Tracy 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.
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.
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? 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.
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
One 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.
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.
There are a variety of ways that we can include assessment within game playing. Here are a few:
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.
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
“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.
We will be in San Francisco next week for NCSM and NCTM. Stop by our booth (204 and 932, respectively), to meet some of our authors and receive 25% off our books and videos and to pick up one of our fancy tote bags!
Lucy West will be signing at the NCSM bookstore at 1 p.m. Monday, April 11, then later that afternoon at 4 p.m. at the Stenhouse booth. You can preview her latest video Adding Talk to the Equation here.
At NCTM you can catch up with Jessica Shumway at 11 a.m. and with Lucy West at 1 p.m. on Thursday, April 14.
Friday is also ShadowCon time, starting at 5 p.m. For details, follow #shadowcon16 on Twitter. Our own Tracy Zager will be there, so be sure to say hello to her!
For more than twenty years, Lucy West has been studying mathematical classroom discourse. She believes that teachers need to understand what their students are thinking as they grapple with rich mathematical tasks and that the best way to do so is through talking and listening. Adding Talk to the Equation helps teachers learn how to skillfully lead math conversations so all students stay in the game, stay motivated about learning, and ultimately deepen their understanding.
This video features five case studies filmed in grades 1–6 and shows teachers at various stages in their practice of generating and managing rich mathematics conversations. Lucy emphasizes the progression that occurs as teachers get more comfortable with new talk moves and as they learn to tune in and respond to the math conversations taking place among their students. Although these discussions occur during math instruction, the strategies used to create an environment for respectful, productive discourse can be applied to any subject area.
The video segments examine the importance of creating a safe learning environment; the value of thinking, reasoning, and questioning; the role of active, accountable listening; and the necessity of giving all students a “You can do this” message. Lucy also emphasizes that slowing down, even in the face of time constraints, is crucial for creating a classroom where all students feel they have something to contribute.
The 84-page companion guide includes transcripts of all of the case studies, with detailed commentary from Lucy that gives you a window into her thinking and the complexities of the work she is doing with teachers, as well as her reflections on missed opportunities.
Students love math games and puzzles, but how much are they really learning from the experience? In the third book of the popular Well Played series authors Linda Dacey, Karen Gartland, and Jayne Bamford Lynch help you engage students in deep mathematical discussions, enhance students’ conceptual understanding, develop students’ fluency with number systems, ratio and proportional relationships, expressions and equations, statistics and probability, and patterns, graphs, and functions.
Each book in the series shows you how to make games and puzzles an integral learning component that provides teachers with unique access to student thinking.
The twenty-five games and puzzles in each of the Well Played books, which have all been field-tested in diverse classrooms, contain:
explanations of the mathematical importance of each game or puzzle and how it supports student learning;
variations for each game or puzzle to address a range of learning levels and styles;
clear step-by-step directions; and
classroom vignettes that model how best to introduce the featured game or puzzle.
Well Played, K-2 is the second in a series of three books by Linda Dacey, Karen Gartland, and Jayne Bamford Lynch that helps teachers make puzzles and games an integral part of math instruction.
Following the same accessible format as the first book in the series (for grades 3-5), 25 field-tested games and puzzles are each introduced with math areas of focus, materials needed, and step-by-step directions. Readers will see how they play out in the classroom and get tips on maximizing student learning, exit cards for student reflection, variations, and extensions. The rich appendix has reproducible directions, game boards, game cards, and puzzle materials, and each chapter includes assessment ideas and suggestions for online games and apps.
Get your K-2 students talking and learning more as they play games and build their thinking as mathematicians. Well Played, K-2 is available now, and you can preview the entire book online!
We are excited to have a guest post from Geoff Krall today who calls on all teachers to take the initiative to find the help, resources, and advice they need to help close the gaps in their students’ knowledge. Check out Geoff’s blog Emergent Math and also follow him on Twitter!
It’s a lot easier to complain that students don’t know, say, their multiplication tables than to actually teach multiplication.
Setting aside the oft problematic mindset of a teacher complaining about what “these kids” don’t know for the time being, consider actually teaching to the gaps you feel are present.
Let’s be clear: not a single one of us have entered the school year 100% satisfied with where 100% of our students are at math-wise. There’s always something that was allegedly “covered” in previous years and for whatever reason was not retained by the students. A couple years ago I was at a relatively well-off suburban school where 99% of their students graduate and go on to college and even those teachers were complaining about what their students did and didn’t know.
Often under the guise of unspecific complaints about students “not knowing their basic math facts” or “numeracy”, teachers sometimes pass blame upon students, The Calculator, or their prior schooling. What I often don’t see happen is addressing those gaps in knowledge. Sometimes a cursory remediation worksheet is handed out, and after-school tutoring is offered, but many times I don’t see teachers actually teach to those gaps. Y’know: teaching kids these lugubrious “basic math facts.” Even more specific complaints about how students “don’t know how to do fractions” (whatever “do” means) are ripe for teaching opportunities, rather than tsk-tsk-ing.
Which is unfortunate because there’s never been a more robust cache of resources to remediate in a healthy, fun way. It reminds me of that bit from Arrested Development where Lucille Bluth brushes off her poor raising of Buster because “kids don’t come with a handbook.”
[Ron Howard voice] In fact, there are countless books that address the very learning gap you’re complaining about. NCTM has so many publications that would probably be perfect. Or go here and click on the grade lower than you. Shoot, just go to amazon and type in what you feel your students are struggling with.
If you feel your students lagging in a particular area of their learning, I’d suggest rather than complaining and sending them to a worksheet or instructional video, consider doing some learning yourself and find a book, blog, text, paper, resource, or teacher to teach you how to teach to this area. I’ve learned so much from my non-grade level colleagues about teaching number sense, rounding, fractions, ratios / proportions, and even an alleged area of expertise of mine: algebra (thanks Andrew!).
I mean, if you need a specific recommendation, I’d consider learning how to facilitate some number talks via Elham and Allison’s Intentional Talk and go from there. Or follow ’em on twitter and get their awesome advice for free! (But seriously, get their book.)
Also, we’re not talking about shutting everything else down classroom-wise, lest you’re worried about losing precious class time. While coverage is overrated, let’s put that aside for now, shall we? We’re talking 10-20 minute activities and discussion here, maybe a couple times a week. Stop complaining and start learning how to teach this stuff. If we want students to learn, we probably ought to do some learning ourselves, no?
Besides, teaching these skills and concepts is fun. This has probably been my biggest takeaway of the year so far: leading number talks is so fun, I’d do it even if I wasn’t addressing learning gaps.
The authors of Well Played sat down with us recently to talk about how puzzles and games are not just fillers and not just a way to practice math. Combined with the teacher’s questioning and assessment, real math learning happens, classroom talk is enhanced, and students become “co-teachers” who support each other. Listen to this conversation with Linda Dacey, Karen Gartland, and Jayne Bamford Lynch and then head over to the Stenhouse website to preview the book online!