MTH 506 by Cheryl Anderson https://padlet.com/cheryl_anderson/cheryllynn6940 My Week 4 Collaboration en-us 2013-05-21 14:26:23 UTC 2016-05-16 03:45:24 UTC hello@padlet.com http://d2s8n7nv9yizdf.cloudfront.net/assets/thmbs/smiley-07af99ec10f16fddbf83e8a7949d6e36.jpg When teaching students to divide fractions, many teachers have offered the following advice, “ours is not to reason why, just invert and multiply”. By stating this, teachers are saying they, themselves, do not understand the concept of dividing fractions.  Students really need conceptual knowledge of fractions to create understanding.  By giving manipulatives to students so they can reason how many ¼’s are in 2, they gain concrete understanding of an abstract concept.  This also helps to build spatial knowledge.  As they are provided concrete models, words should be used to help them “see” these concepts more clearly.  “Asking students to first use words to describe what these equations are asking (e.g., “how many halves in 4⅓”) can help them think about the meaning of division and then develop an estimate” (Van de Walle, 2013).  When we simply teach invert and multiply, students are not expected to understand the why or that they are being asked how many of a certain fraction does it take to make a second number, that being divided.  They must have an understanding of number sense and breaking numbers apart into pieces. When developing number sense with fractions, children should be provided multiple opportunities to use manipulatives and number lines so they can “play” with numbers and come up with their own strategies for computing them.  “The goal is to prepare students who are flexible in how they approach fraction computation” (Van de Walle, 2013).    According to the Rational Number Project, there are four strong beliefs about how children must develop number sense for fractions: (1) Children's learning about fractions can be optimized through active involvement with multiple concrete models, (2) most children need to use concrete models over extended periods of time to develop mental images needed to think conceptually about fractions, (3) children benefit from opportunities to talk to one another and with their teacher about fraction ideas as they construct their own understandings of fraction as a number, and (4) teaching materials for fractions should focus on the development of conceptual knowledge prior to formal work with symbols and algorithms (Cramer et al. 1997).  To help develop knowledge of equivalent fractions, after having given concrete examples, students can be introduced to Tic-Tac-Toe where they must cross off equivalent fractions in rows or columns in order to win.  Bingo would be another game students could play.  In an effort to help develop that number sense, students can be led through the activity from Illuminations at http://illuminations.nctm.org/. In this activity, students are expected to create a number line with fractions, improper fractions, mixed numbers, and integers, and to use estimation to practice combining various numbers.  Further, in this lesson, a string will be stretched across the classroom and various points will be marked for 0, 1, 2, 3, and 4. This classroom number line will be used to show that all proper fractions are grouped between 0 and 1, and that improper fractions or mixed numbers are all grouped above 1. Students clip index cards with various proper fractions, improper fractions, and mixed numbers on the clothesline to visually see groupings. Students then play an estimation game with groups using the same principle. Encouraging students to look at fractions in various ways will help foster their conceptual fraction sense.  All activity sheets and a supply list are available on the website.  Another activity dealing with connections amongst various math concepts, including fractions, and available at the same website above, is Fun With Baseball Stats.  The learning objectives are that students will work with decimals, fractions, and percentages in the context of baseball statistics, and develop skills in mathematical reasoning and computations and apply those skills to everyday life.  This grades 6-8 activity allows students to explore statistics surrounding baseball. They are exposed to connections between various mathematical concepts and see where this mathematics is used in areas with which they are familiar. This lesson plan is adapted from the May 1996 edition of Mathematics Teaching in the Middle School. Another activity comes from the Bits and Pieces Curriculum provided by Connected Math (2006).  In this series, students are exposed to numerous activities dealing with fractions and computation.  They are led through a “sixth-grade fundraiser” where they have to monitor the fraction of the goal they have met throughout the process using fraction bars and thermometers with their established goal at the top.  As they accumulate “money” they continue to calculate the fraction of the goal and use equivalent fractions to establish a base.  They are then required to apply this concept to a real-life situation where they actually conduct a fundraiser for a selected charity or middle school class.   cheryl_anderson https://padlet.com/cheryl_anderson/cheryllynn6940/wish/10134270 2013-05-21 18:37:11 UTC https://padlet.com/cheryl_anderson/cheryllynn6940/wish/10134270 Jesse Miller                                                                                                    jtmiller https://padlet.com/cheryl_anderson/cheryllynn6940/wish/10208647 Hi Cheryl, great post.  I really liked the quote at the beginning of your post.  “Ours is not to reason why, just invert and multiply.”  It seems like many teachers make this mistake when introducing new math concepts.  This has been my experience in most of the math classes that I have taken, and I am guilty of using similar methods in my own classroom.  In my attempts to keep the concepts simple for students to understand, I have inadvertently restricted students from developing a deeper, conceptual understanding of the material.  While the algorithms and procedures will work to produce a solution, they will not be retained if students can not make sense out of them.  Students must be able to recreate, prove, and explain the algorithms or formulas for themselves.  I am able to recreate and re-teach myself the majority of the formulas and algorithms that I know.  Otherwise, I am sure that I would forget them.  I have argued as others have in the past that while teaching for a deeper level of understanding would be ideal, there is no possible way that I will come close to hitting all of my required state standards.  Why spend two weeks on something that I can get them to do in two days?  This is the reality of our current system.  As Van de Walle, Karp, and Bay-Williams (2013) explain “Some teachers may argue that they can’t or don’t need to devote so much time to fraction operations—that sharing one algorithm is quicker and leads to less confusion for students” (p. 316).  This approach does not work according to the authors.  In my opinion, teachers must choose between teaching with a focus on coverage or teaching with a focus on deeper understanding.  Even though we are currently discussing an isolated topic, dividing fractions, we could have the same conversation over almost every math concept.  This course is helping me to redefine my priorities as a teacher.  I would rather hit fewer concepts throughout the year but know that my students truly understand and can conceptualize the material.  If I am teaching fraction division, I want students to be able to model, estimate, draw, explain, and predict common errors.  I want them to be able to explain why the algorithm works and where the concept will be used in real life.  This will take much more time than “invert and multiply”, but it is worth it.

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