Shaping the river: Contexts of mathematical development and their implications for assessment and standardsetting. . Christopher Correa Kevin F. Miller University of Michigan Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure Educational practices The case of fractions Implications for assessment & standards setting The uses of assessment “Quality control inspectors”? How might assessment information lead to increased learning? Interaction between • Kinds of information provided • Models of learning • Instructional practices Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure Educational practices The case of fractions Implications for assessment & standards setting Historical perspectives: “A calculating people” “When a country has a national sport as part of its culture— that is, contests that almost all members find interesting to participate in or watch—the team representing that country tends to be strong, even if its population is small…I think that school mathematics is a national (in the sense used above) intellectual pursuit in Asian countries, but not in the United States. Whereas students cannot escape from it in the former, in the latter they are told (at least implicitly) that, if they are poor at or dislike mathematics, they are free to seek achievement in some other area. Hatano, G. (1990). Toward the cultural psychology of mathematical cognition. Comment on Stevenson, H. W., Lee, S-Y. (1990). Contexts of achievement. Monographs of the Society for Research in Child Development, 55 (1-2, Serial No. 221), 108-115. These things can change… America in the 1830s “We are a traveling and a calculating people,” said Ohio booster James Hall in his book Statistics of the West, with evident satisfaction. “Arithmetic I presume comes by instinct among this guessing, reckoning, expecting and calculating people,” said English traveler Thomas Hamilton, with evident distaste. Each man attached a different value to the idea, but both agreed that Americans in the 1830s had some sort of innate reckoning skill that set them apart from Europeans. Cohen, P. C. (1982). A calculating people: The spread of numeracy in early America. Chicago: U. of Chicago. (pp. 3-4) Geary, D. C., Salthouse, T. A., Chen, G-P, & Fan, L. (1996). Are East Asian Versus American Differences in Arithmetical Ability a Recent Phenomenon? Developmental Psychology, 32, 254-262 Younger (20 year old) and Older (late 65-75 year old) adults Matched for health, education Given Arithmetic and basic cognitive tasks Addition performance Age & Addition Performance in the U.S. & China 50 # problems in 2 minute Addition problems like 19 + 8 + 27 = ? # correctly solved in 2 minutes Interaction between age and culture 40 30 20 U.S. China 10 0 Younger Older Subtraction performance Age & Addition Performance in the U.S. & China 70 60 # problems in 2 minute Subtraction problems like 35-9 = ? # correctly solved in 2 minutes Interaction between age and culture 50 40 30 20 U.S. China 10 0 Younger Older Perceptual speed – No culture effect 60 Age & Perceptual Speed Performance in the U.S. & China 50 # correct in 3 minutes Perceptual speed task Identifying whether number strings are the same or not • 179359821 • 178539821 # correctly solved in 3 minutes ETS kit of factor-referenced tests (Ekstrom et al., 1976) 40 30 20 U.S. China 10 0 Younger Older Mental rotation – no culture effect Rotation of figures in 3dimensional space # solved in 3 minutes # problems in 3 minutes From ETS kit of factor-referenced tests (Ekstrom et al., 1976) 20 Age & Mental Rotation Performance in the U.S. & China 10 U.S. China 0 Younger Older Conclusions Differences are Large Specific Probably recent in origin Where might they come from? Differences in what children learn Differences in educational environment Differences in teaching processes Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure Educational practices The case of fractions Implications for assessment & standards setting Language and Learning to Count Children need to learn a system of number names as they learn to count Not a trivial task Number names in Chinese & English - Part I Counting to Ten Numeral Chinese (written) Chinese (spoken) English 1 2 一 二 3 三 4 四 5 五 6 7 六 七 8 八 9 九 10 十 yi san si wu liu qi ba jiu shi er one two three four five six seven eight nine ten Both languages share an unpredictable list – No way to induce “five” from “one, two, three, four” Linguistically, learning to count to ten should be of equal difficulty in both languages Number names in Chinese & English - Part II From Ten to Twenty Numeral 11 12 Chinese 十一 十二 (written) 13 14 15 16 十三 十四 十五 十六 17 18 19 20 十七 十八 十九 二十 shi wu shi liu shi qi shi ba shi jiu er shi Chinese shi yi shi er shi san shi si (written) English eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty Chinese has a clear base-ten structure – similar to Arabic numerals: 11 = “10…1” English lacks clear evidence of base-ten structure – Names for 11 and 12 not marked as compounds with 10. – Larger teens names follow German system of unit+digits name, unlike larger two-digit number names compare “fourteen” and “twenty-four” Number names in Chinese & English - Part III Above Twenty Language Rule Example 三十七 Decade unit (two,three,four,five,six,seven,eight,nine) + ten (shi) + unit san shi qi Decade names (twen,thir,for,fif,six,seven,eight,nine) + ty + unit thirty-seven Chinese (written) Chinese (written) English Both languages share a similar structure – similar to Arabic numerals: 37 = “3x10 + 7” For Chinese, this extends previous system For English, it represents a new way of naming numbers A longitudinal view 110 110 2-year-olds 100 100 90 US China US 80 80 70 70 70 60 60 50 50 40 40 30 60 40 0 1 2 10 5 6 7 8 9 10 11 12 Month China US 30 20 0 3 4 20 10 50 20 90 80 30 4-year-olds 100 China 90 Median Abstract Counting 110 3-year-olds 10 0 1 2 3 4 5 6 7 8 9 10 11 12 Month 1 2 3 4 5 6 7 8 9 10 11 12 Month Learning difficulties reflect language structure ..and they don’t stop here! The Panda’s snack No language difference for counting-principle errors such as doublecounting • Mastering number list and understanding numerosity not the same • • Producing sets of n items No language difference % Correct • 100 US China 80 60 40 20 0 3 4 5 Age Producing Sets of 12 100 % Correct • Language affects only some aspects of early number knowledge Producing Sets of 4 80 60 US China 40 20 0 3 4 Age 5 Teens numbers Continuing effects % Correct Learning Arabic numerals involves a mapping from verbal number names 100 80 US China 60 40 20 0 K 1 2 Grade Larger 2-digit numbers Teens continue to cause problems % Correct 100 80 US China 60 40 20 0 K 1 Grade * 2 Summary Early mathematical development is a mix of language-dependent and universal factors Sensitivity to symbol structure begins very early Base-ten concepts and “teens” are problematic for speakers of English Foundation for later mathematics Other sources Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure The case of fractions Educational practices Implications for assessment & standards setting Interaction between symbol structure and classroom processes – Rational numbers English terms somewhat opaque Numerator Denominator Chinese terms more transparent 分子 (“fraction child”) 分母 (“fraction mother”) ***** Transparent fraction terms – does it make a difference? Not entirely clear Miura et al. (1999) vs. 1/3 = ? Paik & Mix (2003) Children in both countries likely to pick “numerator + denominator” foil Language support can only take you so far… return Student thinking – components of knowledge In addition to content domains, the most recent TIMSS student assessment also considers different types of thinking: Knowing facts, procedures, and concepts • What students need to know Applying knowledge and conceptual understanding • Applying what the student knows to solve routine problems or answer questions Reasoning • Solving for unfamiliar situations and multi-step problems Thinking about Fractions Knowing Applying Reasoning Thinking about Fractions Knowing Applying Reasoning Thinking about Fractions Knowing Applying Reasoning Fourth-grade students in many high-achieving countries are proficient in problem solving. Fourth-grade student achievement in Japan, TIMSS 2003 Fourth-grade student achievement in Netherlands, TIMSS 2003 U.S. fourth-grade students are more proficient at knowing procedures and knowledge than applying knowledge to solve problems. Fourth-grade student achievement, TIMSS 2003 …though this changes over time. Eighth-grade student achievement, TIMSS 2003 Teacher Knowledge Teachers may be more comfortable facilitating complex problem solving if they have a strong understanding of mathematical concepts Teacher Knowledge in the U.S. Zhou et al. (2006) compare pedagogical content knowledge of U.S. and Chinese teachers Why? K-12 experiences Teacher Training Few content specialists Changes in workforce Few opportunities for learning Cultural Beliefs and Scripts Beliefs about learning Can you get smarter? • Americans more likely to believe that intelligence is fixed and that success is due to ability rather than effort • Can be maladaptive • Teaching & Assessment practices can reinforce this idea Self-Theories of Intelligence Dweck, 1999 Entity Theorists Intelligence is fixed Trait largely determined by nature Incremental Theorists Intelligence is malleable Quality that can be increased through nurture Desire similar outcome • achieving good scores, doing “well” Different motivation for pursuing this outcome • Performance goals seeking to validate abilityitasis “When –I take a course in school, good relative to others very important for me to validate that I am smarter than other students.” • Learning goals – seeking to develop ability “In school I am always seeking opportunities to develop new skills and acquire new knowledge.” Beliefs about Teaching We used multiple correspondence analysis to represent individual teachers and their responses to the interview question, “How do students best learn mathematics?” Beliefs about Teaching …and find within-group similarities Practice “I think that drill is very important… I think that there is just no substitute for going over it and over it and over it until it’s very firmly entrenched in their minds.” (Teacher U4-2) With some students, there can be something said for a lot of rote practice at the very beginning of a concept… there are some students who need 40 problems of it just to get the pattern of what they're doing. (Teacher U4-3) Remembering and Understanding US Students Chinese Teachers 5 US Teachers Mean frequency Attributions for student errors by Chinese & American teachers and college students Remembering vs. Understanding Chinese Students 6 4 3 2 1 0 learn understand remember forget Scripts for teaching Customary teaching practices that are informed by a culture’s underlying values and assumptions of what teaching should be like. National Scripts National Scripts National Scripts Identifying a Script for Chinese Lesson Introductions Data Participants include 19 fourth- and fifth- grade teachers from six schools in Beijing, China Videotaped Classroom Observations Lesson introductions are defined as “what happens between the beginning of class and the final solution of the first novel problem.” Interviews A Proposed Script for Lesson Introductions in China I. Teacher Reviews Old Knowledge (whole- class) II. Students Develop Methods and Solutions (small groups) III.Students Share Results (whole-class) I. Teacher Reviews Old Knowledge (whole-class) II. Students Develop Methods and Solutions (small groups) III.Students Share Results (whole-class) I. Teacher reviews old knowledge Video example I. Reviewing Old Knowledge Review Definitions of Key Concepts • Students, not the teacher, provided definitions • More than one student shares definition Quick Computation • Teacher uses visual aid to present between five and ten review problems to whole class • Teacher elicits whole-class or individual student responses • The types of problems chosen are directly related to new content taught later in the lesson. I. Reviewing Old Knowledge “First, I begin to study the textbook and prepare the lesson carefully. Then, I try to find the connector between the original knowledge and the new knowledge so as to lead them from the original one to the new one. After that, students can get a network of knowledge.” I. Teacher Reviews Old Knowledge (whole-class) II. Students Develop Methods and Solutions (small groups) III.Students Share Results (whole-class) II. Students Develop Methods and Solutions Students ask questions about novel problem II. Students Develop Methods and Solutions Group Discussion I. Teacher Reviews Old Knowledge (whole-class) II. Students Develop Methods and Solutions (small groups) III.Students Share Results (whole-class) III. Students Share Their Ideas “The teacher is just a guide in the class and should leave the initiative to the students. Teacher should give more opportunities for the students to talk. It doesn’t matter if they say something wrong if only they can express themselves. I can just correct them in time or let other students correct them. Anyway, we should get the students actively participate in the learning activities in class.” III. Students Share Their ideas After group discussions, the teacher calls on students to explain their methods III. Students Share Their Ideas Individual students demonstrate their solution in front of class • Teacher asks for volunteers • Students use visual aid, such as overhead projector, to demonstrate their (or their group’s) solution. • Usually, more than one student shows work • Teacher stands to the side and probes students to provide more complete explanations or more precise definitions. Summary Teacher Reviews Old Knowledge (wholeclass) Students Develop Methods and Solutions (small groups) Students Share Results (whole-class) Scripts and Beliefs are hard to change Cohen & Hill (2001) found nearly all teachers endorsed reform beliefs generally, but they agreed with specific ideas that were consistent with their existing models of mathematics instruction. • 89.6% agree that mathematics should “study mathematics in the context of everyday situations” • However, 52.1% also agreed “teachers should make sure that students are not confused a the end of a mathematics period.” Overview Quality control inspectors? Getting from standards and assessment to learning Historical perspectives Poor mathematical performance in the U.S. is a comparatively recent phenomena Sources of differences in mathematical performance between children in China and the U.S. Symbol structure The case of fractions Educational practices Implications for assessment & standards setting Implications What are the qualities of mathematics education in the U.S? What is the role of standards and assessment in influencing student thinking? What is the role of standards and assessment in influencing teaching?
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