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"Galileo Galilei once said: 'Mathematics is the language with which God has written the universe.' Too bad most schools are awful at helping us become fluent in  Universe ." Math-Explorable Explanations In a dream world of some maths teachers, students could play mathematics for its own sake, with childish whimsy of heart, and explore countless possibilities for discovery in our universe. Unfortunately, many of the established systems in education impede this dream from becoming a reality in classrooms. In truth, accountability is necessary due to maths’ unique potential to propel or squelch students’ future educational and career endeavors . Therefore, ...
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Other professions play, why shouldn't we? The idea of work being commensurate with play is not novel. Many a professional athlete, musician, etc., regularly interchange these two words when speaking of their craft. Football players’ ultimate goal is to play and, hopefully win, the Superbowl. A famous rapper once commented he had reached one of his dreams to play his music at the 20,789-seated Madison Square Garden. Similarly, Carnegie Hall is a site where classical music enthusiasts regularly experience energy and awe as they listen to a musician who has likely worked her whole life to play a favorite piece in this prestigious venue. Actors receive ...
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In my last post , I ended with a proposition made by John Van de Walle: “ The traditional whole-number algorithms for addition, subtraction, multiplication, and division should no longer be taught in schools.” Instead, he argued that alternative and/or invented strategies should be encouraged and supported because they are: number-oriented, rather than digit-oriented, flexible, rather than rigid, “left-handed” rather than “right-handed” (starting with the biggest place value first), student-centered, rather than teacher-directed, and focused on number sense and mental math (Van De Walle, 2004). In order to support alternative ...
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If you have been following this blog series you now know more about units coordination , how it relates to fractions , and where it shows up in middle school concepts .  So how do we help students progress in their ability to coordinate units? One thing I have learned from the amazing work of Christopher Danielson is that the best prompts will simply open a dialogue.  In his book How Many ? , he offers pictures to start a conversation about just that.  As the teacher’s edition (which I highly recommend) states, we need to put discussion about unit relationships “front and center for examination and discussion” (40).  These conversations ...
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I advocate for more equitable and just classrooms, schools, and schooling. This work must be addressed at multiple levels of the educational system (e.g., see Aguirre, Mayfield-Ingram, & Martin, 2013, Wager and Stinson, 2012), but in all of the work that I do I always try to highlight the importance of honoring and building on the brilliance of young children’s thinking. Despite decades of evidence, some adults (teachers, administrators, parents, etc.) are surprised, and often a little incredulous, at the idea that young children can understand complex mathematical concepts and even invent strategies to add, subtract, multiply, and divide without ...
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Don’t you love when students have an “aha” moment?  What about your own “aha” moments? I still remember the first time I taught inequalities with algebra tiles and finally understood why the inequality symbol needs to be switched when multiplying or dividing by a negative number.   I have had many “aha” moments since being introduced to units coordination and the work of Anderson Norton and Amy Hackenberg through the US Math Recovery Council .  Incorporating this work into my classroom has been transformational for my thinking.  Here are just a few ways my thinking has shifted. The first blog post in this series introduces an assessment ...
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If you found my first post explaining units coordination with whole numbers helpful, this post will give you the details from my experience working with fractions and how impactful it is to students when understanding units coordination. Fraction problems like this are typical in my 7th-grade classroom: Is ½ of ⅘ more or less than ⅘?   If ½ pound costs $3.50, what is the price per pound? In all fraction problems, there exists one whole of which the fraction is part.  By middle school, most students have the part-whole understanding that ½ is one out of 2 parts of the whole unit.   If a student is struggling with the second question above, ...
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A refresher from my last post:  I am a Learning Support teacher in G1, and along with the G1 teaching team, I look for opportunities to use and elicit evidence of student thinking to in order to design small group lessons, and adjust my instruction based on their thinking. My current small group continued the Basic Facts & Place Value unit. Following the lessons from Days 1-4 (see NCTM Blog Post ), I met with one of our instructional coaches to talk through planning my next instructional steps. We looked at the following areas: Teaching Focus : What math teaching & learning questions do you have about the first few lessons - what are ...
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Thank You!

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Thank you! It has been a great honor and a humbling experience to serve as president of the National Council of Teachers of Mathematics. As I said in my first president’s message, NCTM is its members—a community of professionals committed to ensuring that each and every student learns significant mathematics in equitable environments. I believe that statement more firmly today than ever before. I have had the wonderful opportunity, while attending NCTM conferences and Affiliate meetings, to meet many of you. Every time I do, I come away energized and encouraged for the future of our profession after witnessing not only your positive energy and commitment to ...
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After my son had collected candy at a parade he wanted to eat some.  I told him he could choose one thing from his bag. He chose a clear zipper bag with 2 packages of Starburst in it.  Each package contained 2 Starburst. He chose one, but he actually got to eat 4 candies. If you are familiar with Christopher Danielson ’s work, you have most likely had conversations around units and the variety of units that can exist in one situation, like the example above.  If not, I will discuss his book How Many and the hashtag #unitchat in a future post. It wasn’t until the past few years, however, that I realized there is more to it than just ...
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As a Learning Support teacher, I look for opportunities to connect student thinking to in order to design small group lessons, and adjust my instruction based on their thinking. My role as a Grade 1 Learning Support is typically to push-in to support three classes. At times, however, we decide as a team to identify a small group for short-term intervention work. We did just that on our current unit: Basic Facts & Place Value. Keeping the two Mathematics Teaching Practices highlighted below in mind ( Principles to Actions ), I started our small group work with an exploration. Elicit and use evidence of student thinking. Effective teaching of ...
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As part of our series on implementing a Badging system , we've explored sample tasks and student responses centered around Math, Equity, and Culture and Data Representation .  Because the main purpose of these badges is to encourage middle school students to "fall in love" with mathematics, in the final segment I want to examine a topic strand that makes it easy for students to love and appreciate math:  Math and Beauty .    The name of this topic strand is purposefully vague.  Math and Beauty topics could center around the beauty of mathematics in nature.  They could also represent artists' attempts to make mathematical beauty apparent in their creations.  ...
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So far in this series I've introduced you to the concept of "Badges" (what I call my free-choice activities that are designed to engage kids in how mathematical concepts are used outside of a math classroom) and we've explored one of the topics within my Badging system, activities centered around Math, Equity, and Culture .  Today, we will investigate another topic strand, focused on Data Analysis and Representation . I worry that too often in Middle Grades we focus on Algebra preparation so students are ready for the Algebra -> Trigonometry -> AP Calculus progression that we (and their college counselors...) hope to be their high school experience.  ...
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Next month at our Annual Meeting in Washington, NCTM will release and have an enhanced conference strand on Catalyzing Change in High School Mathematics: Initiating Critical Conversations. I want to thank the writing team, particularly the chair Karen Graham, for bringing Catalyzing Change to publication. I also want to thank all the reviewers, many of whom saw multiple drafts, as well as the NCTM Publications staff who ensured the book was ready by the Annual Meeting. Member feedback was critical at every step of the process. Catalyzing Change seeks to initiate critical conversations around the following challenges: Explicitly broadening the purposes ...
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Too often students see the math that they're learning in school as disconnected from life, impotent to have real meaning and purpose outside of its own confines.  After my last introductory post on the reasons to implement a Badging system , I want to take you through a couple of opportunities I've given my students to see math work in a civic sense.  These badges I've collected fall under a heading I've entitled Math, Equity, and Culture and show mathematics used in powerful ways to those students interested in societal issues. First, I took an old ABC news article about the Electoral College written by mathematician John Allen Paulos and I asked ...
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By Evgeny Milyutin and Lisa Meyer Put yourself in a students’ shoes for a minute. Imagine that throughout your whole life, the only language you’ve heard or spoken is Spanish. You’re from a rural town in Honduras, the only home you know, and then at age six your family moved to the USA. Once there, you were signed up for 1s t grade at the local elementary school, and you were thrown right into an English-speaking classroom. While at school, you received some English language support, but what you heard all day long at school is English. Take another look, you used to love math class in Honduras; yet, in your American classroom you feel bored, exhausted ...
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I want my middle school students to be successful in mathematics, but I also want them to fall in love with mathematics.  Some students are naturally inclined to really enjoy "traditional" mathematics; they are efficient and accurate at arithmetic, easily notice patterns, or might enjoy trying to solve new problems that stretch their knowledge base.  Other students, however, don't feel that same connection to traditional school mathematics.  It might be due to a lack of success in past math classes or a familial disdain for mathematics passed on from their parents, but there exists in every middle school students who are not inclined to love the type of ...
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Why Teach Mathematics?

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As a teacher of mathematics, you have no doubt had multiple students ask what is actually a very complicated question: “Why do I have to learn this?” At one time or another, perhaps while grinding through a series of context-free symbolic manipulation exercises, nearly every one of us has struggled to provide our students with an answer they, and/or we, find satisfactory. But the student “why?” question raises an equally important, but less often-asked question: “Why do we teach math?” Francis Su, past president of the Mathematical Association of America, has argued that this is a very important question because how we answer it strongly influences who we ...
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My Favorite Routines

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Demonstrate kindness Challenge yourself and smile Learn from your mistakes What will I try differently this year? What went well last year? What didn’t go well? How can I get my students to understand ______ better? My personal reflections led to me to make changes in my teaching, such as asking my students to reflect often, revamping my assessment practices, and incorporating more activities focused on building empathy. After trying so many new ideas every year, I find there are a few that I revisit and establish as tradition in my classroom. I’m going to share with you my favorite activities I use to improve classroom culture and student learning. ...
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Calendar Time Must Die

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I spend a lot of time in early childhood classrooms, and, as the headline suggests, I have gotten to a point where I cannot sit through one more moment of Calendar Time. It is difficult to select what makes me most crazy about this practice: ● Is it that too often Calendar Time takes up prime instructional real estate -- those precious morning minutes when children are fresh and ready for a new day? ● Is it that PreK learners, who have been doing Calendar Time daily for months, still shout out every potential answer to the question “What day is it today?” ● Maybe it is the way so many children put their heads in their hands or turn to poke at their ...
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