Posts filed under 'math'

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

See you at NCSM and NCTM

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!

 

 

 

Add comment April 9th, 2016

PREVIEW NOW: Lucy West’s new video Adding Talk to the Equation

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.

Add comment March 29th, 2016

Now Online: Well Played, 6-8

1033Students 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.

You can preview the entire book online now!

Add comment March 18th, 2016

Now Online: Well Played, K-2

Well PlayedK2Well 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!

1 comment December 15th, 2015

Complaining about what students don’t know vs. addressing the situation

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.

amazon_com__number_sense

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.

3 comments November 3rd, 2015

Author conversation: Well Played

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!


Add comment August 25th, 2015

Comparisons: A Little Bit More Older

In this fun guest post from Tracy Zager, you can follow along as her daughter tries to figure out just how much older some of her friends are, and as she does, you can get an excellent insight into how Tracy guides her mathematical thinking. Tracy’s new book Becoming The Math Teacher You Wish You’d Had will be published in 2016.

Ask any elementary school teacher, and she or he will tell you that comparison problems are much harder for most kids than operations with other actions. For example, fourth-grade-teacher Jennifer Clerkin Muhammad asked her students to draw a picture of this problem from Investigations:

Darlene picked 7 apples. Juan picked 4 times as many apples. How many apples did Juan pick?

Her students are pros at representations and skillful multipliers, but we saw a lot of this:

apples unsure

Kids who were used to thinking of multiplication as either groups, arrays, or repeated addition had some productive, furrowed-brow time ahead of them. How can you represent comparisons? How is this multiplication? How do you draw the 4?

Then they started to get somewhere.

apples groups

apples array

With each representation they discussed, Jen asked the excellent question:

“Where do we see the 4 times as many in this representation?”

Seriously. Jot that one down. It’s a keeper.

Jen and her class played with a series of problems involving multiplicative comparison, (height, quantity, distance, etc.). For each problem, she asked student to create a representation. And then she’d choose a few to discuss, asking, “Where is the ______ times as ______ in this representation?” Lightbulbs were going off everywhere as students broadened and deepened their understanding of multiplication.

Teachers often see the same struggle with comparison situations that can be thought of as subtraction or missing addend problems, like:

Marta has 4 stuffed animals. Kenny has 9 stuffed animals. How many more stuffed animals does Kenny have than Marta?

If kids have been taught “subtract means take away” or “minus means count backwards,” then this problem doesn’t fit the mold. And if kids have been taught keyword strategies, they’re likely to hear “more,” pluck those numbers, and add them together.

Knowing this as I do, I jump on opportunities to explore comparison problems with my own kids. I want them to have a fuller sense of the operations from the beginning, rather than developing too-narrow definitions that have to be broadened later. So, subtract doesn’t mean take away in our house. They’re not synonyms. I actually don’t worry about defining the operations at all: we just play around with lots of different types of problems, and I trust that my kids can make sense of them.

Yesterday, I was talking with Daphne about a neighbor visiting tomorrow. Our neighbor has a little girl, whom I’ll call Katrina. She’s about 2. We also have more awesome girls who live across the street. Let’s call them Lily, 15, and Gloria, 13. Gloria helps out once a week, and my girls idolize her. Here’s how the conversation kicked off:

Me: “If Katrina’s mother comes over to help me on Sunday, could you play with Katrina for a bit so her mom and I can work?”

Daphne: “Yes!!! We love playing with her. And she loves us. She cries when she has to leave us. We’re her best friends.”

Me: “I think Katrina feels the way about you and Maya that you and Maya feel about Lily and Gloria. It’s fun to hang out with older kids. You learn a little about what’s coming for you.”

Daphne: “Yeah. But for Katrina, Lily and Gloria would be TOO old. Like, she’d think they were grown-ups because they’re SO much older than she is. We’re older than Katrina, but we’re less older than Lily and Gloria are.”

So there’s my opportunity. She was comparing already. With a little nudge, I could encourage her to quantify and mathematize this situation. I thought I’d start with the simplest numbers so she could think about meaning first.

Me: “How much older are you than Katrina?”

Daphne: “I think she’s 2. So…”

There was a pretty good pause.

Daphne: “Are we counting on here?”

That’s paydirt.

Me: “What do you think?”

Daphne: “I think so. Because I want to know how many MORE. So, 2, 3, 4, 5, 6. I’m 4 years older than she is. Yeah. That makes sense.”

Me: “OK. So how many years older is Maya than Katrina?”

Daphne: “6.”

Me: “You didn’t count on that time. What did you do?”

Daphne: “Well, I knew that I was 4 years older than Katrina, and Maya is 2 years older than me, so 4, 5, 6. Because Maya is OLDER than me, I knew they had to be farther.”

Me: “What do you mean farther?”

Daphne: “Like, it had to be more. I was starting with 4, and then it had to be more farther apart.”

Me: “Oh! So you were trying to figure out whether to go 2 more or 2 fewer?”

Daphne: “Yeah. And it had to be 2 more, because Maya’s more older than Katrina than I am. I mean, I’m older than Katrina, and Maya’s older than me, so she’s more older than Katrina than I am. It had to be 2 MORE.”

Hot damn!

Daphne’s life might have been easier right here if she had some tools she doesn’t have yet. I’m thinking about number lines, symbols, and a shared mathematical vocabulary including words like distance or difference. Those tools might help her keep track and facilitate communicating her thinking to somebody else. But I love that we were able to have this conversation without those tools, while driving in the car where I couldn’t even see her gestures. Her thinking is there, and clear. The tools will come as a relief later on, when the problems are more complicated.

We kept going.

Me: “OK. So you’re 4 years older than Katrina and Maya’s 6 years older than Katrina. I wonder how that compares to Gloria? Is she about the same amount older than you two as you two are older than Katrina?”

Daphne: “Well, she’s 13. This is gonna be hard.”

Long pause.

Daphne: “She’s 7 years older than me.”

Me: “How’d you figure that out?”

Daphne: “I went 6. 7, 8, 9, 10, 11, 12, 13.” She held up her fingers to show me.

Me: “So how much older is Gloria than Maya?”

Daphne: “Umm…so Maya’s 8…umm…5. She’s 5 years older than Maya.”

Me: “How do you know?”

Daphne: “Because Maya’s 2 years older than me, so it’s 2 closer.”

This is the kind of problem where kids lose track of their thinking easily. Do we add or subtract that 2? In the last problem, she’d added. This time she subtracted. And she knew why!

When I glanced at Daphne in the rear view mirror, she had this awesome look on her face. She was looking out the window, but not seeing what was outside because she was so deep in thought. I wish I could have peeked inside her mind, and seen what she was seeing. Because, to use Jen’s question, where do we see the 2 years older? How did she see it? There was no time to ask, because she was onto pulling it all together.

Daphne: “So, Gloria’s a little bit more older than us than we are older than Katrina, because Gloria is 5 and 7 years older, and we’re 4 and 6 years older. But it’s pretty close.”

We pulled into the school parking lot on that one.

In our family, we count and manipulate objects all the time. Counting more abstract units like years, though, gives me a chance to open the kids’ thinking a little bit. In this conversation, Daphne was both counting and comparing years. Tough stuff, but oh so good.

(This post first appeared on Tracy Zager’s blog.)

Add comment August 11th, 2015

Blogstitute: Got Common Core?

Today’s Blogstitute post comes from Christine Moynihan, whose latest book is Common Core Sense: Tapping the Power of Mathematical Practices. In this post Christine introduces the GOLD framework that helps make the Standards for Mathematical Practice more accessible to elementary teachers. Be sure to leave a comment or ask a question for a chance to win 12 Stenhouse books! On Twitter you can follow along using #blogstitute15.

Got Common Core?
By Christine Moynihan

FC FINAL CommonCoreSense.inddSomething I hear from many teachers is that it is challenging to be up-to-date on everything that teachers should and must know in order to be effective practitioners. This is especially true for elementary teachers, who are asked to be content experts in reading, writing, grammar, spelling, science, social studies, and, of course, mathematics. Not only do they need to have expertise in these curriculum areas in terms of content, but they must also be experts in the best instructional practices that will support their students in learning in each of these areas. (I’m not even going to go into how they also have responsibility for social and emotional growth, health and wellness, behavior management, and the list goes on. . . . )

So, as a former classroom teacher, I get it. As a former curriculum specialist, I also get it. As a former principal, I most certainly get it. As a current educational consultant, not only do I get it, I hear it all the time—there is just so much to know, so much to learn, so much to do. As a result, when I ask a variation of the “Got Common Core?” question, many teachers respond that although they “get” the basics of the Common Core in terms of the standards for mathematical content for their specific grade levels, they believe that they have a somewhat light understanding of the standards for mathematical practice. Most teachers report that what they know about the MPs has been by way of an introductory look at them at a professional development session and/or staff meeting, with little or no follow-up.

My major purpose in writing Common Core Sense: Tapping the Power of the Mathematical Practices emanates from my desire to help teachers gain a foothold in understanding the MPs and how they can affect their practice. The book is meant to be a vehicle for making the eight Standards for Mathematical Practice more accessible to elementary teachers, for I see them as the core of mathematical proficiency. As I wrestled with how to do that, I defaulted to something that has always worked for me as a learner—to devise some kind of a framework, a mnemonic of sorts, to aid in understanding and then activating that understanding. Because I had been saying over and over again that “the gold of the Common Core really lies within the mathematical practices,” I constructed the GOLD framework to help teachers see some of the major components of each MP and then think about what they may look and sound like in classrooms, and what might need to be done to support the incorporation and implementation of the MPs into daily practice.

The Framework:
Go for the goals—What are the major purposes of the practice?
Open your eyes & observe—what should you see students doing as they utilize the practice? What should you see yourself doing?
Listen—What should you hear students saying as they utilize the practice? What should you hear yourself saying?
Decide—What do you need to do as a teacher to mine the gold?

I identified three major goals for each mathematical practice, fully aware that there are many more goals to be found within each. In the link you will find what I have identified as the second goal of Mathematical Practice #3: Construct viable arguments and critique the reasoning of others. What’s not to love about MP3? When you can analyze your thinking enough that you can clarify it, defend it, justify it, and represent it, you have learned something that will be valuable in all areas of life. In terms of mathematics, that ability leads you straight to the path of being mathematically proficient—a goal we all have for our students. I hope that the chart for the second goal I identified for MP3 can help in your work to make this MP come alive for the students in your classrooms.

Accept that viable explanations of mathematical thinking must be organized, reasonable, and justifiable/laden with proof.

8 comments July 6th, 2015

Now Online: Well Played

well-played-grades-3-5In addition to fun, math games can be an important part of your curriculum and give teachers unique insights into student thinking. In their new book, Well Played, Linda Dacey, Karen Gartland, and Jayne Bamford Lynch offer 25 field-tested games and puzzles that go beyond computational fluency to engage students in grades 3-5 with key mathematical concepts.

Each game or puzzle is introduced with math areas of focus, materials needed, and step-by-step directions. Readers will see how it plays out in the classroom and get tips on maximizing student learning, exit cards for student reflection, variations, and extensions. The extensive appendix has reproducible directions, game boards, game cards, and puzzle materials, and each chapter includes assessment ideas and suggestions for online games and apps.

Well Played will help you get students talking and learning more as they play games and build their thinking as mathematicians. You can now preview the entire book online, order the e-book version, and preorder the print version which ships early next month.

1 comment June 19th, 2015

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