards are grade-by-grade‒specific and hence are more detailed than the NCTM 2000 standards, but they do resemble them in setting their sights lower than our international competitors, by, for example, locking algebra into the high school curriculum.

And they contain inexplicable holes even when compared to the much shorter NCTM Curriculum Focal Points, the major one being the absence of fraction conversion among their multiple representations (simple, decimal, percent). Other puzzling omissions include geometry basics such as derivation of area of general triangles or the concept of pi. One can argue those can be inferred, but the same can be said regarding all those state standards we acknowledge as "bad"-that all those missing pieces "can be inferred."

What to make of such obvious deficiencies and omissions? Unfortunately, the main authors of the Common Core mathematics standards had minimal prior experience with writing standards, and it shows. While they may have had a long and distinguished list of advisers, they did not seem to have sufficient experience to select the wheat from the chaff. How, otherwise, can one explain their selecting an experimental approach to geometry, teaching it on the basis of rigid motions, that has not been successfully tried anywhere in the world? Simple prudence and an ounce of experience would tell them either to stick to what is known to work or to recommend a trial phase before foisting it sight-unseen on a nation of 300 million.
EN: How do the Common Core math standards compare to those in use in the world's highest-performing nations? Crucially, on what do you base that assessment?
ZW: It is not difficult to show that the Common Core standards are not on par with those of the highest-performing nations.

Here is what Professor R. James Milgram of Stanford, the only professional mathematician on the Common Core Validation Committee, wrote when he declined to sign off on the Common Core standards:

This is where the problem with these standards is most marked. While the difference between these standards and those of the top states at the end of eighth grade is perhaps somewhat more than one year, the difference is more like two years when compared to the expectations of the high achieving countries-particularly most of the nations of East Asia.

And here is what a non-American member of the Validation Committee wrote to the Council of Chief State School Officers when declining to validate the standards:

I cannot in all conscience, endorse statements 2 and 3 [(2) Appropriate in terms of their level of clarity and specificity; (3) Comparable to the expectations of other leading nations] The standards are, in my view, much more detailed, and, as Jim Milgram has pointed out, are in important respects less demanding, than the standards of the leading nations.

We also have it straight from the horse's mouth, so to speak. Professor William McCallum, one of the three main writers of the Common Core mathematics standards, speaking at the annual conference of mathematics societies in 2010, said,

While acknowledging the concerns about front-loading demands in early grades, [McCallum] said that the overall standards would not be too high, certainly not in comparison [with] other nations, including East Asia, where math education excels.

Jonathan Goodman, a professor of mathematics at the Courant Institute at New York University, found exactly that: "The proposed Common Core standard is similar in earlier grades but has significantly lower expectations with respect to algebra and geometry than the published standards of other countries."

It is also worth mentioning that the standards, in addition to being "[c]omparable to the expectations of other leading nations," were also supposed to be "[r]eflective of the core knowledge and skills in ELA and mathematics that students need to be college- and career-ready." That is, at least, what the other Common Core Validation Committee members certified when they signed off on the standards in 2010.

College readiness is defined by what colleges require as prerequisites from their incoming freshmen. The enrollment requirements of four-year state colleges overwhelmingly consist of at least three years of high school mathematics including algebra 1, algebra 2, and geometry, or beyond. Yet Common Core's "college readiness" definition omits content typically considered part of algebra 2 (and geometry), such as complex numbers, vectors, trigonometry, polynomial identities, the Binomial Theorem, logarithms, logarithmic and exponential functions, composite and inverse functions, matrices, ellipses and hyperbolae, and a few more.

What should we make, then, of a recent study purporting to "validate" that Common Core standards indeed reflect college readiness? The study, led by David Conley, was published more than a year after Common Core standards were already certified as college-ready by...David Conley as a member of the Common Core Validation Committee. Paraphrasing Shakespeare, he doth attest too much.

In summary, the Common Core mathematics standards fail on clarity and rigor compared to better state standards and to those of high-achieving countries. They do not expect algebra to be taught in grade 8 and instead defer it to high school, reversing the most significant change in mathematics education in America in the last decade, supported by the 2008 recommendations of the National Mathematics Advisory Panel, and contrary to the practice of our international competitors. Moreover, their promise of college readiness rings hollow. Its college-readiness standards are below the admission requirement of most four-year state colleges.

WSW: When you are so far behind, comparing the United States with better-performing countries through the incredibly narrow lens of standards doesn't make a lot of sense. I think Common Core is in the same ball park, certainly not up there with the best of countries, but Common Core isn't up there with the best state standards either, and what does that mean? Look at California's standards for example. They are great standards and have been unchanged for over a decade, but many in math education hate them. They think they are all about rote mathematics, but I think such people have little understanding of mathematics.

So, let's just pretend for a moment that Common Core is just as good as the very best. Who, in education circles, will agree with that enough to put it all in practice? The standard algorithm deniers will teach multiple ways to multiply numbers and mention the standard algorithm one day in passing. Korea will say "no calculators" in K-12, a little extreme perhaps, but some in the U.S. will say "appropriate tools" means calculators in 4th grade. We, in this country, are still not on the same page about what content is most important, even if everyone says they'll take Common Core. Without a unified, concerted effort to teach real mathematics, there isn't much chance of catching up.

In other countries, if you say "learn to multiply whole numbers," no one questions how this should be done; students should learn and understand the standard algorithm. In the U.S., even if you say "learn to multiply whole numbers with the standard algorithm," some people will declare wiggle room and try to avoid the standard algorithm.

There is one big hope for our international competitiveness. Other countries see that their best STEM students come to the U.S. for graduate school-more than half of our STEM graduate students are foreign-and to start high-tech companies. Instead of thinking that this is possible because of their strong K-12 mathematics education, they erroneously conclude that they should adopt our version of K-12 mathematics education. We just might catch up with these countries without any effort on our part.
EN: What, then, are your main areas of disagreement?
WSW: Ze'ev refers to Andrew Porter's work to support his argument that Common Core lacks focus. In the corrected version of Porter's paper, he says that 39.55 percent of grades 3‒6 coarse-grained topics for the states are on Number Sense and Operations, but Common Core gets 55.47 percent. To me, that says that Common Core focuses on arithmetic in grades where arithmetic should be the focus, and that the states did not focus on arithmetic.

My only serious disagreement with Ze'ev is his summary that "analyses of Common Core standards find them to be mediocre and not obviously better than many sets of state standards." If Common Core is mediocre, then mediocre is being set at a high standard. There are many states that set a very different, and much lower, standard for mediocre.

ZW: Steve sees the benefit of having Common Core standards that are better than those of "more than 30 states," while I see the disadvantage of confining the whole nation to mediocre standards that are worse than those of highly rated states and high-achieving countries.

Taking this a step further, I believe the Common Core marks the cessation of educational standards improvement in the United States. No state has any reason left to aspire for first-rate standards, as all states will be judged by the same mediocre national benchmark enforced by the federal government. Moreover, there are organizations that have reasons to work for lower and less-demanding standards, specifically teachers unions and professional teacher organizations. While they may not admit it, they have a vested interest in lowering the accountability bar for their members. With Common Core, they have a single target to aim for, rather than 50 distributed ones. So give it some time and, as sunset follows sunrise, we will see even those mediocre standards being made less demanding. This will be done in the name of "critical thinking" and "21st-century" skills, and in faraway Washington D.C., well beyond the reach of parents and most states and employers.

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The Common Core Math Standards - Education Next

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More than 40 states have now signed onto the Common Core standards in English language arts and math, which have been both celebrated as a tremendous advance and criticized as misguided and for bearing the heavy thumbprint of the federal government. Assessing the merits of the Common Core math standards are Ze'ev Wurman and W.

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I am not a mathematics teacher, but I have a degree in mathematics and an intense interest in how the subject is taught. When I retire, I would like to teach math, which is why I started tutoring high school students in my spare time three years ago. My first student was a 9th grader having difficulty with geometry. He stated his problem succinctly: "I don't know how to do proofs." Confronted with what I thought could be a common problem, I was still unaware that what I was really seeing was a national crisis in mathematics education.

Here's some of what I would soon learn:

-Only 55 percent of 8th graders taking the National Assessment of Educational Progress (NAEP) exam in math correctly answered the question, "How many pieces of string will you have if you divide 3/4 yard of string into pieces each 1/8 yard long?"

-In an international math test taken by students worldwide in 1995 (the Third International Mathematics and Science Study, or TIMSS), U.S. student math proficiency for 8th graders fell below the international average (28th out of 41 countries). For 12th grade, U.S. math performance was among the lowest (18th of the 21 countries participating).

-In 1989 the National Council of Teachers of Mathematics (NCTM) published its Curriculum and Evaluation Standards for School Mathematics -an extensive set of mathematics standards for grades K-12 which de-emphasized memorization of number facts, the learning of proofs, and algebraic skills, but encouraged the use of calculators and "discovery learning."

-The National Science Foundation (NSF) promoted the NCTM standards beginning in 1991 and awarded millions of dollars in grant money for the writing of math texts that embraced them and to state boards of education whose math standards aligned with them (see Figure 1).

Not knowing about these significant events (all of them before last December's release of the Program for International Student Assessment report ranking American 15-year-olds 24th out of 29 in math among industrial countries (see Figure 2), I had no idea that our children were being deprived of a math education, thanks in no small part to a dubious education theory, watered-down standards, and a well-meaning but intellectually bankrupt federally subsidized program of math illiteracy.

What I did know was that my 9th grader didn't know how to do proofs.
I looked through his textbook, one of whose authors was a recent president of NCTM, and I was surprised to find very few proofs of anything. More troubling, most theorems in the book were stated as postulates-that is, propositions stated without proof-and students were told to memorize them. The problems at the end of the chapter required students to do only a few simple proofs.

Proofs in geometry class have been a mainstay of mathematics. In fact, proofs were always considered an essential part of high school geometry, not only because of their importance in higher math, but because learning the rules of logical argument and reasoning has applications in science, law, political science, and writing. To see proofs being shortchanged in a geometry textbook was shocking.

Algebra texts were in no better condition, in terms of presentation and content-or, rather, their lack of content. Even if you accept the argument that geometry in general, and proofs in particular, are unnecessary for students to learn, at least algebra should be taught properly, since algebra is the common language of, and gateway to, all of higher math. The absence of clear explanation and logical development left students I later tutored in algebra as lost as my geometry student. Their textbooks (and, probably, their teachers too) encouraged them to use a graphing calculator. Operations with algebraic fractions, like a/b + c/d, were given little attention, to say nothing of quadratic equations, once the pinnacle of any first-year algebra course. Instead, the quadratic formula is presented for the students to memorize and apply-if it is even mentioned at all.

At the time I started tutoring, my daughter was in 2nd grade. I was concerned that she was not learning her addition and subtraction facts. Other parents we knew were saying the same thing. Teachers told them not to worry because kids eventually "get it."

One teacher told me her understanding of the new method. "It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren't dependent on kids' mastering a previous lesson."

In a world where it doesn't matter when you learn something, because you'll get it eventually, there seem to be few if any critical junctures, no mastery of procedure, no building on what you've learned-no learning.
The Hell-Hath-No-Fury Postulate
Coincidentally, I had the opportunity to find out how my experiences related to the politics of math education. I work for the federal government, which has a program that gives employees a chance to work on Capitol Hill to gain experience and knowledge of legislative and congressional procedures, which is valuable information when working in government. I applied for and received a six-month detail to work in a Democratic senator's office. Senator X (so called, in keeping with mathematical convention to describe a class of variables, because, as I was also to learn, both the good intentions and the shortcomings of Congress are institutional) was interested in establishing a science project to nurture a "homegrown" breed of scientists and engineers who would then support that state's burgeoning technology industry. Since I thought a likely place to start would be math education, the staffers working the education issue asked me to see what I could come up with.

I compiled a list of questions that I sent by e-mail to various mathematicians involved with the math education issue. The questions focused on the quality of textbooks and teaching, with emphasis on algebra and geometry. I also wanted to know whether K-6 texts taught arithmetic well enough to prepare students to learn algebra.

The nice thing about working on the Hill is that you almost always get responses to e-mails and phone calls. Fifteen minutes after I sent an e-mail to Harvard mathematics professor Wilfried Schmid, he called. I found out that his initiation into the world of K-12 math education was similar to mine-through his daughter. He explained how she was not being taught her multiplication tables. He was shocked at the math instruction she was receiving in the 3rd grade. Its substance was shallow, memorization was discouraged, students were kept dependent on mental crutches (her teacher made her work with blocks or count on her fingers), and the intellectual level was well below the capability of most of the kids in his daughter's class.

Schmid's reaction to the problems of math texts and teaching was similar to that of other mathematicians I talked to in the course of my Capitol Hill assignment, particularly those with children. Those dialogues led me to develop an ad hoc theory that I will postulate (this means I don't have to prove it): Hell hath no fury like a mathematician whose child has been scorned by an education system that refuses to know better. In Schmid's case, he talked to parents, school boards, and ultimately with the Massachusetts commissioner of education. Along with others, he succeeded in revamping Massachusetts's math standards, much to the dislike of the education establishment and textbook publishers.
Framing the Debate
The controversy over K-12 math education has come to be known as the "math wars." Like Schmid, mathematicians have been active in this debate, as has the "mathematics community" at large, including not only mathematicians at the university level, but teachers and others involved in the education establishment. They believe that students must master basic skills (the number facts, standard algorithms for adding, subtracting, multiplying, and dividing) in tandem with larger concepts about mathematics.

On the other side of the debate are the followers of an education theory that promotes discovery learning, minimization of both teacher instruction and repetitive drills, and a disdain for standard procedures (algorithms). The math being protested-by the mathematics community-is called a variety of things: "reform math," "standards-based math," "new new math," and, most commonly, "fuzzy math."

Although the education theories on which much of fuzzy math is based are promoted in many education schools in the United States, it is not accurate to say that everyone in the education arena buys into these ideas. Therefore, for purposes of clarity, and to be consistent with a vocabulary used by others describing the math wars, I will use the term "educationist" to refer to those who promote the contested theory of math education known generally as discovery learning.
Early Skirmishes
The math wars revolve around a four-part problem: A disputed theory of education that informs NCTM's standards; state boards of education that base their standards of learning for mathematics on the NCTM standards; textbooks written to incorporate these standards; and teachers and others in the education establishment who are indoctrinated in the disputed education theory and who may not possess enough knowledge of mathematics to overcome the first three factors.

The education theory at the heart of the dispute can be traced to John Dewey, an early proponent of learning through discovery. But for all practical purposes, the story begins on October 4, 1957, when the Soviet Union launched Sputnik. This event signaled the American malaise in science and mathematics education and the need to overcome it if we weren't to fall further behind the Soviet Union and lose the cold war. Thus the U.S. Congress was motivated to pass appropriations that triggered the development of the "new math," a national effort, spurred by the National Science Foundation, that eventually found its way into most of our schools.

The curriculum, designed primarily by mathematicians, had problems, but it introduced to algebra, geometry, and trigonometry a long-missing formalism, logic, and consistency, and it resulted in calculus being taught in high school. The problem, however, was that a similar formalism was introduced into K-6 texts and curricula, with the result that students (and elementary school teachers, who were caught totally off guard) were exposed to number bases, set theory, and axioms long before they were ready for them. And soon enough the new math was being blamed for not teaching basic arithmetic: it was often said that new-math students could tell you that 5 + 3 = 3 + 5, but didn't know that it was equal to 8.

Mathematicians have agreed for years that emphasizing sets and number bases in math programs designed for the lower grades was a horrendous mistake. Notwithstanding these errors, however, the difference between the current slew of textbooks and those from the new-math days of the 1960s is definitely worth noting: Accomplished mathematicians wrote many of the texts used in that earlier era , and the math-though misguided and inappropriate for the lower grades and too formal for the high school grades-was at least mathematically correct. Some of the high school texts were absolutely first-rate, and new-math-era textbooks like Mary Dolciani's "Structure and Method" series for algebra and geometry continue to be used by math teachers who understand mathematics and how it is to be taught. (They usually use them on the sly, since most teachers are required to use the books that the schools have adopted.)

During the new-math era, which spanned the period between Sputnik to the early 1970s, mathematicians dominated the design of math texts and curricula for the first, and almost last, time. Up to that point mathematicians had been kept out of the math education picture, and K-12 mathematics tended not to include any examination of the logical structure of mathematics itself, with the single exception of Euclidean geometry. Students in the first half of the 20th century had instruction in practical matters: consumer buying, insurance, taxation-everything but algebra, geometry, or trigonometry, which, when they were taught, were frequently lacking in depth.

Significantly, the new-math era was one of the only times that mathematicians were given an opportunity to make proper math education available to the masses. (Not until the past few years, working with several state education departments, would they be allowed back into math education decisions.) And some believe that had certain prominent mathematicians who had started working with the development of the new-math programs managed to maintain their influence on those programs, the math education that would have emerged from new math-both lower grades and high school-would have been on par with the best of the math programs overseas.

Eventually, however, the problems with K-6 formalism and the logic and formalism of the program in general doomed new math. The general public, the education community, and even mathematicians themselves judged the new-math programs a failure. Mathematicians were assigned the blame, and the education establishment took back the reins. That establishment received an inadvertent boost in 1983 with the publication of A Nation at Risk, the shockingly pessimistic assessment of the nation's schools by the National Commission on Excellence in Education. The report sounded another alarm about student math performance, and the NCTM, increasingly dominated by educationists, took advantage of this new education crisis to write revised math standards. The Curriculum and Evaluation Standards for School Mathematics, published in 1989, purported to put the country back on the math track. But because it was, in part, a reaction to the new math and those believed responsible for it, NCTM did not, as mathematicians point out, promote a lively public debate, as had the creators of the new math, but suppressed it.
Some Secrets about Discovery Learning
The NCTM standards were a brew of progressivism-a nod to the 1920s when math was supposed to be practical-and constructivism, which was progressivism that adapted research from cognitive psychology to the task of teaching and called it discovery learning. The standards were based on theories of learning that assumed that children had an innate ability to understand math. The group's math curricula were thus structured to allow children to discover math concepts rather than to be given them, through direct instruction. The standards also expanded their reach to include, in addition to basic arithmetic, algebra, geometry, and trigonometry.

The NCTM's view was that traditional teaching techniques, known as "drill and kill," numbed student minds, turned them off math, and taught them nothing. And so the new standards recommended that students learn "strategies" for learning number facts rather than memorize those facts. It emphasized the use of calculators in all grades. Most important, however, the standards recommended certain areas that should receive "decreased attention" in grades K-4, including "complex paper-and-pencil computations," "long division," "paper-and-pencil fraction computation," "use of rounding to estimate," "rote practice," "rote memorization of rules," and "teaching by telling." This last item, teaching by telling, is a reference to direct instruction (telling students what they need to know), which NCTM believed should be replaced by "discovery."

Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students' not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering-and taking ownership of-the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.

Texts that are based on NCTM's standards focus on concepts and problem solving, but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches "mathematical thinking" rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo without first teaching him to swim.

The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics ( PSSM 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I've seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student's going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method-one that is held in disdain because it uses rote memorization and is not discovered by the student-the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.
Enter Stage Left: The National Science Foundation
The NCTM standards received a boost of credibility in 1991 when the Education and Human Resources Division of the National Science Foundation funded two grant programs related to math education. The first was for state education departments that aligned their math standards with NCTM's and school districts that adopted constructivist math programs aligned with NCTM's standards; the second, for the development of commercial mathematics texts that also followed the NCTM party line, a spending program that now took NSF into the textbook business.

Eventually, NSF supplemented grants to school districts by funding "distribution centers" that promoted the very programs NSF had helped to create. This "Invasion of the Body Snatchers" technique has worked well in enhancing the NCTM constructivist ethic and the texts that NSF helped develop. Meanwhile, NSF continues to fund revisions to some of those texts.

The imprimatur of the National Science Foundation on both NCTM's standards and the various textbooks they funded and NSF's grant program caused states to revise their standards of learning for mathematics. Case in point: In 1992, the California State Board of Education adopted the California Mathematics Framework-a set of standards based prominently on NCTM's standards of 1989.
And Stage Right: the Right
By this time, people were beginning to notice. NSF's embrace of NCTM's philosophy of education, among other problems with the teaching of mathematics, became grist for Lynne Cheney, then an active senior fellow at the American Enterprise Institute. With well-articulated essays in leading media, Cheney took out after the educationists and won the respect of mathematicians and scientists as she helped raise awareness among a wider audience across the United States. Isolated math revolts began to occur. One of the first was in Silicon Valley, where parents are engineers, scientists, programmers, and mathematicians-and they didn't like the way their children were being taught mathematics. Two local mathematicians, Jim Milgram of Stanford and Hung-Hsi Wu of Berkeley, became key players in the rewriting of California's math standards and the elimination of NSF-funded books from the state's curriculum. Several years later, as I mentioned earlier, Wilfried Schmid from Harvard became active and helped rewrite Massachusetts's standards. Two other states, Minnesota and Michigan (Milgram and Wu went to the Wolverine State as well), also just recently revised their math standards. Minnesota's standards were changed due in large part to the efforts of Dr. Larry Gray, chair of the mathematics department at the University of Minnesota, and of concerned and outraged parents.

But the education bureaucracy did not roll over, and in the fall of 1999 the U.S. Department of Education released a list of ten recommended math programs, designated as "exemplary" or "promising," all of them aligned with the NCTM standards and based on texts funded by NSF.

The reaction was swift. More than two hundred university professors-including Milgram and Wu, Schmid, and several winners of the Fields Medal, the highest international award in mathematics-wrote an open letter to Secretary of Education Richard Riley, calling on the Department of Education to withdraw the recommendations. The open letter was also published as a full-page ad in the Washington Post, paid for by the Packard Humanities Institute, long a critic of constructivist education. The Department of Education did not withdraw the recommendations, but instead added two more NCTM-aligned books to the list.
The Education Department's 1999 list, as it represented a more visible-and seemingly overtly political-assertion of political will, was a critical point in the new-math wars. And when Lynne Cheney slammed the Clinton administration as she criticized math textbooks and the NSF in her editorials, she helped ensure that the math wars would become (as they remain) partisan. Republicans tended to be sympathetic to the issues, and some hearings were held. But nothing came of them, and no investigations into NSF and its funding practices were launched. No questions were asked about why the Department of Education didn't rely on mathematicians in the review of proposals for these programs, nor was anyone in the department ever questioned about the NCTM's education philosophy and the millions of tax dollars spent on texts that were the subject of fierce objections from 200 prominent mathematicians and scholars.
An Evidentiary Interlude
In the various arguments about how best to teach math, educationists make the point that research shows that their approaches work best. I tend to be suspicious of that research, and apparently others are as well. Ironically, a recent study by the National Academy of Science's Mathematical Sciences Education Board, sponsored by the NSF itself, was also skeptical. "Evaluations of mathematics curricula provide important information for educators, parents, students and curriculum developers," concluded the NAS about 19 specific mathematics curricula, including all 13 NSF-sponsored curricula, that it analyzed. They all "fall short of the scientific standards necessary to gauge overall effectiveness."

While the proponents of these NSF-sponsored math programs may be able to claim that the research shows no evidence that the programs are "ineffective," the mathematics community, and parents who are protesting to the various school boards across the United States, can now claim that the research cannot be used to support claims of superior effectiveness-or any effectiveness at all.

Is there a common ground in this war? It seems that for now both sides agree that subject matter should be as important as pedagogy and that improving the math education of teachers is probably the most important weapon in the battle. What constitutes a proper math education for prospective teachers, however, is subject to debate. A mathematician I know who teaches at a small college told me that a graduate of his education school caused much embarrassment when, during the interview for a job in an elementary school, the job seeker was unable to add two fractions when asked to do so. The college's education department subsequently decided to put its math content courses under control of the math department.
Meanwhile, Back on the Hill
Though academic debate about mathematics curricula will no doubt continue, the field of argument is increasingly muddied by politics. It was in this context that I began my investigation into math education in 2002. I recall meeting with Senator X's deputy chief of staff and two other staffers not long after completing my research on math curricula and the battles that had shaped-often, misshaped-them. "So what are your ideas on how math and science education can be enhanced?" they asked. My answer was something like, "You can enhance a car by painting it, but if the car has no engine, it's not going to do much good." This was not what they were expecting to hear. Nor were they expecting to hear that Lynne Cheney had also taken up the cause of anti-fuzzy math. At that point, the discussion took a decidedly troubling turn. These staffers-Democrats-now worried that they could not support policies that were also advocated by the wife of a powerful Republican.

I told them about the open letter from the two hundred mathematicians and urged them not to confuse the message with the messenger. "This is a real issue," I said. "Kids aren't learning the math they need to learn."

I had discussions and sent e-mails in the hopes that I would at least get a chance to brief Senator X on the issue and, perhaps, persuade him to ask some tough questions of NSF when it came time to fund their programs. But I felt that at any moment everything was going to be whisked away.

And one day it was. The staffers in my office talked with other Democratic staffers on the Hill, who told them that it would be wise to stay away from the "fuzzy math/Lynne Cheney/Bush agenda" issue. Ultimately the staffers I was working with told me they couldn't take a chance on having Senator X "come off like Lynne Cheney."

This development was not surprising to any of the mathematicians with whom I had been working-most of them Democrats, like me. The senator was never briefed, and no investigation into NSF was launched. I was thanked for my hard work. I went back to my regular job and started tutoring middle school students in math at a school in D.C. while continuing to work with high school students in my neighborhood. That year a 9th-grade girl was having problems in geometry and came to me for help. "What seems to be the problem?" I asked. "I don't know how to do proofs," she said.

"I know," I replied. "Don't worry. It isn't you."

All politics is local, I decided.
Barry Garelick is an analyst with a federal government agency in Washington, D.C. The opinions expressed in this article are those of the author.

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An A-Maze-ing Approach To Math - Education Next

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I am not a mathematics teacher, but I have a degree in mathematics and an intense interest in how the subject is taught. When I retire, I would like to teach math, which is why I started tutoring high school students in my spare time three years ago.

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It was another body blow to education. In December of 2004, media outlets across the country were abuzz with news of the just-released results of the latest Trends in International Mathematics and Science Study (TIMSS) tests. Once again despite highly publicized efforts to reform American math education (some might say because of the reform efforts) over the past two decades, the United States did little better than average (see Figure 1). Headquartered at the International Study Center at Boston College and taken by tens of thousands of students in more than three dozen countries, TIMSS has become a respected standard of international academic achievement. And in three consecutive TIMSS test rounds (in 1995, 1999, and 2003), 4th- and 8th-grade students in the former British trading colony of Singapore beat all contenders, including math powerhouses Japan and Taiwan. United States 8th graders did not even make the top ten in the 2003 round; they ranked 16th. Worse, scores for American students were, as one Department of Education study put it, "among the lowest of all industrialized countries."

During the clamor over the TIMSS results (released in December 2004), I heard Robyn Silbey, a math "content coach" from a Rockville, Maryland, public school, being interviewed by Ira Flatow for his Science Friday program on National Public Radio. Silbey worked at College Gardens Elementary. She explained that her school was one of four in the Montgomery County Public Schools (MCPS) district experimenting with Singapore's math program. And, according to Silbey, it was working. The Singapore texts and methods were so effective in College Gardens that the scores of students there on the math computation portion of the standardized Comprehensive Tests of Basic Skills (CTBS) rose from the 50th and 60th percentiles to the low 90s in the first 4 years they were used.

I later learned that an evaluation of the pilot program conducted by MCPS found that in the schools where Singapore Math (SM) was being used as a pilot program, students typically outperformed their peers in other district schools. Yet despite these positive results, three of the four pilot schools dropped out of the program after fewer than four years. Why, I wondered. If the county's own evaluation found benefits from Singapore Math, why not continue using it? In view of America's disappointing rankings in math and Singapore's record of success, why wasn't the Singapore Math program given a serious and extended try?
In the Beginning
While the story of Singapore Math in Montgomery County does not answer all the questions about the persistently poor math literacy of American students, the failure of the program to take hold there does provide disturbing clues about some of the institutional and governmental practices that impede improvement in education-and not just in Montgomery County.

In my early research into what happened in Montgomery County, I met John Hoven, then co-president of the Gifted and Talented Association of Montgomery County and now a national advisor to NYC HOLD (New York City Honest Open Logical Decisions on Mathematics Education Reform), a nonpartisan advocacy organization that provides information to parents, teachers, and others on math education issues. Hoven, an economist in the Antitrust Division of the U.S. Department of Justice by day, had discovered Singapore Math while waging a successful battle to get MCPS to forgo a National Science Foundation (NSF) grant worth $6 million. The grant would have trained teachers to use a middle-school program called Connected Math, one of several, Hoven learned, that was funded by the Education and Human Resources Division of the NSF and based on standards developed by the National Council of Teachers of Mathematics (NCTM). Though the NCTM is a private organization, it exerts enormous influence over the math standards and texts used by most states and districts in the United States-standards and texts that, in Hoven's view, were failing.

During the campaign against the Connected Math grant, Hoven discovered Singapore Math. He learned that Singapore, whose population is half that of New York City, had begun modifying its education policies in the early 1980s to build up its labor force in such a way as to create technical skills unavailable elsewhere in the Third World. The Curriculum Development Institute of Singapore (now called the Curriculum Planning & Development Institute) had created the math program and the accompanying texts, called Primary Mathematics (which were published in English, Singapore's official administrative language, in 1982), to help boost that technological prowess. The Primary Mathematics series was at the heart of Singapore's national math curriculum as it achieved its successive TIMSS victories.

Many professional mathematicians, concerned with the decline of math education in the United States, took a hard look at the Singapore Math methods and texts and liked what they saw. The texts have been distributed in the United States by an Oregon company since 1998 and are used by many home-school parents and promoted by Internet-based parent and professional organizations. In addition, the private nonprofit Rosenbaum Foundation helped fund the implementation of Singapore Math programs in scattered sites around the United States and in Israel. A study of Singapore Math conducted by the American Institutes for Research (AIR) for the U.S. Department of Education (released in January 2005) concluded, "What the United States needs overall are the sound features of the Singapore Mathematics system." In studying several different American school districts that were experimenting with the program, including the Montgomery County Public Schools, AIR researchers found that "Singapore Mathematics textbooks can produce significant boosts in achievement." But the AIR report also cautioned that making Singapore Math work in the United States "will require the same sustained commitment to developing a quality mathematics system that Singapore gave to its reform efforts."

That was a lesson still to be learned in Maryland.

A Long Way from Singapore to Montgomery County

John Hoven and his allies persuaded the Montgomery County Public Schools to try Singapore Math instead of pursuing the Connected Math grant, but instead of receiving $6 million, the district would have to spend its own funds. And that wouldn't be $6 million. "The initial plan was for a $50,000 pilot spread out over two years," Hoven recalls. "It was $50,000 in a $1 billion budget. The money would pay for textbooks and nothing else-no teacher training, nothing." Hoven knew it was just a drop in the bucket for the district, but he was sure that the Asian math program would sell itself.

MCPS selected four middle-class, ethnically diverse, suburban schools-College Gardens in Rockville, Charles R. Drew and Highland View in Silver Spring, and Woodfield in Gaithersburg-to participate in the pilot. But few teachers at the schools realized how different Singapore's approach to math was from what they had been used to; it was nothing less than a total shock to the schools' systems.

Unlike many American math textbooks, such as Math Thematics , published by Houghton Mifflin, which are thick, multicolored, and multicultural, Singapore's books are thin and contain only mathematics. There are no graphics (other than occasional cartoons pertaining to the lesson at hand), no spreadsheet problems, and no problems asking students to use a calculator to find the mean number of dogs in a U.S. household. With SM, students are required to show their mathematical work, not explain in essays how they did the problems or how they felt about them. While a single lesson in a U.S. textbook might span two pages and take one class period to go through, a lesson in a Singapore textbook might use five to ten pages and take several days to complete. The Singapore texts contain no narrative explanation of how a procedure or concept works; instead, there are problems and questions accompanied by pictures that provide hints about what is going on. According to the AIR report, the Singapore program "provides rich problem sets that give students many and varied opportunities to apply the concepts they have learned."

Another key difference is the number of topics covered by Singapore's texts for a single grade. The AIR study frequently criticizes American math texts for being an inch deep and a mile wide, covering a great range of topics with little time spent on developing the material, including mastery of math facts. (One of the texts with which the AIR study compares Singapore's Primary Mathematics series is Everyday Mathematics, a program developed with NSF funding and used widely in Montgomery County.) The MCPS 1st-grade curriculum goals, for instance, contain a number of nonessential topics, such as sorting concrete objects (like Post-its with names of favorite pets on them) into categories, activities that take up instructional time which, critics of the MCPS curriculum argue, could be better spent laying the foundation for algebra in 8th grade.

Singapore's texts also present material in a logical sequence throughout the grades and expect mastery of the material before the move to the next level. In contrast, mainstream American math texts and curricula frequently rely on a "spiral" approach, in which topics are revisited and reviewed. The expectation of that approach is that not all students achieve mastery the first time around. One Ohio school teacher familiar with the spiral approach summed up much of the criticism of the method on an Internet math forum, saying, students "can't remember how to do it when [they] do return-or if they do remember it, it's now being taught in a different way."

The most important feature of Singapore's texts is an ingenious problem-solving strategy built into the curriculum. Word problems are for most students the most difficult part of any mathematics course. Singapore's texts help students tackle them through a technique called "bar modeling," in which students draw a diagram to help them solve the problem. Typically, in U.S. texts, students are taught to use a method called "Guess and Check"-trying combinations of numbers until the right numbers are found that satisfy the conditions of the problem-a method that many professional mathematicians consider inefficient (see sidebar). The bar-modeling technique not only provides a powerful method for solving problems, but also serves as a link to algebra. Symbolic representation of problems, the mainstay of algebra, emerges as a logical extension of the bar-modeling technique.

What Happened in Montgomery County?

Given all of the mathematical strengths of the Singapore program, why was the pilot abandoned so quickly in Montgomery County? The simplest answer is that where Singapore Math worked the best, in College Gardens, it is still being used; where it didn't work as well, it was dropped. But that does not begin to explain what happened.

All four Montgomery County schools used the Singapore Math texts in 2000-01 and 2001-02, but only College Gardens and Highland View kept the program in 2002-03. The "math computation" scores at College Gardens show a dramatic improvement for both 2nd and 4th grades (see Figures 2a and 2b), but in "general math" there is no discernible pattern; all four schools had either no change or a decrease in scores.

Additional results from the pilot were detailed in the evaluation conducted by the MCPS Office of Shared Accountability after the second year of the experiment. The county evaluators found that students in the four Singapore Math pilot schools generally progressed through the curriculum at an accelerated pace compared with their peers in control schools. But while the school district's evaluation was positive in tone (Singapore Math "helped prepare students for higher-level math placements in middle school"), it reported mixed results and offered no recommendation for expansion.

Because the effectiveness of a program as sophisticated and multidimensional as Singapore Math cannot be thoroughly evaluated in just two years of testing, the story of its failure in Montgomery County says more about school politics and finances than about math programs. (It would help, for instance, to track students who went through Singapore's program through their 8th-grade tests to ascertain how well they were prepared for algebra.) The mixed math results of the county's evaluation should have been seen for what they were: an interim assessment. Instead, the county ended the funding for the program after the second year. If schools wished to continue, they had to pay for the materials out of their own budgets, which they didn't need to do if they used district-approved texts such as Everyday Mathematics .

Detailing the many reasons for dropping support for the pilot by Montgomery County without waiting for long-term results would take more space than is available here. But we can get a sense of the thing by examining some of the reasons that the three schools gave for quitting Singapore Math and those given by College Gardens for staying with it.

The first problem was lack of planning and preparation. The depth and breadth of the differences between Singapore Math and American math were not appreciated. The decision to use Singapore Math was made in 1999, for instance, but textbooks and other teaching materials did not arrive at the four schools until late spring the following year, giving teachers just three months to prepare to introduce the program to students in the fall of 2000.

The Singapore Math manuals were another problem: they provided very little guidance on how to teach a particular lesson-because they are written for teachers who, for the most part, have a deeper understanding of mathematics than most U.S. teachers do. That dilemma was compounded by the lack of experience with Singapore's program by Montgomery County and its delay in training teachers to use it. The Montgomery County Public Schools eventually developed a training program, but some people believe it was too little, too late. College Gardens and Highland View found funds to hire Singapore Math specialists (like Robyn Silbey) to help get the program off the ground and coordinate the training within their schools. Scott Baldridge, a Louisiana State University mathematician who provides professional training to teachers in implementing the Singapore program, believes that such training helps. "Some teachers get it on their own," he says, "but many need professional development to see how the curriculum interacts with the students over several years."

Even with adequate training, the two-year span of the pilot resulted in three of the pilot schools (all except College Gardens) introducing Singapore Math all at once, across all grades, which put older children at a severe disadvantage, since Singapore Math concepts build on one another. This helps to explain the difference in the math test results.

Another complaint expressed by teachers and administrators in all four schools was that Singapore Math was not in line with state standards. Indeed, the state's academic standards include data analysis, statistics, and probability, which Singapore's texts do not address. But it is more complicated than that since, by state law, each school district has authority over its own curriculum. Although required to administer the state tests, school districts are not required to align their curriculum with the state standards. That means, says Hoven, that "they can choose to aim for world-class standards instead, which at one time was one of the goals of the MCPS Long-Range Plan." But the plan was revised to be aligned with state standards instead-a move, say some, that was, in effect, a decision to lower Montgomery County's standards.

This brings us to the question of money. Eileen MacFarlane, principal at Drew, said that her school initially supplemented Singapore Math with additional material on statistics, data, and probability to cover the misalignment, but the cost of purchasing Singapore's materials from their own budget became a problem.

Joanne Steckler, now retired as principal of Highland View, said, "No one told us to drop the program." But being required to buy its own materials had the same effect. "For one year we did purchase our own Singapore materials," said Steckler. "But we did not want other curriculum areas to suffer because of lack of funds to purchase materials, so we gave up Singapore Math."

Shawn Miller, principal of Woodfield Elementary, also cited the cost of the texts. Once the county stopped funding Singapore Math, Miller made the decision to go with the county-approved program, Everyday Mathematics, which, he said, "was better aligned with the state curriculum."

With such standards-alignment and budget concerns, the schools had a safe way to bow out of the pilot.

A Culture Shock to the System

But budget questions were hardly fundamental. Taking on a program like Singapore Math meant going against what many teachers believed math education to be about; surely, it was not what they were trained for. Since the success of Singapore's programs relies in many ways on more traditional approaches to math education, such as explicit instruction and giving students many problems to solve, in some ways its very success represented a slap in the face to American math reformers, many of whom have worked hard to eliminate such techniques from the teaching canon.

Gail Burrill, a former president of the NCTM, suggests quite bluntly that the success of Singapore Math cannot be imported. "These are books used by a different culture, a culture that is more homogeneous, and a culture that has a consistent way of thinking about mathematics." And Cathy Seeley, a former president of the NCTM, hints as much by arguing that Singapore's success (as well as that of other Asian countries) is not about the textbook. "We have to look beyond their textbooks to determine what these lessons are."

The logic of the argument that it is the Asian culture or something "beyond their textbooks" that produces math success leads to the conclusion that, as NCTM adherents often contend, content doesn't matter nearly as much as the teacher or the culture that produces the "proper conditions for learning." Eileen MacFarlane maintained that the teachers in the Singapore pilots drew from the texts, but then quickly added, "The text is a resource, not a curriculum." She said this despite her enthusiasm for Singapore's program.

But the belief that the difference between Singapore Math and American math is just in the teaching or, as some suggest, the culture, is a rationalization, says David Klein, a mathematician at California State University, Northridge. "Math reformers assume that math education is bad in the United States because the NCTM reforms were not properly implemented nor understood by teachers," he continues. "They never consider the possibility that the NCTM standards themselves and the textbooks written for those standards are one of the causes of poor math education in this country."

The only person I heard openly disagree with the "teacher not text" argument was Dr. Sherry Liebes, then the principal of College Gardens, the only school that kept the program. While she said that teacher training is important, she added that Singapore's texts provide a structured curriculum, and thus "It's one less thing for teachers to worry about." This notion was echoed in the AIR study, which quoted a teacher in one of the pilot schools in Montgomery County: "Having to explain Singapore Mathematics made me understand that I never really understood the mathematics I was teaching."

Another stumbling block for the Maryland teachers was their concern that the Singapore Math program did not contain "real-world" activities. The term, as used by those who follow the ideas supported by the National Council of Teachers of Mathematics and education schools for teaching math, generally means a problem for which American students have not received much instruction or preparation. This is intentional, it turns out, because it is believed to be good for students to learn to approach problems for which they have not received explicit preparation. The National Education Association (NEA), for instance, in its online version of "A Parent's Guide to Helping Your Child with Today's Math," gives an example of a "real-world" problem:

A farmer sends his daughter and son out into the barnyard to count the number of chickens and pigs. When they return the son says that he counted 200 legs but the daughter says she counted 70 heads. How many pigs and chickens does the farmer have?

The NEA then suggests that some students may solve the problem using algebra (those who know how to do so, that is), while others might solve it using Guess and Check. Still others may choose to draw pictures to solve it. The NEA admits that some methods might be considered more efficient, but points out that the correct answer can be found using multiple methods and that "by allowing students to think flexibly about numbers, we encourage them to 'own' the math forever, instead of 'borrowing' until class is over." That this real-world problem depicts an approach that no sensible person would use in counting pigs and chickens is beside the point.

This kind of real-world math is indeed missing from Singapore's program-apparently, if TIMSS tests mean anything, without much harm. Rather than waste students' time with inefficient methods for solving problems, Singapore's texts provide instruction that eliminates trial and error, one of the goals of mathematics. Bar modeling is a powerful pictorial technique that results in one answer, deduced by using mathematical principles that students have learned rather than by employing the haphazard trial-and-error method of Guess and Check.

For One Brief Shining Moment

An exact description of which differences in math instruction matter most is perhaps impossible. For instance, an emphasis on sequential mastery of skills that builds on previously acquired skills is a key component of the Singapore Math program and not important in the American approach, where activities don't require such skills. While the latter creates the illusion of equal achievement, international tests like the TIMSS would seem to provide a reality check on that illusion.

The struggle to make math instruction work, of course, is not limited to Montgomery County. In the state of Washington, parent protests against the adoption of several standard math curricula (like Connected Math) led a state representative to introduce a bill earlier this year to put Singapore Math in all the state's elementary schools. And in New York City, Elizabeth Carson, who heads NYC HOLD, has led a battle for years to rid the city's schools of programs like Everyday Mathematics. She calls it a "tragedy for our children and our nation" that American attempts at math reform "bear no resemblance to the programs and standards of the highest-achieving nations."

Having watched as three of the four schools dropped Singapore's program in Montgomery County, John Hoven shared Carson's concern. Discouraged, he resigned a year ago from the county's Gifted and Talented Association. "I had stopped believing I could make a difference," he says. "I felt it was time for someone else to try."

In the meantime, the decline in the numbers of U.S.-trained scientists and engineers, compared with the increasing numbers of those trained in Asian countries, has not gone unnoticed. In this year's State of the Union address, President George W. Bush stated: "We need to encourage children to take more math and science, and make sure those courses are rigorous enough to compete with other nations." He proposed "to train 70,000 high-school teachers to lead Advanced Placement courses in math and science ... bring 30,000 math and science professionals to teach in classrooms ... and give early help to students who struggle with math, so they have a better chance at good, high-wage jobs."

A few months later, President Bush created the National Mathematics Advisory Panel to advise the White House and the secretary of education on the best use of scientifically based research to advance the teaching and learning of mathematics. The panel includes several people who have actively fought against the NCTM-led "fuzzy math" trend in this country.

While the goal of bolstering high-school math is a laudable one, the success of high-school students in math depends on what they've learned in the lower grades. If those foundations are weak, the addition of Advanced Placement courses in math and science in high schools will prove to be a weak enhancement. Unfortunately, changing the way math is taught in the lower grades appears to threaten an education philosophy and method that is pervasive in our schools, and does not move us towards academic excellence.

Barry Garelick is an analyst for the federal government and lives in the Washington, D.C., area. He is a national advisor to NYC HOLD, an education advocacy organization that addresses mathematics education in schools throughout the United States.

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It was another body blow to education. In December of 2004, media outlets across the country were abuzz with news of the just-released results of the latest Trends in International Mathematics and Science Study (TIMSS) tests.

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Math tutor Christopher Seber has racked up more than 5 million views for Math Meeting, his YouTube channel, and he's part of a trend that's seeing education videos surpass pet videos in popularity. The South Florida Sun Sentinel has a video feature on the phenomenon.

-Education Next

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What We're Watching: Math Videos Overtake Cute Pets in YouTube Popularity - Education Next

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Math tutor Christopher Seber has racked up more than 5 million views for Math Meeting, his YouTube channel, and he's part of a trend that's seeing education videos surpass pet videos in popularity. The South Florida Sun Sentinel has a video feature on the phenomenon. -Education Next

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If you are seeking a new way to add relevancy to your classroom math activities, the answer may be right in your school library! Literature is the ideal vehicle to help your students see the importance of numbers in their daily lives. Included: Author Marilyn Burns is one educator who says, "Math and literature together? Why not!" She shares her thoughts with Education World. Plus more ideas for integrating math and literature!

"I've found throughout my teaching career that students are my best teachers, that the more I understand how children think and reason, the better I'm able to find ways to help them learn," author Marilyn Burns told Education World. "I believe that we can't teach what we don't understand, and we can't teach well what we don't enjoy. I think that children's literature offers a wonderful vehicle for helping teachers teach math well."

In Books by Marilyn Burns, Burns uses traditional and original literature to address mathematical concepts. Her efforts show students that those subjects, like most classroom topics, are interrelated.

"Combining math and literature in classroom activities is a way for teachers to invite children into the world of math," stated Burns. "Reading books that weave mathematical ideas into engaging stories helps dispel the myth that math is dry, unimaginative, and inaccessible. Children's books can not only generate interest in math but also provide contexts that help bring meaning to abstract concepts. Using children's literature is a win-win -- for children and for teachers."

CHILDREN'S LITERATURE OPENS DOORS TO MATHEMATICS

An educator who began teaching secondary mathematics in 1962, Burns describes the start of her writing career as a "fluke." A friend asked Burns whether she would be interested in writing a book about math for what became the Brown Paper School series. She did so, creating The I Hate Mathematics! Book, published in 1975. That publication jump-started her parallel career. Burns has since written about a dozen books for children and the same number for teachers. Through offering workshops for other educators, she developed her love for teaching younger students, an experience she found delightful in a different way.

"From presenting many workshops to teachers, I realized that many elementary teachers, particularly teachers of young children, aren't comfortable with mathematics," Burns explained. "For many, their own learning of mathematics was difficult and often unpleasant, and they took only what was minimally required." She added that when math becomes an elective, more than 50 percent of high school students elect not to study it any longer.

"Teachers who aren't comfortable with math typically prefer teaching reading and language arts, drawn to the many beautifully illustrated children's books available and how they spark children's interest and imaginations," continued Burns. "I began creating math curriculum materials that involve literature to show teachers how to connect their interest in children's literature with helping children experience the wonder and delight of mathematics."

HOW TO BEGIN? DIVE RIGHT IN!

"My advice to educators just beginning to incorporate literature into math activities is to choose a book and dive in," said Burns. "Read the story aloud to the class and discuss it as you would any other book. Then introduce an activity. As with all math lessons, keep the emphasis on children's reasoning, ask students to communicate their thinking and solutions, and encourage discussion among students."

Burns recommends two resources from Math Solutions Publications for K-3 teachers, books one and two of Math and Literature (Grades K-3). Those resources introduce more than 50 children's books that are useful for teaching math ideas and present vignettes of actual classroom lessons that use them, along with samples of student work. Some of the literature selections may be familiar to classroom teachers -- Rooster's Off to See the World, Little House in the Big Woods, Ten Black Dots, -- and some may be new. For teachers of older students, Burns suggests Math and Literature (Grades 4-6), also from Math Solutions. That resource presents instructions for using 20 age-appropriate books in mathematics classes.

"In 1994, I launched the series of Brainy Day Books, books for children, published by Scholastic, specifically designed to help learn math," Burns elaborated. "I wrote the stories for two of the five books in the series. For the other books, I wrote the section at the end, 'For Parents, Teachers and Other Adults,' explaining the math underlying the story and providing ways to involve children with the mathematical ideas."

Because of beginning the Brainy Day Books, Burns began to edit Scholastic's Hello Math Reader series, which produces easy readers for children in Pre-K through grade 3. Each book offers activities at the end of the stories. There are now more than 30 books in that collection. Some of her favorites are Stay in Line, One Hungry Cat, A Quarter from the Tooth Fairy, and Even Steven and Odd Todd.

See reviews for some of Burns's books in an Education World BOOKS IN EDUCATION article, Math and Reading Do Mix.

MAKING MATH 'REAL' FOR REMEDIAL LEARNERS

"My students tend to be very uninterested in school and continually deny that whatever subject we do has any value in their lives," said high school teacher Sharon Powell. "I was hoping that by tying everything together, I could make them see that the things we learn in class do relate to their lives."

Powell knows well the struggles a classroom teacher can face in trying to link subjects such as math and literature. Her Northwestern High School class in Rock Hill, South Carolina, is part of a remedial program for students in grades 9 through 12 designed to improve the test scores of students who fail the school's "exit exam." Those students need extra help to pass the exam so they may receive a diploma rather than a certificate.

"Last year after the exit exam, I wanted to do something that would combine all the subject areas that I teach into one set of lessons," Powell explained. "I had been teaching seven different combinations of reading, writing, and/or math. I used to be an elementary teacher, and for years I felt that all subjects should be combined at the elementary level in order to be able to spend more time teaching reading."

Calling upon her two only real resources -- 25 years in the classroom and a determination to get kids to pay attention -- Powell set about locating literary works that could be used to support her math curriculum.

BOOKS GET TO THE "MEAT" OF MATHEMATICS

One of Powell's big successes in marrying math and literature in her classroom has been with the book The Crazy Horse Electric Game, by Chris Crutcher. "Some of the things I did with the book had to do with figuring out the cost of the trip," she said. "The students had to read bus schedules and discover the cost of medical treatments by using percentages based on different rates of insurance payment. They computed statistics for the players. I had them find a job in the classified ads that Willie was qualified to do and plan a budget for him. They did maps of his trip. We did distance problems based on how fast the boat was traveling and how far it had to go if the family took him to the hospital. Basically, I looked for ways that math was used in the book and expanded on it. This can be done with any book."

Powell has even borrowed from Hollywood to make her point! "As a treat, I show a movie occasionally. I have a three-day lesson plan that takes Good Burger from movie to lesson. The students figure out how much the repair costs. I give them an estimate and have them adjust it by percentages for different repair shops. They have to figure out the profit on burgers based on different prices for materials. They have to compute how many hours have to be worked at different salaries to pay the repair bill. I give them raises, and they have to solve how much less time will have to be worked than before the raise. I compare the profit from the two restaurants based on the cost of the food. I give quantity discounts and compare the profit based on certain sales. It takes a little while to plan the lesson, but it is not horrible. I make up the salary schedules and prices for items rather than researching them. This saves time, but I do have to create my own answer keys.

"Books are stories about people and their lives, and these lives involve numbers," Powell continued. By keeping that in mind, she believes, any teacher can join math and literature activities. "My kids like this kind of work. They see it as real or fun, but either way, it is something they are willing to do. My own children actually think that math problems are harder when there is no story to go with them."

RELATED WEB RESOURCES

Math Solutions A division of Marilyn Burns Education Associates, Math Solutions offers in-services, workshops, and publications designed to improve mathematics instruction.

Math and Children's Literature This site contains articles and sample ideas from resources to help teachers integrate math and literature lessons.

Math in Daily Life From the Annenberg/CPB Project, this resource explains the mathematics involved in daily activities such as cooking and home decorating.

Cara Bafile
Education WorldÂ®
Copyright Â© 2001 Education World

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Math and Literature -- A Match Made in the Classroom! | Education World

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If you are seeking a new way to add relevancy to your classroom math activities, the answer may be right in your school library! Literature is the ideal vehicle to help your students see the importance of numbers in their daily lives. Included: Author Marilyn Burns is one educator who says, "Math and literature together?

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Read an excerpt from From Play to Practice

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An Educational platform for parents and teachers of pre-k through 5th grade kids. Support your kids learning journey with games, worksheets and more that help children practice key skills. Download, print & watch your kids learn today!

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An Educational platform for parents and teachers of pre-k through 5th grade kids. Support your kids learning journey with games, worksheets and more that help children practice key skills. Download, print & watch your kids learn today!

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