My picture book Penguins on Parade uses a purely fanciful problem to illustrate students making sense of counting by tens from a number that doesn’t have zero in the ones place. The characters in the book enjoy imagining the penguin parade starting with 8 penguins followed by groups of 10 penguins. Their teacher Ms. Green asks them to figure out how they could count the penguins by tens starting at 8. Share this book with your class and show them characters deepening their understanding of tens and ones as they tackle an unfamiliar problem.

Trevor is frustrated by this problem since he’s never been told how to do it. But his partner Zoe perseveres and engages him in using Ten-Frame Cards with dots on them to help them figure out the subtotals for the count. Then they examine their list, 8, 18, 28, 38, with their classmates and begin to understand why adding on a ten increases the number in the tens place while the number in the ones place remains the same.

Two other classmates Ellie and Kayla use an open number line (or empty number line) to help them figure out what they get when they add tens onto 8. They break up each ten into a 2 and an 8 to make the fewest jumps they can to add on a ten. They know it is efficient to jump to the next multiple of ten and then add on the amount left in the ten. They are applying their knowledge of the number pairs that make ten and their understanding of counting up and down on a number line.

Explorations like this are much more powerful than just remembering which place is the ones place, which place is the tens place and adding ones to ones and tens to tens. Experiences like this help students understand that the 4 in 47 means 4 because it is 4 tens, but also means 40 because 4 tens are 40.

The characters in the book model how they are learning to use the Standards for Mathematical Practice. Your students will see this class sharing ideas and helping Trevor gain confidence in his ability to make sense of new problems!

Elementary students can find it challenging to solve  problems with more than one solution, especially if they have never worked through this type of problem before.   Some students have the idea that every math problem has only one solution.   So working with  the Common Addition and Subtraction Situations: Put Together- Both Addends Unknown  (http://www.corestandards.org/Math/Content/mathematics-glossary/Table-1/) can help students learn more about persevering to solve unfamiliar problems. My picture book, Monkeys for the Zoo presents a model of students engaging in such a problem.

Working with this problem, Ms. Green’s students find out that some problems do have more than one possible solution. Mia sees that if you can have 3 howler monkeys and 10 spider monkeys, you could also have 10 howler monkeys and 3 spider monkeys. Then she discovers that you could change that combination to have 9 howler monkeys and 4 spider monkeys. And Jayden uses 9 + 4 to come up with 8 + 5. This problem pushes students to see the relationships between combinations and use one combination to find another.

Present your students with one of these problems. On the last page in Monkeys for the Zoo there are several suggestions, or you might make one up about another context that your students would enjoy thinking about. Students will find different entry points and enjoy working together to see how many solutions they can find. At first, some students may not accept that more than one combination of addends will work. They may be satisfied with only a few of the possible solutions. Involve them in modeling the problem and discussing solutions with their peers to help them see that there are many possibilities. Ask questions like:
“Who used one combination to help you find another? How did it help you?”
“If this combination works, how can we use it to find another?”
“What are some other ways you can use one solution to help you find another?”
For students struggling with this idea, reexamine Monkeys for the Zoo, and ask, “How do you think Rosie used the solution 3 howlers and 10 spiders to get 3 spiders and 10 howlers? How do you think Mia used 3 spiders and 10 howlers to help her find 4 spiders and 9 howlers?”

EXTENDING THE PROBLEM
When students are ready, see if they can find all of the possible solutions. Have students discuss how they know they have all the combinations, what strategies helped them find all the combinations, and which combination of addends are preferable in particular real world situations. For another challenge, you can add conditions to situations where both addends are unknown. For example, consider this problem: Juan has 8 books. He wants to put more books on his shelf than next to his bed. How many might he put next to his bed and how many might he put on his shelf? Your students may also enjoy the challenge of making up their own unknown addend problems.

I  have written lesson plans to help teachers use the UYMP books to engage students in  understanding and use the practices.  These lessons and the introduction are available here as pdfs.  The introduction is pasted below to give you more information about the lessons.  Here are the pdfs for the introduction and the lesson plans.

Teacher Guide Introduction UYMP bks

Teacher Guides Penguins on Parade,

Teacher Guides Monkeys for the Zoo,

Teacher Guides Hatching Butterflies

Use Your Math Power Books: A Model for Mathematical Practices and Opportunities for Student Reflection

 Goal: The following lessons guide the teacher in using the Use Your Math Power books to model mathematical practices and give students opportunities to reflect on how they can enhance their use of the practices, focusing on participation, perseverance, and communication.

Thinking behind the lessons:  By reading and reflecting on the Use Your Math Power books, math students can reflect on how they can use best practices and engage in a learning community. In the lessons, students focus on practices that help them succeed in math. The lessons can be used early in the year to set expectations, or during the year to reinforce best practices.

Though set in a primary classroom, these books provide a tool for students of various ages to think about mathematical practices, especially perseverance and communication. Older students listening to the books may find it easier to focus on the practices since the math content will be less challenging.   When used in first grade, teachers may wish to wait to use each books until the time of year when they are working on the math content in the book’s problem.

The Lessons:  There are 3 lessons for each book, a first reading, a lesson focusing on a common struggle in classroom participation, and one focusing on effective communication.

Hatching Butterflies   1. A First Reading, 2. Working with Partners,  3. Focusing on Communication

Monkeys for the Zoo  1. A First Reading,  2. Gaining Confidence to Share Your Ideas,  3. Focusing on Communication

Penguins on Parade  1. A First Reading,   2. Making Sense of Unfamiliar Problems,  3. Focusing on Communication

I love this quote from Cathy Seeley, shared today on twitter from an NCSM talk.

“Protecting students from (the right kind of) struggling is one of the worst ways we treat students inequitably.”

I so agree! All of our students need opportunities to work through struggles.  If we ask questions, listen carefully to students’ ideas, and have them discuss their thinking with each other, we can provide prompts that help them access their prior knowledge and make sense of unfamiliar problems.

As an example, I am sharing some pages  from Penguins on Parade.   The teacher, Ms. Green, helps her primary students learn to work through struggles with unfamiliar tasks.  First she asks her students to think about an imaginary parade of penguins.

In a few pages, Jayden has a suggestion.  Ms. Green asks her students to try to work with partners and figure out how to count in an unfamiliar way.

Trevor reluctantly works with Zoe, trying to use the ten frames to count the penguins by tens starting from 8.  As Zoe adds tens they count by ones and she records each subtotal in a list.  Then we see them talking about whether they are counting by tens.

Once they share their work with the class, another student sees their pattern and figures out that 48 will be next.  The students work together to check and see that 48 is correct.  Then they continue to try to use the pattern and discuss why it works.

The discussion continues.  These primary students are learning to reason about mathematics, providing a model for students and teachers.  They are learning to work through accessible (the right kind of) struggles without their teacher giving them too much protection!

In a math talk community students learn together and help each other. Partners share responsibility for understanding problems they solve.

What if partners aren’t sharing the responsibility for their work? Sometimes one person’s ideas dominate and the talk becomes a monologue with little opportunity for learning together.

Can we help students learn to work well with a variety of people?

In a recent Teaching Children Mathematics article, a fourth grade teacher explains how she has encouraged more productive math talk in her class.  (Click here for article.)  She lists 5 steps that helped her.

The author explains how her class explicitly discussed the importance of math discourse throughout the year and how they gradually learned the skills of active listening, revoicing, responding, and justifying ideas.

When students see that discussing ideas helps them all learn, they engage in more productive discussions in large groups, small groups, and in pairs.  Just as the article’s author reminds her students more and more frequently to use active listening and revoicing, we need to regularly remind students of our math talk expectations.

Then surprising things can happen!  Here’s an example from  Use Your Math Power: Hatching Butterflies.  In the beginning of the story, Carlos does not ask his partner Hannah questions about her strategy, even though he doesn’t understand it.   He just copies her work.  Once their teacher reminds them to discuss their ideas, Carlos gets Hannah to think about a different strategy.  

This led to  the students reasoning about why they could use addition or subtraction to solve the problem.  As a math specialist recently observed after reading Hatching Butterflies, “they both grew. Hannah couldn’t even describe why she did take away. Carlos knew where he wanted to go with his method, and he forgot to add up the jumps. He grew in his understanding as he shared in front of the class.”

When students share ideas, they often get to think about refining their ideas and connections between different strategies, deepening their math understanding.  This is the richness of math discourse and the power of paired work!

In my last post, I talked about giving students the opportunity to puzzle about unfamiliar problems in the curriculum.  I offered the table of Common Addition and Subtraction Situations  as a source for problems.  In my book,  Use Your Math Power: Monkeys for the Zoo,  the teacher, Ms. Green, asks her students to think about one of these types of problems          

As her students talk about the problem in pairs, their questions and misunderstandings suggest that this is an unfamiliar problem for them.  Here is part of one of the conversations about the problem.  

After their “Turn & Talk”, Ms. Green has the class come together to share some of their ideas.

Once Carlos has retold part of the problem, one student asks a question about how they can begin to solve it.

 “Other kids might be wondering about that too,” Ms. Green replied.  “Can anyone help us find a way to start?”

This leads to Ellie’s idea of having 13 children stand up with 3 on one side to be howlers and 10 on the other side to be spider monkeys.  When Ms. Green asks her class if this will work, the discussion takes off.  Some think it should be solved another way.  This leads to a discussion about whether this problem has many possible solutions, which leads to them finding more combinations that work.

By discussing the meaning of the problem and how they might solve it, students develop problem solving strategies and make sense of important math ideas.  They use retelling and acting out to help them make sense of the problem.  Ms. Green’s questions and prompts move the discussion along  productively.

It is so powerful to help students use discussions, retelling, acting out, and visualizing to make sense of and solve problems.  I am wondering about your students’ experiences using these strategies.  How can we help all of our students feel confident about using strategies like these to persevere to make sense of a problem instead of wanting to be told which procedure to use?