Want to Engage Your Students? Engage Them in the Math Practices
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It’s one of the most important questions math teachers ask every day: how do we engage students in meaningful, enjoyable mathematics? In this webinar for the Adaptive Math Learning community, presenters Zachary Champagne, Researcher at the Mathematics Formative Assessment Project at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM), and Tim Hudson, former Math Curriculum Coordinator for Missouri’s Parkway School District, and DreamBox’s Senior Director of Curriculum Design, shared useful insights about the Mathematical Practices that will help deepen students’ understanding, enjoyment, and success in math class. Zachary and Tim discussed how to stop teaching ‘tricks’ and instead engage students in thinking like a mathematician. They also shared insights about the power of formative assessment, the importance of uncovering students’ intuitive thinking, and how technologies such as adaptive learning can support the Mathematical Practices. Topics included: understanding equality and precision, observing students engaged in sense-making, and designing learning experiences that empower students to “look for” important mathematics. Additionally, Julie Benay, Principal of Malletts Bay School in Vermont, shared how her school implemented DreamBox and the outcomes they experienced. View the webinar to learn how to make math more engaging for your students.
![Mathematics Practice Standards
“These standards describe the varieties of expertise
that mathematics educators at all levels should seek
to develop in their students.”
“[They] describe ways in which developing student
practitioners of the discipline of mathematics
increasingly ought to engage with the subject matter
as they grow in mathematical maturity and expertise
[K-12]”](https://image.slidesharecdn.com/07-15-14edwebadaptivemathlearningzacharychampagnejuliebenaydr-140721181330-phpapp01/85/Want-to-Engage-Your-Students-Engage-Them-in-the-Math-Practices-9-320.jpg)
![What Does Precision Look and
Sound Like in Mathematics?
“[Students] state the meaning of the symbols
they choose, including using the equal sign
consistently and appropriately.”](https://image.slidesharecdn.com/07-15-14edwebadaptivemathlearningzacharychampagnejuliebenaydr-140721181330-phpapp01/85/Want-to-Engage-Your-Students-Engage-Them-in-the-Math-Practices-31-320.jpg)
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Want to Engage Your Students? Engage Them in the Math Practices
- 1. Want to Engage Your Students? Engage Them in the Math Practices
- 2. Zachary Champagne Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) Email: [email protected] Twitter: @zakchamp Julie Benay Principal, Malletts Bay School, Colchester VT Email: [email protected] Twitter: @CSDCommunity Moderator: Tim Hudson Senior Director of Curriculum Design, DreamBox Learning Email: [email protected] Twitter: @DocHudsonMath
- 3. Exit Slip on the First & Last Day of School: What is Mathematics? What do Mathematicians Do?
- 4. From a 5th grade teacher in NY: “I had a lot of good people teaching me math when I was a student – earnest and funny and caring. But the math they taught me wasn’t good math. Every class was the same for eight years: ‘Get out your homework, go over the homework, here’s the new set of exercises, here’s how to do them. Now get started. I’ll be around.’ p. 55, Teaching What Matters Most, Strong, Silver, & Perini, ©2001
- 5. Design Limitation “They were so concerned with making sure we knew how to do every single procedure we never learned how to think mathematically. I did well in math but I never understood what I was doing. I remember hundreds of procedures but not one single mathematical idea.” p. 55, Teaching What Matters Most, Strong, Silver, & Perini, ©2001
- 7. Zachary Champagne Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) Email: [email protected] Twitter: @zakchamp
- 8. MFAS-CCSS Project • Approximately 1300 K – Geometry Tasks and Rubrics developed between 2011 – 2013 and are now available via CPALMS http://www.cpalms.org/Resource/mfas.aspx • K – 3; Algebra and Geometry Lesson Study Toolkits developed between 2011 – 2013 are now available via CPALMS
- 9. Mathematics Practice Standards “These standards describe the varieties of expertise that mathematics educators at all levels should seek to develop in their students.” “[They] describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise [K-12]”
- 10. Elaborations The Elaboration Document can be downloaded at the following link: http://commoncoretools.me/2014/02/12/k-5- elaborations-of-the-practice-standards/
- 11. Math Practice One Make sense of problems and persevere in solving them.
- 12. What does laying ceramic tile have to do with making sense of mathematics?
- 13. 3 inches 4 inches 5 inches
- 14. Why Mathematics?
- 15. What is Sense Making in Mathematics?
- 16. 198 + 37 = ? Consider This Problem…
- 17. How a fourth grade student may solve it…
- 18. What About the Content Standards? 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. • Use strategies such as counting on; • making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); • decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); • using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); • and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
- 19. Consider This Context.. At one very lucky elementary school there were exactly 15 students in every class. At the school there were 19 classrooms. How many students attended the school?
- 20. Standard Algorithm 4 1 9 x 1 5 1 9 5 + 1 9 0 2 8 5
- 21. Partial Product Algorithm 1 9 x 1 5 (9 x 5) = 4 5 (10 x 5) = 5 0 (9 x 10) = 9 0 (10 x 10) = 1 0 0 2 8 5
- 22. The Standard to Achieve is “Make Sense” Which Makes More Sense? 1 9 x 1 5 (9 x 5) = 4 5 (10 x 5) = 5 0 (9 x 10) = 9 0 (10 x 10) = 1 0 0 2 8 5 4 1 9 x 1 5 1 9 5 + 1 9 0 2 8 5 Standard Algorithm Partial Products
- 23. © DreamBox Learning Partial Products on the Number Line
- 24. Composing Partial Product Arrays © DreamBox Learning
- 25. Multiplying with Partial Products © DreamBox Learning
- 26. Optimal Partial Products © DreamBox Learning
- 27. Making Sense of the Algorithm Visually © DreamBox Learning
- 28. Algorithm with Estimate First © DreamBox Learning
- 29. Distributive Property with Variables © DreamBox Learning
- 30. Math Practice Six Attend to Precision
- 31. What Does Precision Look and Sound Like in Mathematics? “[Students] state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.”
- 32. The Equal Sign • Children in the elementary grades generally think that the equal sign means that they should carry out the calculation that precedes it and that the number after the equals sign is the answer to the calculation. • Children must understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value. Faulkner, K., Levi, L., & Carpenter, T. 1999
- 33. A Research Study Faulkner, K., Levi, L., & Carpenter, T. 1999
- 34. Common Core - Equality in K- 2 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?. K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
- 35. Equality Read each equation aloud and say whether it is true or not true. Then say why you think so. 8 = 4 + 4 3 + 2 = 4 + 1 7 = 7 3 + 3 = 8
- 36. A First Grader’s View on Equality (Take 1 and 2)
- 37. Equality with Virtual Manipulatives © DreamBox Learning
- 38. Thanks For Your Time! Zachary Champagne Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) Email: [email protected] Twitter: @zakchamp Formative Assessment Tasks: www.cpalms.org/Resources/mfas.aspx
- 39. Using Adaptive Learning Technology to Support Mathematical Development Julie Benay Principal Malletts Bay School Colchester Vermont Email: [email protected] Twitter: @CSDCommunity
- 40. About Malletts Bay School • On the shores of Lake Champlain, minutes from Burlington • Public school district of about 2,000 students • Five Buildings • Two K-2 Schools • Malletts Bay School (PreK and 3-5) • 1 Middle School • 1 High School. • Malletts Bay divides the community into distinct areas. • District Hallmark • Leadership in implementing differentiated instruction
- 41. Malletts Bay Students • Many of our families live in Colchester & work elsewhere • High percentage of divided or single parent families • Over the past 10 years, grown to nearly 40% of students who qualify for Free or Reduced Price Lunch • Small, but growing, population of English Learners because Colchester is close to the Vermont Refugee Resettlement Center • ~17% of our students qualify for special education or a Section 504 plan. • Majority of our families have Internet access
- 42. Math at Malletts Bay School • CCSS-based Curriculum • Core program: Everyday Math (EDM) • All teachers follow a District pacing guide for EDM • Each unit of EDM focused on a “big idea” aligned with standards and using supplemental resources • Locally designed assessments determine the focus of instructional planning in each classroom • “Workshop” model enables teachers to focus on instructing small groups of students based on learning needs • Our District employs one math coach who works across the three elementary settings with 45 teachers. • When we began using an adaptive math program (DreamBox), we did not have a Title I funded math intervention program.
- 43. Math Achievement • For the past three years, we have exceeded Vermont state average • Vermont’s No Child Left Behind (NCLB) test has been the New England Common Assessment (NECAP). • We meet Adequate Yearly Progress for “All” students • We are “Identified” for: • subgroups of students with disabilities • particularly concerning, with only about 1/3 of students meeting the standard • Subgroups of students from lower income homes
- 44. Access to Technology At MBS • All classrooms have at least four devices: • laptops • netbooks • desktops • Students access DreamBox in the classroom during independent practice or skills time. • Small dedicated “mini-lab” reserved for students on IEPs to access specialized programs • 21 Classrooms share 1 laptop cart and 1 ipad cart • WiFi throughout the building. • All classrooms have an interactive whiteboard.
- 45. Sample Daily Schedules TIME Teacher A TIME Teacher B 8:30-8:40 Attendance and announcements 8:30-8:40 Attendance and announcements 8:45-9:30 Unified Arts (PE, Music, Art, etc) 8:40-9:10 Math Intervention 9:30 – 10:25 Writing/Word Study 9:10-10:10 Math Instruction 10:25-11:25 Math 10:10-11:05 Writing/Word Study 11:25-11:45 Math Intervention 11:10-11:50 Lunch/Recess 11:50-12:30 Lunch/recess 11:50-12:40 Science or Social Studies 12:30 – 1:25 Science or Social Studies 12:40-1:40 Reading Instruction 1:25-2:25 Reading Instruction 1:40-2:10 Reading Intervention
- 46. Adaptive Learning at MBS • We learned about DreamBox through a workshop attended by one of our special educators • After exploring the program (playing in the “sandbox”) and talking with DreamBox, we purchased a limited number of seats for students with disabilities in grades K-5 • All special educators attended a free training session to learn how to: • manage rosters • utilize and interpret the rich data provided by the software. • Parents were engaged through DreamBox parent letters • All students assigned a “seat” in the program had access both at home and at school
- 47. Early Feedback • We saw results immediately • The process of placing students in the program: • gave us good information about critical gaps in the learning progression • helped us tailor instructional support in the classroom and in special education instructional sessions. • Students really enjoyed DreamBox. • Other programs we used required a great deal of practice and repetition, and students resisted being assigned to use them • Conversely, students looked forward to using DreamBox and did not want to sign off when their sessions ended! • DreamBox provided learning experiences well matched to students development of mathematical thinking. • Students are provided with just enough challenge to keep the sessions interesting, and subtle prompts and direction for strategies when they were “stuck.” • Students are engaged by the games personalizing their “rooms.”
- 48. Adaptive Learning in the Math Workshop • Once other students in the classroom observed their peers using DreamBox, they asked if they could have access! • Our K-5 team considered the results of DreamBox • We are now working to implement a math workshop model that will allow teachers time to balance whole group instruction with small guided math groups. • Our workshop model encourages the use of the eight Math Practices so key to the Common Core math standards. • We have expectations for instruction to ensure that the adaptive learning program supplemented instruction, not replacing it • Ongoing data updates are shared among teachers, special educators, and parents to ensure coordinated efforts to help students grow in their mathematical thinking and achievement.
- 49. Math Workshop Model: Malletts Bay School Part I: Mini-lesson (Full Class) • Post learning target • Warm up: Mental Math, Math Message • Direction Instruction: Big Ideas • Guided practice and gradual release (modeling, partner work, small groups) Part II: Small groups and Independent Practice (connected to Big Idea) May include: • Teacher led small groups • Planned, differentiated stations (games, problem solving) • Pairs, student-led groups • Seatwork and independent practice (Everyday Math “math boxes”) Part III: Summary and Closing (Full Class) • Questions, comments, observations • Reflections, exit tickets • Homework explanation Part IV: Intervention (Additional Dedicated Practice) May include: • Additional practice related to this lesson’s Big Idea • Practice with basic facts and skills • Enrichment • Dedicated time for students to leave for supplemental or specialized instruction
- 50. Tiers of Support & Universal Access • Our school uses a tiered model of support. A key feature is to ensure that all students benefit from “first instruction” in grade level standards. • To accomplish this, we set aside a specific time in the daily schedule for supplemental instruction. • Students who leave the classroom for Tier II or Tier III instructional support leave during these periods. • We see the potential for DreamBox to serve as an engaging and perfectly tailored “anchor activity” within our workshop model. • With an eye toward prevention, we used DreamBox with students who were having difficulties in math but who were not identified as needing special education. • Beginning in 2014, all students K-5 will have access to this adaptive learning program.
- 51. Julie Benay Principal, Malletts Bay School, Colchester VT Email: [email protected] Twitter: @CSDCommunity
- 52. Q & A
- 54. DreamBox Combines Three Essential Elements to Accelerate Student Learning 54
- 55. DreamBox Lessons & Virtual Manipulatives Intelligently adapt & individualize to: • Students’ own intuitive strategies • Kinds of mistakes • Efficiency of strategy • Scaffolding needed • Response time
- 56. Robust Reporting
- 57. Strong Support for Differentiation
- 58. DreamBox supports small group and whole class instructional resources Interactive white-board lessons www.dreambox.com/teachertools
- 59. Free School-wide Trial! www.dreambox.com
Editor's Notes
- #12: Young students want to make sense of mathematics. They want to uncover the world through mathematics. But by third grade, many students hate mathematics. We steal that from them…my hope is that today we can spend some time talking and examining how the CCSS can foster that sense of wonder and awe through the elementary school experience and throughout their lives!
- #13: Justin story
- #14: ( 3, 4, 5) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17) Most of us
- #15: Mathematics is widely hated among adults because of their school experiences, and most adults avoid mathematics at all costs. Even if it is what they are doing for their job, or something they really want to know the answer too.
- #17: Talk to someone near you about how you may solve this problem
- #24: Because we started with small numbers like 6 x 6 and deliberate sequences of problems that focus on relationships and partial products, students have strategies for mentally multiplying 17 x 4 and using partial products. Even though 13 x 4 isn’t an ideal partial product, the lesson is not about ideal partial products at this point. It’s about the idea and the relationships.
- #25: In other lessons, students don’t select from pre-made arrays, but instead they build the correct array using the ruler and the arrow buttons. Here again, the lesson is designed so that students understand the structure of the distributive property and can represent it with an array..
- #26: As students “unfold the map,” the current piece being created shows the blueprint grid that becomes part of the map picture once the partial product is created. This helps support the Common Core Practice Standard: Reason Quantitatively & Abstractly. Because students weren’t explicitly told at the start of this lesson to always multiply by ten, this manipulative is designed to elicit genuine evidence of each student’s own thinking.
- #27: All of the work begun with multiplying on an array in 3rd grade pays off now that we’re multiplying fractions. Students mostly learn “just multiply the top numbers and multiply the bottom numbers” but have no real idea why that works. Here, students actually build fractions and fractions of fractions on an array. Just like back in 3rd grade, we choose the problems and problem sequences strategically. Problems are randomly generated, but only within the defined parameters set by DreamBox teachers. The relationships are the focus. This array shows that 5/12 x 2/3 = 10/36. But we also go ahead and ask students a question that can’t be represented on this array: 13/12 x 2/3.
- #28: All of the work begun with multiplying on an array in 3rd grade pays off now that we’re multiplying fractions. Students mostly learn “just multiply the top numbers and multiply the bottom numbers” but have no real idea why that works. Here, students actually build fractions and fractions of fractions on an array. Just like back in 3rd grade, we choose the problems and problem sequences strategically. Problems are randomly generated, but only within the defined parameters set by DreamBox teachers. The relationships are the focus. This array shows that 5/12 x 2/3 = 10/36. But we also go ahead and ask students a question that can’t be represented on this array: 13/12 x 2/3.
- #29: All of the work begun with multiplying on an array in 3rd grade pays off now that we’re multiplying fractions. Students mostly learn “just multiply the top numbers and multiply the bottom numbers” but have no real idea why that works. Here, students actually build fractions and fractions of fractions on an array. Just like back in 3rd grade, we choose the problems and problem sequences strategically. Problems are randomly generated, but only within the defined parameters set by DreamBox teachers. The relationships are the focus. This array shows that 5/12 x 2/3 = 10/36. But we also go ahead and ask students a question that can’t be represented on this array: 13/12 x 2/3.
- #30: Lastly, we get to the generalized distributive property lesson – a 6th grade Common Core Standard that actually is a challenge for many Algebra 1 students. We bring in variables and students realize that “FOIL-ing” – which we never call it in the product for a number of good reasons – is nothing more than the partial products they’ve been doing since 3rd grade – it’s the same as the multiplication algorithm, too. It’s a natural progression with connections to much of their prior knowledge. When you think of middle and high school teachers showing students how to FOIL – and maybe wondering why kids struggle with it – we should think about all of these many lessons, models, and very strategic lessons that have been built into DreamBox for students to work with over the course of 4 years. When we talk about gaps in student understanding or holes in prior knowledge, we oversimplify the complexity of what’s lost by thinking “skill gaps” are easily remedied. Students need to access great models and manipulatives over the course of many years as they develop into mathematicians.
- #31: Young students want to make sense of mathematics. They want to uncover the world through mathematics. But by third grade, many students hate mathematics. We steal that from them…my hope is that today we can spend some time talking and examining how the CCSS can foster that sense of wonder and awe through the elementary school experience and throughout their lives!
- #37: http://www.cpalms.org/Public/PreviewResourceAssessment/Preview/36526
- #38: All of the work begun with multiplying on an array in 3rd grade pays off now that we’re multiplying fractions. Students mostly learn “just multiply the top numbers and multiply the bottom numbers” but have no real idea why that works. Here, students actually build fractions and fractions of fractions on an array. Just like back in 3rd grade, we choose the problems and problem sequences strategically. Problems are randomly generated, but only within the defined parameters set by DreamBox teachers. The relationships are the focus. This array shows that 5/12 x 2/3 = 10/36. But we also go ahead and ask students a question that can’t be represented on this array: 13/12 x 2/3.