Report

Middle School Math Concepts: “Is this reasonable?” Esther Kim [email protected] 2011 API By Subgroup Subgroup API % of school population All Students 736 596 100% 5% 963 895 730 2% 3% 84% 784 726 5% 69% English Language Learners 668 19% Students with Disabilities 546 African American Asian Filipino Hispanic White Socio-economically disadvantaged 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 c) 3 b) -2 d) 6 Over 50% or under 50%? 2–6+3–1 a) -3 c) 3 b) -2 d) 6 81.5% of the students responded correctly. 2–6+3–1 a) -3 c) 3 b) -2 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 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 c) 12/13 b) 6/7 d) 17/25 Over 50% or under 50%? Which of the following is the largest? a) 3/4 c) 12/13 b) 6/7 d) 17/25 31% of the students chose A. Which of the following is the largest? a) 3/4 c) 12/13 b) 6/7 d) 17/25 Over 50% or under 50%? (5/6)(30) a) 25 c) 150 b) 36 d) 156 70.8% of the students responded correctly. (5/6)(30) a) 25 c) 150 b) 36 d) 156 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 3/8, 1/3 b) 0.36, 0.39, c) 1/3, 0.36, 3/8, 0.39 0.39, 1/3 d) 0.36, 3/8, What has worked? Multiple Representations: One size does not fit all – – – – Pictures, Charts, Tables Verbal Algebraic Numeric Connecting concepts within lessons Math 7 Subgroup Black, not Latino Hispanic, not White Special Education % proficient or advanced (whole school) 29% % proficient or advanced (my students) 60% 48% 64% 18% 36% Need Challenge Possible Solution More opportunities Teachers did not for students to learn this way engage conceptually and build automaticity through relatable experiences or manipulatives Teachers need training, spaces, and time to practice Conceptual games and apps Forums for professionals in different sectors to work together Availability and limitations of expertise Critical Concepts in Middle School Mathematics Mark Ellis CSU Fullerton [email protected] *Note: If you have an iPhone or Andriod phone with data access, please install the free Socrative Student Clicker app. My room # is 51016 Once 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? 1 1 1 2 3 Learning Trajectories K -2 3-6 Rates, proportional and linear relationships Equal Partitioning Unitizing in base 10 and in measurement Number line in Quantity and measurement Properties of Operations From Phil Daro, CMC-S 7 - 12 Rational number Fractions Rational Expressions 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 sensemaking, reasoning, problem solving, and communication http://www.skill-guru.com/gmat/wpcontent/uploads/2011/02/thinking-cap.gif (NCTM Position Statement, 2011) http://www.nctm.org/about/content.aspx?id=31734 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 Ten Number and OperationsFractions Ratios and Proportional Relationships Number & Quantity The 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/BigId eas_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%2 0report.pdf • Math Reasoning Inventory, https://mathreasoninginventory.com/ • Seigler, R. (2012). Knowledge of Fractions and Long Division Predicts LongTerm Math Success http://youtu.be/7YSj0mmjwBM and http://www.psychologicalscience.org/index.php/news/releases/knowledge-offractions-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