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Fine-grained, continuous assessment for the diverse classroom: A key factor to increase performance in mathematics

Authors:

Abstract

The increasing focus on formative assessment has permeated through educational policy internationally. Yet, scholars call for more empirical data to support the promise of formative assessment, as well as the development of theory. In this paper we propose that continuous assessment for everyone during instruction— a focus not commonly addressed in the literature—can be a key factor for students' achievement in mathematics. We present results from an intervention at one school with long history of poor performance in mathematics. Preliminary results from a longitudinal study showed that the intervention impacted positively on students' basic mathematical skill. However, differences in teaching practices suggest that instant, continuous assessment for all students is a key factor for improving mathematical performance.
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Paper Title
Fine-Grained, Continuous Assessment for the
Diverse Classroom: A Key Factor to Increase Performance in
Mathematics
Armando Paulino Preciado Babb, University of
Calgary; Martina Metz; Soroush Sabbaghan, University of
Cagary; Brent Davis, University of Calgary; Sharon Friesen,
University of Calgary
Author(s)
Understanding Structures That Support
Mathematics Teachers and Effective Practices
Session Title
Roundtable Presentation
4/12/2016
Presentation Date
Washington, D.C.
Presentation Location
Mathematics Education, Teacher Education - In-
Service/Professional Development, Teacher Knowledge
(cultural/content/pedagogical)
Descriptors
Mixed Method
Methodology
Division K - Teaching and Teacher Education
Unit
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Fine-grained, continuous assessment for the diverse
classroom: A key factor to increase performance in
mathematics
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Abstract—The increasing focus on formative assessment has
permeated through educational policy internationally. Yet, scholars call
for more empirical data to support the promise of formative
assessment, as well as the development of theory. In this paper we
propose that continuous assessment for everyone during instruction—
a focus not commonly addressed in the literature—can be a key factor
for students’ achievement in mathematics. We present results from an
intervention at one school with long history of poor performance in
mathematics. Preliminary results from a longitudinal study showed
that the intervention impacted positively on students’ basic
mathematical skill. However, di#erences in teaching practices suggest
that instant, continuous assessment for all students is a key factor for
improving mathematical performance.
Keywords*+,
 
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Introduction
 -&.'(&./&,01
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Continuous, all students’ response formative assessment
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<") 
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6
     ) A " 7!43
-$%&;29:"   "
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9  -7!4  $%&;  ")  =C2)            
            +            
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       A  +   "  "      
+ 3?"
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""3)6
-$%&=2""micro-sca#olding*
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" 
    ?  "    
 -
$%&=2        <  ")  
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+G+
< +  ) 4  G  
<"" -&./&2
 ")
Mathematics teacher knowledge and curricular resources
4      A          "    -2
      -2        -2  
") #   " + 3 "
+          3          
knowing more mathematics -&./$&./.,&..;2knowing
mathematics di#erently -  4  1  "  $%%(,    1    $%%C,
 1 #$%&;2)  ""    "   +
)! ""
/
" E
+- 1 ! &..',1LM $%%=2)  LM
-$%%=2educative curriculum materials*
"" ->14 $%&;,01
!$%&$, ":14$%&C,4+$%&$2) A
+"3
+")
A+"D+ -1!
&..', 6 1  &.(., " 1 0  $%%&, # &...
$%&$2)   intended  enacted   #  -$%%=2
            5  ""    "
5)6++ 
 adopts+"
+") 0+ + + <"  
D+ ""
""+ " ) A"
  + &delity       <    +    +  
 "")
    #3  -$%%=2  D  +      
        -$%&=2    +   intended
object of learning    enacted object of learning) 4 "  
intended object of learning+<"M
*+ "")A+
(
  "    +    M          
M)@
+<"
)AM 
             *  H<"    "  
 
      "J  -  ")  ($2)  4  M      
            3  -$%&=2 variation
theory"+"")
 
+3")
Methodology
4 A?""-!
! >1$%%C2 3+
  "  +    "          "  
3            ""    
)!G 
)4+L?'
+    "" - &=% 2  
  " "  ) A $%&C    
" B  - 1L $%&%2 ") 4
B-$%&=2"5"
      
.
  )        ?
++" 
-$%&&2)
"+"M+
"B"
)""++
  )A$%&C?$%&; ""
 ""
)
4         " 
 !4-!47$%&'2+
+          + 
) 4      
  ""      "        "
" "", 
  "")       
)4$%&CN$%&; +
"+ ) +
+ 
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!4?!"+
 +        ) 4  +
$%&C$%&;
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 ):'+"$%&;
 )
A""+++"*4+C'
4+&&)!
!4 +
" ")+ +  
    "      "        8  
 B)
    +        +    D    +
*          
G<)
Findings
      ?      "    -7#2    !4    
    $%&C    $%&;  +    8  "    (
":&=-=0.004, t(66)=-2.952)4
3  : ' $%"-"O %)%%'
-&&2O?$)((2*4" )A
3 7# 4 + 
D    $%&C    $%&;  -O% ) 2  )  4
""""
"+)
6      +    + 
)6 
&&
  " B    +  
  "    )   +
 "   3 ) 4   " 
 +     +    
        M    )  4      +
"              "  
+"G)6 
  +      B      "      
++ +
??   +  "  
)
    8    D      "      +
)  4      D  +  4  3      
 )+
   ? "     ) A 
4    +      +
+"<")
  D  +          G    )  4  
+  G   ) 0+
++G)4
GH??F )J
                     "
++ + "
&$
G   "     + <) 6 
++M"
B")4
   " +) 4
+++
+  ) These differences might be reasons for the difference in performance in the two
groups. We conclude that while both teachers adopted the curricular material, Teacher B did not
follow its principle of continuous assessment.
Conclusions
4""    A  "" 
       " + 3
+    "          "    3
) 4   !4 7#  $%&C N $%&; + 
8          "    "  "  
3)> + 
$%&;N$%&=++""")
4"D+  +
 "")8+++
" 3 7#   !4      < ) 4
" "      
+D+3
"      )  4 "      
              "  
&C
G         "          D    
3")   
 +      < " + *
4 " +  +
+++ )
 " "  
"     )      
                  +       
" )<
   F 3 ")    
+)
 ) 
M$%&;"
 )8"
3 
D")
<""+"
+ """
")
@8+""""
G"
"5""+
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"+
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References
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+)A)1)-)2Proceedings of the
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... While a focus on systematic variation has been widely reported in the literature on learning study as critical for task design and research analysis of mathematics lessons, we assert that such focus can also have a significant impact on how the tasks are enacted in the classroom. Like Kullberg et al. (2014), who reported how the same task can be enacted in different ways and offer different possibilities to learn, we have observed (Preciado-Babb et al. 2016a) how the use of the same lesson plans by different teachers yielded contrasting results in terms of students' learning (as measured by standardized tests and as observed in terms of student engagement in lessons). In this chapter, we stress the importance of both continuous assessment during the enactment of a lesson and teachers' appropriate responses to student feedback. ...
... The results were not the same for all groups (Preciado-Babb et al. 2016a). By contrasting results and observations from different classrooms, we were able to identify teaching approaches associated with higher rates of improvement in CTBS scores, particularly conceptual understanding, and higher levels of student engagement in mathematical activities, as observed during classroom visits. ...
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The use of systematic variance and invariance has been identified as a critical aspect for the design of mathematics lessons in many countries where different forms of lesson study and learning study are common. However, a focus on specific teaching strategies is less frequent in the literature. In particular, the use of systematic variation to inform teachers’ continuous decision-making during class is uncommon. In this chapter, we report on the use of variation theory in the Math Minds Initiative, a project focused on improving mathematics learning at the elementary level. We describe how variation theory is embedded in a teaching approach consisting of four components developed empirically through the longitudinal analysis of more than 5 years of observations of mathematics lessons and students’ performance in mathematics. We also discuss the pivotal role of the particular teaching resource used in the initiative. To illustrate, we offer an analysis of our work with a Grade 1 lesson on understanding tens and ones and a Grade 5 lesson on distinguishing partitive and quotitive division.
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The Math Minds Initiative began with the goal to improve mathematics instruction at the elementary level, with an eye to all learners continuously extending their mathematical understanding. The project has integrated findings from cognitive sciences and six years of empirical data to develop both a teaching model (the RaPID model) and a framework for lesson observation and analysis. The model identifies essential elements of effective mathematics lessons, and the framework offers a lens for both teachers and researchers to attend to and analyze the interplay of resource, teacher, and learner in the context of individual lessons, with consideration of how they are woven into a broader learning trajectory. When project teachers have used the model/protocol in collaboration with a resource that clearly identifies and weaves together key mathematical ideas, we have seen a powerful impact on student learning. The chapter elaborates on the development of this framework, with particular attention to contemporary literature from the cognitive sciences that has oriented the project; it also includes a critical review of hundreds of theories of learning, their assumptions on the nature of mathematical knowledge, and their advice for teaching. We conclude the chapter by elaborating on implications for teacher professional development.
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For four years we have invested in improving mathematics teaching at the elementary level. By drawing from diverse research emphases in mathematics education and by considering the impact of lessons in terms of student engagement and performance, we have identified four key elements impacting learning in mathematics. Here, we describe the protocol currently used to structure feedback for teachers in the Math Minds Initiative. The key elements that comprise the protocol are: (1) effective variation, (2) continuous assessment, (3) responsive teaching, and (4) engagement.
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The Handbook of Formative Assessment in the Disciplines addresses current developments in the field, with a focus on domain dependency. Building from an updated definition of formative assessment, the book covers the integration of measurement principles into practice; the operationalization of formative assessment within specific domains, beyond generic strategies; evolving research directions including student involvement and self-regulation; and new approaches to the challenges of incorporating formative assessment training into pre-service and in-service educator training.
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The value and effectiveness of formative assessment in the classroom has gained an increasing amount of attention during the past decade, especially since the publication of seminal work by Black and Wiliam titled Assessment and Classroom Learning. Since that time, there has been a renewed interest in describing and evaluating teacher practices related to formative assessment. Based on evidence of its effectiveness in the classroom and on improving standardized test scores, many prominent educational entities have initiated reform efforts to promote the use of formative assessment, yet these practices have not been embraced by classroom teachers. This case study investigated internally constructed and externally imposed contextual elements that constrained or facilitated the use of formative assessment by three high school science teachers. Cornett’s curriculum development model of personal practice theories was modified to include assessment, termed personal practice assessment theories (PPATs), and chosen as a framework for the study. This research revealed distinct differences among the three teachers’ PPATs and several different factors that constrained or facilitated the use of formative assessment in their instruction. Most notable of these factors were the forms of teacher knowledge that played a critical role in shaping their assessment practices and had a bearing on their ability to convert espoused theories about assessment into actual classroom practice. Other externally imposed barriers that constrained the use of formative assessment included expectations, habits, and dispositions of students; the pressure that teachers felt to “cover” all of the curriculum in order to prepare students for the end-of-year, high-stakes exam; and an instructivist rather than constructivist approach to teaching and learning. Results from this study add to the growing body of knowledge about the complex terrain teachers negotiate in making teaching and assessment decisions and provides a framework for future studies.
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How can you keep all students engaged in deepening their mathematical understanding without overwhelming the weakest students or boring the strongest? Teachers in the Math Minds project design lessons around structured sequences that seek to engage all students with questions on which they can succeed, and to then proceed through increasingly sophisticated variations. Teachers attend closely to student responses so that they can adjust difficulty in a manner that allows success and challenge for all. In this paper, we describe key principles that have emerged from the Math Minds initiative. We draw particular attention to variation theory (Marton, 2015) and consider how it plays out in interaction with the other principles.
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