Posts filed under 'math'

Blogstitute 2017: Which Comes First in the Fall–Norms or Tasks?

In this last post of our Summer Blogstitute series, Tracy Zager, author of Becoming the Math the Teacher You Wish You’d Had, shares her ideas for kicking off the school year in your math classroom ready to notice, imagine, ask, connect, argue, prove, and play.

Which Comes First in the Fall–Norms or Tasks?
Tracy Johnston Zager

I periodically hear discussion about whether it’s better to start the new school year by establishing norms for math class or to dive right into a rich mathematical task. I’m opinionated, and I’m not shy about my opinions, but in this case, I’m not joining one team or another. They’re both right.

The first few weeks of math class are crucial. You have a chance to unearth and influence students’ entrenched beliefs—beliefs about mathematics, learning, and themselves. You get to set the tone for the year and show what you’ll value. Speed? Curiosity? Mastery? Risk-taking? Sense-making? Growth? Ranking? Collaboration? You get to teach students how mathematics will feel, look, and sound this year. How will we talk with one another? Listen to our peers? Revise our thinking? React when we don’t know?

In Becoming the Math Teacher You Wish You’d Had, I wrote about a mini-unit Deborah Nichols and I created together. We called it, “What Do Mathematicians Do?” and we launched her primary class with it in the fall. We read select picture-book biographies of mathematicians, watched videos of mathematicians at work, and talked about what mathematics is, as an academic discipline. We kept an evolving anchor chart, and you can see how students’ later answers (red) showed considerably more nuance and understanding than students’ early answers (dark green). [Figure 2.1]

Figure 2.1

Throughout, we focused on the verbs that came up. What are the actions that mathematicians take? How do they think? What do they actually do?

In the book, I argued that this mini-unit is a great way to start the year if and only if students’ experiences doing mathematics involve the same verbs. It makes no sense to develop a rich definition of mathematics if students aren’t going to experience that richness for themselves. If professional mathematicians notice, imagine, ask, connect, argue, prove, and play, then our young mathematicians should also notice, imagine, ask, connect, argue, prove, and play—all year long.

In June, I saw this fantastic tweet in my timeline.

It caught my eye because Sarah’s anchor charts reminded me of Debbie’s anchor chart, but Sarah had pulled these actions out of a task, rather than a study of the discipline. I love this approach and am eager to try it in concert with the mini-unit. The order doesn’t matter to me.

We could (1) start with a study of the discipline, (2) gather verbs, (3) dig into a great task, and (4) examine our list of mathematicians’ verbs to see what we did. Or, we could (1) start the year with a super task, (2) record what we did, (3) study the discipline of mathematics, and (4) compare the two, adding new verbs to our list as needed. In either case, I’d be eager for the discussion to follow, the discussion in which we could ask students, “When we did our first math investigation, how were we being mathematicians?”

Whether we choose to start the year by jumping into a rich task on the first day, or by engaging in a reflective study about what it means to do mathematics, or by undertaking group challenges and conversations to develop norms for discourse and debate, we must be thoughtful about our students’ annual re-introduction to the discipline of mathematics.

How do you want this year to go? How can you invite your students into a safe, challenging, authentic mathematical year? How will you start?

1 comment August 1st, 2017

A bold choice for a math methods course

When I wrote Becoming the Math Teacher You Wish You’d Had, I wrote directly to readers, and I had specific readers in mind: real teachers, in various stages of their careers, who were ready to learn how to teach math so much better than how they were taught. Before writing it, I’d worked with preservice teachers and their inservice mentors for seven years in a variety of schools. I wanted to write a book that would be useful to both groups, knowing full well that some parts would resonate more with teachers who are just starting out and other parts would grab the attention of experienced teachers. I’ve been hearing from experienced teachers who are finding the book motivating, thought-provoking, and practical, which makes me so happy. I still wondered how it would go over with preservice teachers, though. Would it inspire them, or overwhelm? When Christine Newell decided to use it as the central text in her math methods class last term, I asked her to keep me posted, and we’ve had conversations throughout the semester. I’m so grateful that she took the time to reflect on her experience because it may help other math methods instructors. I have loved reading every one of her students’ letters, and it’s clear Chrissy nurtured a safe climate and taught a wonderful course. She’s started them off beautifully, and I can’t wait to hear how these teachers grow throughout their careers.

-Tracy Zager, author of Becoming the Math Teacher You Wish You’d Had

A bold choice for a math methods course
Christine Newell

“I didn’t learn math this way” and “I wish I had learned math this way” have become common refrains in the professional development I facilitate. Somewhere in there is generally an underpinning of feeling totally cheated out of this “new math” that feels exciting and rich and actually makes sense. Veteran teachers are being asked to change not just the way they teach math, but their whole understanding of what mathematics is, and preservice and beginning teachers are facing the challenge of teaching in a way they were never taught. Regardless of years of experience, teachers are looking for support to become the math teacher they never had and are being asked to be. Tracy Zager’s powerful book, Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms, is the answer to this. After my first read, it’s already dog-eared, tabbed, and annotated, and I’ve been back and forth from favorite concepts to ideas and resources countless times. This is pretty remarkable considering it was released just six months ago.

Becoming the Math Teacher You Wish You'd Had, by Tracy ZagerI made the pretty bold decision to choose Tracy’s book as the required text for the Math Methods course I taught for preservice teachers this past semester. It was a departure from the content-rich texts that the other instructors were using for this course, Van de Walle’s Teaching Student-Centered Mathematics, and Chapin & Johnson’s Math Matters. To be clear, I love both of these books and find them invaluable resources as I work with teachers, but I wanted to try something different. I wanted my preservice teachers to learn not just about content and pedagogy, but also about the importance of redefining math for themselves and creating “favorable conditions” for all students to see themselves as mathematicians.

Even before the first chapter, Tracy frames the experience for readers by saying that when reading this book, “there is no wrong way, as long as reading it is useful to you.” (p. xv) This is not a trivial statement. It sets the stage for the message throughout the book that math is flexible and creative, that mathematicians explore and believe in their intuition and revise their thinking. This was new thinking for my students. Each chapter zeroes in on an important attribute of mathematicians (read: all students) and offers snapshots from real classrooms where teachers and students are engaging in math in meaningful ways. Balancing content and pedagogy is a constant negotiation for math methods instructors, and Becoming the Math Teacher You Wish You’d Had offers jumping-off points for conversations around both. For my students, it was an approachable introduction to teaching elementary mathematics for this reason. It enhanced our content conversations by opening up my students’ ideas about what elementary students think and can do, and challenged what they thought was the role of the teacher.

In addition to the mathematical merits of the bookTracy writes in a way that makes you feel like you’re having a one-on-one conversation with her. Many of my students commented that they felt like they “knew” Tracy and the teachers she featured by the end of the book. This gives me hope that once they land in their own classrooms, my students will pull this resource off their shelves early and often. I’ll let my students say the rest. They were asked to write a letter to Tracy explaining the impact her book had on them in this course. The verdict? The book shaped our experience together this semester in profound, positive, challenging, inspiring ways. (Excerpts below printed with permission.)

The impact that reading your book this semester has made on my teaching has been huge. Every single chapter has given me tools, interesting scenarios, and great advice as to how I should teach mathematics in my very own classroom.

Thank you for writing such an insightful book, a book that challenged the norm and made us pre-teachers think “outside the box.”

Your book has taught me so many ways to teach math effectively but, most importantly, how to love math.

I cannot express enough how much I enjoyed each page of your book. Not only did you share such powerful and influential messages, but you inspired me.

Thank you for writing this wonderful book and inspiring teachers to feel more confident in math! It was wonderful to have read this before going to teach first grade because I feel better prepared to teach math.

Add comment June 26th, 2017

Which One Doesn’t Belong? Wins Mathical Award

Which One Doesn't Belong w awardChristopher Danielson, a mathematics author, teacher, and curriculum developer from Minnesota, has won the Mathical Prize for his book, Which One Doesn’t Belong? A Shapes Book.

The award will be presented to Danielson on April 22 by the Mathematical Sciences Research Institute (MSRI) at the National Math Festival in Washington, DC. Danielson won the award in the Grades 3-5 category.

“For a number of years I have longed for a better shapes book,” said Danielson. “I wanted a shapes book that gives space for noticing relationships, asking questions, and thinking together,” said Danielson. “I designed Which One Doesn’t Belong? to be an invitation to a mathematical conversation.”

The book–which is intended to be used by children, parents, and teachers–features sets of four shapes with the recurring question, “which one doesn’t belong?’ Any of the shapes can be the right answer; the key is getting kids to justify their answer in their own language. The school version comes with an extensive teacher’s guide, including an “answers key” that describes one possible argument that can be made for each shape in the book. Which One Doesn’t Belong? and the teacher’s guide can both be ordered from Stenhouse.

Which One Doesn’t Belong? encourages children to use mathematical thinking to explore new concepts,” wrote the committee who awarded the prize. “The layout is brilliant and in classroom testing, children were active readers, enthusiastic to share their insights and justifications in the discussion. Perhaps the best feature is that questions have no single, simple answer!”

Danielson has worked with math learners of all ages—12 year-olds in his former middle school classroom, Calculus students at Normandale Community College, teachers in professional development, and young children and their families at Math On-A-Stick at the Minnesota State Fair. He designs curriculum at Desmos. He is the author of Common Core Math For Parents For Dummies, the shapes book Which One Doesn’t Belong?, and the forthcoming counting book How Many? He blogs about teaching on Overthinking My Teaching, and for parents at Talking Math with Your Kids. He earned his B.A. in mathematics from Boston University, his M.A. in Education from the University of Michigan, and his Ph.D. in Mathematics Education from Michigan State University.

The Mathical Book Prize is organized by MSRI in partnership with the National Council for Teachers of English (NCTE) and the National Council for Teachers of Mathematics (NCTM).

Add comment April 21st, 2017

Read, Apply, Learn

At Stenhouse, we spend a lot of time thinking about how to create resources that are useful for teachers. We are always eager to hear how teachers, coaches, and administrators use our books, videos, and courses in practice. That’s why we’re especially excited about Jill Gough and Jennifer Wilson’s upcoming NCSM preconference. In it, they’ll be talking about how they use professional literature to grow their teaching practice. How do they apply what they’ve read? How do they collaborate, both in-person and online, to reflect on that application with their colleagues? What new learning and productive changes in teaching practice result from that work?

We asked Jennifer and Jill for a sneak peek of their session, and we’re happy to share it with you here. We hope you can join them in San Antonio, or follow along online.

Read, Apply, Learn
By Jill Gough and Jennifer Wilson

In Kindergarten Reading Workshop this week, the teaching point was when we want to learn new things, we first read what experts say. Now, it is clear that we are preparing our young learners for a unit on nonfiction reading and on research.  What if we transfer that simple, direct teaching point to our own work?

We set three goals this year as a team of teachers committed to narrowing the achievement gap for our learners. These goals are to learn more math, to scale what we learn across our schools, and to more deeply understand the Standards for Mathematical Practices.  With these goals, we have to ask, what do experts say?

We have been reading a lot lately, and we have been considering how to share what we are trying and learning in both our home communities and in a more global community. We are now studying and strongly recommend 5 Practices for Orchestrating Productive Mathematics Discussions by Mary Kay Stein and Margaret Smith, NCTM’s publication, Principles to Action,  The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. by Daniel Coyle,  Beyond Answers: Exploring Mathematical Practices with Young Children from Mike Flynn, and Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms by Tracy Zager and more.

In Beyond Answers, Mike Flynn suggests “We need to give students the opportunity to develop their own rich and deep understanding of our number system. With that understanding, they will be able to develop and use a wide array of strategies in ways that make sense for the problem at hand.” How might we slow down to afford our students the opportunity to develop their own deep understanding and grow their own mathematical flexibility? What will be gained when our young learners have acquired a deep foundation of understanding, confidence, and competence?

BeyondAnswersFlynn

In Becoming the Math Teacher You Wish You’d Had, Tracy Zager encourages us to engage our learners in productive struggle so that they are “challenged and learning”. She writes “As long as learners are engaged in productive struggle, even if they are headed toward a dead end, we need to bite our tongues and let students figure it out. Otherwise, we rob them of their well-deserved, satisfying, wonderful feelings of accomplishment when they make sense of problems and persevere.”

BecomingMathZager

So what does productive struggle look like in the classroom with students? What does productive struggle look like in professional learning communities with teachers? How do we learn to bite our tongues and give students time to figure it out? What stories can you share about students engaged in productive struggle?

What if we take ideas and apply them in our learning and teaching? What might we learn about our students, ourselves, and mathematics? What is to be gained by reflecting on our learning and sharing our thinking with our PLN here, there, and everywhere?

We look forward to considering these questions Sunday at our NCSM pre-conference session. And we look forward to sharing what we learn and discuss with those who can’t attend in real-time on Twitter and later through our blogs.

Jill (@jgough)  – Sneak Peek on Flexibility: Experiments in Learning by Doing

Jennifer (@jwilson828) – Sneak Peek on Empowering Learners: Easing the Hurry Syndrome

#NCSM17 #LearnAndShare #SlowMath

Flynn, Michael. 2017 Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, ME: Stenhouse Publishers.

Zager, Tracy. 2017. Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Portland, ME: Stenhouse Publishers.

 

Jill Gough learns, serves, and teaches as the Director of Teaching and Learning at Trinity School. Previously, she taught in the Westminster Schools, after 14 years of teaching in public schools in Mississippi and at the Kiski School of Pennsylvania. Jill received the Presidential Award for Excellence in Mathematics and Science Teaching in 1998 and Mathematical Association of America’s Sliffe Award in 2006 for excellence in teaching junior high.
Jennifer Wilson has been an educator for 24 years, spending 20 of those years teaching and learning mathematics with students at Northwest Rankin High School in Flowood, Mississippi. She currently teaches Advanced Placement Calculus and Geometry and also serves as a Curriculum Specialist with the Rankin County School District. Jennifer is an advocate for #slowmath, in which students and teachers take the time to enjoy mathematics.

 

 

Add comment April 1st, 2017

See you at NCSM and NCTM

We are looking forward to seeing you at this year’s NCSM and NCTM conferences in San Antonio.

At NCSM we will be exhibiting our books at booth #404.

At NCTM you can find us at booth #1325. Stop by to meet our authors:

Thursday
9:30-10: Anne Collins (Accessible Algebra)
10-10:30: Lucy West (Adding Talk to the Equation)
11:30-Noon: Chris Moynihan (Math Sense)
12:30-1: Mike Flynn (Beyond Answers)
1-1:30: Jessica Shumway (Number Sense Routines)
1:30-2: Nancy Anderson (What’s Right About Wrong Answers)
3-3:30: Christopher Danielson (Which One Doesn’t Belong?)

Friday
10-10:30: Chris Confer (Small Steps, Big Changes)
11-11:30: Kassia Omohundro Wedekind  (Math Exchanges)
3-3:30: Tracy Zager (Becoming the Math Teacher You Wish You’d Had)

Stop by at both conferences to browse and purchase our latest titles, pick up our free tote bag, and for a chance to win $1,000 in Stenhouse titles! Download a full schedule of Stenhouse authors presenting at both conferences.

Add comment March 31st, 2017

Now Online: Accessible Algebra

Accessible AlgebraAccessible Algebra is for any pre-algebra or algebra teacher who wants to provide a rich and fulfilling experience to students as they develop new ways of thinking through and about algebra.

Each of the thirty lessons in this book identifies and addresses a focal domain and standard in algebra, then lays out the common misconceptions and challenges students may face as they work to investigate and understand problems.

Authors Anne Collins and Steven Benson describe classroom scenarios in each lesson and also suggest ways teachers may assign a problem or activity, how to include formative assessment strategies, and suggestions for grouping students.

Each lesson includes sections on how to support struggling students as well as additional resources and readings.

We just posted the full preview online!

Add comment March 27th, 2017

Now Online: Becoming the Math Teacher You Wish You’d Had

becoming-the-math-teacher-you-wish-youd-hadTracy invites you on a journey through this most magnificent book of stories and portraits…This book turns on its head the common misconception of mathematics as a black-and-white discipline and of being good at math as entailing ease, speed, and correctness. You will find it full of color, possibility, puzzles, and delight…let yourself be drawn in.

— Elham Kazemi from the foreword

While mathematicians describe mathematics as playful, beautiful, creative, and captivating, many students describe math class as boring, stressful, useless, and humiliating. In Becoming the Math Teacher You Wish You’d Had, Tracy Zager helps teachers close this gap by making math class more like mathematics.

Tracy spent years observing a diverse set of classrooms in which all students had access to meaningful mathematics. She partnered with teachers who helped students internalize the habits of mind of mathematicians as they grappled with age-appropriate content. From these scores of observations, Tracy selected and analyzed the most revealing, fruitful, thought-provoking examples of teaching and learning to share with you in this book.

Through these vivid stories, you’ll gain insight into effective instructional decision making. You’ll engage with big concepts and pick up plenty of practical details about how to implement new teaching strategies.

All teachers can move toward increasingly authentic, delightful, robust mathematics teaching and learning for themselves and their students. This important book helps us develop instructional techniques that will make the math classes we teach so much better than the math classes we took.

Add comment December 15th, 2016

Now Online: Which One Doesn’t Belong?

Danielson’s book reveals the wonder and freedom of expression that many children don’t often experience in mathematics. A single, simple question puts children in a position to speak mathematically even at early ages. Ask students of all ages “Which one doesn’t belong?” and revel in the reasoning and conversation that results.
—Dan Meyer

How can I recommend this highly enough? Christopher Danielson emphasizes the stimulation of curiosity and that math is about making precise things that we—and children—can informally observe, without having to learn any mathematical language first. Which One Doesn’t Belong? is a glorious book for adults and children to explore together, and the Teacher’s Guide makes it into a profound mathematical resource.
—Eugenia Cheng, pure mathematician, University of Sheffield and School of the Art Institute of Chicago, and author of How to Bake Pi

Which One Doesn't Belong student bookWhich One Doesn’t Belong? is a children’s book about shapes. More generally, it’s a book about mathematics. When children look for sameness and difference; when they work hard to put their ideas into words; when they evaluate whether somebody’s else’s justification makes sense—in all of these cases, children engage in real mathematical thinking. They build mathematical knowledge they can be proud of. They develop new questions. They argue. They wonder.

In the accompanying teacher’s guide, author Christopher Danielson equips teachers to get maximum benefit from Which One Doesn’t Belong? Through classroom stories, he models listening to and finding delight in students’ thinking about shapes. In clear, approachable language, Danielson explores the mathematical concepts likely to emerge and helps teachers facilitate meaningful discussions about them.

You can preview portions of the teacher’s guide online now!

2 comments August 4th, 2016

Blogstitute 2016: Unleashing the Power of Games

In the next post in our Blogstitute 2016 series, we turn our attention to math and math games. Linda Dacey, Karen Gartland, and Jayne Bamford Lynch, authors of the Well Played series share with us what they learned when they examined how to make math games increase student learning. Be sure to leave a comment, ask a question, or tweet about this post using #blogstitute16!

Unleashing the Power of Games

Linda Dacey, Karen Gartland, and Jayne Bamford Lynch

Last week, we overheard a conversation between a second grader and his mother as he climbed into her car. His mother greeted him warmly and then added, “We need to stop for groceries on the way home.” The boy responded, “Oh no, I really want to get home and play the math game we learned today. Do we have to go shopping? I really want to play some more!”

More and more classrooms are offering opportunities for students to play math games, and students appear to enjoy them. Most textbooks now incorporate games into their lesson plans and, when teachers set up stations, a math game almost always is included. As we noticed this increased attention to games, we began to think about their use. We wondered about changes we could make in the games that were played or in the ways they were played in order to increase student learning. As a result, we spent eighteen months thinking and writing about math games, and we’d like to highlight a few things we learned.

Play in Partners

pairsOne of the most important insights we gained was that games offer more powerful learning opportunities when students play in teams. When one student plays against another, they rarely discuss strategy or what they are learning, perhaps because they do not want to give away an advantage. As a team of two or three players, students state their reasons for what move they want to make next. As they play, they coteach, practice vocabulary, create mathematical arguments, and critique their teammates’ suggestions. Over and over again, teachers have told us that this simple change has transformed the playing of games in their classrooms.

One of our favorite conversations occurred in a game that required students to match cards with equal values. Students could find matching cards in their hand or trade one of their cards for one in their opponent’s hand (cards are placed faceup) to make a match. Elly and Quinn were partners, and Elly wanted to take their opposing team’s card showing 5 x 9 + 3 x 9 to match their card showing 8 x 9.

Elly: We should take that card to match this one.

Quinn: Wait, wait—how do you know they match?

Elly: It’s that distribution thing.

Quinn: Huh?

Elly: You know, you split the eight nines into five and three of them.

Quinn: Oh yeah, it’s a property or something. We should look it up, but after we win.

They actually did go to the word wall after the game and identify the distributive property. Elly remarked, “I don’t think that’s what I called it, but good to know.”

Increase Time on Task

One of our least favorite games is Around the World. In this game two students are shown a math fact. The student who identifies the correct answer first moves to compete against the next student. The goal is to make it “around the world” by beating each and every classmate. As a result, the student who needs the least practice gets the most, and the student who needs the most practice likely considers only one fact. Most games are not nearly as problematic, but many can be altered to increase time on task. Sometimes we can adjust game rules so that the following occur:

  • There is an opportunity to trade cards (such as in the game described earlier), which increases attention to opponents’ decisions.
  • Points are awarded for finding a move worth more points than opponents found.
  • Both teams respond to a roll of the dice simultaneously and then compare their decisions.
  • Students play cooperatively, with both teams involved in all moves.
  • Students decide on a reward for finding an error in their opponent’s play.

Include Assessment

There are a variety of ways that we can include assessment within game playing. Here are a few:assessment

  • Think about what players might say or do to indicate their mathematical ideas, and make a list of these “look-fors” to focus our observations of students’ play.
  • Create recording sheets for students to complete as they play that we can look at later and that help students recognize that they are held accountable for their learning while playing.
  • Have students complete exit cards after they play a game that can help us decide who might need further instruction or who might need additional challenge. We can offer questions such as If you land on 24, what number would you like to roll? Why? or You were dealt cards showing the numbers 2, 4, 5, and 7. Where would you place these numbers in the equation ___ − ___x = ___ + ___x, to get the greatest value for x? We can also ask questions such as What did you learn from your partner as you played the game?

Games often engage students. With some simple changes we can greatly increase their educational value. We hope these suggestions lead you to identify other ways to unleash the power of games so that they are Well Played.

9 comments June 28th, 2016

Taking apart the “key word” strategy

Today we have a guest blog post from Christine Moynihan, author of the recent math books Common Core Sense and Math Sense. Christine takes a close look at the often-used “key word strategy” and explains why it might not be the best way to help your students get to the right answer.

Taking apart the “key word” strategy
By Christine Moynihan

Christine Moynihan

Christine Moynihan

“Does your answer make sense?” Can you estimate how many times you have asked your students this question in regard to word problems? Even if you are a beginning teacher, I am willing to bet that it has been, on a daily basis, several times a day, five days a week, 180 days a year! Do the math and you will find that you have asked that question of your students a staggering number of times.

Even if your students answer in the affirmative — “Yup, my answer makes sense”!what may pass for their answers making sense is that they’ve  checked to see if their computation is correct; if it is, they assume they are good to go. Tucking their answers back into the context of the problem after they have completed the computation still may not come naturally to them, even though you and their previous teachers have tried to build it into your practice, their practice, and the practice of mathematics in general.

In my pre-service math methods classes, I was told that when I was helping students learn to solve word problems, I had to support them first in making sense of the problem so that they could then determine which operation they had to perform. One way to do this, considered good teaching practice, was to tell them to search for key words.  The next step was to get them to underline those key words, underline the numbers in the problem, and then to use the key words to write a number sentence to solve the problem.  My guess is that many of you may have had the same experience in pre-service as well as having been taught that way when you were in elementary school yourselves.  A few of the key words/phrases you most likely learned are: for addition – in all, altogether, total, increase, combined; for subtraction – difference, left, fewer, decrease, gave away, take away; for multiplication – of, times groups, twice, double; for division – per, groups, share, out of.

So what’s wrong with the key word strategy?  I have had to answer that question countless times in my previous roles of mathematics curriculum specialist and building principal, as well as in my current role of educational consultant. The short answer is that using the key word strategy has limited scope and value.  Further, it can be incredibly misleading and steer students to wrong answers that make no sense at all within the context of the problem.  Additionally, it sends students the wrong message in terms of them thinking that solving problems can be done routinely, formulaically, and without much deep thought.

Let me share a recent experience I had with one of the fourth grade students with whom I have worked for two years.  We were going over a sheet of problems he had done in class, as he had to make corrections to the ones he had gotten wrong.  Now Jared is a very bright fourth grader who excels in reading, is highly engaging, and knows a great deal of information about a wide variety of subjects.  He likes to work fast, think fast, move fast, and complete his work fast.  Mathematics is not always his favorite subject and he often shuts down his thinking if he cannot see a solution path quickly.  Here is one of the problems he had done:

Mr. Smith bought 14 new globes that his students could

use in class as they studied Earth.  Each globe cost $76.

How much did the new globes cost in all?

Jared’s answer?  I bet many of you can predict what his answer was.  If you thought that he had written $90, you would be correct.  I had him read the problem aloud and asked why he thought his answer made sense.  His response was, “I did the right thing because 14 plus 76 equals 90.  I did it in my head, but I know it’s right and I even put a dollar sign in front!”

I asked him to draw a picture that represented the problem.  He drew 14 circles to show the globes. When I asked what the cost was for the globes, he wrote 76 under the first one and then did the same under the others.  He then said, “This won’t work.  Look at the problem – it says ‘in all’ and that means you have to add, so the real answer has to be $90.”

My next question was, “How much did two globes cost?”   He immediately replied, “Well, it could be $152 but that can’t be right!  They told me to add!”  I had him continue to work and after he had successfully solved the problem, he looked at me and said with utter conviction and even a bit of indignation, “It’s not my fault!  They put in ‘in all’ so how I am supposed to know I shouldn’t add?”

I know for certain that there isn’t anyone who teaches key words as the only and/or best problem-solving strategy. But the rigidity with which Jared applied the strategy reinforced for me just how limiting and misleading it can be. I continue to work with him with one of my goals being that he understands that the responsibility for making sense of a problem rests with him and that he needs to add to his repertoire of problem-solving strategies to ensure his answers make sense.

1 comment April 27th, 2016

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