Cultivate Algebraic Thinking with Daily Routines
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Creating opportunities to develop algebraic thinking and enhancing conceptual understanding of mathematics is essential at every grade level. In this webinar, Math/Technology Curriculum Specialist Aubree Short explored the use of problem solving methods and hands-on manipulatives to guide students in the discovery of algebraic concepts at all levels of learning.
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Cultivate Algebraic Thinking with Daily Routines
- 1. Inspire and Cultivate Algebraic Thinking Presented by Aubree Short & Kelly Urlacher
- 2. Today’s Outcomes ● Learn strategies to create algebraic thinkers in your classrooms ● Explore ideas for student discourse ● Discuss the integration of algebraic thinking through daily routines
- 3. Warm Up Pinch Cards Primary:Jenny has 13 pennies in her pocket. Some of her pennies fell out. If Jenny had 22 pennies to begin with, how many pennies fell out? Intermediate: Manny ate ½ of the pizza and Nate ate ⅓ of the pizza. How much of the pizza is left?
- 6. Algebraic Thinking vs. Algebra Class The main purpose of algebra is to learn how to represent general relationships and procedures; for through these representations, a wide range of problems can be solved and new relationships can be developed from those known… However, students tend to view algebra as little more than a set of arbitrary manipulative techniques that seem to have little, if any, purpose to them. L.R. Booth. Difficulties in Algebra. 1986
- 7. Algebraic Thinking Algebraic thinking is not just about an Algebra class, but encompasses how to think throughout a student’s math journey. Algebra is often being referred to as the “gateway” to higher education. Elementary school is the necessary time to start thinking about symbolic representations, the explanations of the students, and what are the underlying issues, misconceptions, and general ideas.
- 8. Why Algebraic Thinking Integration is Necessary Over 60 percent of all students entering community colleges must take what are called developmental math courses … [that] are algebra-based and focus on linear and quadratic equations. – Ginia Bellafante, nytimes.com, 2014 Because Algebra has come to be regarded as a gatekeeper … the high failure rate in Algebra, especially among minority students, has rightfully become an issue of general social concern. – H. Wu, math.berkeley.edu Algebra I is the key — and the barrier — to students’ ability to complete a challenging mathematics curriculum in high school. – Southern Regional Education Board, publications.sreb.org
- 9. Reason Abstractly and Quantitatively Mathematically proficient students are able to: ★ represent quantities in a variety of ways ★ remove the problem context to solve the problem in an abstract way (equation) ★ refer back to the problem context, when needed, to understand and evaluate the answer -O’Connell & SanGiovanni, Putting the Practices Into Action (2010)
- 10. Match It Work with a partner to match each expression or equation to the corresponding problem. Justify each of your matches. -O’Connell & SanGiovanni, Putting the Practices Into Action (2010)
- 11. Can tech support algebraic thinking? Blended learning is no longer a distant idea. Technology can be a symbiotic element in the classroom. Build conceptual understanding using their own strategies. Recognizing transfer between conceptual and abstract.
- 12. Manipulatives to Explore Abstract Ideas Research supports the use of manipulatives. Students gain deep understanding by using manipulatives in their math learning. Exploration comes to life. Manipulatives are a means, not an end.
- 13. Teacher Tools as Tech Manipulatives www.dreambox.com/teachertools
- 14. Mathematical Concepts and Misconceptions Rhett Allain, Associate Professor of Physics at Southeastern Louisiana University, rightly points out that confusion is the sweat of learning; deeper learning could be considered an invigorating mental workout.
- 15. Misconceptions ★ In the organic process of learning, mistakes are natural and beneficial. ○ Positive vs negative ★ Providing insight for teachers, but can be used by the students too. ○ How can we learn from this? ★ Misconceptions are a new learning opportunity. ○ How can we build an environment that supports growth? ★ Conversation: ○ How do you address misconceptions in the classroom?
- 16. How can you integrate algebraic thinking? Exploring symbolic representations of thinking. All students can draw or write this out: using notation when suitable. Question It: Tell the answer. Ask students to write the question (in the form of a problem.) -O’Connell & SanGiovanni, Putting the Practices Into Action (2010) Answer = 10 The problem is… Answer = 3 ½ The problem is… Answer = $4.50 The problem is…
- 17. Contextualize & Decontextualize Discuss appropriate operations to solve problems model building appropriate equations use diagrams to model math situations Ask, “What operation makes sense?” Ask, “How should we build an equation to match this problem?” Ask students to write word problems to go with a situation. Consistently ask students to explain the equations or diagrams. Ask students to label answers by referring back to the problem to determine what a quantity represents. Ask students if the quantity makes sense. X + 4 =10
- 18. How can you integrate algebraic thinking? Math Talks: How are students thinking about the algebraic idea? What does it sound like when they talk about their thinking? True or False 80÷4=(80÷2)+(80÷2) https://www.teachingchannel.org/videos/common-core-teaching-division
- 19. 80÷4=(80÷2)+(80÷2) 20 20 20 20 40 40 4040= +
- 20. Conceptual Understanding for All! Setting students up to be active learners using algebraic thinking. Because [low-achieving students] are less likely to have acquired the basics on the same schedule as more advanced learners, struggling learners are often confined to an educational regimen of low-level activities, rote memorization of discrete facts, and mind-numbing skill-drill worksheets… [They] have minimal opportunities to actually use what they are learning in a meaningful fashion. - Wiggins & McTighe, Schooling by Design
- 22. Constructing Meaning to Mastery Grant Wiggins, Educational Leadership 2014 Mastery is effective transfer of learning in authentic and worthy performance. Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject, as measured against valid and high standards. Students learn from constructing meaning. This can not be given to them, they need to have the power to try different ideas, different methods, and have the chance to build their cognitive map.
- 23. Empowering Students to Learn As teachers, we often teach algorithms too soon and assume understanding. We should be providing examples of problems that help the students realize that their informal procedures are beneficial. By giving a thought-provoking question in new and unfamiliar situations, encourage: ▪ Independent, Critical Thinking ▪ Problem Solving ▪ Design own Solutions
- 24. Teaching Strategies By presenting mathematical concepts through multiple examples and problems, we can foster understanding with several different methods. Examples: ▪ Recipe: [y = 2k + a] To make y, you need to double k and add a to it. ▪ Stories: [3(s+2) = 12] There is a number. You add 2 to the number, then multiply by 3, and the number becomes 12. ▪ Function machine: Inputs and Outputs, find the rule. ▪ Balance model: Two Expressions are seen to be equal.
- 25. Today’s Outcomes ● Learn strategies to create algebraic thinkers in your classrooms ● Explore ideas for student discourse ● Discuss the integration of algebraic thinking through daily routines
- 26. Reflection What did you learn? How are you feeling? What are your next steps?
- 27. DreamBox Learning® K–8 Math Available in English & Spanish
- 28. DreamBox Lessons & Virtual Manipulatives Intelligently adapt & individualize to: • Students’ own intuitive strategies • Kinds of mistakes • Efficiency of strategy • Scaffolding needed • Response time
- 29. Insight Dashboards Tailored insights for every participant of the learning experience.
- 30. ISTE is right around the corner… Catch us at @DreamBox_Learn Seeing is believing! www.DreamBox.com/request-a-demo
- 31. We value your feedback! Let us know how we’re doing: https://www.surveymonkey.com/r/BXMMK59
Notes de l'éditeur
- Aubree
- This will include “how you came to use DreamBox and why” and other technology partnerships
- In kindergarten through grade five (K–5), the focus is on addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals, with a balance of concepts, skills, and problem solving. Arithmetic is viewed as an important set of skills and also as a thinking subject that prepares students for higher mathematics. Measurement and geometry develop alongside number and operations and are tied specifically to arithmetic along the way. In middle school, multiplication and division develop into the powerful forms of ratio and proportional reasoning. The properties of operations take on prominence as arithmetic matures into algebra. The theme of quantitative relationships also becomes explicit in grades six through eight, developing into the formal concept of a function by grade eight. Meanwhile, the foundations of deductive geometry are laid in the middle grades. Finally, the gradual development of data representations in kindergarten through grade five leads to statistics in middle school: the study of shape, center, and spread of data distributions; possible associations between two variables; and the use of sampling in making statistical decisions. In higher mathematics, algebra, functions, geometry, and statistics develop with an emphasis on modeling. Students continue to take a thinking approach to algebra, learning to see and make use of Mathematics is a logically progressing discipline that has intricate connections among the various domains and clusters in the standards. Sustained practice is required to master grade-level and course-level content. Further, table OV-1 (adapted from Achieve the Core 2012) summarizes an important subset of the major work in kindergarten through grade eight, as the progression of learning in the standards leads toward Mathematics I or Algebra I.
- Kelly Conceptual, general relationships. Use that knowledge to create meaning in future situations. But what kids think of algebra are: formulas, algorithms, techniques that do not connect with the student.
- Kelly
- kelly Failure: 2004 LAUSD 48000 9th graders took algebra 44% failed, then 75% of those who repeated in the spring failed (LA Times 2004) Algebra is to gain access NOT screen kids out Retake algebra because they obviously don’t recall the basics. But if we are talking recall, we aren’t talking about true understanding. True understanding isn’t recalled if you made sense of it for yourself in the first place.
- We have to build a strong understanding of numbers (quantities). When faced with a problem, students must be able to represent the problems using abstractions (e.g. numbers, symbols, and diagrams). Students must see the connection between the problem situation and the abstract representation (equation) that stands for the problem.
- Kelly DreamBox - how does it encourage algebraic thinking? We are no longer in the position for us to be convincing you that technology belongs in the classroom. Now we are on to phase two, how can technology be a symbiotic element in the classroom, research supports the use of technology, learning, personalization, and blended learning. Concrete thinking - on the surface Abstract thinking - in depth Abstract - mastery of ideas, theories, and notions and how they work together.
- •Research suggests to start with concrete manipulatives •Manipulatives do not encompass the mathematical idea, however students gain deep understanding by using manipulatives in their math learning. •Are a means, not an end.
- Kinder Numbers To Ten on the MathRack (Count to tell the number of Objects) Sixth grade Coordinate Grids: Location and Measurement (Graph points in the coordinate plane to solve problems) When you evaluate digital curricula, some things to think about. Conceptual understanding, exploration, hints to guide (not solve), adaptivity, personalization www.dreambox.com/teachertools TECH IN GENERAL: what it does well is adaptivity and providing helpful models (like 2d array) for students to explore. where it is limited is allowing students to flexibly explore any direction they want to go. For example, it is hard to allow students to explore a function with a graph and an equation and a table all of which they can modify, adjust, explore, and get targeted feedback. usually tech is just designed to focus on one thing, and programmers have to anticipate the direction students will go. which is difficult to do.
- Open Middle Problems - the more tries, the more points. Value the process, not just the answer. Robert Kaplinsky Downy SD? Openmiddle.com Common Student Idea: Confusion is bad Common Student Idea 2: The Instructor is the Source of Knowledge How to Integrate Algebraic Thinking: Identify math concepts and misconceptions http://www.wired.com/2014/09/two-common-misconceptions-about-learning/
- Confront their own misconceptions Many students have such a negative association Value mistakes
- Ideas: Student work with headline stories pg 36 pg 33: symbolism, what is the meaning based on the context
- Discuss appropriate operations to solve problems model building appropriate equations use diagrams to model math situations Ask, “What operation makes sense?” Ask, “How should we build an equation to match this problem?” Ask students to write word problems to go with a situation. Consistently ask students to explain the equations or diagrams. Ask students to label answers by referring back to the problem to determine what a quantity represents. Ask students if the quantity makes sense.
- Visual representation at the end: see the arrays at the end. Show some of the video - the true and false Have the audience do the activity. Pose the problem. 3:34
- I love the way she makes the learning visible to all of her students by using a model to show the problem.
- We want students to all experience the full spectrum of conceptual understandings.
- Lesson design cohort: launch, explore, summarize. Back mapped. Aubree. Mold this into a piece to change thinking, ponder on their own, audience involved.
- Communication, independent thinking, fluency of basic skills, and fluency with symbols and variables Algebra is just as much about constructing arguments as it is simplifying and solving. DIfferent methods of solving Different means of writing symbols Explanation of understanding
- Will need to give more time to make sense of problems experience productive struggles think for themselves instead of trying to remember and regurgitate someone else’s procedures, formulas, ideas. Refer back to the other activities
- Recipe: connects formulas to algebraic equations. Procedures can happen on either side of the equation (particularly the right side, since they mostly see it on the left) Story: This helps with single variable equations with no variable on the other side. It can reinforce the misconception that you do a procedure to what is on the left and the answer on the right side of the equals sign. Function Machine: works well for data provided in a table format. Balance Model: If you operate on one, you must operate equally on the other. SOME ARE NOT READY FOR THAT, developmentally. Example for each. Table groups, take an equation, find a way to use one example in their classroom.
- Write on a piece of paper something you learned, how you are feeling after our session and what are your next steps. I am going to give you 1 min of think time. Then stand up and choose either what you learned, how you are feeling or your next steps to share. Place you hand on your head, your heart, or down at your sides to show what you are going to share and find someone in the room with the same signal and share your learning, feeling, or next steps with them.
- DreamBox Learning provides a new class of intelligent adaptive learning technology is the true game changer in education. Combines 3 essential elements 1) Mathematics- CCSSM & Standards for Mathematical Practice- unlike other programs that provide drill and practice DreamBox builds both conceptual understanding and procedural fluency 2) Motivating (persist and progress) 3) Powerful intelligent adaptive learning engine providing millions of personalized learning paths—each one—tailored to a student’s unique needs. Notes: DreamBox curriculum aligns with these Common Core Standards: Counting and Cardinality, Comparing, Operations and Algebraic Thinking, Number and Operations in Base Ten, and Number and Operations in Fractions.