Iowa Council of Teachers of Mathematics Featured Articles
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Iowa Council of Teachers of Mathematics blog postsIowa Council of Teachers of MathematicsWild Apricot - membership management software and moreenSat, 19 Jun 2021 01:16:49 GMTSat, 19 Jun 2021 01:16:49 GMTThu, 04 Mar 2021 20:17:08 GMTLet's Factor! by Suzanne Riehl<p>Divisibility, factoring, composing, decomposing -- these are important themes in mathematics. This article discusses those topics, but the reason I am writing about methods to factor is simply for the fun of it. I mentally factor numbers into primes as a pastime while waiting in line or confined in car. (Yes, I am a math nerd.) As a bonus, I learn as I play. I'll start with basic techniques and then share the <a href="https://iowamath.org/resources/Documents/divtests.pdf" target="_blank">150 Method</a>.</p>
https://iowamath.org/Articles/10164511
https://iowamath.org/Articles/10164511Wendy WeberThu, 07 Jan 2021 17:07:37 GMTThe Case to Axe State Testing by Dr. Clayton Edwards<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">Like most everyone else, my school year last year “ended” due to COVID-19, and I was very sad that I wasn't able to see my students again and properly celebrate how successful they were. I still provided optional assignments during that time. I got a hundred emails a day asking for feedback or to answer questions. That was great and I was happy for the communication, whether it was school related or not (I had a student email me at midnight one night with a screenshot of a movie she was watching just to let me know the main character looked like me...I love middle school), but I never realized how much I missed the student to student and teacher to student communication in the classroom until the pandemic. It was very difficult to simulate that form of communication online.</font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial"><img src="https://lh3.googleusercontent.com/27YRcP0FNdjCqU0TvpqmGG2NyBHBDXWAcvdOoWfZiHIEfr2a0CqbBLj5SUQjhmeIdb0hL1BlE975PC16ZXiRkCEDp_M1G3mWTeVH88CVvwlToZx7B6erMjIIywJJYLEeCpoFNbCs" width="258" height="251"></font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">While there was a lot I missed about being at school, the one thing that I didn’t miss was having my students take our state assessment, ISASP (or Iowa Assessments previously). Not taking the test may have actually been one of the greatest successes to come out of last year. This is what I generally think about once our testing is finished…except replace “today” with “this week.”</font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial"><img src="https://lh5.googleusercontent.com/qB-8zTyEiZsIEM988YLL-WeXe15WmAgN3p7v-kthFGS6lL8nGgtN7YOw4uGWq_HdGJe4fsch0H6_IFjxxlSOdB1abvK40ejyxTbzrw-_9Q6bbLORlp8Q9HTmBSf2gsjpm-FnVlI3" width="273" height="235"></font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">If you don’t know me very well, you may be saying to yourself that I don’t like my students taking the test because we do poorly, which may reflect negatively on me. Maybe some sour grapes towards the test? If that was your initial reaction, you would be wrong. My students always do well on state assessments, and it isn’t because we prep or prepare for the test, but because we do things in class like collaborate, revise, justify, and struggle in a productive manner. In fact, if you look at my previous article on our initial ISASP experience, my 7th and 8th grade students scored the highest in the state of Iowa.</font></p>
<p style="line-height: 19px;"><a href="https://www.iowamath.org/Articles/8985018" target="_blank"><font style="font-size: 15px;" color="#1155CC" face="Arial">https://www.iowamath.org/Articles/8985018</font></a></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">So why would I prefer not to take the test if we do so well?</font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">I design my class to be one big formative assessment. All of the work my students do in class helps me to make informed decisions about where each individual student should move next in his or her mathematical journey. It’s a process. I provide a task. The students experience the task. I give the students feedback along the way. Students use that feedback to improve and have a better understanding of the mathematics. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">I benefit because I know exactly what my students know. I can hear their conversations. I can see their justifications. I can identify any misconceptions and intervene appropriately. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">The students benefit because they have an opportunity to gather feedback and learn from the experience. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">The problem with our current state test, any state tests that we have ever taken, and I assume any type of high stakes test that other states take, is that it is lacking any of these beneficial qualities. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">I gain nothing from the results of these tests. I gain nothing from the experience of proctoring these tests. My students gain nothing from the results of these tests. My students gain nothing from experiencing these tests. It’s great to know that my students are “doing well.” I know I get many emails from parents congratulating us for “doing well.” I would rather be “doing well” than not, but what does “doing well” really mean? I have no idea because I don’t have the questions the students did on the test and I don't have their work and justifications from the test. Without these two items, I get the vaguest of vague feedback which is not helpful in the slightest to anyone. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">I’ve heard of districts trying to analyze the data returned to them. What are they looking at? At best I can see a very general mathematical category and the number of questions my students got right or a percentile in that category. Not helpful. These categories reported are far too broad to pin down any real misconceptions. About the best I can do is peak my students’ interest and go to the Iowa School Performance Profile (https://www.iaschoolperformance.gov/ECP/Home/Index) and show them individually all the schools they “beat.” It’s awesome to be able to say that you did better than other schools, but is that really the goal? Doing this does nothing to inform my practice or help students improve. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">Many districts then turn around and make arbitrary decisions based on these vague scores because using a score is simple, even if it doesn’t have a lot of meaning behind it. The MTSS process can suffer when students are placed in remedial time because of a score. Many of these students end up working on random skills and don’t make progress because you are using scores that have no substance other than a student is “good or bad” at math. What are the students to be working on? Fractions for example...that is too vague to effectively help a student improve. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">This selection process based on scores goes much deeper than MTSS. For example, many years ago, we placed students into high school Algebra as 7th and 8th graders based on a single number (as I know many schools currently do). A single score. To this day I feel horrible for being a part of that decision. Sure, most of the students who were advanced did fine in high school Algebra, but I can only imagine the learning gaps that haunt them to this day that resulted from bypassing one or two full grades. If you haven’t checked the Iowa Core recently, there is a lot to process, and in hindsight, skipping that much material was an awful idea. I feel much better about the process today, as we do not look at test scores at all, but use a more standards based approach to make sure students that are advanced have shown a high level of proficiency in all mathematical standards that would be skipped. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">High stakes testing has been around forever, but why does it continue? Just because we have always done it? As a method of accountability? As a way to compare schools? Maybe, although in my mind, none of those reasons has anything to do with helping students grow and prosper mathematically, which is what it should be about. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">I am sure these tests aren’t going anywhere. I am not that naïve to think that something that runs so deep throughout education will go away just like that. If our students have to take the tests, could they be created to be useful and not just a time waster; something that is taken and never looked at again? Here are my recommendations:</font></p>
<blockquote>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">1. Use the bare number of questions possible to feel like the standards were “covered.” The ISASP assessment had 50+ questions for math alone, and by the time my students justified everything to the best of their abilities, it took them almost three hours to complete. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">2. Release the questions immediately after the test and let teachers hang on to student work. Teachers can go through the student work matched with each question to help make informed decisions, much like they would anyway in their own classrooms. Make new questions for the next year if you are worried about the questions floating around. </font></p>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">3. If the data that is typically collected on high stakes assessments to compare and judge is still necessary, you can still collect it regardless of the changes from items 1 and 2, although I still think this is counterproductive. Instead of pushing educators apart, work on teachers collaborating together on these assessment questions once the test is finished to better help their students improve. You can have a lot of great discussion when you have a math question and varying student work to accompany it. </font></p>
</blockquote>
<p style="line-height: 19px;"><font style="font-size: 15px;" color="#000000" face="Arial">If the powers that be keep mandating high stakes assessments as is, we will keep doing well. I am always proud of my students and what they are able to accomplish. Each of my students has enough personal pride and school pride to do well, even though they may not see the purpose of taking state assessments (I don’t either). Even though they will fight and claw to do the best they can, the point is they shouldn’t have to unless it will help them understand mathematics better in the long run. </font></p>
https://iowamath.org/Articles/9747843
https://iowamath.org/Articles/9747843Wendy WeberSun, 18 Oct 2020 23:26:08 GMTHave you Heard of MTBoS and the Global Math Department? by Sarah Martin<p>Have you ever heard of MTBoS?</p>
<p>MTBoS stands for “Math Twitter Blog-o-Sphere'' and is a community of math teachers who blog and tweet. However, it’s way more than that! The <a href="https://mtbos.org/" target="_blank">MTBoS website</a> has a directory of community members you can follow via Twitter. There is a search engine to search blogs on specific topics written by members of the community. There is also a Desmos bank to search for Desmos activities that have been made by members of the community. Another great thing about MTBoS is if you need help you can tweet with the hashtag #MTBoS. You will likely have an answer or suggestion in a matter of minutes, and if not then, at least by the end of the day.</p>
<p><a href="https://www.bigmarker.com/communities/GlobalMathDept/conferences" target="_blank">The Global Math Department</a> is another helpful resource. This group began with teachers who knew each other through Twitter, blogs, and Twitter Math Camp. It has since grown into a wide range of math educators who love to share their ideas with others. They have a weekly newsletter with bits of information that mainly come from Twitter. They also have a free weekly math webinar. I encourage you to explore their site and subscribe to their weekly newsletter.</p>
<p>Both the MTBoS and the Global Math Department have been great providers of information for me as a math teacher. They have also helped shape me into the math teacher I am today.</p>
<p>Sarah Martin<br>
7th Grade Math Teacher<br>
Shenandoah Middle School</p>
https://iowamath.org/Articles/9311614
https://iowamath.org/Articles/9311614Wendy WeberFri, 03 Jul 2020 15:57:34 GMTOh, Subtraction Hurts me! by Ed Rathmell<p>After a teacher asked me to talk with a second grade boy about subtraction facts, my goal was to determine how he was thinking to solve the problems. I soon found out that his strategy was to guess. He felt that he was off the hook as soon as he said an “answer.” The answer might be incorrect, but so what, …he didn’t really try. He had just guessed. Sometimes you guess right, sometimes you guess wrong. </p>
<p>But, his feelings became evident even before that. When I informed him that his teacher had asked me to talk with him about subtraction, he had a pained expression as he said, “Oh, subtraction hurts me!” That was one of the most uncomfortable interviews I have ever conducted. Subtraction really did hurt him, …and it was obvious during each problem I presented. He knew that he didn’t understand, and guessing was the only strategy that he had figured out yet.</p>
<p>Primary grade teachers have students like this every year. Unfortunately, the materials they have do not provide the kind of help that is needed for these students. Textbooks simply don’t provide enough time for most students to make sense and develop flexibility and fluency, that is, to deeply understand. The pandemic has exacerbated this problem, especially for those students. </p>
<p>Teachers have had an impossible job to recreate ways to teach math this past spring. It is not their fault. How do we help students make sense when we can’t be in the same room? How do we keep them actively involved? How do we know what they are thinking? How do we help them make connections? How do we know what they have learned? How do we know if some students are being left behind? How do we prepare them for success in school math next fall? …</p>
<p>It doesn’t make any difference if students are in our classroom, on-line, homeschooling, or on vacation. There are no shortcuts. They will not understand unless they make sense of the concepts and the reasoning strategies they can use in everyday life. Drill and practice seem like the best solution to many people, but over 75 years of research has clearly demonstrated that there are no long-term effects for most students. The focus is on the answer, not on how you can get the answer. In the 1940s, Brownell found that about 40% of all students did not even get any immediate effect. And drill and practice did nothing, for any student, to promote what we now call number sense.</p>
<p><strong>Helping Students Understand Math</strong></p>
<p>To make sense of math and be able to use it effectively, students need repeated experiences: </p>
<ul>
<li>to make sense of a variety of ways to represent each concept so they have a better opportunity to recognize when that concept can be used in everyday life,</li>
<li>to make sense of a variety of reasoning strategies that can be used with each concept so they can efficiently use that concept with different numbers and in different contexts, </li>
<li>with those representations and reasoning strategies so they can be used flexibly and fluently, and</li>
<li>using those concepts and reasoning strategies to solve problems they will encounter in a variety of everyday situations.</li>
</ul>
<p>These recommendations are all consistent with national and state standards. The one thing that differs slightly is the additional suggestion for repeated experiences. That comes from well-documented research on memory and learning. The reality is that students do need repeated experiences to develop flexibility and fluency in their thinking. It takes time for students to internalize new thinking so they spontaneously use it in appropriate situations. </p>
<p>For example, after two weeks of brief daily lessons on using ten to add and subtract in the spring of grade 2, less than half of them spontaneously used that thinking when provided the opportunity. Even though they could explain that thinking when specifically asked, they resorted to much less efficient counting in other situations. Students can receive huge benefits from extended opportunities to make sense of new concepts and new thinking. Practice in the use of new thinking is essential, if we expect students to actually use that thinking. Just because symbolic drill is not effective doesn’t mean that repeated experiences with the use of concepts and reasoning strategies is not needed.</p>
<p><strong>Meaningful Distributed Instruction</strong></p>
<p>Thirty-three years ago I had the pleasure of observing Marsha Bachman’s second grade math class in Grinnell. She used brief daily conceptual previews to help prepare her students for success with subtraction. These were not drill and not symbolic practice. They involved helping students make connections among concepts, manipulatives, and symbols. When I asked why she did that, she simply said, “I’ve found that it’s much easier for the kids when we get to subtraction.” Two weeks later after observing her students during the first day of instruction on subtraction, their understanding impressed me. By coincidence, I had just recently taught the same lesson using the same textbook, but with considerably more student confusion. That really got my attention! </p>
<p>Over the next few years, some of my undergraduate pre-service teachers and I tried similar approaches in action research studies. Altogether we covered about 20 different topics at grade levels ranging from K through grade 8. These conceptual previews led to overwhelming success. In every instance students had at least a 20% achievement advantage over students without the previews. Several of my graduate students also did action research projects for their MA papers. In each case using conceptual previews enhanced achievement with similar results. </p>
<p>I want to highlight one of these studies. Tammy Boeckman, a sixth grade teacher in Ft. Dodge at the time, got amazing results. After using daily conceptual previews for fractions and decimals for the entire year (no symbolic practice), her students, including more than her share with learning problems, earned a class average score of over 90% on a very comprehensive fraction and decimal assessment—two years in a row. Nationwide, eighth grade students averaged about 20% lower on very similar National Assessment of Educational Progress (NAEP) items. Despite not practicing computation with fractions or decimals, her students performed over 10% higher on computation than eighth graders typically did on similar NAEP items.</p>
<p>Since the late 1980s, everything I have written, both articles and curriculum, has been based on using brief daily conceptual experiences to help students make sense and enhance their math achievement. And I stressed the importance of using similar approaches in each of my teacher education classes. Since no instructional materials are organized like that, I decided to retire from teaching so I could create what I had been promoting for years. For a thorough discussion of meaningful distributed instruction, see Chapter 5, Number and Operations: Organizing Your Curriculum to Develop Computational Fluency in Achieving Fluency: Special Education and Mathematics (NCTM, 2011).</p>
<p>Now I have nearly completed an integrated and comprehensive collection of on-line lessons for addition and subtraction for students in grades K-3. They are currently over 1500 lessons that are:</p>
<div style="margin-left: 4em">
<ul>
<li>daily,</li>
<li>supplementary,</li>
<li>brief (about 5-minutes),</li>
<li>conceptual, </li>
<li>animated, </li>
<li>planned with pauses after each question, and</li>
<li>accompanied by brief formative assessments for each expected outcome.</li>
</ul>
</div>
<p>The pauses are designed to provide students the opportunity to think, solve, explain, and discuss their solutions, …before one animated illustration of a reasoning strategy that could be used to solve the problem is presented. The lessons are designed for teachers to use a problem solving approach to instruction. And there are enough repeated experiences for students to have time to make sense and to develop flexibility and fluency. </p>
<p>Brief 5-item paper-and-pencil assessments will quickly inform teachers about student progress towards expected content outcomes. A complete list of outcomes is listed at the bottom of our web site home page. Additionally, there are on-line assessments for each reasoning strategy designed to inform teachers about progress with basic facts, but more importantly, about progress on actually using the reasoning strategy being assessed. Immediately after a class has used the on-line assessment, teachers will have access to a list of students who are not yet using that strategy.</p>
<p>The topics include:</p>
<div style="margin-left: 4em">
<ul>
<li>Counting and Comparing,</li>
<li>Numbers and Partitions,</li>
<li>Exploring With Word Problems, and </li>
<li>Reasoning Strategies.</li>
</ul>
</div>
<p>The lessons are designed to help students make sense of different representations for the understandings and skills needed to use addition and subtraction. These representations include animated objects with five frames, ten frames, number lines, open number lines, tree diagrams, and part-part-whole diagrams. The animations also illustrate the step-by-step thinking that can be used with each of the reasoning strategies.</p>
<p>The counting and comparing lessons, not only help students learn these skills, they also address all of the common students errors. This is the underlying knowledge needed to be successful with addition and subtraction. Most of this has been created for pre-K children.</p>
<p>The numbers and partitions lessons help students learn to use the structure of the five frame or ten frame to solve partition problems without counting. Students will understand part + part = whole and whole – part = other part in ways that connect their knowledge about addition and subtraction, something that students often lack. This lack is partially the result of subtraction language that does not connect to addition knowledge. Also thinking of subtraction only as “take away” does not help students make those connections.</p>
<p>There is a section on each of the Cognitively Guided Instruction problem structures. Most of these lessons have students solve or create a word problem. The others are animated illustrations of each problem structure. </p>
<p>They also provide enough lessons to help students make sense of different reasoning strategies while using addition and subtraction. Each of seven different strategies has at least four weeks of lessons followed by six weeks of practice for that thinking—far more than most students will need. These strategies include counting on, counting back, counting up to subtract, using tens to add and subtract, using known facts (including doubles) to add and subtract, using nice numbers, and changing the problem to one that is easier. Additionally, estimation strategies include using front-end numbers, using nice numbers, using bounds, and using rounding.</p>
<p>These often overlooked reasoning strategies are crucial in helping students make progress in achievement. In Australia, bumps in achievement have been attributed to students developing new ways of thinking. For example, shortly after students learned to use ten to add and subtract, that group of students made a big jump in their achievement, as evidenced on tests. Those same reasoning strategies help with retention of basic facts. In three schools, all first and second grade students were interviewed to identify which reasoning strategies they could explain. Students who could explain a strategy beyond counting dropped about 10% in performance over summer vacation. Other students dropped over 50% in each school.</p>
<p>These on-line lessons are currently being provided free to anyone who registers so they can log in. You can examine the lessons and try them with your students by registering at: <a href="http://www.thinkingwithnumbers.com" target="_blank">www.thinkingwithnumbers.com</a></p>
<p>Please encourage primary grade classroom teachers, special education and resource teachers, and perhaps most important now, parents of young children in the primary grades to try our web site with their children. It is free; it just takes a commitment to spend 5 minutes a day with your child. More importantly, it will make a difference in success with school math next fall.</p>
<p>Enjoy listening to your child. You can’t believe how much fun it is to hear new, but confident, and unexpected explanations. </p>
<p><br></p>
<p>Ed is an Emeritus Professor of Mathematics Education at the University of Northern Iowa. He is a former Iowa Council of Teachers of Mathematics President and long time member of the ICTM Executive Board. </p>
https://iowamath.org/Articles/9075980
https://iowamath.org/Articles/9075980Wendy WeberMon, 01 Jun 2020 19:48:40 GMTPAEMST: What’s that? by Deb Little<p style="line-height: 19px;"></p>
<p style="line-height: 19px;"><em><font face="Ubuntu" style="font-size: 18px;">Reflections from Deb Little, the 2018 PAEMST elementary math awardee</font></em></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><font>It sure is a strange sounding acronym. And this acronym is necessary, it stands for a mouthful of words! PAEMST, pronounced</font> <em><font>pam-st,</font></em> <font>stands for Presidential Award for Excellence in Mathematics and Science Teaching. Read on to learn more about PAEMST, my journey as the 2018 PAEMST elementary mathematics awardee, and why you should consider nominating yourself or someone else for this award. </font></font></p>
<p style="line-height: 19px;"><font face="Ubuntu"><font style="font-size: 18px;"><strong>What is PAEMST? </strong></font></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><font>The PAEMST</font> <span style="background-color: rgb(255, 255, 255);"><font>is the highest recognition that a kindergarten through 12th grade science, technology, engineering, mathematics, and/or computer science teacher may receive for outstanding teaching in the United States.</font></span> <font>I had never heard of it before I got nominated for it. Teachers do not go into teaching to get awards. When we read about someone getting an award we think something like, “Oh! Good for them! They must be a really gifted educator.” And then we continue about our work with children trying to do our best. </font></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><strong>How did I become nominated for the PAEMST?</strong></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><font>In November of 2017, Sandra (Sandy) Ubben had written and asked me if it would be okay to nominate me. Sandy is currently an</font> <span style="background-color: rgb(255, 255, 255);"><font>Illustrative Mathematics Certified Facilitator</font></span><font>. When I worked with her, she was a math consultant for Central Rivers AEA. To say that I was flattered to be nominated by my mentor and someone whom I consider a math educational genius is an understatement. I said, “Yes,” and didn’t think anything more about it. Then the emails from the PAEMST organization started to flood my inbox. </font></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><strong>What happened next?</strong></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><font>As I read over the emails from the PAEMST organization, I became a bit overwhelmed. I quickly realized that applying for this award would not be as simple as filling out a one-page application form and attaching my resume. It</font> <font>includes a written narrative and video recording of one’s class in action. The narrative would be a reflection of my lesson and a description of my contributions to education focusing on Five Dimensions of Outstanding Teaching:</font></font></p>
<ul>
<li><span style=""><font style="font-size: 18px;">Mastery of the content being taught in the lesson.</font></span></li>
<li><span style=""><font style="font-size: 18px;">Use of instructional strategies that support student learning.</font></span></li>
<li><span style=""><font style="font-size: 18px;">Effective use of assessment to support student learning.</font></span></li>
<li><span style=""><font style="font-size: 18px;">Reflective practice and lifelong learning to improve teaching and student learning.</font></span></li>
<li><span style=""><font style="font-size: 18px;">Leadership in education outside of the classroom.</font></span></li>
</ul>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;">I also learned that the PAEMST program is organized and run by the National Science Foundation (NSF) on behalf of the White House Office of Science and Technology Policy (OSTP). If chosen as an awardee, I would receive a certificate signed by the President of the United States, a trip to Washington D.C. to attend a series of recognition events and professional development opportunities, and a $10,000 award from the National Science Foundation. To be honest, I was not sure if I was worthy of such an award and decided at that point not to apply. </font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;"><strong>Why didn’t I think I was worthy?</strong></font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;">Let’s face it. The words “excellence” and “highest honor” are a bit daunting. As a teacher, I’ve strived for excellence in my practice, but there are many lessons that I’ve reflected on that I would not describe as “excellent.” My journey as a teacher is constantly evolving. I could be described as Christina Tondevold says, “a recovering traditionalist.” In my practice of 27 years, I’ve moved from an “I do, we do, you do” approach to one that’s inverted. Now, my students grapple with a problem first and reason to find solutions in their own way. Next, class time is spent with students sharing their reasoning and sense making, asking questions of their classmates’ solutions, comparing what’s similar and different and most efficient. Conjectures are made and recorded and reflected upon in subsequent lessons. It is a flip of what went on for many years in my mathematics classroom. In January of 2015, I had taken a course offered by my AEA that gave me my first taste of the meaning of cognitively guided instruction. During the 2015 -2016 school year, Sandy Ubben worked closely with me to help me get better at facilitating this flipped version. At her request, I opened my classroom so that other educators could see these sense making learning sessions. I really felt like I was at an infant stage with facilitating this kind of mathematical learning. </font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;"><strong>What changed my mind about applying?</strong></font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;">Right before my school’s winter break in December, I had a discussion with UNI professor, Amy Lockhart at a school event. I had taught both of Amy’s sons when they were in fourth grade. She encouraged me to apply for the PAEMST if for no other reason than to “honor the person who had nominated me.” I decided to apply and give the application process my best effort. I wanted to honor my nominator, Sandy Ubben.</font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;"><strong>What happened next?</strong></font></span></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;">I’m not going to lie. The application takes a considerable amount of time to complete. From January to the May 1 deadline, I spent many hours choosing a lesson video and then reflecting, writing, and revising my written narrative. I consulted with former PAEMST Awardee Annette Louk for advice on my application. I attended the online question and answer informational sessions held by the PAEMST organization. I spent hours reading research from math education leaders from sources like the National Council of Teachers of Mathematics (NCTM). Through this whole process, I learned so much about myself as a teacher because I was constantly asking myself why I do what I do in my mathematics work with students. It inspired me to dig deeper into my understanding of the practices that I had implemented in my classroom. By the time I submitted my application, I had grown as an educator.</font></span></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><span style="background-color: rgb(255, 255, 255);"><font>After submitting my application,</font></span> <font>I learned that the applications are reviewed first at the state level. The Iowa State Selection Committee spends countless hours to choose 2 to 3 applicants to move on to the national level. The NSF convenes a national-level selection committee composed of scientists, mathematicians, education researchers, school and district administrators, and classroom teachers. Recommendations are sent to the White House Office of Science and Technology Policy (OSTP) for final selection.</font></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;">In August of 2018, I found out that I was a state finalist along with Natalie Franke and Chris Mathews. In October, the three of us were formally recognized at the Iowa Council of Teachers of Mathematics Conference for being state finalists. In March of 2019, we were honored by Governor Kim Reynolds at a reception and luncheon in Des Moines. When I learned of the exceptional math leadership of my fellow finalists, I thought that there would be no way I would be chosen for the national award. I felt so honored to be standing alongside educators of their caliber.</font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><strong>When did you find out you had been selected as the national PAEMST awardee?</strong></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;">I had an inkling in late June of 2019. I was contacted by the PAEMST organization that they needed the FBI to do a background check. They were clear that this did not mean that I was the finalist and that saying anything about it to anyone could disqualify my application.</font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;">Then on September 27, 2019, I received an email from the PAEMST organization that I had been selected as the 2018 PAEMST Awardee and was given the steps to make flight and hotel arrangements for the recognition ceremonies and professional development events to be held from October 14 - 18. I was stunned, humbled, and so honored! </font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><strong>Why do I encourage others to apply for the PAEMST?</strong></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><font>It is an </font> <span style="background-color: rgb(255, 255, 255);"><font>excellent opportunity to reflect on the work you're doing in your classroom, to network, build relationships, learn with outstanding educators and other professionals across the country, and to be recognized for the skill and leadership with which you approach mathematics and/or technology, and science learning. You also will receive feedback from both the state selection committee and the national committee about your work. Rarely do we receive feedback from STEM leaders outside of our own districts. In addition, there is the $10,000 reward money. Unlike</font></span><span style="background-color: rgb(255, 255, 255);"><font>most monetary awards that are earmarked for school-related use, this money can be used at your discretion. I am using it to help cover the costs to attend mathematical conferences and to take additional mathematics courses. </font></span></font></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;"><strong>So how do you nominate yourself or someone else?</strong></font></span></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><span style="background-color: rgb(255, 255, 255);"><font>If you're interested in applying or nominating someone, you can find more information by visiting the</font></span> <a href="https://paemst.org/award_process/view" target="_blank"><span style="background-color: rgb(255, 255, 255);"><font>PAEMST site</font></span></a><span style="background-color: rgb(255, 255, 255);"><font>. Every other year the award is open to either a K-6 or 7-12 teacher. The current 2020 nomination cycle is for K-6. Because of COVID-19, the application dates were adjusted. Part of the K-6 cycle application was due on May 1, and then those applicants have until October 26, 2020 to submit final applications. That would mean that the next cycle for applications will be for 7-12 STEM teachers.</font></span></font></p>
<p style="line-height: 19px;"><span style="background-color: rgb(255, 255, 255);"><font face="Ubuntu" style="font-size: 18px;"><strong>What should I do while I wait for the next nomination cycle?</strong></font></span></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><span style="background-color: rgb(255, 255, 255);"><font>In the meantime, keep growing as a professional in the area of mathematics. If you haven’t already, sign up and become an active member of</font></span> <a href="https://www.iowamath.org/membership" target="_blank"><span style="background-color: rgb(255, 255, 255);"><font>Iowa Council of Teachers of Mathematics</font></span></a> <span style="background-color: rgb(255, 255, 255);"><font>(ICTM). The membership is very affordable and offers excellent resources. Become a member of</font></span> <a href="https://www.nctm.org/" target="_blank"><span style="background-color: rgb(255, 255, 255);"><font>NCTM</font></span></a> <span style="background-color: rgb(255, 255, 255);"><font>so that you have cutting-edge research articles at your fingertips. Attend ICTM and NCTM conferences to learn in person from leading educators. Consider presenting at the next ICTM conference or writing a journal article for</font></span> <a href="mailto:ICTMJournalEd@gmail.com" target="_blank"><span style="background-color: rgb(255, 255, 255);"><font>ICTM</font></span></a> <span style="background-color: rgb(255, 255, 255);"><font>or NCTM. Invite other educators into your classroom. Learn and grow by honing your practice by working with a district math coach, a math consultant from your AEA, or a mathematics education professor from a local college. Get on</font></span> <a href="http://twitter.com" target="_blank"><span style="background-color: rgb(255, 255, 255);"><font>Twitter</font></span></a> <span style="background-color: rgb(255, 255, 255);"><font>(I call it my nightly P.D.) and connect with math educators across our state, nation, and world. I believe that we grow the most when we share our classroom practice and reflections with others. </font></span></font></p>
<p style="line-height: 19px;"><font face="Ubuntu" style="font-size: 18px;"><span style="background-color: rgb(255, 255, 255);"><em><font>Deb Little is a fourth grade teacher at Denver Community School District. She can be reached at dlittle@denver.k12.ia.us and on Twitter:</font></em></span> <span style="background-color: rgb(255, 255, 255);"><em><font>@Mrs_Little_17. </font></em></span></font></p>
https://iowamath.org/Articles/9007734
https://iowamath.org/Articles/9007734Wendy WeberMon, 06 Jan 2020 20:51:54 GMTSuccess on ISASP by Dr. Clayton Edwards<p>With all the fanfare leading up to ISASP, taking ISASP in April and May, and then waiting for the ISASP results, the process was draining on myself and the students. We didn’t do anything different to prepare for the assessment, but there were a lot of small events that brought on some unneeded anxiety. For example, I could tell my students were feeling the heat, as many teachers, including myself had mentioned a new and more expansive assessment throughout the school year to the students. When we took the test, students even had to deal with the Pearson person coming in and observing them testing. I have visitors from other districts come into my room regularly to check out what we are doing, so the students are used to it, but I can imagine as a student, that there is nothing like having a random stranger sit in during one of the more nerve wracking days of the year watching you like a hawk. I’ve got no problem with anyone having tattoos, but this guy had 100 of them, and my students commented that the tats were a little distracting! Students have plenty to worry about in their lives, and it was unfortunate that anything associated with ISASP stressed any of them out. The stress for myself and my students certainly wasn’t worth it, especially with the relative letdown that came when we took the test and realized that it really wasn’t a lot different than the old Iowa Assessments, and finally getting the results that didn’t help me to inform my teaching at all. All I learned is that my students did “well.”</p>
<p>Current 8th Grade (As 7th Graders With Me)<br>
Proficient 66%<br>
Advanced 22%<br>
Not Yet Proficient 12% <br>
State Average 60/10/30</p>
<p>Current 9th Grade (As 8th Graders With Me)<br>
Proficient 42%<br>
Advanced 49%<br>
Not Yet Proficient 9% <br>
State Average 61/11/28</p>
<p>Before you say that I have the best students already (I do love my students), both of these classes had been right at or below the state average as far back as the 6th grade. My students are improving because of what we do in class on a regular basis, and my high expectations for their learning. In this post, I will outline what I believe we do in class that makes us successful on ISASP and otherwise, and my strong recommendation for an appropriate amount of time to take the ISASP assessment, as I think this is one of the biggest misconceptions about ISASP that needs to be explained, especially to those who don’t have a mathematics education background.</p>
<p>I had a lot of curious people come up to me at a recent State Mathematics Leadership Team meeting and asked how my students did on the test. I let them know how well we did, and I got a lot of questions that made it sound like I had some sort of secret formula for success. Here was my “secret formula.” Avoid test prep. Avoid trying to dumb down material to work on “basics.” Avoid cramming before the test. None of these things work. Everything in this category promotes surface level learning. Students also sense when they are doing something that isn’t as rigorous as it probably should be, and worse, students know if you are giving them different material because they haven’t been successful with whatever everyone else is working on. If you think that is building confidence, it does the opposite and makes students feel worse about themselves. Special education students are no different, and I feel our special education population was successful on ISASP because I have similar expectations with them as I would any other student. </p>
<p>Here are some of my “secrets” I do that I would contribute to our success on this test. I am not into telling anyone how to do things in their own classrooms. Teachers are unique and should be allowed to let their uniqueness shine through. This may look different to you in your classroom than it does for me, but if you would like to talk about how this looks in my classroom, I would be happy to discuss. </p>
<p><strong>Assess EVERYTHING, giving feedback to help make revisions</strong></p>
<p>Anything my students do is assessed by me personally. I leave personal feedback on every assignment no matter how small or how large. I use class time to give verbal feedback as well. I want my students to have a good understanding of where they are, and I want to have a solid understanding myself of what my students know. A lot of the more informal ways of assessing don’t give me the information I need as a teacher to make informed decisions about my students’ mathematical understanding. Things like thumbs up/down, fist to five, completion points, and going over the homework as a whole class doesn't give me enough information about what a student really knows. Sure, assessing everything is time consuming, but it is worth it. Have you ever gotten to a final assessment and were surprised that your students did poorly? You may have been using one of those methods above. You should always know how your students will do on an assessment prior to the assessment because you have been giving feedback and assessing along the way. By the way, this has nothing to do with grades, just feedback. </p>
<p><strong>Revisions</strong></p>
<p>After I assess an assignment/assessment/task, it is always given back to a student to revise. The expectation is that revisions are made. I even work in time during class to allow for these revisions. It doesn’t make any sense to let a student move on who has only mastered 80% of an assignment. I can go over the answers with the class and pretend that the students thought through their mistakes in the 2 seconds I would give them, but I would be kidding myself. I allow for ample time for students to revise, and I will reassess after the revisions for full credit. Almost all of my students get 100% in my class because of this, and I absolutely think this is fine. I am only concerned that the content is understood and mastered. This is one reason we do so well on the ISASP assessment, or the NWEA MAP. We tie up any misconceptions right away. </p>
<p><strong>Tasks to promote productive struggle</strong></p>
<p>Some people shy away from 3-Act type tasks because they may take up large chunks of time, and most of the ISASP questions are more general multiple choice, which wouldn’t fit the format of a 3-Act task. I will give you an opposite viewpoint. My students excel on long and grueling tests because they have multiple chances throughout the year to develop productive struggle. They are willing to stick with something and not give up. They know how to make adjustments and persevere through tough situations. Part of fostering productive struggle is through the teacher’s questioning techniques, which obviously isn’t happening during ISASP, but another part is students utilizing strategies that they have learned to help themselves move forward when stuck. How many students have you had who have started out on fire on one of these large tests working hard and reading each question, only to revert and finish the last 30 questions in four minutes? Working on tasks that require some productive struggle can help students stick with the test for the long haul, because they are used to doing this daily in class. </p>
<p><strong>Requiring justification for EVERYTHING</strong></p>
<p>For anything my students do, they turn in accompanying work that gives me an idea of their level of understanding. For me, the point of assigning something is to see what they know, and if my students turn in pages of answers, I really don’t learn anything from that. I am looking for things like charts, tables, numbers with context, paragraphs, pictures, etc. I can diagnose misunderstandings much easier when students justify everything, and students feel better with their levels of understanding because they can explain it as well. </p>
<p>We are a 1:1 school, and while we use a lot of computer programs, we also probably use more paper than any school in Iowa. We use common programs like IXL, Buzzmath, and Desmos, and while answer are often typed onto a screen, I always check the justification to go with these question and answer screens. If you don’t require justification, 1:1 mathematics can quickly turn into a button mashing event. And you wouldn’t believe how easily students can hit buttons on computer programs and eventually get something correct….</p>
<p>This helps with ISASP as my students are used to showing this level of understanding for everything, including a standardized test. My students each averaged 5 pieces of scratch paper each during ISASP. Having students justify answers slows down the thought process causing students to think deeper about whether an answer is correct instead of just picking what first pops into their heads. </p>
<p><strong>Longer time limits for assignments</strong></p>
<p>Anyway you can build this into your classroom would be ideal. I have students that understand the mathematics at much different rates, and I have learned that it is silly to go with the one section of a book, and then move onto the next section the next day format. Some students will figure things out in a day, some will take three. When I bring this up, people always say that maybe the universal instruction needs to improve, and I wholeheartedly disagree with that opinion. When more time isn’t built in, oftentimes students move on without understanding. This happens multiple times over a student’s career putting them further behind each time until they are years behind. This takes the teacher being flexible and having multiple activities going on at once, but since I have changed to being more of a facilitator/questioner anyway, this works well. My students do well on ISASP because we have actually taken the time to make sure everyone understands each standard at a high level without moving on and leaving gaps in learning.</p>
<p>On the surface, ISASP appears to be untimed and very supportive of my students' everyday experience in class where they have time to think through and process problems and tasks. However, once I figured out how many questions were on the assessment, I knew I had to lobby for more time. The untimed moniker of ISASP was frustrating because it really wasn’t untimed. We had to finish the mathematics test in a day. While our district is very supportive of our needs as educators, it was probably hard for an administrator to understand that a middle school test could take an average of three hours to complete. Believe me, it sounds crazy just typing it. Couple that with administrators from our district hearing other districts say it took an average of 40 minutes for their students to take the mathematics portion on ISASP, and I could see how people could be skeptical about my three-hour time frame. </p>
<p>Here is my thought process. In my classroom, my students are used to analyzing a problem, thinking about their next steps to solve the problem, solving the problem, and providing justification for each problem showing their understanding. To do this for a standard mathematics problem, it may take 2 to 4 minutes. That doesn’t sound at all far fetched to me since I live it everyday in my classroom. My students have a deep understanding of mathematics because that is what I encourage. That is what the expectation is. Rewind back to ISASP. 52 questions. Each question different than the next. 2 to 4 minutes per question mimicking what we would do in class. About 500 sheets of scratch paper later. There is my average estimate of three hours to complete the test. If we averaged 40 minutes for the test, that would be less than a minute per question. I can only imagine the effort and thought going into each question would be low in that circumstance. 40 minutes seems like the more far-fetched option to me. Needless to say my middle school students were extremely successful. I am so proud of their efforts. The level of buy-in from my students was incredible. </p>
<p>Reading this post you may think I love standardized assessments and think about them often. I actually despise them if we are being honest. They take a long time to finish, and the results that I get don’t help me make better decisions in my classroom. Unless I can have the test questions after the test is over, and I can keep the scratch paper to match up with each question so I can examine my students’ thinking, the results don’t help me. </p>
<p>After participating with my role within ISASP the first year, I know I won’t be as stressed out the next time around. I plan on keeping things simple. I will continue to do the things that have worked in my classroom, that consequently translate to a deeper level of mathematical understanding, which will yield better results on these types of assessments. If I were to make a recommendation for ISASP, cut the number of questions in half. I think the state could still get the measurements they are looking for with far fewer questions, especially since the results received don’t help teachers make meaningful decisions anyway. Until the next iteration of ISASP, I will be saving my money to make an even bigger Cheez-It pyramid so the students’ will have plenty of testing fuel, for our ultra-fancy scratch paper pencils, and for the massive roll of paper it will take to cover all the walls in my room! I certainly will not be living for the next test, but rather the genuine excitement my students feel on a daily basis when a level of understanding is achieved!</p>
<p><strong><font color="#5BC7FF">UPDATE</font></strong><br>
The Iowa School Performance Profiles were recently released to the public, and our students got some outstanding news. The mathematics growth category is measured by matching students across the state by percentile on the old Iowa Assessment and comparing what they got on ISASP. For instance, all students in the state scoring at the 74th percentile on the old Iowa Assessments were compared using ISASP. We were very successful in those comparisons. We received the highest mathematics growth score in the state for both 7th grade (current 8th graders) and 8th grade (current freshman).</p>
<p align="center"> <img src="https://iowamath.org/resources/Pictures/ISASPresults.png" alt="" title="" border="0"><br></p>
<p data-pm-slice="1 1 []">My students were extremely excited and proud to hear the news. There are a lot of middle schools in the state, and it is an honor to be considered the best of the best.</p>
<div align="center"><img src="https://iowamath.org/resources/Pictures/cheez-it.jpg" alt="" title="" border="0"></div>
<p align="center"><br></p>
<div align="center"><img src="https://iowamath.org/resources/Pictures/pencils.jpg" alt="" title="" border="0"></div>
<p align="center"><img src="https://iowamath.org/resources/Pictures/student.jpg" alt="" title="" border="0" style="margin-left: auto; margin-right: auto; display: block;"><br></p>
<p align="center"><img src="https://iowamath.org/resources/Pictures/class.jpg" alt="" title="" border="0"><br></p>
https://iowamath.org/Articles/8985018
https://iowamath.org/Articles/8985018Wendy WeberTue, 14 May 2019 05:00:00 GMTBefore Basic Fact Instruction... by Angie Shindelar<p data-pm-slice="0 0 []">Hello ICTM members! My name is Angie Shindelar and I serve on the ICTM Board as the Vice-President for Elementary. I previously taught elementary and middle school math at Nodaway Valley CSD. I am currently a Math Consultant for Green Hills AEA. </p>
<p>My recent articles have discussed effective instruction for addition and subtraction basic facts. You can find the articles on the ICTM website in the three previous newsletters if you are interested and missed them. In this article I continue with the basic fact theme but turn the focus to thinking about the learning progression <strong>prior</strong> to basic fact instruction. Understanding this progression and being intentional about building a foundation for basic fact instruction can ensure our students’ success in achieving fluency for addition and subtraction basic facts in a timely way. These critical understandings are important for K-2 teachers but also for teachers working with older students that are struggling to make sense of basic fact strategies fluency. </p>
<p>Prior to instruction in basic facts students should develop understanding of parts and wholes. There are three critical components to this work: </p>
<p> 1) subitizing</p>
<p>2) knowing plus 1 and minus 1</p>
<p>3) composing and decomposing numbers</p>
<p>Below is a description of each component and a list of Iowa Core Math Standards that encompass the learning for each component.</p>
<p>The first critical component of developing understanding of parts and wholes is subitizing. Subitizing can be described as instantly seeing how many without the need to count. The human brain can instantly tell “how many” of up to 5 objects without having to count. Beyond 5, our brains have to break a quantity into smaller chunks. Subitizing experiences provide students opportunities to think about quantities in different arrangements and using different models. A typical subitizing experience is when a teacher quickly flashes a quantity for students to see and ask students, “How many do you see?” The teacher asks several students to tell how many they saw and how they knew how many there were. Students will describe everything from attempts to count the items individually, relating to arrangements they are familiar with, and describing smaller parts within the whole.</p>
<p>The quantity might be represented with a dot arrangement, a five/ten frame, or a rekenrek (number rack). Below are examples of different arrangements of a quantity. To build understanding of parts and wholes, students should experience visualizing quantities in various arrangements and with the different models.</p>
<p><img src="https://iowamath.org/resources/Pictures/dotpatterns.png" alt="" title="" border="0"><br></p>
<p data-pm-slice="1 1 []">When subitizing is a regular routine, students first become familiar with arrangements up to 5. Once the arrangements move beyond 5 the goal shifts to looking for the parts within the whole. Students learn to resist the temptation to count and instead focus on the arrangement looking for the smaller parts. For example, when shown the quantity of 6 in the arrangement below, students may agree there is 6 but describe how they knew differently. Some may see 3 and 3, while others noticed 4 and 2. By working on subitizing as a regular routine, students learn to look for parts they recognize in quantities typically up to 10. </p>
<p data-pm-slice="1 1 []" align="center"><img src="https://iowamath.org/resources/Pictures/6reddots.png" alt="" title="" border="0" width="235" height="133" style="height: 133px;"><br></p>
<p data-pm-slice="0 0 []">One of the best resources for subitizing is <em>Number Talks</em> by Sherry Parrish. There is an entire section on subitizing with many examples of arrangements and models for quantities up to 10. This book really helped me understand the importance of providing different arrangements for each number and varying the model.</p>
<p> While subitizing is not specifically referred to in the Iowa Core Math Standards, it lives within the Counting and Cardinality standards. You can find references to subitizing when you read further about the standards and unpack the specific learning within them. This standard relates most closely to the subitizing work: </p>
<p> Understand the relationship between numbers and quantities; connect counting to cardinality. <strong>(K.CC.4.)</strong></p>
<p> The second critical component of developing understanding of parts and wholes is plus 1 and minus 1. This can be tricky, because we often think of it as facts like 5 + 1 or 4 -1. However, prior to working on basic facts in this format, it is critical to build understanding of what plus 1 and minus 1 actually mean. Students in early grades spend a considerable amount of time counting and thinking about what number comes next, what is one more than, what number comes before, and what is one less than. They develop this understanding by connecting it to the count sequence and are typically successful and confident. However, when given problems like 5 + 1 or 4 - 1, too often we see students not make the connection of what is one more than, what number comes before, and what is one less than to addition and subtraction. It is often baffling to teachers as they see students counting fingers to solve.</p>
<p> As students are learning about these important understandings, it is critical to make explicit the connections to the language of addition and subtraction and representing with equations.</p>
<p>Explicitly discussing and modeling that one more than and what number comes next is written as +1 is critical to students making connections. They have to see visual models, hear the language relationship to addition and subtraction, and have opportunities to use mathematical symbols by writing equations to represent the plus 1 and minus 1 concepts.</p>
<p> Word problems also help students make connections to addition and subtraction. Posing addition and subtraction word problems as a regular part of students’ learning provides opportunities to use the language of one more or one less and make connections to addition and subtraction. Including connections to equations is key to this work. Here are examples of intentionally connecting the concept of plus 1 and minus 1 in word problems:</p>
<p>Sam has 6 toy cars. His brother gave him 1 more. How many toy cars does Sam have now?</p>
<p>Mei has 7 cookies. Rob has 1 less cookie than Mei. How many cookies does Rob have?</p>
<p>Students work on finding ways to represent the problem and solve. While sharing ways to represent, the teacher brings forward the discussion of what 1 more or 1 less means and how to record mathematically with an equation. </p>
<p>While plus 1 and minus 1 is not specifically referred to in the Iowa Core Math Standards, it lives within both the Counting and Cardinality and the Operations and Algebraic standards. You can find references when you read further about the standards and unpack the specific learning within them. The standards that encompass plus 1 and minus 1 are listed here:</p>
<ul data-rte-list="default">
<li>
<p>Count forward beginning from a given number within the known sequence instead of having to begin at 1. (<strong>K.CC.2.</strong>)</p>
</li>
<li>
<p>Understand the relationship between numbers and quantities; connect counting to cardinality. <strong>(K.CC.4.)</strong></p>
</li>
<li>
<p>Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. <strong>(K.OA.1.)</strong></p>
</li>
<li>
<p>Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. <strong>(K.OA.2.)</strong></p>
</li>
<li>
<p>Fluently add and subtract within 5. <strong>(K.OA.5.)</strong></p>
</li>
</ul>
<p><strong> </strong>The third critical component of developing understanding of parts and wholes is composing and decomposing numbers. Understanding that any number can be composed and decomposed in different ways is essential to later fact fluency work. For example, think about all the ways to make 6. Through exploring ways to make 6, for example, students build understanding of the quantity 6 as well as patterns and properties. Discussions of the combinations can highlight patterns like when we have 3 and 3, we can take away 1 from one group of 3 and give it to the other group of 3. So now have 2 and 4. If we again take away 1 from the group of 2 and give it to the group of 4, we now have 1 and 5, etc. Or we might highlight that 4 and 2 is a way to make 6, and we also see that 2 and 4 are shown. Exploring this brings forward discussion of the commutative property for addition.</p>
<p>Providing regular opportunities for students to engage with composing and decomposing numbers is essential. Discussion of students’ findings and connections to patterns and properties will deepen understanding of parts and wholes. Resources that highlight activities for parts and wholes understanding are <em>Student-Centered Mathematics</em> by John Van de Walle and <em>Developing Number Concepts, Book 2</em> by Kathy Richardson. Here are a few activities I have assembled in this <a href="https://docs.google.com/document/d/1LRl9VPJKnebOAumUW66ZsjnYFFGVsfHbCdWkH56jReQ/edit?usp=sharing">document</a> based upon activities I have used from these resources.</p>
<p> In addition, the Put Together Take Apart problem type that requires finding both addends is a perfect opportunity for students to work on composing and decomposing. Below are examples:</p>
<ul data-rte-list="default">
<li>
<p>There are 6 animals in the zoo. Some are tigers and some are bears. How many are tigers? How many are bears?</p>
</li>
</ul>
<ul data-rte-list="default">
<li>
<p>There are 7 chairs. Some are red and some are blue. How many are red? How many are blue?</p>
</li>
</ul>
<p> Composing and decomposing numbers is specifically referred to in the Iowa Core Math Standards in the Operations and Algebraic standards. You can find more specific references when you read further about the standards and unpack the specific learning within them. The standards that encompass composing and decomposing numbers are listed here:</p>
<ul data-rte-list="default">
<li>
<p>Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. <strong>(K.OA.1.)</strong></p>
</li>
<li>
<p>Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. <strong>(K.OA.2.)</strong></p>
</li>
<li>
<p>Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). <strong>(K.OA.3.)</strong></p>
</li>
<li>
<p>For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. <strong>(K.OA.4.)</strong></p>
</li>
<li>
<p>Fluently add and subtract within 5. <strong>(K.OA.5.)</strong></p>
</li>
</ul>
<p><em>Note: There are first grade standards that also could be included, however I chose not to include them because the intent is to describe essential learning prior to working on basic fact strategies.</em></p>
<p>These three critical components for developing understanding of parts and wholes set students up for success as they begin formal work in fact strategies. Students that are learning fact strategies with ease and steadily working toward achieving fluency have a solid understanding of parts and wholes. Contrastly, when students are struggling with learning fact strategies and are not progressing as expected toward achieving fluency, you can almost guarantee the gap in their learning centers around unfinished learning around parts and wholes.</p>
<p> I am always interested in your thoughts and feedback for any of the topics I discuss in the ICTM newsletter. You can reach me at <a href="mailto:ashindelar@ghaea.org">ashindelar@ghaea.org</a>.</p>
https://iowamath.org/Articles/8998927
https://iowamath.org/Articles/8998927Wendy Weber