Which One Doesn’t Belong? In the chat box, share your choice – and your reason – for which one doesn’t belong. Example of an “open” fraction question – a correct answer requires a mathematical argument, and there are many valid arguments that can be made. Great way to get students to practice making mathematical arguments and a way to informally assess students fraction understanding (what words do they use, do they bring in benchmarks, to they convert to other representations, can they see equivalence, etc) By show of hands: teachers, directors, other How many teach fractions? From http://wodb.ca/numbers.html

Presented by the Adult Numeracy Center and TERC Melissa Braaten Rethinking Fractions COABE Webinar July 24, 2017 Intro self. Ask about intro and branding. Presented by the Adult Numeracy Center and TERC Melissa Braaten

Objectives After this workshop, you will be able to… informally assess students’ understanding of fraction concepts explain the value of beginning fraction instruction with benchmarks add to your bank of resources to teach fractions for understanding describe how fraction concepts are introduced developmentally across several levels

Agenda Which One Doesn’t Belong? Research Teacher Video Fraction Progression in the CCRS Number Lines and Fractions

What are benchmark fractions? February 10, 2011 What are benchmark fractions? Benchmarks are common fractions that are easy to calculate and visualize. ½ (.5, 50%) ¼ (.25, 25%) ¾ (.75, 75%) 0 and 1 (0%, 100%) Tenths, thirds, fifths, hundredths… Anyone familiar with this study? Mention that it is in the resource list. EMPower Professional Development, TERC, 02.10.11

My back hurt 21 out of 31 days last month. February 10, 2011 We use benchmarks to make sense of less intuitive fractions. I have read 65 out of 120 pages…is that more or less than half of the book? My back hurt 21 out of 31 days last month. Anyone familiar with this study? Mention that it is in the resource list. EMPower Professional Development, TERC, 02.10.11

College Ready Students? February 10, 2011 College Ready Students? What Community College Developmental Mathematics Students Understand about Mathematics James W. Stigler, Karen B. Givvin, and Belinda J. Thompson University of California, Los Angeles October, 2009. Anyone familiar with this study? Mention that it is in the resource list. EMPower Professional Development, TERC, 02.10.11

Which is the largest: 4/7, 1/3, 3/8, or 4/9? 33% Got this type of question right? 33% How can benchmarks help? I think there is more information about this question in the original slide? If you know that 4/7 is the only fraction larger than one half, you can solve this easily. EMPower Professional Development, TERC, 02.11.2011

Add a simple fraction and a decimal, like ¼ + .6 February 10, 2011 Add a simple fraction and a decimal, like ¼ + .6 Got this type of question right? 19% What was the most common error? Converted decimal to a fraction, then added numerators and denominators How could a benchmark have helped students recognize that their answer doesn’t make sense? There were three criteria upon which our definition of difficulty was based. First, we included all items for which fewer than 25 percent of participants marked the correct answer. We also included items for which more students selected an incorrect answer than selected the correct answer. Finally, we counted those items for which there were two incorrect answer options selected by at least 20 percent of students. .6  6/10 1/4 + 6/10 = 7/14 How can benchmarks help? EMPower Professional Development, TERC, 02.10.11

Which is larger, 4/5 or 5/8? How do you know? How can benchmarks help? [S]tudents did everything from using division to convert the fraction to a decimal, to drawing a picture of the two fractions, to finding a common denominator. What was fascinating was that although any of these procedures could be used to help answer the question, students using the procedures were almost equally split between choosing 4/5 or choosing 5/8. This was often because they weren't able to carry out the procedure correctly, or because they weren't able to interpret the result of the procedure in relation to the question they were asked. Only 6 percent of the students produced an explanation that did not require execution of a procedure: they simply reasoned that 5/8 is closer to half, and 4/5 is closer to one. No one who reasoned in this way incorrectly chose 5/8 as the larger number. Stigler, pg 19 Seventy-one percent correctly selected 4/5, 24 percent, 5/8 (4 percent did not choose either answer).

Haphazard application of half-remembered procedures Unbothered by results that don’t make sense Lack of disposition to reason about mathematics General findings from the Stigler study

February 10, 2011 “Because the procedures were never connected with conceptual understanding of fundamental mathematics concepts, they have little to fall back on when the procedures fade…[S]tudents conceive of mathematics as a bunch of procedures, and one often gets the sense that they might even believe it is inappropriate to use reason when memory of procedures fails…” What Community College Developmental Mathematics Students Understand about Mathematics Stigler et al. EMPower Professional Development, TERC, 02.10.11

How do I design lessons around a benchmark? Reasoning about one-half Measure to the nearest ½ inch Decimal: .5 Percent: 50% Fraction: ½ Probability 1 out of 2 chances Part of a Whole Part of a Set Reasoning about one-quarter Measure to the nearest ¼ inch Decimal: .25 Percent: 25% Fraction: ¼ Probability 1 out of 4 chances Part of a Whole Part of a Set How do you design a whole lesson or a series of lessons around ½ or ¼?

Teaching Fraction Concepts

As you watch the video, consider…. What big idea is the teacher focusing on, and what types of questions does she ask?

Based on the video… What stood out for you? Small groups to discuss (2 mins) share out After this slide and discussion, participants should take out fraction progression. Explain what it is. What stands out for them? How long does it take in K-12 to teach fraction concepts?

In looking at the fraction progression… What do you notice? Where along the continuum of fraction standards was that video clip?

1 4 1 2 2 4 3 4 1 1 4 3 4

1.5 3 2 3 3 3 1 2 2 3 2 4 3 4 1 3 12 4 12 5 12 6 12 7 12 8 12 9 12 10 12 11 12 1

Number Line Activity What is the point of this activity?

Rigor Conceptual Understanding Procedural Skill and Fluency Application Procedural Skill and Fluency

Next Steps What did you learn and what are you going to do with that knowledge?

For more information about the Adult Numeracy Center at TERC, check out: http://adultnumeracy.terc.edu/ Thanks for being part of this session! Melissa Braaten: melissa.braaten@gmail.com