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Utilizing fraction manipulatives or symbolic manipulations may not support students in developing a relational understanding of equivalence. Instead, the use of basic manipulatives or unstructured tools, such as linking cubes, or having students create their own models, can help them develop a deeper understanding of equivalence as well as show their thinking about fractions.

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The example of the 4th grade student showed how the fraction bars took away the fraction language in her explanation

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Based on the video clip, is CGI what he is speaking about in terms of how to teach math that is not just about the achievement gap? CGI works to provide a method of socialization INTO mathematics that he speaks about in the full message?

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When students make their own interpretations, drawings to find a fractional answer, instead of using manipulatives, the learning is more powerful because they own it.

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Using fraction manipulatives doesn't help as much as teacher think it does

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The video reminds me that we need to always look at our students with the lens that they come into our classroom with so much knowledge already and we need to build off of that vs. looking at them with a deficit lens and thinking about what they don't know (yet!). The reading pointed out that some manipulatives, like fraction bars, can hinder students actual conceptual understanding of math. I am looking forward to learning more about how we can teach fraction equivalency beyond manipulatives this week.

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The "prove it" phrase is problematic when asking black children to show their brilliance in math vs. explain how you got your answer.

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relational thinking is important to understanding fraction

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Sometimes students are using the tools provided by the teacher incorrectly and not necessarily thinking about ways to solve the problem or explain their thinking.

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The idea of having students make their own fraction manipulative really helps them understand what a fractions. Another idea that helps students is using equal sharing problems to introduce equivalent fractions.

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The video reminded me that we need to teach via a cultural lens that accepts the axiomatic principle that black children are brilliant; and that we need to reflect on our own practices and approach to teaching.

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1. What resonated with me from the video is that when Dr. Martin mentions that black children are brilliant, he is asked to prove it, which is not necessarily the case for children of other racial backgrounds.  2. From the reading, it is interesting to see how many times children use manipulatives for working with fractions, but many times, they don't really know/understand the mathematical concept for using them. Children learn fractions at a superficial level.

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Part 2: Even though manipulatives are great tools for students to use, it can "eliminate the need for children to think mathematically".

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Teaching fractions with the use of  manipulatives can be a detriment because it doesn't help children explicitly learn the mathematical concepts, doesn't encourage them to use reasoning and critical skills to understand the physical relationship between fractions. We really only need: pencil, paper, colored pencils/markers and linking cubes.

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What stood out to me was that it is important for students to come up with their own thinking about sharing equally (fractions).  Use of manipulative such as fraction bars isn't helpful if they haven't had opportunities to think about and make sense of fractions themselves.

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I am glad that I am here learning from teachers who are mindful and just letting students simply just say what they see.

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From Part 2, what stood out to me is the importance of giving students open-ended materials for developing and representing their understanding of equivalent fractions. These tools help students to work with fractional quantities that helps them deepen their understanding of fraction relationships. Relationships are essential to understanding fractions.

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Things that resonated with me:    - Prove it ideas can be problematic  - Look for the brilliance students bring with them into the classroom and use that as a fund of knowledge to help enhance and develop their thinking and learning.  - Too much research on deficit thinking and what students can't do.  - Interventions and teaching strategies should include utilizing student strengths and understandings to promote new learning. - relational understanding of equivalence can promote comprehension, competence, and independence during tasks. - Overuse of structured manipulatives and procedures can limit student creativity and learning. - Use of structured manipulatives and procedures mirror neglecting letting the student show their brilliance through solving problem their way, using what they know and understand.

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The video is a reminder that we need to focus on the CGI principles that  Every student comes to math class knowing some mathematics and Every student is capable of extending their mathematical ideas.

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Part I:  This reminds me of how math education created its proficiency based structural bias. I think CGI's model addresses the current challenges.   Part II: Children need opportunities to manipulate the tools and problem solve in order to make the connections (relational thinking and reasoning).  Pre-made fraction pieces or similar manipulatives may hinder students to understand the the concept or even create more misconceptions.

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What stood out to me in the reading was how so many teachers give the fraction strips to the students (as most math programs tell us to) yet it seems to do more harm than good to their mathematical thinking and understanding of fractions.

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Part 2: Teaching Equivalence:   The only tools children need to solve  problems involving fractions are: paper and pencil for annotating, cutout pieces of paper for folding & cutting, colored pencils/markers for allocating shares and linking cubes for representing discrete quantities in a problem

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What resonated the most from the reading is the fact that some types of manipulatives, such as fraction bars, "can eliminate the need for children to think mathematically." I've used fraction bars and never thought of that. I want my students to have good foundational skills when it comes to fractions. I see now that some manipulatives do not teach them to have reasoning skills which helps them with more difficult concepts.

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Sad to hear, "Never enough examples..."

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Part 1:  It is unfortunate that our math programs are geared towards certain ethnic groups.  I think CGI will bridge that gap.                                                                                                                                                                                                                     Part 2:  What stuck me in the reading was how students can simply identify the color of the manipulative and tell you the answer instead of saying the fraction.  As a teacher this is where you will need to ask them, what is the fraction?  Explain your thinking.

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Par 1:  Dr. Martin stated, "Building effective and meaningful mathematics education based on so-called racial achievement gaps and propositions about how black children differ from white and Asian children. The vast majority of pedagogical curricular interventions are based on the white and Asian children." That's a problem because there aren't enough examples about black children.  Part 2:  In the article, it is mentioned that only a few manipulatives are needed to solve fraction problems.  Fraction strips only give an estimate answer.  It's important to make mathematical relationships between fractions and using manipulatives.

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Dr. Martin is one of the co-authors of a great book for K-8 teachers called The Impact of Identity in K-8 Mathematics. See the link to learn more about the book. I highly recommend.

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By Julia Aguirre, Karen Mayfield-Ingram, Danny Martin

Each teacher and student brings many identities to the classroom. What is their impact on the student's learning and the teacher's teaching of mathematics?

This book invites K-8 teachers to reflect on their own and their students' multiple identities. Rich possibilities for learning result when teachers draw on these identities to offer high-quality, equity-based teaching to all students. Reflecting on identity and re-envisioning learning and teaching through this lens especially benefits students who have been marginalized by race, class, ethnicity, or gender. The authors encourage teachers to reframe instruction by using five equity-based mathematics teaching practices:

Going deep with mathematics

Leveraging multiple mathematical competencies

Affirming mathematics learners' identities

Challenging spaces of marginality

Drawing on multiple resources of knowledge

Special features of the book:

Classroom vignettes, lessons, and assessments showing equity-based practices

Tools for teachers' self-reflection and professional development, including a mathematics learning autobiography and teacher identity activity at nctm.org/more4u

Suggestions for partnering with parents and community organizations

End-of-chapter discussion questions

About the Authors:
Julia Maria Aguirre is assistant professor of education at the University of Washington-Tacoma. Her work focuses on mathematics teaching and learning, teacher education, and culturally responsive mathematics pedagogy, with a primary goal of strengthening access and advancement in mathematics education for historically marginalized youth. The National Science Foundation, Carnegie Foundation, and Spencer Foundation have funded her work.
Karen Mayfield-Ingram is associate director for the EQUALS programs at the Lawrence Hall of Science at the University of California, Berkeley. Her work focuses on mathematics and equity professional development, teacher leadership, and parent involvement. She is the author of The Journey--Through Middle School Math (EQUALS, 2005) and co-editor of the EQUALS Middle School Investigations series.

Danny Bernard Martin is professor of education and mathematics at the University of Illinois at Chicago. His research focuses on understanding the salience of race and identity in black children's mathematical experiences. He is the author of Mathematics Success and Failure among African Youth (Erlbaum, 2000), editor of Mathematics Teaching, Learning, and Liberation in the Lives of Black Children(Routledge, 2009), and co-editor of The Brilliance of Black Children in Mathematics: Beyond the Numbers and Toward New Discourse(Information Age, 2013).

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The Impact of Identity in K-8 Mathematics Learning and Teaching: Rethinking Equity-Based Practices (Download) - National Council of Teachers of Mathematics

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Each teacher and student brings many identities to the classroom. What is their impact on the student's learning and the teacher's teaching of mathematics? This book invites K-8 teachers to reflect on their own and their students' multiple identities. Rich possibilities for learning result when teachers draw on these identities to offer high-quality, equity-based teaching to all students.

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Part 1: Dr. Martin notes that currently math education is based on that of lack and what black students can't do, versus their brilliance. Math education (for ALL students) needs to be based on what students can do (their brilliance) rather than what they lack. This really makes me think about how we must really center instruction around CGI's principled ideas. To me this video specifically speaks to principles 1, 2, and 7.

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Part 1: In the video Dr. Martin mentions that the majority of math intervention programs are designed based on the so called racial achievement gaps and propositions about how Black children differ from white and Asian children BUT  instead effective and meaningful math education should be designed based on the brilliance of Black children

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Part 1:What thoughts do you have about Dr. Martin's speech, "The Brilliance of Black Children?"  Part 2: What struck you about, "Some Issues With Common Approaches to Teaching Equivalent Fractions?"