Why bother with the Rule of 3 to begin with?: A comprehensive multimedia look at the varied rationale behind the exploration of the Rule of 3
Tim Boerst
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In this section I will portray a number of angles from which learning about and taking instructional action in light the Rule of 3 seemed prudent. I do not pretend to connect all educational agendas into one neat package, for to do so would be practically (if not theoretically) impossible. It is however possible to see investigation and implementation of instructional practices guided by the Rule of 3 as warranted in multiple ways. The following are considerations I made when deciding to use the Rule of 3 with students in my 5th grade classroom.
The latest version of the NCTM Standards (2000), Principles and Standards for School Mathematics (PSSM), introduced representation as a mathematical process with importance across mathematical topics and across grade levels k-12. Initially the distinction between representation and the "communication" standard that was a hold over from the original standards (NCTM, 1989) was admittedly hazy. My understanding at this point is that instruction needs to ensure that students can communicate mathematical ideas with clarity and precision, but also that they can construct, connect, and decipher multiple representations of mathematical ideas. Certainly I had engaged students in learning about different representations in the past, but not in the integrated/intensive way advocated by the PSSM. Also the PSSM have a much more pronounced emphasis on algebraic thinking across the k-6 spectrum than before. For these reasons investigating/ utilizing the Rule of 3 in classroom instruction aligned strongly with the recommendations in national standards.
For more information, do a quick search on MEAP here.
...In this light using the Rule of 3 to help students explore multiple representations serves educational and larger practical issues of teachers in Michigan....
In 2002 Michigan revised its curriculum frameworks for mathematics (pdf) and articulated individual grade level benchmarks for the first time. These documents make plain an increased attention to algebra and continued emphasis upon data analysis. According to these benchmarks students must not only know about algebraic, numerical, and graphic representations, they must be able to use them in sophisticated ways. Charged with operationalizing these benchmarks, the state recently revamped the Michigan Education Assessment Program (MEAP), the state high stakes standardized test. There are very few if any "raw" computational problems for students to complete. Instead most items require students to select and/or interpret a broad array of representations. Whatever may be thought about the test itself (since its format and uses are problematic in a number of ways), without diverse experiences with numerical, graphical, algebraic, and verbal representations, students face the prospect of being ill-prepared to demonstrate mathematical proficiency. Furthermore, with heavy accountability consequences in place for districts, schools, and teachers, student performance on standardized tests can no longer be brushed aside as "one measure" (2). In fact if thoughtful instructional action is not undertaken to address student knowledge of mathematical representations, we the run the very real risk that ill-conceived ways to address "problems" highlighted by standardized test results will be forcefully advocated. In this light using the Rule of 3 to help students explore multiple representations serves educational and larger practical issues of teachers in Michigan.
...Focus on representation and the Rule of 3 is a way to connect elementary school mathematics learning with the content and practices of mathematics learning in later grades....
Below: two recent examples of articles describing & supporting the exploration of "higher level" mathematics in the elementary grades, appearing with increasing frequency in professional journals.
"Developing Elementary Teachers’ 'Algebra Eyes and Ears'" by Maria L. Blanton and James J. Kaput
"Algebraic Problem Solving in the Primary Grades" by Robert B. Femiano
Math Trailblazers, Second Edition, grade five Student Guide (2004). Copyright by Kendall/Hunt Publishing Company. Reprinted with permission.
District
Curriculum Rearticulation
My school district has recently decided to explore the adoption a new
mathematics text series in hopes to better support teaching and learning
aligned with state and national standards. All of the NSF endorsed texts
for elementary students were explored. We noticed many differences between
the new texts and those that we had been using for the previous decade.
New topics were introduced, more difficult concepts were explored at younger
ages, the order of topics was different, and the texts layered mathematical
ideas throughout the text instead of lumping them in separate sections.
If district teachers are to use the new curricular materials, we will
need to think about mathematics in a far more integrated fashion and be
able to help students (who will have little prior experience for doing
so) utilize algebra, graphs, as well as numbers to make sense of problem
solving contexts. Investigation of the Rule of 3 will support understanding
of the instructional issues that teachers will face as they attempt to
teach in ways that interrelate multiple representations and diverse mathematical
ideas.
Furthermore, this integration of representations is aligned with a growing movement to expand attention K-12 to topics previously reserved for higher level mathematics, such as algebraic thinking. Developing instructional strategies that allow students to access higher level mathematical ideas in ways that connect with prior knowledge and current mathematical proficiency is challenging work. Focus on representation and the Rule of 3 is a way to connect elementary school mathematics learning with the content and practices of mathematics learning in later grades.
Notice the tight clustering of all representational types across just a few pages in one of the texts that our district piloted (click for larger view).
K-12 attention to algebra is evident in the latest version of the NCTM Standards.
...In this case use of the Rule to 3 was not only professionally responsible, it was also understood and accepted by parents as a challenging track of engagement for their advanced student....
Involving parents with new curricular materials: what one NSF endorsed mathematics text has to offer (pdf)
My classroom phone rang on the day before school started. After a short friendly exchange the parent on the other end of the line inquired as to whether I had developed a mathematics curriculum that would meet the accelerated mathematical needs of his child. While it is currently lost in fray of heightened concerns about students who will not meet grade level benchmarks, many parents do not worry that their children will attain the grade level benchmarks, but rather that their children will not be encouraged to meet higher more individually appropriate standards. Parents can be important sources of information about student learning needs and also critical partners in education. The challenge of meeting the learning needs of students (advanced, typical, or remedial) is coupled with the challenge of gaining parents confidence that what is being done is educationally warranted. Creating separate curricula for individual students is fraught with difficulties (intellectual, social, psychological, environmental,…) and yet meeting the needs of advanced students is a perennial issue that must be addressed in thoughtful fashion. As will be seen elsewhere in this snapshot students can be challenged through development of algebraic, graphic, numerical, and verbal representations of problem solving solutions. Furthermore, connection between representations, translation across them, and critical reflection upon their relative strengths is demanding work worthy of advanced student engagement. In this case use of the Rule to 3 was not only professionally responsible, it was also understood and accepted by parents as a challenging track of engagement for their advanced student.
View a video excerpt from a Teacher Reflection Group meeting excerpt describing a student who was able to use a graphic representation to access an initially difficult problem (X minutes, requires QuickTime Player). External speakers recommended.
In elementary mathematics the disproportionate focus of tasks and time upon numerical/computational work make the subject as a whole appear out of reach to children whose strengths lie elsewhere. Finding instructional approaches that nurture student perception of mathematics as a discipline where there are routinely multiple paths of productive engagement is essential to improving access to the subject. It has been the case that students who were confused and frustrated by numerical attempts to solve problems can utilize graphical approaches to arrive at initial solutions and then look back with a greater willingness and fresh insight at numerical approaches. The Rule of 3 can improve student access to problem solving experiences and be a means of improving mathematical dispositions.
View student work samples with commentary
...The Rule of 3 can be thought of as a way to encourage new and deeper exploration of the mathematics involved in problem solving tasks....
Subject
Matter Rigor
Quite often problem solving experiences appear to be about finding particular
answers or applying/practicing previously learned techniques in mathematical
scenarios. In the text that our district has abandoned there were typically
4-6 "story problems" at the bottom of each page where students
would apply the skill they had practiced in 20-30 decontextualized items
at the top of the page. Sometimes the problems on the mathematical menus
that I assigned were approached in the same way. Some students appeared
to believe that solving menu problems would merely be about a search to
match known strategies and comfortable representations to problem scenarios
with which they seemed to fit. This approach often led to minimalist problem
solving attempts, surface level solutions, and a lack of justification.
Simply put, the work lacked mathematical rigor. Greater rigor could be
encouraged through an increased emphasis upon algebraic and graphic representations
since they facilitate examination of the "big picture" beyond
the specifics of particular problems. The Rule of 3 can be thought
of as a way to encourage new and deeper exploration of the mathematics
involved in problem solving tasks.
Notice the increasingly varied and detailed use of multiple mathematical representations within one student's math menu problem solving work over the course of the year. The top two samples from early in the year are based in numerical and linguistic representations
...The Rule of 3 is an impetus to reexamine instructional tasks and spur exploration of the mathematics within them.
View a TRG
"rough case" meeting excerpt where the group discusses the professional
challenge of looking in new ways at old instructional tasks.
(Video coming soon.)
Even though no two classes and no two school years are ever the same, it is possible to become comfortable with worthwhile instructional tasks after using them over a period of time. I had designed and revised the sets of problems on the math menu over a period of 5 years so that, while they did change, they became a familiar instructional landscape. Introducing a new approach like formulating multiple representations for each menu problem could prove professionally challenging in the sense that I would not only need to think through each representational possibility, but also consider instructional implications, possible student difficulties, and dimensions of assessment (3). The Rule of 3 is an impetus to reexamine instructional tasks and spur exploration of the mathematics within them.