Recognizing Misconceptions as Opportunities for …Comparing Decimal Numbers Compare 0.5 and 0.17...

Recognizing Misconceptions as Opportunities for

Learning Mathematics with Understanding

#LearnFromMisconceptions

Mark Ellis, Ph.D., NBCTProfessor, College of Education,

CSU Fullerton @EllisMathEd

Danielle Curran, M.A.AVP, Mathematics Instruction

and ImplementationCurriculum Associates

@danigirl1216

Agenda

• Understanding Misconceptions

• Things Adults Do or Say That Create or Reinforce Students’ Misconceptions

• Common Misconceptions and Things to Do and Say

• Productive Strategies for Eliciting and Addressing Misconceptions

Look at these two responses to the same addition problem.

What is different about the errors?

Mistake vs. Misconception

What distinguishes a misconception from a mistake or misapplication is the root of the error; it is a student’s understanding of the concept or relationship that is incomplete. In other words, it is a conceptual misunderstanding, not an error in calculation or notation.

What Is a Misconception?

Look at these two responses to the same addition problem.

What is different about the errors?

Mistake vs. Misconception

What Is a Misconception?

What distinguishes a misconception from a mistake or misapplication is the root of the error; it is a student’s understanding of the concept or relationship that is incomplete. In other words, it is a conceptual misunderstanding, not an error in calculation or notation.

Misconceptions Are Part of the Learning Process

“. . . erroneous understandings are termed

alternative conceptions or misconceptions (or

intuitive theories). Alternative conceptions

(misconceptions) are not unusual. In fact, they are

a normal part of the learning process.”

(Lucariello, J., & Naff, D., 2014)

Misconceptions Are Part of the Learning Process… When Learning Is Focused on Sense Making!

The Mathematical Practices Are All about LEARNING WITH UNDERSTANDING

Mathematically proficient students . . .

Make sense of problems and perseverein solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critiquethe reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

• Mathematics ability is primarily a function of opportunity, experience, and effort—not innate intelligence. • All students think mathematically.• Effective mathematics teaching cultivates students’ mathematical identities.

21st Century Research Tells Us:Every Person Thinks Mathematically!

(Boaler & Staples, 2008; Gutierrez, 2013; Kisker, et al., 2012; Malloy & Malloy, 1998; NCTM, 2000, 2014; National Research Council, 2009; Razfar, Khisty, & Chval, 2009.)

Things Adults Do or Say That Create or Reinforce Students’ Misconceptions

Things Adults Do/Say That Create or Reinforce Students’ Misconceptions

•What might teachers and other adults be saying/doing that contributes to student misconceptions?

Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: EQUAL SIGN

3 x 7 = ____ is read aloud to students as:

“Three times seven gives . . . ?”

“Three times seven equals . . . ?”

3 x 7 = ____ is more precisely read aloud to students

as, “Three times seven is equivalent to ____?” or

“Three times seven is the same as _____?”

Two strategies to reinforce the correct meaning of the equal sign:

A. Placing the missing expression to the left rather than the right

of the equal sign (e.g., ____ = 3 x 7).

B. Posing True/False statements that students must evaluate and

explain. For example: True or False? 3 x 7 = 7 + 7 + 7

Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: MULTIPLYING BY 10

17 x 10

“When you multiply by 10, just add a zero.”

1.7 x 10

“When you multiply by 10, just add a zero.”

1.7 x 10 = 1.70

1 7 x 1 0 = 1 7 010s 1s 100s

10s 1s

“When you multiply

by 10, each digit

graduates to the next

higher place value.”

Why does this

happen?

Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS

“Two point eight”

“Two and eight-tenths”

2.8 is read as, “Two and eight-tenths.”

2.8 is equivalent to 28

How might this change the way you think about:

2.8 x 15

10x 15

= (2 x 15) + (8

10x 15)

= 30 + (4

5x 15)

2.8 x 15

10x 15

= (2 x 15) + (8

10x 15)

= 30 + (4

5x 15)

2.8 x 15

= (28 x 0.1) x 15

= 28 x (0.1 x 15)

= 28 x 1.5

= 28 x 11

= 28 + 14

Questions?

Common Misconceptions and Ideas for Supporting Student Thinking

• What is the misconception?

• Why do students develop it?

• What are strategies to support student thinking?

Common Misconception: Counting Teen Numbers

For early learners, it is natural and common to read a number like 14 as “one four.”

What is the conceptual issue here?

Counting past 10 requires learners to understand place value. Reading 14 as “one four” tells us that there is not a good understanding of place value.

What sort of supports will help students to revise their understanding?

What if a student reads this value as “ten four”?

What if a student reads this value as “ten four”?This is perfectly fine as it demonstrates an understanding of place value! The vocabulary will come with repetition.

Common Misconception: Comparing Decimal Numbers

Compare 0.5 and 0.17

0.17 is larger than 0.5 because it has more digits.

Compare 0.5 and 0.17

0.17 is larger than 0.5 because it has more digits.

Working with whole numbers, more digits always gives a larger value. For example, a two-digit number is always

larger than a one-digit number.

Representing Decimal Numbers with a Hundredths Grid from Ready Classroom Mathematics (Grade 5)

Representing Decimal Numbers on a Number Line from Ready Classroom Mathematics (Grade 5)

Starting with the number 23.5, place a 0 anywhere you like (even between other digits) so that the new number is:

a) Equivalent to 23.5b) Greater than 23.5c) Less than 23.5

For each response, explain why your reasoning makes sense.

Common Misconception: Division by a Fraction

Describe a situation that is represented by 6 ÷ 1/2

Draw a model that represents 6 ÷ 1/2

6 people each get ½ of a candy bar. [multiplication by ½, not division]

6 pieces of candy are shared among 2 people.[division by 2, not ½]

Examining Patterns in Division by a Fraction from Ready Classroom Mathematics (Grade 6)

Questions?

Productive Strategies for Eliciting and Addressing Misconceptions

➢ Similar to what is done with early versions of a writing assignment in English or history classes, students’ rough draft thinking in mathematics offers a place to begin exploring, sense making, and revising their understanding.

➢ Such an approach communicates to students that learning mathematics is a generative process of developing, refining, and extending understanding rather than a static state of knowing or not knowing.

Reacting to Misconceptions: The Teacher Sets the Tone

“It is by the way that a teacher

responds to what a pupil offers that he

or she validates–or indeed fails to

validate–that pupil’s attempts to join in

the thinking.”(Barnes, 2008.)

Creating a Positive Climate for Errors

Students need a Positive Climate for Errors in the mathematics classroom.• Ideas are shared more often• Strong sense of

collaboration• Focus on analysis of errors

and sense making

(Steuer, Rosentritt-Brunn, & Dresel, 2013)

Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)

Evaluative Response

Stepping In

To learn more, see: https://www.mathagency.org/status-mindset-resources.

Lack of Sincerity or Specificity

Final Reflections

The Teacher’s Perspective Matters

“The best teachers focus on

how to look rather than

what to see.”

—Beau Lotto, Deviate

Changing What We Look for Changes What We Do

• Look for and value students’ ideas and assetsrather than seeing deficits.

• Look for ways to invite students into mathematics rather than diminishing their mathematical identities.

• Look for students’ brilliance rather than seeing the results of the failure to recognize and value this!

We Must Move Away from Teaching Mathematics as Rote, Teacher-Directed Compliance . . .

This Photo by Unknown Author is licensed under CC BY-NC-ND

https://journals.tdl.org/jume/index.php/JUME/article/view/273

And Move toward Making Learning Mathematics CENTERED ON STUDENT THINKING!

From Ready Classroom Mathematics and Reimagining the Mathematics Classroom.

Questions?

Mark Ellis, Ph.D., NBCTProfessor, College of Education,

CSU Fullerton @EllisMathEd

Thank you!

Danielle Curran, M.A.AVP, Mathematics Instruction

and ImplementationCurriculum Associates

#CurriculumAssoc

References

• Barnes, D. (2008). Exploratory talk for learning. In N. Mercer & S. Hodgkinson (Eds.), Exploring talk in school: Inspired by the work of Douglas Barnes (pp. 1–16). Thousand Oaks, CA: Sage.

• Cioe, M., King, S., Ostien, D., Pansa, N., & Staples, M. (2015). Moving students to “the why?” Mathematics Teaching in the Middle School, 20(8), 484–491.

• Cockburn, A. D., & Littler, G. (Eds.) (2008). Mathematical misconceptions: A guide for primary teachers. Thousand Oaks, CA: Sage.

• Ellis, M. W. (2018). Culturally responsive mathematics teaching: Knowing and valuing every learner. Billerica, MA: Curriculum Associates. Available online:

https://www.curriculumassociates.com/products/ready-classroom-mathematics/culturally-responsive-mathematics-teaching.

• Ellis, M. W. (2008). Leaving no child behind yet allowing none too far ahead: Ensuring (in)equity in mathematics education through the science of measurement and instruction. Teachers College Record,

110(6), 1330–1356.

• Ellis, M. W., & Berry, R. Q. (2005). The paradigm shift in mathematics education: Explanations and implications of reforming conceptions of teaching and learning. The Mathematics Educator, 15(1), 7–

• Gutiérrez, R. (2018). The need to rehumanize mathematics. In I. Goffney and R. Gutiérrez (Eds.), Rehumanizing mathematics for Black, Indigenous, and Latinx students (pp. 1–10). Reston, VA: NCTM.

• Jansen, A. (2019). Inviting rough-draft thinking to humanize the math classroom. Webinar presented for Association of Maryland Mathematics Teacher Educators. Available online:

https://www.youtube.com/watch?v=o2PfGyiWJK4.

• Jansen, A. (2020). Rough draft math: Revising to learn. Portsmouth, NH: Stenhouse.

• Jansen, A., Cooper, B, Vascellaro, S., & Wandless, P. (2016). Rough-draft talk in mathematics classrooms. Mathematics Teaching in the Middle School, 22(5), 304–307.

• Kalinec-Craig, C. (2017). The rights of the learner: A framework for promoting equity through formative assessment in mathematics education. Democracy & Education, 25(2), 1–11.

• Lucariello, J., & Naff, D. (2014). How do I get my students over their alternative conceptions (misconceptions) for learning? Removing barriers to aid in the development of the student. American

Psychological Association Coalition for Psychology in the Schools. http://www.apa.org/education/k12/misconceptions.aspx.

• National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education,

Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

• Steuer, G., Rosentritt-Brunn, G., & Dresel, M. (2013). Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate. Contemporary Educational Psychology, 38(3),

196–210.

• Yeh, C., Ellis, M., & Hurtado, C. (2017). Reimagining the mathematics classroom: Creating and sustaining productive learning environments, K–6. Reston, VA: National Council of Teachers of

Mathematics.

Recognizing Misconceptions as Opportunities for …Comparing Decimal Numbers Compare 0.5 and 0.17...

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