3rd Graders Apply Their Knowledge about Area

The third grade classes at our school recently finished their unit about measurement.  During this unit, one of the topics they studied was area.  As a way to help them apply their knowledge, we brought in floor tiles and carpet samples.  (If you tell the store these samples are for a school project, they will often donate them to you!)  Students are working to determine how they would like to remodel the classroom - choosing the tiles and carpet they like best.  The tiles are of different sizes - 6"x6", 12"x 12", and 16"x16".  We were purposeful in having only samples.  Students do not have enough tiles to lay them across the floor and count them to find the area.  They will have to find another method to calculate the area.

Students have a variety of tools to help them find the area of the classroom: rulers, yardsticks, grid paper, posterboard (to cut out squares congruent to the tiles they have selected), and chart paper.  The students are to create a diagram of the classroom detailing what sections of the room will be tile and what sections will be carpet.  Once they have this plan, they will calculate the costs of the materials and write persuasive letters to our principal in an attempt to convince her to remodel their classroom.

One "a-ha" moment for students was the difference between square feet and square yards.  We had a discussion today about the importance of using correct units.  If, for example, we order 100 square yards of carpet when we really need 100 square feet of carpet, we will spend a lot of money and have a lot of carpet left.

Students have been very motivated by this project!  They are working to be precise in their measurements and detailed in their design.  This is a great example of students engaging in the mathematical practices!

Lunch conversations

I have the pleasure of spending my lunch duty with our fifth grade students.  They are a great group of kids but have a tendency to get a little noisy and sometimes lack appropriate conversation topics.  To help them have a topic for conversation at their tables, I found some "conversation starters" (i.e., math problems) from the Investigations series for them to discuss.  I love these problems because there are multiple correct answers and they were easy for students to try without paper and pencil to record their thinking.

At first, the students wanted to share their answers with me.  As I asked whether others at their table agreed with their possible solution, students began to ask one another about the answers.  This was fantastic!  Rather than viewing me as the keeper of the answers, the students began to reason about the problem with one another and build their understanding together.

Another benefit was that students in other grades talked about the problems as well.  Students were practicing counting skills, math reasoning, and many of the mathematical practices all while eating lunch!  I hope you will consider trying to find a way to involve more conversations about math at your school!

Here are some examples of the problems we used:

Vocabulary

During a math data team meeting last week, a grade level noticed that vocabulary might have caused some difficulty on the recent math test.  We discussed the importance of building vocabulary with students - especially our English Language Learners and low-SES students.  The big idea is that students can communicate using math vocabulary rather than merely give a definition when a word is provided.  The vocabulary needs to help students make sense of the mathematics.  Here are some ways to incorporate more vocabulary into your math classroom!

The teachers at our building are already implementing the use of Frayer models to help students use and understand important math words.  I love that Frayer models provide students with both examples and nonexamples to help students understand what a vocabulary word means and what it does not mean.  If you have not used a Frayer model before, here is an example:

In their book Putting the Mathematical Practices Into Action: Implementing the Common Core Standards for Mathematical Practice K-8 (a GREAT read by the way!), O'Connell and SanGiovanni provide helpful suggestions for using vocabulary to increase precision.  Some of the ideas include:

• The use of word webs to explore math ideas and expand vocabulary.  For example, when studying measurement, you could web tools, attributes, units, etc.
• Sort and Label will help students categorize ideas and concepts.  Students organize the words and assign a label.  This can reveal a lot about student thinking and uncover misconceptions!  For example, how might students sort the following: sum, minus, join, compare, subtract, add, take apart, plus

Another idea that students love is to print labels with a math vocabulary word on each label.  Put a label on the back of each student.  Students play 20 Questions - asking questions that can only be answered with "yes" or "no" - to figure out what word is on their back.  This promotes conversation and students frequently use additional vocabulary in their questions to determine their word.  For example, if a student has triangle on their back, they might as, "Am I a polygon?" or "Do I have parallel sides?".

Base 10 Blocks on your iPad!

I discovered the iPad app "Number Pieces" this past week.  It is a great way to have students work with base 10 blocks and it is FREE!

There are a couple of features that make this app really useful for students:
1.  Students can decompose or recompose numbers using the same blocks.  For example, if you want to break apart 1 ten into 10 ones, you click on the piece you want to separate and then touch the separate icon.  Alternatively, if you want to put 10 ones together to make 1 ten, lasso the blocks (a blue dotted line will appear) and then touch the "join" icon.  You could also lasso a large number of pieces to rearrange them into larger groups (such as moving 32 ones into 3 tens and 2 ones).
2.  If you want to subtract, you can change the color of the blocks.  This allows students to maintain the original amount without having to remove pieces, allowing them to see the part-part-whole relationship.  With the app, students don't have the confusion of trading the pieces and keeping track of their counting.  The app also includes a marker feature so students could draw a part-part-whole organizer or represent the computation.  See the picture for an example.

I'm sure you have many additional ways this app could be used with students.  I would love to hear your ideas!

Using data to generate conversations about strategies

Recently, as a part of my evaluation, I gave students in a 5th grade class the following two problems to solve:

1. OfficeMax sells pencils in large boxes.  Mrs. Bailey bought some pencils for the students at school.  There were 150 pencils in each box.  She bought 8 boxes.  How many pencils did Mrs. Bailey buy?
2. 86 x 23 = ____

Students were able to solve the problem in any way they wanted during this pre-assessment.  In past years, I felt like I had involved students in discussing their strategies and analyzing both their own work and the work of others.  Because I am not a classroom teacher this year, I was more explicit in presenting the data to students and in this process we noticed some interesting patterns about strategies.  Here is a chart of the accuracy and strategies students used to solve the problems above:

For the first problem:  OfficeMax sells pencils in large boxes.  Mrs. Bailey bought some pencils for the students at school.  There were 150 pencils in each box.  She bought 8 boxes.  How many pencils did Mrs. Bailey buy?

Strategies                                                       Accuracy

Repeated Addition	10/11 students answered correctly	91%
Standard/Traditional Algorithm	6/9 students answer correctly	67%
Doubling/Halving	1/1 students answered correctly	100%

For the second problem: 86 x 23 =

Strategies                                                       Accuracy

Repeated Addition	0/1 students answered correctly	0%
Standard/Traditional Algorithm	6/15 students answered correctly	40%
Partial Products with Boxes	4/4 students answered correctly	100%
Partial Products	0/1 students answered correctly	0%

Students were engaged in the conversation and shared great thinking!  For example, students explained why someone might use repeated addition in the first problem but not in the second problem since the numbers became too large to easily keep track of the addition. 

We WANT for students to have multiple strategies for solving problems and to know WHEN and WHY to use them.  It might make more sense to use repeated addition when solving 175 x 3 because the numbers are reasonable for addition.  Students should recognize, however, that when solving 175 x 83 that repeated addition is not a reasonable strategy to use.  This activity also led to a great conversation about efficiency.  Which algorithms are most efficient?  In what situations?  I'm sure the students in your classrooms have great ideas to share!

Applying Powers of 10

Our district has a new superintendent this year.  In recognition of his 100th day since joining the district on July 1, 2013, our 5th grade students created a sign that read "The Power of 10".  We took a photo as students stood in front of the sign holding white boards with various representations of 10 x 10, including 10^2 and 100.  We then did a second photo with students standing in front of a sign that read, "We can't wait to see what happens by the next power of 10" with students again holding white boards with representations of 10 x 10 x 10, including 10^3 and 1,000.  We e-mailed these pictures to the new superintendent and he was appreciative of the message.

This was a great activity that allowed students to apply their understanding of powers of 10.  They were finishing a unit about powers of 10 and were preparing to take their unit test.  I challenged the students to figure out on what date that 1000th day would fall.  The classroom teacher said the students were engaged and talked about the problem for at least two days.

One student had an elegant solution.  He started with October 8, 2013 being the 100th day.  The student added 365 days to find October 8, 2014 and then an additional 365 days to find October 8, 2015.  From this date, he added by months until he reached 1,000 days on March 26, 2016.

When the superintendent next visited our building, he stopped in the classroom of this student and asked where the student would be on March 26, 2016.  Our superintendent then put a reminder on his phone to take this student out to lunch on March 26, 2016 to celebrate his 1,000th day.  What a great way to acknowledge the great mathematical thinking that occurred in this classroom!  Thank you to our new superintendent for encouraging such excitement about mathematics!

Using assessment to guide instruction

Assessment has been on my mind lately.  Part of my job is to be our building test coordinator and I have helped to administer on-line assessments in reading and math to all of our students in grades 2-5.  This has taken a tremendous amount of time on my end - and at least an hour of lost instructional time from each classroom.  I have concerns about how students interact with these tests, including:

1.  Students do not have much experience in "close" reading on a computer.  They typically have not had experience with needing to read ALL of the information on the screen.  Rather, they often play games that provide feedback for incorrect responses.  The online tests we are giving do not provide feedback as to whether or not they are answering correctly as they take the test.

2.  Students need stamina so they do not rush through the test.  Again, students have had experiences where the response time is quick on the computer.  In contrast to those experiences, when taking these on-line tests, students need to read and think carefully before choosing a response.

3.  Adaptive testing makes assumptions about what is "less challenging" or "more challenging".  The next test item the child sees is based on whether their previous response was correct or incorrect.  Who makes these decisions about the difficulty of content?  If a child answers an addition problem correctly, should they see an item that probes more deeply about their understanding of addition?  Or should they see a multiplication problem - where they may or may not make the connection to addition?

Any time students are spending time testing, they are losing instructional time.  If we are going to take time from instruction to gather assessment information, then we need to be sure this information can inform our practice and benefit students.  How much time are we willing to sacrifice for assessment?  How will the new Common Core assessments impact our instructional practices?

It is a lot to consider....

So, where do you find some rich mathematical tasks?

It can feel like a daunting task - finding engaging lessons that help students understand the conceptual foundations of mathematics.  I'm often asked where to find ideas for lessons.  One of my favorite resources is the Ohio Resource Center (ORC).

 www.ohiorc.org 

The ORC has more than just mathematics - it is also a great resource for language arts, social studies, and science!  However, mathematics is near and dear to my heart and when I use the ORC it is typically for math ideas.

http://www.ohiorc.org/for/math/

When you visit the Mathematics page of the ORC, you can click on the first link at the center of the page.  This will take you to the standards - and you will find both the "old" Ohio Academic Content Standards/NCTM's Principles and Standards as well as the "new" Common Core State Standards for Mathematics.

http://www.ohiorc.org/standards/math/

If you click on the Common Core link, it will take you to the ORC's resources that have been aligned for Common Core.  There are suggestions for activities to help develop the Mathematical Practices as well as content-specific lessons.  Many of the lessons come from NCTM - the National Council of Teachers of Mathematics.

A wonderful feature about the ORC is that all of the lessons that are on the site have been reviewed by mathematics educators in Ohio.  With so many resources available on the internet, I find that the ORC helps narrow the resources so that I am looking at high-quality suggestions for teaching.  It is also helpful to look at assessment items (from NAEP and others) that have been aligned to standards so that you get a sense of what sort of content is specific to that mathematics standard.

There is also a Mathematics Bookshelf and Problem Corner as well as some other features on the main page for mathematics.  If you haven't had a chance to explore the ORC, take time to check it out!

Trusting the kids....

In order to be a facilitator in a mathematics classroom, a teacher needs to be willing to give up control in their classroom.  This does not mean that "anything goes".  Rather, there should be parameters for how classroom discussions will take place with a focus on respect for all participants.  The teacher also needs to be willing to let the students' conjectures and thinking drive the conversation.  While the teacher has an ultimate destination in mind, the path to that destination is determined by the students and their ideas.

A beautiful example of this occurred last week.  A fifth-grade class was working on the idea of how numbers change when multiplying by 10 (or 100 or 1,000).  The teacher had students explore the magnitude of 1, 10, and 100 using base-10 blocks.  Students then predicted what it would look like if they were to have a 2-D version of 1,000, 10,000, and 100,000.  Groups used paper to build these models.  After building their models, the class had a discussion about their observations.  They built the rule that when multiplying by 10, a digit moved one place to the left.  This conjecture was tested using calculators.  When students were confident in the rule, they wrote it on the chart paper.  Students were eager to explain their thinking and were engaged throughout the lesson.

Rather than teaching the students to "add 0 when multiplying" (I'll need another post to talk about why that is confusing language for students), this teacher build upon students' observations and reasoning to help them develop the mathematical rule.

In order to facilitate this lesson, the classroom teacher had to trust that the students would discover the rule - and they did!  :-)

Partnering with Parents

Parents often want to be partners with schools and support their child's education.  However, these same parents might suffer from their own math anxiety and they are uncertain about how to help their child - especially since so much of the ways in which students are solving problems seem "new".  When responding to parents about "new" math, I explain that it isn't that the math is new but rather that we have learned more about how to better teach math.

One particularly confusing aspect to many adults is why students would work left to right when solving a computation problem when many of us were taught in school that you should start in the right column and then work your way left.  Research has shown that, when given the opportunity, a majority of  students invent strategies or algorithms by working left to right.  Encouraging students' invention of strategies and having them talk about their thinking provides an essential foundation as they move into expanded algorithms and traditional algorithms - allowing students to see the place value that becomes "hidden" by the traditional algorithms.

To help parents understand some of the ways that their students might be solving problems, I created a guide for parents.  Parents can refer to the guide as they help their students with math at home - and hopefully remove some of the anxiety parents face about the "new" math.

http://bit.ly/13x5LYD

I hope that in discussions with parents, you can be encouraging, help parents to be less anxious about mathematics, and forge strong partnerships to support our future mathematicians!

Classroom Conversations

I have been reading Caitlin Tucker's Blended Learning in Grades 4-12: Leveraging the Power of Technology to Create Student-Centered Classrooms.  It is an easy read and gives many useful ideas to use in your classroom.  While the focus of the book is using technology as a tool in teaching, I am struck by the suggestions she emphasizes that are just good teaching practices.  Many of her ideas can be applied to primary grades as well.

One of my favorite quotes so far is:
"Online work frees teachers from their role as the only source of information and feedback.  When students engage in dynamic online discussions and collaborative group work, they become valued resources in the class.  They ask each other clarifying questions, compliment strong ideas, provide suggestions for improvement, and offer alternative perspectives.  This also allows for improved student engagement and immediate peer feedback."

Tucker's description could certainly apply to online learning but it just as easily could describe the classroom conversations that should occur during our mathematics teaching.  Teachers can help students make sense of the mathematics and discuss them with students by acting as a "guide on the side".  Introduce a problem and allow students to work through the problem together.  Try to avoid answering questions directly - instead, ask other students in the group what they think or ask students how they might test a theory.  How will they decide if they are correct?  Where might they find needed information?

By having a strong understanding of the mathematical content to be taught, the teacher can guide students towards understanding through questioning rather than telling.  I have often told students it is not enough to trust something if it does not make sense.  It is important for them to continue thinking about it, asking questions, and trying until they have made sense of the topic in their own way.

Tucker provides great suggestions for how we might use technology to enrich these mathematical conversations - but regardless of whether the conversations are in person or online, it is important that the students are involved in making sense of the mathematics!

Differentiation Part 2

We watched this TED talk today at our ODE Network of Regional Leaders meeting.  http://bit.ly/1diEpei

I LOVED what Dan says and think the way in which he poses problems ties well with my previous post about differentiation.  When we allow students to be part of the process of creating the problem and engaging in the parameters of the problem, the problem will differentiate itself.  Check out more at Dan's blog: http://blog.mrmeyer.com/

Differentiation

Last week, I attended a training for our district's new vendor assessments.  While I'm reserving my opinion about the vendor assessments and the data we will obtain from the testing, I found a comment by the trainer to be particularly troubling.  The assessment provides a report with a number assigned for the student's skill level with mathematics that the teacher can then use to find activities to help differentiate instruction in the classroom.  My concern with this process is that the majority of the materials provided were worksheets with little instruction involved.  Differentiation is NOT providing more work for students but rather helping to tailor instruction to the learning needs of the student.

In my math classroom, I prefer that all students are working on the same problem.  A rich mathematical task has multiple entry points so that most students are able to begin working on the problem.  An excellent example is NCTM's House Numbers problem:  http://illuminations.nctm.org/LessonDetail.aspx?id=L225

It takes a few minutes to introduce the problem and be sure students understand any associated vocabulary.  Once I have made sure all students understand the parameters of the problem - for example, that the sum of the digits is 4 and that all of the digits must be different - they are ready to begin!  Because there are many correct answers for this problem, students are successful if they find one correct answer or if they find many.  The challenge is to find as many house numbers as possible.  Questions such as, "How do you know you've found them all?" and "How did you decide what numbers would work?" help students to reason and justify their mathematical thinking.  Students are practicing adding in a much more engaging way rather than solving math facts on a worksheet.  You can further differentiation by giving students a calculator to check the sums, pairing them with a student working just above their level, or providing a framework to help them organize their thinking (such as an organized list).  The idea is to see what students need before providing too much support.

The best part of this type of differentiation is that when you discuss the problem, all of the students were involved in the problem and can participate and contribute to the discussion!

Homework

If school has not already started for you, it is about to begin.  As a teacher with two busy children who both play sports, our family time is limited.  I dread the nights when the kitchen table becomes a battleground for homework.  I'm sure many parents feel the same!

Homework should be engaging for students, help connect families with what students are learning in schools, and be PURPOSEFUL!  Purposeful does not necessarily mean that you are practicing mathematical skills taught that day in class.  The purpose can be broad - to help students talk about mathematics!  Here are two ideas for homework that can help you engage students:

1.  Give students a problem that they will be solving in class the following day.  The homework is to read and begin the problem. By not requiring completion of the problem, you take away the pressure students feel for a "right" solution.  Some students will discuss the problem with friends and family members - and talking about mathematics is something we want students to do!  Students can jot down the ways they might begin the problem.  Again, the tentative nature of the assignment means that students don't have to worry about having the "right"answer.

When students come to class the next day, have them work in groups to share how they would begin the problem.  Give them sentence strips or index cards to record their strategies.  After a few minutes, stop the class and ask them to share some of their strategies.  You can group their cards into similar strategies.  If they wrote their name along with their strategy, students will know who is thinking in a way similar to them and who might have some new ideas to share.

Then students can begin working on the problem!  If they are stuck, they could try one of the strategies you charted or talk to people using those strategies.  Rather than spending time in class to have students understand the problem, they can come ready to begin the work!

2.  Find an interesting pattern or number and ask students to find out more about this number.  For example, the number 29 is special.  Why?

• It is a prime number.
• 29 plus 2X^2 is a prime number for every value of x up to 28.  29, 31, 37, 47, 61, and so on.

Students can discuss these patterns and test whether or not they work with other numbers.  Again, students are thinking deeply, making connections, and communicating about their mathematical thinking!

Have a wonderful start to the school year!