The Math Learning Center
http://www.mathlearningcenter.org/blog
enWhat Was Wrong with the Old Way?
http://www.mathlearningcenter.org/blog/what-was-wrong-old-way
<p>We’ve all had those conversations in which someone laments that math isn’t taught the same way it was “in the good ole days.” Our understanding of best practices in mathematics has changed, and change can be difficult for everyone.<!--break-->And it can be especially difficult to a parent who just worked a long day and is now trying to help their student with homework, using strategies they never learned in school. </p>
<p>So how can we help them understand why we’ve changed our strategies for teaching math? I have two key points I try to share with families that have helped to make these conversations productive: comparing math literacy with reading literacy, and the evolution of best practices. </p>
<p>For many years we taught and learned math on a surface level through drills and timed tests. While this practice may have appeared to develop fluency, it didn’t establish a firm math foundation upon which students could build throughout their education. However, in other subjects we provide students with building blocks to help them understand how to construct and deconstruct the big picture and how to apply those skills to different situations. In reading we don’t expect children to memorize every word in the dictionary. That’d be absurd! Instead, we teach them phonics so that they know how words are built and they can apply those deeply understood principals to new words they encounter. This firm foundation supports newer and higher levels of reading with a great level of success.</p>
<p>The way we teach math now, in small but rigorous steps, is giving students the means to take on new and varied applications of mathematics with confidence. In short, we are teaching math phonics.</p>
<p>I recently had the opportunity to hear Rick DuFour speak about how education has changed. He shared a great anecdote about his sister that illustrates why change is needed. Around 40 years ago his sister went to have her vision corrected through surgery. At the time, the procedure required razor blades, an exhaustive number of follow-up trips to the surgeon, and about a year to fully heal. This practice was cutting edge at the time. </p>
<p>Today, if you go to have your vision corrected, they’ll shoot you in the eye with a laser, you’ll go home and take a nap, and usually within 72 hours you’ll have 20/20 vision. Forty years ago we didn’t even know this was possible. The razor blade procedure was not only acceptable, it was considered best practice. If you went to get your vision corrected today and they pulled out a box of blades, it would be considered malpractice. In like fashion, we’ve changed the way we teach math because we know better now. As we continue to research and seek out the best ways to instruct our students, we will encounter change. </p>
<p>I can’t help but wonder, what will replace the laser?</p>
<p></p>
<p><em>Justin Brown is a 2nd grade Bridges teacher</em></p>Tue, 20 Dec 2016 11:38:07 -0800Justin Brown484 at http://www.mathlearningcenter.orgBridges First Edition to Be Retired in 2018
http://www.mathlearningcenter.org/blog/bridges-first-edition-be-retired-2018
<p>The second edition of Bridges and Number Corner launched in 2013, and since then The Math Learning Center has continued to offer materials and support for first edition users. <br class="Apple-interchange-newline" /><!--break--> We will continue to produce kits upon request through December 31, 2018, after which we will consider the product line retired. Consumable student books will also be discontinued at that time.</p>
<p>Note that we currently offer first edition users a 30% discount on Bridges Second Edition classroom kits and a 15% discount on Number Corner Second Edition kits. These discounts will also end December 31, 2018.</p>
<p>After first edition is retired we will do our best to fill requests, depending on remaining inventory. We hope that communicating our plans well ahead of time will allow you to plan for a smooth transition.</p>
<p><em> Rick Ludeman is president of MLC. </em></p>Fri, 07 Oct 2016 11:26:25 -0700Rick Ludeman483 at http://www.mathlearningcenter.orgWhat Do You Notice?
http://www.mathlearningcenter.org/blog/what-do-you-notice
<p>I was recently at a workshop where the leader brought up the base-ten area pieces on the document camera. She asked the teachers seated in the audience, “What do you notice?” </p>
<h3>Area models and the place value number system</h3>
<p><img src="/sites/default/files/images/B4-AUG-1.png" border="0" alt="What Do You Notice?" width="450" height="240" /> </p>
<p>Once you notice one thing, notice something else. Keep going. What’s the pattern? What will the next-larger one look like? What will the next-smaller one look like? Take a moment or two—just sit with the pattern.</p>
<p>Maybe you notice that moving left to right, each rectangle is 1/10th the size of the previous rectangle. Maybe you notice that from right to left, each rectangle is 10 times the size of the previous one (CCSS 4.NBT.A). What relationship among operations does this demonstrate? I am blown away by the things I see when I slow down and take more time to notice. I love this subtle demonstration of inverse operations.</p>
<p>You may also notice the geometric shapes as they change from right to left (or vice versa), alternating between square and non-square rectangles. Why is this? Does this always happen… in all bases? Why does it always happen?</p>
<p>Though this series of arrays models the base-ten place value system and captures the concept of unitizing, it is conceptual in nature. The actual place value system is abstract and is difficult to model conceptually. How, after all, are we able to use only 10 numeric symbols to represent any number? I may be going out on a limb here, but I made an interesting connection after going through the fourth grade Number Corner September Calendar Grid, which takes students on a mysterious journey through an ancient Egyptian number system that does not involve place value. In this Egyptian number system, the combination of base and factor was not employed. In other words, you have an additive system that does not group “like” place values. It struck me that this is how base-ten pieces work.</p>
<p>In the ancient Egyptian number system 187 is one scroll, eight heel bones, and seven staffs. These symbols represent (1 x 100) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1). It takes 16 ancient Egyptian numerals to write it! It also takes 16 base-ten pieces. In our modern system, with place value, it takes only 3 numerals. You can see the difference clearly in expanded notion: (1 x 100) + (8 x 10) + (7 x 1). Eights and sevens don't have their own symbols in the Egyptian system or the base-ten area pieces. For more information <a href="https://en.wikipedia.org/wiki/Egyptian_numerals" target="_blank">check this out.</a></p>
<p>Perhaps this is obvious, but it is interesting to note that when the area model shows 100, it literally shows 100 ones. In the 10x10 array we actually see 100 little squares. Place value, on the other hand, hides this 100 by putting the “one” in a new place. This is why we can represent 10 or 1,000 or 100,000 with different combinations of the same two symbols. Even if you consider the 10x10 a symbol, the base-ten pieces only have "symbols" for powers of 10. It seems that the base-ten area pieces function more like the Egyptian number system than the modern system.</p>
<h3>Looking for a challenge?</h3>
<p>Let’s consider what this model would look like for base-six. And for the sake of simplicity we can call the unit the zero term.</p>
<p><img src="/sites/default/files/images/B4-AUG-2.jpg" border="0" alt="Base 6 table" width="600" height="202" /></p>
<p>What does this tell us about the shape of base <em>n</em> when <em>n</em> is even? Odd? Does this pattern continue into negative exponents?</p>
<p>Now for an algebraic exercise… what is the pattern for the perimeter of these rectangles? Is there a general pattern that can be expressed for all bases? </p>
<p><em>Spencer is a grade 4 Bridges teacher. </em></p>
<p> </p>Thu, 22 Sep 2016 12:09:32 -0700Spencer Olmsted478 at http://www.mathlearningcenter.orgWhy Curriculum Still Matters in the Age of Free Instructional Materials
http://www.mathlearningcenter.org/blog/why-curriculum-still-matters-age-free-instructional-materials
<p>In his August <a href="https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Curricular-Coherence-in-the-Age-of-Open-Educational-Resources" target="_blank">2016 President’s Message</a>, current NCTM President Matt Larson examines the relationship between the growing availability of open educational resources and curricular coherence. He explains that “[a] critical feature of high-quality curricular materials is that they are coherent. Coherence, with respect to mathematics curriculum, generally means that connections are clear and receive emphasis from one year to the next, from one concept to another, and from one representation to another.” </p>
<p>We agree. Coherence is not just a matter of ensuring the consistency of conventions, vocabulary, models, and concepts within and across grade levels. Rather, a curriculum demonstrates coherence by helping students develop an understanding of mathematics as a unified body of knowledge, not as a collection of disparate problems and exercises. A coherent curriculum helps students build a strong foundation of skills and understanding, and it regularly invites them to make their own connections between topics and between what they are currently learning and what they have learned before.</p>
<p>Larson warns that “the danger in online curricular selection is the undercutting of curricular coherence by the introduction of disjointed tasks that are of questionable quality, do not fit within the mathematical learning progression, and are incoherent.” We know that many districts have provided teachers with curriculum maps and invited them to select from among the free resources available online to create curricula. When curricula are assembled in this way, the absence of explicit connections among activities and ideas undermines students’ mathematics learning.</p>
<p>Larson summarizes the resulting dilemma for mathematics educators as follows:</p>
<blockquote>
<p>The dilemma is that while districts, schools, and teachers have greater access than ever to tools and resources for selecting and developing instructional materials, the skill required to develop a high-quality curriculum is both complex and often underappreciated. The widespread availability of online tasks therefore makes having and working with a coherent curriculum at the school and district level even more important because it is the curriculum that establishes the learning goals in a coherent progression and helps teachers see and understand the multiple pathways that students might take through the progression.</p>
</blockquote>
<p>We know from firsthand experience that creating rigorous, focused, coherent curriculum for teachers is incredibly demanding and fascinating work. However, we also understand that creating high-quality curriculum materials is only part of the picture. Because most of us on the Math Learning Center curriculum development team are mathematics teachers and have continued to work in classrooms, we also know there is great skill and artistry in bringing that curriculum to life with students. When teachers can rely upon a coherent curriculum, they are free to spend their time doing what they do best and for which there is simply no substitute or shortcut: developing strong relationships with students and tailoring their instruction every day to facilitate mathematics learning for all students.</p>
<p><em>Martha Ruttle is the senior curriculum editor for MLC.</em></p>Tue, 13 Sep 2016 13:57:06 -0700Martha Ruttle477 at http://www.mathlearningcenter.org Word Resource Cards Support the Math Practice Standards
http://www.mathlearningcenter.org/blog/word-resource-cards-support-math-practice-standards
<p>Math vocabulary equips students to understand concepts, extend their learning, and engage in deeper communication. <a href="http://catalog.mathlearningcenter.org/search?term_node_tid_depth=All&field_searchkeys_value=&model=WRC#results" target="_blank"> Word Resource Cards </a> help you create a rich environment for mathematical exploration. Here are examples from my own classroom: <!--break--></p>
<p><strong> MP.3 </strong> Construct viable arguments and critique the reasoning of others — Vocabulary enhances student discussions. Just yesterday, students in my fifth grade classroom pulled out Word Resource Cards to compare definitions and visual representations as they debated the similarities and differences in the words <em> multiple </em> and <em> product </em> . </p>
<p><strong> MP.4 </strong> Model with mathematics — As students model, they apply mathematical terms to situations. Last week, a student proudly showed me how she is able to subtract using a number line. Her mom remarked on the number line's strength as a tool for finding the difference. Our short conversation was rich in mathematical thinking and vocabulary.</p>
<p><strong> MP.6 </strong> Attend to precision — Mathematical precision begins early. Even young students can learn that the word and does not belong in the context of counting whole numbers. As they counted numbers to 1,000, my second graders learned to say nine hundred nine instead of nine hundred and nine.</p>
<p><img src="/sites/default/files/images/WRC.jpg" border="0" alt="Word Resource Cards" width="450" height="240" /></p>
<h3><strong> Vocabulary Resources </strong></h3>
<p>Word Resource Cards help students develop the mathematical language they need to communicate their reasoning accurately and precisely. Each card portrays a word or term along with an illustration of its meaning. The working definition appears on the back of each card for your reference. You can use Word Resource Cards to gradually develop a classroom math word wall. Post the cards on the word wall (or in a classroom pocket chart) when the terms arise during the course of a lesson—and let it empower students' math practice skills all year long. </p>
<p>The North Carolina Department of Education supports teachers by organizing CCSS math vocabulary in a <a href="http://maccss.ncdpi.wikispaces.net/file/detail/2014+Building+Vocabulary.doc" target="_blank"> <strong> simple document </strong> </a> by grade level and domain.</p>
<p>Check out our free <a href="http://catalog.mathlearningcenter.org/apps/math-vocabulary-cards" target="_blank"> Math Vocabulary Cards App. </a></p>
<p><em> Cynthia Hockman-Chupp is a curriculum specialist for MLC. </em></p>
<p> </p>Thu, 01 Sep 2016 11:40:41 -0700Cynthia Hockman-Chupp471 at http://www.mathlearningcenter.orgInteractive Paper Plate Fractions
http://www.mathlearningcenter.org/blog/interactive-paper-plate-fractions
<p>While preparing to teach a fifth grade lesson using clock fractions, I ran across a photo on Pinterest that inspired me to try a little paper plate math. <!--break--> <img src="/sites/default/files/images/Plates2.jpg" border="0" alt="Fractions on a Clock" width="500" height="375" /></p>
<p><strong> How to Assemble: </strong></p>
<ol><li>Get two plates in two different colors.</li>
<li>Mark the center and make one straight cut from the edge to the center point. I stacked the plates on top of each other and cut them at the same time.</li>
<li>Slide the plates together so that the cuts match up in the middle and you’re able to rotate one plate to reveal the other. </li>
<li>Add incremental numbers to one of the plates. For this example, I numbered one plate to resemble a clock.</li>
</ol><p><strong> Model with Many Uses: </strong> <br /> Use these handy little visuals in a variety of ways:</p>
<ul><li>Modeling fractions – Ask students to show you 1/12, 2/3, or 1/4 of an hour.</li>
<li>Adding fractions – Ask students to spin the model to show 1/2 + 1/4 and other addition expressions. A pair of students might even hold two plates to add fractions with a total greater than 1.</li>
<li>Subtracting fractions – Students can model 9/12 – 1/3 and other subtraction problems.</li>
<li>Fraction stories – Write and demonstrate fraction stories on the interactive model.</li>
<li>Homework – Send this interactive visual model home.</li>
</ul><p><em> Cynthia Hockman-Chupp is a curriculum specialist for MLC. </em></p>
<p> </p>Wed, 24 Aug 2016 14:21:26 -0700Cynthia Hockman-Chupp470 at http://www.mathlearningcenter.orgA Tribute to Dr. Robert Sylwester
http://www.mathlearningcenter.org/blog/tribute-dr-robert-sylwester
<p>Ask people to recall a person who impacted their life, and there's a good chance they will name a teacher. It's not hard to understand why. Great teachers care about the lives of their students, they inspire, encourage, and give more than they take.<!--break--></p>
<p>If you asked me who was influential in my life, I would tell you Dr. Robert Sylwester. He was a professor, a writer, a speaker, a mentor. And at his core, he was a teacher. Bob passed away recently, about six months short of living 90 years. He left behind a legacy many aspire to achieve. He was well known for his books on cognitive science and for his ability to capture an audience with insights and facts about learning and the human brain. </p>
<p>I could sit and listen to Bob all day long. Because of him, I became a better teacher. I learned that learning was more than memorizing and following procedures. I learned that teaching was more than telling. I learned that our greatest gift is our ability to grow our brains, to think more deeply, ask more questions, explore more ideas, make more attempts, and celebrate more “aha!” moments. </p>
<p>I will miss Bob's frequent newsletters, his humor and stories, his love for the Oregon Ducks, and his passion for learning. But because of who he was and what he did, he will continue to live on through me and countless others. </p>
<p><em>Dan Raguse is the executive director of MLC.</em></p>Wed, 17 Aug 2016 03:57:11 -0700Dan Raguse468 at http://www.mathlearningcenter.orgEffective Parent Education
http://www.mathlearningcenter.org/blog/effective-parent-education
<p>Looking for resources to educate parents?<!--break-->Bridges leaders Barb Blanke, Kimberly Kelly, and Jessica Djuric have compiled a wealth of slides, articles, books, and handouts that could be used for family gatherings, student-led conferences, back-to-school sessions and more. Visit their site, <a href="https://sites.google.com/site/effectiveparenteducation" target="_blank">Effective Parent Education.</a></p>
<p>For additional ideas, see Lori Bluemel and Pia Hansen’s posts about Family Math Night <a href="https://bridges.mathlearningcenter.org/implementation/blog/family-math-night-0" target="_blank">here</a> and <a href="https://bridges.mathlearningcenter.org/implementation/blog/family-math-night" target="_blank">here</a>.</p>
<p><em>Cynthia Hockman-Chupp is a curriculum specialist for MLC.</em></p>Wed, 10 Aug 2016 04:03:43 -0700Cynthia Hockman-Chupp467 at http://www.mathlearningcenter.orgSpanish Components Kits Now Available
http://www.mathlearningcenter.org/blog/spanish-components-kits-now-available
<p>We have good news for Bridges and Number Corner Second Edition classrooms with Spanish-speaking students: All language-based game boards, spinners, card decks, and other similar items are now available in Spanish. <!--break--></p>
<p>These new Spanish Component Kits contain only the pieces that require translation. Thus they are designed to be used as an add-on to a full Bridges or Number Corner Second Edition kit.</p>
<p>For K–2 classrooms there are both Bridges and Number Corner kits available. (Note the Bridges kits include all the Number Corner items as well.) Word Resource Cards in Spanish for K–2 are sold separately.</p>
<p>The situation for grades 3–5 is somewhat different. Because there are very few Bridges components with terms to translate, both Bridges and Number Corner users simply need to purchase the Number Corner Spanish Components Kit. Again, Word Resource Cards are sold separately.</p>
<p>Click one of the following links to see more information about the Spanish Components Kits:</p>
<p><span style="text-decoration: underline;"> Bridges Spanish Components Kits </span></p>
<p> <a href="http://catalog.mathlearningcenter.org/store/product-8875.htm" target="_blank"> Kindergarten </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8878.htm" target="_blank"> Grade 1 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8879.htm" target="_blank"> Grade 2 </a></p>
<p><span style="text-decoration: underline;"> Number Corner Spanish Components Kits </span></p>
<p> <a href="http://catalog.mathlearningcenter.org/store/product-8880.htm" target="_blank"> Kindergarten </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8881.htm" target="_blank"> Grade 1 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8882.htm" target="_blank"> Grade 2 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8883.htm" target="_blank"> Grade 3 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8884.htm" target="_blank"> Grade 4 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8885.htm" target="_blank"> Grade 5 </a></p>
<p><span style="text-decoration: underline;"> Spanish Word Resource Cards </span></p>
<p> <a href="http://catalog.mathlearningcenter.org/store/product-8873.htm" target="blank"> Grades K-2 </a> , <a href="http://catalog.mathlearningcenter.org/store/product-8874.htm" target="blank"> Grades 3-5 </a></p>
<p>To learn more about Bridges in Mathematics 2nd Edition and request preview access to the curriculum, visit the <a href="http://www.mathlearningcenter.org/bridges/overview" target="_self"> Bridges Overview page </a> .</p>
<p><em> Collin Nelson is a marketing coordinator for MLC. </em></p>Wed, 03 Aug 2016 11:41:42 -0700Collin Nelson466 at http://www.mathlearningcenter.orgIntroducing Fractions, a New App from MLC
http://www.mathlearningcenter.org/blog/introducing-fractions-new-app-mlc
<p>We’re proud to announce the release of our newest free math app, <a href="http://catalog.mathlearningcenter.org/apps/fractions" target="_blank"> Fractions </a> for iPad, Web and Chrome! With this app, students can use a bar or circle to represent, compare, and perform operations with fractions with denominators from 1 to 100. They can choose the fraction model and number of equal parts or use a color to select specific parts to show a fraction of the whole. <!--break--></p>
<p>What’s especially cool is that the Fractions app creates opportunities to investigate and compare fractions. If, for example, the white bar has a value of 1, what might be the value of the colored bars that follow?</p>
<p><img src="/sites/default/files/images/Blog%20Post%20Images/fractionsapp-5.png" border="0" alt="Introducing Fractions, a New App from The Math Learning Center" width="600" height="294" /></p>
<p>Students can check and justify their predictions by coloring in or counting the remaining sections in each line. As they identify the value of the pieces relative to the white strip, the fractions can be labeled.</p>
<p><img src="/sites/default/files/images/Blog%20Post%20Images/fractionsapp-1.png" border="0" alt="Introducing Fractions, a New App from The Math Learning Center" width="600" height="263" /></p>
<p>One idea is to ask students to pair-share any mathematical observations they can make about the pieces and invite volunteers to share their thinking. Be sure to note along with them that each row of fraction pieces is equivalent to 1. Then work with students’ input to write a set of equations representing these equivalencies on the board. Have students build and report equivalencies for ½, ¼, and ¾ as you record equations to match. </p>
<p><img src="/sites/default/files/images/Blog%20Post%20Images/fractionsapp-2.png" border="0" alt="Introducing Fractions, a New App from The Math Learning Center" width="600" height="370" /></p>
<p>If they have access to the app, students can continue the exploration by building their own representation of the following fractions, sharing observations with each another and the class as they work.</p>
<p>3/16, example:</p>
<p><img src="/sites/default/files/images/Blog%20Post%20Images/fractionsapp-4.png" border="0" alt="Introducing Fractions, a New App from The Math Learning Center" width="600" height="62" /></p>
<p></p>
<p>5/8</p>
<p>3/4</p>
<p>5/16</p>
<p>7/16</p>
<p>Can they then record equations that result in the sum shown on the model?</p>
<p><img src="/sites/default/files/images/Blog%20Post%20Images/fractionsapp-3.png" border="0" alt="Introducing Fractions, a New App from The Math Learning Center" width="600" height="137" /></p>
<p>Enjoy the new Fractions app! And let us know on Twitter how you’re using it in the classroom by tagging <a href="https://twitter.com/MLCmath" target="_blank"> @MLCmath </a> or using <a href="https://twitter.com/search?q=%23bridgesmath&src=typd" target="_blank"> #BridgesMath </a> !</p>
<p>The free Fractions app is available to use as a web app or can be downloaded from iTunes or the Chrome Web Store. Download Fractions and learn more <a href="http://catalog.mathlearningcenter.org/apps" target="_blank"> here </a> . For step-by-step instructions for using Fractions, check out the info button from inside the app.</p>
<p><em> Cynthia Hockman-Chupp is a curriculum specialist for MLC. </em></p>Wed, 20 Jul 2016 09:40:48 -0700Cynthia Hockman-Chupp465 at http://www.mathlearningcenter.org