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Marilyn Burns: Mathematical Theorist 1

Marilyn Burns: Mathematical Theorist

Student Name

Austin Peay State University

Marilyn Burns: Mathematical Theorist 2

Marilyn Burns: Mathematical Theorist

Introduction

“My vendetta is against the ‘yours is not to question why, just invert and multiply’ and

other rule-bound teaching…” (Burns, 1998, p. 77). Marilyn Burns, the author of this quote, is at

the forefront of the new paradigm in mathematics education. Her five decades of work, which

continues today, emphasizes the importance of student content understanding and mastery,

instead of the traditional memorization of procedures (Burns, 1998; Burns 2007). Her work

involves an implementation of the work of Jean Piaget and individual constructivism (Burns,

1982; Burns, 2007; Ormrod, 2014). Her emphasis on student needs, learning style and student-

constructed knowledge is a new frontier in math education.

Major Strategies of Theory

Student-Based Inquiry

Allowing students to construct their own knowledge, both individually and in social

contexts, is critical to Burns’ theory. She says, “That’s the major part of learning: figuring things

out in order to understand them for yourself” (Burns, 1998, p. 71). Whether instruction derives

from a book, video, teacher, or any other source, the only way a student can learn is if he or she

processes for understanding. To best serve student needs, teachers must enable them to develop

personal understandings in meaningful ways. This can be accomplished both through individual

and group-based inquiry. Burns (1998) continues “Immerse children in doing mathematics by

involving them in activities, explorations, and experiments in which they use mathematics and,

by doing so, learn mathematical concepts and skill” (p.69). The group-based model also engages

all achievement levels in learning. The low performing students will have an opportunity for

more individualized time with other students who can offer new insights (Burns, 2007). Also,

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high performing students are able to gain mastery of requisite concepts and skills by teaching.

Burns (1998) explains, “It’s said that you know something best when you teach it. That’s

because teaching requires figuring out why something makes sense and then thinking about how

to present the idea so that someone else can make sense of it, too” (p.72). By engaging students

in constructing understanding, Burns argues those better serve their needs in learning.

Manipulatives

Piaget argues there are four criteria necessary for students to learn: maturity, physical

experience, social interaction, and equilibration (Burns, 1982; Burns, 2007; Ormrod, 2014).

Burns argues that students gain better mathematical experiences by interacting with physical

objects (Burns, 1998; Burns, 2007). The use of manipulatives, or any physical object, engages

students in discovery and understanding. As an example, when calculating perimeter, area, or

volume students should be given everyday items to measure (Burns, 2007). Burns (2007) says,

“The more experiences children have with physical objects in the environment, the more likely

related understanding will develop” (p. 31). Burns (2007) also notes, “…manipulative materials

help students make sense of abstract ideas” (p.33). In another article, Burns (1982) says, “The

more experiences your students have with physical objects in the environment, the more likely

that related understandings will develop” (p. 48). She then concludes, “manipulative materials

make mathematics learning more engaging and interesting by lifting mathematics off textbook

and workbook pages” (Burns, 2007, p. 33). These ideas demonstrate that student needs are best

served by using physical materials because hey give real world context to mathematical

abstraction.

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Perseverance and Mastery in Assessment

“There is no compelling reason to solve most problems on the spot. Mathematicians

sometimes work on the same problem for years before they arrive at a solution” (Burns, 1998, p.

47). Burns highlights the failure of traditional math assessments. Timed assessments, notably

about arithmetic skills, are commonly used to ensure memorization of mathematical skills.

These assessments give no emphasis to content (Burns, 1998; Burns 2007). Burns (1998)

continues, “…being stressed negatively affects our ability to think clearly” (p. 46). She also has

several examples of student responses to a prompt concerning how they approach timed math

tests. All of these responses are negative (Burns, 1998). The emphasis of math assessments

should, as Burns (1998) says, “…in mathematical problem solving on thinking and reasoning,

not on speed” (p. 48). She continues, “…the purpose of giving tests in school is to find out what

children understand so that we can build from there. Timed tests don’t measure what children

understand, but what they can recall” (Burns, 1998, p. 53). The emphasis on timed tests does not

account for what teachers and students need. Burns engages the idea of focusing on content

master not regurgitation of fact.

Appropriate Development of Academic Language

Burns (2007) notes that the development of academic language is critical to proper

mathematical understandings. She however insists that content understanding must precede

vocabulary. She says, “The vocabulary itself will not reveal the meaning of the concept, nor will

it serve any useful purpose without the prerequisite mathematical understanding” (Burns, 2007,

p. 43). Use of words like slope, equation, inequality, dependent variable, and derivative are

meaningless unless the appropriate concepts are first understood. Burns (1998) suggests

allowing students to initially label concepts using personal terminology, develop the significance

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of the content, and then define the mathematical terminology. She gives the example that words

like slanty-ness or steepness of a line are appropriate when developing the idea of slope (Burns,

2007). Burns continues, “…after introducing new vocabulary, it’s important to use the words

often and in the context of experiences… keep in mind that language acquisition takes time and

occurs from connecting words to experiences” (Burns, 2007, pp. 373-374). The appropriate

development of academic language engages students in connecting concepts to vocabulary

instead of rote memorization.

Develop Mathematical Thinking

Connections to real-world examples are important in engaging students in their math

education. Burns (1985) insists, “…it is essential that the development of arithmetic

understanding be embedded in the context of helping children become powerful mathematical

thinkers, able to solve problems, see relationships and patterns, and use number confidently to

come to decisions” (p. 14). Simple rote memorization of procedures is ineffective in engaging

students in mathematics learning. The goal of mathematics is to take real world problems,

abstract those into mathematical concepts, and then apply mathematics to make reasonable

inferences about the findings (Burns, 1998). Burns (1985) notes, “Teachers need to emphasize

to students that the foremost goal of their mathematics learning is thinking mathematically, not

merely getting right answers” (p. 15). Often, good mathematics problems do not have a right or

wrong answer (Burns, 1985). An important tool is to select a best answer, which can vary from

student to student. Burns (1985) concludes her article, “The Role of Questioning,” with the

thought, “Classroom experiences must extend beyond the goal of arriving at correct answers…

To expect less is pointless and demeaning” (p. 16). Students that can think mathematically can

apply concepts in varied ways to construct meaningful responses to situations.

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Rationale of Selection

“Even in the face of widespread failure in learning mathematics, we seem to want to cling

to educational methods with a nostalgia for them that has long outlasted their usefulness and has

perpetuated failure” (Burns, 1998, p. x). Burns’ methodology focuses on how students engage

with mathematics and approach problems (Shaughnessy, 2012). The Math Reasoning Inventory

(MRI), developed by Burns’ organization Math Solutions, contains interviews of students

explaining how they think about math. This online database is free of charge, and is an example

of how Burns has approached math education (Shaughnessy, 2012). By understanding how

students learn, Burns is able to design a methodology that engages students in meaningful

learning. The higher level of concept mastery also mirrors the emphasis on high-level thinking

of Bloom’s Taxonomy (Ormrod, 2014). Burns emphasizes that it is insufficient to rely on

procedural knowledge, but instead to focus on content understanding and application (Burns,

1985; Burns, 1998; Burns, 2007). By engaging in deeper mathematics, students gain better

understandings of concepts.

Burns also realizes that all students can learn and be successful in mathematics education.

Burns (1998) insists, “…most of us are perfectly capable of learning mathematics with

understanding and with pleasure. It all depends on how we’re taught” (p. 78). The idea that all

students can learn math is refreshing. Instead of labeling students as good and bad, the

recognition that different students need non-traditional approaches engages all without

abandoning some (Burns, 1998). Burns finds ways to engage every student in meaningful ways

that allow for individuality despite universal appropriateness. Burns sincerely desires for every

student to be successful, and this desire leads her to find new, effective methods that enable that

student success

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Implementation of Theory

Group Tasks

An emphasis found in Common Core and the upcoming implementation of the PARCC

Assessment is constructed response questions. A primary form to develop these skills is an

assessment called a task. These typically have a prompt that explains the situation to be

examined. Following the prompt are two to three questions that build upon one another that

require deeper exploration of the topic. I plan to utilize a task at least once a week. These will

typically implemented in small student groups of between three and five students. This form of

assessment is designed for students to reach beyond their current understanding and apply

several concepts in crafting a meaningful response. They require multiple representations, like

graphs, equations and expressions, tables, graphic organizers, etc. that help to foster the

interconnectivity of mathematical concepts. These assessments address Burns’ theory by

addressing student-based inquiry, perseverance and mastery, as well as to develop mathematical

thinking.

Non-Traditional Math Assessments

When prompted to consider what types of tests, quizzes and homework one might expect

in a math class, the vast majority of responses are procedural problems (Burns, 1998). Tests and

quizzes emphasize working large number of abstract problems with few, if any, application

problems. For my class, I intend to offer different assessments to try to engage as many students

as possible. A key part of the different style I intend to employ is the ability for students to

receive feedback from myself, or their peers, before turning their work in for a grade. If I will

review homework for a grade, I want to discuss it in class and allow the student to go home to fix

errors before turning submitting. In most real-world applications, major projects are reviewed,

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reworked and edited repeatedly before completion. There is no reason a math class cannot be

modeled in the same way. For the upcoming Residency I assignment, I will use a portfolio as

my assessment. This will contain several artifacts with different types of work. These will be

turned in after the completion of the unit with students having an opportunity to review and

revise their work prior to submission. This implementation encourages perseverance and

mastery, developing mathematical thinking, student based inquiry, and could be utilized in the

development of academic language.

Classroom Environment

When asked what a typical math classroom looks like, the common response is neat little

rows with student desks only touching back to front (Burns, 1998; Burns, 2007). Burns (1982)

counter, “…there is no need for children to have to work individually as a general rule” (p. 47).

By engaging in collaborative learning with student groups taking the lead in developing content

understanding. I want my class to emphasize this idea. I want a classroom where there is a dull

roar because students are engaged, laughing, learning and having fun. I also want students to

feel comfortable respectfully critiquing other student’s work. The ability to identify errors in

logic and understand where mistakes arise is critical in developing mathematical thinking

(Burns, 1982; Burns, 1998; Burns, 2007). I want my class to be like the one Burns (1982)

described as “…one where children are working together, encouraged to share their ideas with

each other and with you, and dealing with concrete materials from which conceptual

understandings can be created” (p. 48). This environment would enable the application of

student-based inquiry, development of academic language, and development of academic

thinking.

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Manipulatives/Technology

Many consider manipulatives to be acceptable only in an elementary classroom; however,

Burns (1998) argues that every developmental level can benefit from the use of physical objects.

These need not be colored tiles or blocks, though they have their uses, but instead can be soda

cans to talk about volume, everyday round items to measure circumference and diameter, or a

sequence of wooden cubes used to teach geometric series. Students learn best when interacting

with physical objects that model abstract mathematical concepts (Burns, 1998; Burns 2007). I

will use everyday items to engage students in class activities. These objects make the abstract

formulas meaningful. However, manipulatives may be extended beyond the traditional concept

of physical objects. If the goal is to engage students in abstract modeling of real-world

phenomena, then the appropriate use of technology also achieves that goal. Many teachers do

not wish for students to use calculators. However, the modern graphing calculator is an

impressive tool that permits quick repetition and representations to be created. However, the

tools available over the Internet far exceed even the graphing calculator. Using Microsoft Excel,

Mathematica, MatLab and other educational tools, bring math into the hands of students. By

using physical manipulatives and available technologies, students develop mathematical

thinking, use manipulatives, and persevere in new ways.

Academic Language

The terminology and symbols used in math classes are some of the most abstract and

difficult for students to learn. They are in essence their own language, or at least a new dialect.

In order for students to acquire the requisite skills, the language, as Burns (1998; 2007) notes,

must be appropriately developed. The rote memorization of definitions is ineffective. Simply

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telling students to write, or fill in the blank will not develop understanding, as Burns (2007)

notes. I want to make math vocabulary and concepts accessible by playing games like Jeopardy

and Bingo. By hints over general concepts about application, the students can mark the

appropriate terminology or symbol to indicate mastery. This can be done repeatedly and engages

full class engagement. Other games like “I have/ who has” enables students to connect concepts

to definitions and terminology in meaningful ways. This academic language development also

can be used to effectively develop mathematical thinking.

Discovery and Problem-Solving

In statistics units and courses, students are asked to draw inferences from teacher

provided data. Burns(1985) insists, “…children’s classroom experiences need to lead the to

make predictions, formulate generalizations, justify their thinking, consider how ideas can be

expanded…those insights…open up new areas to investigate” (p. 16). Instead of abstract

meaningful data, I intend to have my students make predictions about a question, collect data on

the question, and then evaluate whether their prediction fits the data and why there is

inconsistency. An example would be, when asked to pick a random number 1-10, are all

numbers equally likely? Most students would predict that the answer is yes, or nearly yes.

However, data typically indicates that about 40% of people select the number 7. This approach

engages students in data collection, theoretical thinking, a discussion on probability distributions,

and critical analysis. Instead of simply asking students to define a uniform distribution and what

the expected value is, they instead engage in meaningful mathematical inquiry and discourse.

This engages students in more rigorous mathematical discussions, and leads to deeper content

understanding.

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Conclusion

Marilyn Burns’ efforts create an effective methodology for all mathematics classrooms.

She engages students by forcing them to construct their own mathematical understandings under

the tutelage of an effective teacher. By engaging students in deeper content understandings,

opening up dialogue between students, and establishing better classroom management, Burns

method clearly demonstrates a situation that ensure student success. This method has been an

inspiration to me as a perspective math teacher and I hope that I can create an atmosphere like

Burns describes. The best part is that students are no longer simply receptacles that a teacher

simply dumps knowledge into, but rather become an instrument in their own education.

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References

Burns, M. (1982). How to teach problem solving. The Arithmetic Teacher, 29(6), 46-49.

Burns, M. (1983). Put some probability in your classroom. The Arithmetic Teacher, 30(7), 21-22.

Burns, M. (1985). The role of questioning. The Arithmetic Teacher, 32(6), 14-16.

Burns, M. (1998). Math: Facing an American phobia. Sausalito, CA: Math Solutions

Publications.

Burns, M. (2007). About teaching mathematics: A K-8 resource (3rd ed.). Sausalito, CA: Math

Solutions Publications.

Mitchell, C.E. (1999). Math: Facing an American phobia by Marilyn Burns [Review of the book

Math: Facing an American Phobia, by M. Burns]. Teaching Children Mathematics, 5(7),

446.

Ormrod, J.E. (2014). Educational psychology: Developing learners (8th ed.). Boston, MA:

Pearson.

Shaughnessy, M.F. (2012, Feb. 13). An interview with Marilyn Burns: Informal math inventory.

[Web log comment]. Retrieved from http://www.educationviews.org/an-interview-with-

marilyn-burns-informal-math-inventory/.

Viebranz, G. (1999). Math: Facing an American phobia by Marilyn Burns [Review of the book

Math: Facing an American Phobia, by M. Burns]. Mathematics Teaching in the Middle

School, 4(4), 271-272.