Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
#LearnFromMisconceptions
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
#LearnFromMisconceptions
Look at these two responses to the same addition problem.
What is different about the errors?
Mistake vs. Misconception
#LearnFromMisconceptions
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?
#LearnFromMisconceptions
Look at these two responses to the same addition problem.
What is different about the errors?
Mistake vs. Misconception
#LearnFromMisconceptions
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.
#LearnFromMisconceptions
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)
#LearnFromMisconceptions
Misconceptions Are Part of the Learning Process… When Learning Is Focused on Sense Making!
#LearnFromMisconceptions
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.
#LearnFromMisconceptions
• 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.)
#LearnFromMisconceptions
Things Adults Do or Say That Create or Reinforce Students’ Misconceptions
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions
•What might teachers and other adults be saying/doing that contributes to student misconceptions?
#LearnFromMisconceptions
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 . . . ?”
or
“Three times seven equals . . . ?”
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: EQUAL SIGN
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 _____?”
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: EQUAL SIGN
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
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: MULTIPLYING BY 10
17 x 10
#LearnFromMisconceptions
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.”
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: MULTIPLYING BY 10
1.7 x 10
“When you multiply by 10, just add a zero.”
1.7 x 10 = 1.70
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: MULTIPLYING BY 10
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?
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS
2.8
“Two point eight”
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS
2.8
“Two and eight-tenths”
28
10
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS
2.8 is read as, “Two and eight-tenths.”
2.8 is equivalent to 28
10
How might this change the way you think about:
2.8 x 15
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS
2.8 x 15
= 28
10x 15
= (2 x 15) + (8
10x 15)
= 30 + (4
5x 15)
#LearnFromMisconceptions
Things Adults Do/Say That Create or Reinforce Students’ Misconceptions: DECIMAL FRACTIONS
2.8 x 15
= 28
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
2
= 28 + 14
#LearnFromMisconceptions
Questions?
#LearnFromMisconceptions
Common Misconceptions and Ideas for Supporting Student Thinking
#LearnFromMisconceptions
Common Misconceptions and Ideas for Supporting Student Thinking
• What is the misconception?
• Why do students develop it?
• What are strategies to support student thinking?
#LearnFromMisconceptions
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?
#LearnFromMisconceptions
Common Misconception: Counting Teen Numbers
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?
#LearnFromMisconceptions
Common Misconception: Counting Teen Numbers
#LearnFromMisconceptions
Common Misconception: Counting Teen Numbers
What if a student reads this value as “ten four”?
#LearnFromMisconceptions
Common Misconception: Counting Teen Numbers
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.
#LearnFromMisconceptions
Common Misconception: Comparing Decimal Numbers
Compare 0.5 and 0.17
0.17 is larger than 0.5 because it has more digits.
#LearnFromMisconceptions
Common Misconception: Comparing Decimal Numbers
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.
#LearnFromMisconceptions
Common Misconception: Comparing Decimal Numbers
Representing Decimal Numbers with a Hundredths Grid from Ready Classroom Mathematics (Grade 5)
#LearnFromMisconceptions
Common Misconception: Comparing Decimal Numbers
Representing Decimal Numbers on a Number Line from Ready Classroom Mathematics (Grade 5)
#LearnFromMisconceptions
Common Misconception: Comparing Decimal Numbers
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.
#LearnFromMisconceptions
Common Misconception: Division by a Fraction
6 ÷1
2
Describe a situation that is represented by 6 ÷ 1/2
Draw a model that represents 6 ÷ 1/2
#LearnFromMisconceptions
Common Misconception: Division by a Fraction
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]
#LearnFromMisconceptions
Common Misconception: Division by a Fraction
#LearnFromMisconceptions
Common Misconception: Division by a Fraction
Examining Patterns in Division by a Fraction from Ready Classroom Mathematics (Grade 6)
#LearnFromMisconceptions
Questions?
#LearnFromMisconceptions
Productive Strategies for Eliciting and Addressing Misconceptions
#LearnFromMisconceptions
➢ 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.
#LearnFromMisconceptions
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.)
#LearnFromMisconceptions
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)
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
Evaluative Response
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
Stepping In
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
#LearnFromMisconceptions
Principles and Practices of Rough Draft Thinking (Jansen, et. al., 2016)
To learn more, see: https://www.mathagency.org/status-mindset-resources.
Lack of Sincerity or Specificity
#LearnFromMisconceptions
Final Reflections
#LearnFromMisconceptions
The Teacher’s Perspective Matters
“The best teachers focus on
how to look rather than
what to see.”
—Beau Lotto, Deviate
#LearnFromMisconceptions
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!
#LearnFromMisconceptions
We Must Move Away from Teaching Mathematics as Rote, Teacher-Directed Compliance . . .
58
This Photo by Unknown Author is licensed under CC BY-NC-ND
https://journals.tdl.org/jume/index.php/JUME/article/view/273
#LearnFromMisconceptions
And Move toward Making Learning Mathematics CENTERED ON STUDENT THINKING!
59
From Ready Classroom Mathematics and Reimagining the Mathematics Classroom.
#LearnFromMisconceptions
Questions?
#LearnFromMisconceptions
Mark Ellis, Ph.D., NBCTProfessor, College of Education,
CSU Fullerton @EllisMathEd
Thank you!
Danielle Curran, M.A.AVP, Mathematics Instruction
and ImplementationCurriculum Associates
#CurriculumAssoc
#LearnFromMisconceptions
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–
17.
• 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.