middle school math concepts: “is this reasonable?” esther kim [email protected]
TRANSCRIPT
2011 API By SubgroupSubgroup API % of school
population
All Students 736 100%African American 596 5%
Asian 963 2%Filipino 895 3%
Hispanic 730 84%White 784 5%
Socio-economically disadvantaged
726 69%
English Language Learners
668 19%
Students with Disabilities
546
Percent of students scoring proficient or advanced
Parent Education
6th grade math
7th grade Pre-algebra
8th grade General Math
8th grade Algebra 1
Not a high school graduate
8% 36% 0% 35%
High school graduate
22% 49% 10% 34%
Some college (includes AA degree)
24% 51% 25% 34%
College graduate
39% 55% No students available in this category
44%
Over 50% or under 50%?
Thumbs up if you think over 50%
Thumbs down if you think under 50%
Over 50% or under 50%?
2 – 6 + 3 – 1
a)-3 b) -2c) 3 d) 6
Over 50% or under 50%?
2 – 6 + 3 – 1
a)-3 b) -2c) 3 d) 6
Over 50%
81.5% of the students responded correctly.
2 – 6 + 3 – 1
a)-3 b) -2c) 3 d) 6
Over 50% or under 50%?
45% of 61.4 is a number between
a) 3 and 30 b) 30 and 60
c) 60 and 240 d) 240 and 2800
Over 50% or under 50%?
45% of 61.4 is a number between
a) 3 and 30 b) 30 and 60
c) 60 and 240 d) 240 and 2800
Under 50%
Approximately 20% of the students chose C or D.
45% of 61.4 is a number between
a) 3 and 30 b) 30 and 60
c) 60 and 240 d) 240 and 2800
(0.45)(61.4)=27.63
Over 50% or under 50%?
Which of the following is the largest?
a)3/4 b) 6/7c) 12/13 d) 17/25
Over 50% or under 50%?
Which of the following is the largest?
a)3/4 b) 6/7c) 12/13 d) 17/25
Under 50%
31% of the students chose A.
Which of the following is the largest?
a)3/4 b) 6/7c) 12/13 d) 17/25
Under 50%
Over 50% or under 50%?
(5/6)(30)
a)25 b) 36c) 150 d) 156
70.8% of the students responded correctly.
(5/6)(30)
a)25 b) 36c) 150 d) 156
Over 50%
35.8% of the students responded correctly, and 32.5% of the
students chose A.
Order the following from least to greatest.
3/8, 1/3, 0.36, 0.39
a) 1/3, 0.36, 0.39, 3/8 b) 0.36, 0.39, 3/8, 1/3
c) 1/3, 0.36, 3/8, 0.39 d) 0.36, 3/8, 0.39, 1/3
What has worked?
Multiple Representations: One size does not fit all– Pictures, Charts, Tables– Verbal– Algebraic– Numeric
Connecting concepts within lessons
Math 7Subgroup % proficient
or advanced (whole school)
% proficient or advanced(my students)
Black, not Latino
29% 60%
Hispanic, not White
48% 64%
Special Education
18% 36%
Need Challenge Possible Solution
More opportunities for students to engage conceptually and build automaticity through relatable experiences or manipulatives
Teachers did not learn this way
Teachers need training, spaces, and time to practice
Conceptual games and apps
Availability and limitations of expertise
Forums for professionals in different sectors to work together
Critical Concepts in Middle School Mathematics
Mark EllisCSU Fullerton
*Note: If you have an iPhone or Andriod phone with data access, please install the free Socrative Student Clicker app. My room # is 51016Once you’re in the “room” feel free to text in your thoughts and questions.
Shifting Focus
• Traditional U.S.– How can I teach my kids to get
the answer to this problem?– Evaluate answers to determine
proficiency.• Common Core
– How can I use this problem to teach the mathematics of this unit?
– Examine responses to uncover student thinking and inform next steps pedagogically as all students move toward big idea(s) of a unit.
Reasoning about Division
• Divide 365/4 (by hand) using three methods.
Reasoning about Division
Sense Making: Fraction Division
1. What question might this expression answer?
2. Find the quotient in a way that makes sense.
3. Share your reasoning with your neighbor(s) and/or send to Socrative #51016– What justifies your method?– What prior knowledge is
needed?
3
1
2
11
Number line in Quantity and measurement
Equal Partitioning
Fractions
Rational number
Properties of Operations
Rational Expressions
K -2 3 - 6 7 - 12
Unitizing in base 10 and in measurement
Rates, proportional and linear relationships
From Phil Daro, CMC-S
Learning Trajectories
Two “Big Ideas”
• Equivalence• Proportionality
Misconception about Equality
If you learned to interpret the equal sign as an operation,• 3 + 8 =• 23 x 7 =how would you make sense of these?• 4 x 97 = 4 (90 + 7)• 2x – 7 = x + 11
Reasoning about One-half
Euclid’s Algebra Challenge
• Model this geometrically, then with algebraic symbols.
• If a straight line segment be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
(Euclid, Elements, II.4, 300 B.C.)
How Not to Learn Proportional Reasoning
• What is not developed when students “learn” this first?
• Why does this algorithm work?
Reasoning Proportionally
• John’ s mixture was 3 spoonfuls of sugar and 12 spoonfuls of lemon juice. Mary’ s mixture was 4 spoonfuls of sugar and 13 spoonfuls of lemon juice. Whose lemonade is sweeter, John’ s or Mary’ s? Or would they taste equally sweet?
• Eva and Alex want to paint the door of their garage. They mix 2 cans of white paint and 3 cans of black paint to get a particular shade of gray. They then add one more can of each color. Will the new shade of gray be lighter, darker, or the same?
Role of Technology
Support and advance mathematical sense-making, reasoning,
problem solving, and communication
(NCTM Position Statement, 2011)http://www.nctm.org/about/content.aspx?id=31734
http://www.skill-guru.com/gmat/wp-content/uploads/2011/02/thinking-cap.gif
Rhythm Wheel: Exploring Multiplicative Reasoning
After 4 loops of this wheel:1. How many individual
sounds will be played? How do you know this?
2. How many times will the “open” hand drum sound be played? How do you know?
3. How many times more will the “open” hand drum sound be played than the “slap” drum sound? How do you know?
Rhythm Wheel (part 2)
1. How many loops would the 1st and 2nd wheels each need to make so they stop playing at the same time? Why?
2. If the 2nd wheel does 18 loops, how many loops will the 1st wheel need to make so it stops at the same time?
3. If the neck cowbell sound gets played 30 times, how many times will the open hand drum sound be played? Prove it! (Assume the wheels stop at the same time.) What if the neck is played x times?
Proportionality and Linear Functions
• If two quantities vary proportionally, that relationship can be represented as a linear function.
• If two quantities vary proportionally,– the ratio of corresponding terms is
constant,– the constant ratio can be expressed in
lowest terms (a composite unit) or as a unit amount, and
– the constant ratio is the slope of the related linear function.
• When you graph the terms of equal ratios as ordered pairs (first term, second term) and connect the points, the graph is a straight line.
CCSS-Math: Content Domains and Conceptual Categories
K 1 2 3 4 5 6 7 8 HS
Counting &
Cardinality
Number and Operations in Base TenRatios and
Proportional Relationships Number &
Quantity
Number and Operations-
FractionsThe Number System
Operations and Algebraic Thinking
Expressions and Equations Algebra
Functions
Geometry
Measurement and Data Statistics and Probability
Two Categories of Tools
Content Exploration
Communication & Collaboration
Resources
• Charles, R. (2005). Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics. http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr05v7.pdf
• Learning Progressions/Trajectories in Mathematics– http://ime.math.arizona.edu/progressions/– http://www.turnonccmath.com/– http
://www.cpre.org/images/stories/cpre_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf
• Math Reasoning Inventory, https://mathreasoninginventory.com/
• Seigler, R. (2012). Knowledge of Fractions and Long Division Predicts Long-Term Math Success http://youtu.be/7YSj0mmjwBM and http://www.psychologicalscience.org/index.php/news/releases/knowledge-of-fractions-and-long-division-predicts-long-term-math-success.html
• To Half or Not lesson, http://www.pbs.org/teachers/mathline/lessonplans/pdf/esmp/half.pdf