Monday, August 13, 2018

Virtual Concrete Manipulatives

We know how important concrete manipulatives are for our students as they build a deeper understanding of mathematical concepts. Being able to see and manipulate objects enables students to visually represent problems, see patterns, and make connections.

Hopefully, you have a variety of manipulatives in your room that students can easily access at all times.  Virtual manipulatives are important to have easily accessible, too!  Now is a good time to download some of these apps onto your iPad, bookmark on your computer, links in your SMART Notebooks or PowerPoints and/or add to your Symbaloo!

There are a lot of great FREE virtual manipulatives out there, but today I want to focus on the apps by Math Learning Center.  I have good success with many of them. The site has a lot of other good resources, so when you have some time, check them out as well!


Fractions: This app not only allows you to create bar or circle models of fractions, but it also to layer fraction models to see if they how they compare to one when you add them.
3/5 + 1/3 < 1
The app allows you to place number models or write directly on the screen as well. Take some time to explore the capabilities of this app!


Vocabulary Cards: Another great app that you can probably find multiple uses for! There is a large database of words in this app divided into grade bands K-2 and 3-5.  You can choose to see them all or only certain words.  Each card has 3 parts: the word, examples, and a definition.  You can choose which part you want hidden.  You can also choose the language for the card. I can see these being used as individual review for students, but I can also envision one of these on the board as an intro activity or even a quick exit task.

Money pieces:   The money pieces app allows you to display money with or without the accompanying base ten blocks depending on student needs.  It also has a variety of tools similar to the games Bears in the Cave or Pennies in the Hand where you can put your coins up and then hide some.  For example.  I have a pocket in the above screen.  If I told you that I have 35 cents altogether, can you tell what I have in my pocket?


With the click of a button, I can remove the pocket to show that there is a quarter in it.  You can do this similarly with a hand or a bank.  

Money seems to be difficult for children anymore because they have less interaction with it than we did. Our students don't get the same opportunities we did to spend cash, but it is still important to understand, and this app will give them some basic experiences with it.


Number Frames:  This app not only allows the important 5 frame, 10 frame, and Double 10 frame, it allows you to customize the frame you want up to 120. As with many of the apps, it allows you to enter number sentences and to write on the screen.  This app is not just for Littles!  Teachers in 3-5 can use it to explore place value as well as patterns in multiplication.


Number Lines:  I love this app!  It has so many options for you to customize the number line including fractions, decimals, hidden numbers, hidden tick marks... I think it really helps students to represent their work, and it marries nicely with the beaded number line for moving from concrete to representational.


Number Pieces:  There are two versions of this app.  The one I am showing above is more advanced than the version called Number Pieces Basic. It is base-ten blocks, but you can break the large pieces apart to show number relationships. You have choices in color and orientation of the pieces.  Again, you can enter number sentences or write on the screen.


Number Rack:  You know how much I love this one!  It comes in handy as a teacher model on the SMARTBoard as students manipulate their own Rekenreks.  It is customizable by sets of 10 up to 100. It allows for teacher annotations like the others, but it also allows for teachers to hide beads as below.  Well worth your time to explore this one--not just for Littles!

Do you know how many beads I have hidden?



Pattern Shapes: It is so important that we give students time to play with pattern blocks!  This offers a blank mat for students to create their own patterns.  It also has templates (as above) for students to fill in with shapes.  (Very good for sharpening visual skills)  For older students, it has two different grid backgrounds to allow exploration of area and perimeter.


Geoboard:  The geoboard app has different sizes of geoboards and allows for customization in many ways.  Another great one to have up on the SMARTBoard as students manipulate their own geoboards.



Partial Product Finder: While still in Beta form, this is an awesome app to help your students better understand partial products as well as the distributive property.  You can customize the rectangle up to 30 x 30 and then decompose one or both sides.  The matching equation shows up at the bottom of the screen. I have blogged about this one before--it is awesome!

I highly recommend that you take some time to explore these!  Hopefully you will find some that you make available for student use just as you do other manipulatives.  Maybe you'll find ways to use them within your instruction.  Whatever works best for you and your students!  

Let me know if you have other virtual manipulative apps that you would recommend!



Saturday, August 4, 2018

Culture-Building for the New School Year

How do you plan to transform the culture of math (teaching and learning) in your classroom or building this year?  Have you had a chance to think about this?  It can be easy to get caught-up in the content that will be covered and forget the importance of building strong mathematical communities in our classroom.  However, the time spent building your students into teammates in math will be worth it.  Go slow to go fast.



Let's think about Ron Ritchhart's Cultural Forces that Define a Classroom and how they can impact our building of a math community.

1.  Physical Environment:  Is your classroom arranged in a way to promote collaboration?  Are the spaces clear where students can gather? Have you thought about the places where students will be able to visually share thinking?

2.  Interactions & Relationships: What steps can you take to build a feeling of respect which will allow students to be confident enough to share their ideas and strategies?  What will you do to be sure all students and their ideas are valued in your classroom? How will you encourage collaborative inquiry for your students?  What methods will you employ to build  a growth mindset in students? How will emphasize growth and celebrate success?

  • This sample chapter from Thinking Together: 9 Beliefs for Building a Mathematical Community by Rozlynn Dance and Tessa Kaplan supports these concepts.
  • More ideas can be found in this sample chapter of Count Me In!: Including Learners with Special Needs in the Inclusive Classroom by Judy Storeygard.

3.  Expectations: What are the cornerstones for your mathematics community?  As your class determines the classroom norms, which ones do you feel MUST be part of the list? How will these be enforced?


4. Time:  How will time be structured during your workshop?  What will you put in place so that students don't feel pressure to work through concepts quickly? On the other hand, how will  you build efficient use of time for your students?  What will you do to be sure you offer enough thinking time for students?  In what ways will you support your students to show perseverance?

5. Language: What do you think is the key mathematical language for your students to learn during the year?  What will you do to build their mathematical discourse?  What language will be modeled for them to use when working with partners or small groups?  Will your classroom contain a word wall or other location where students can easily refer? What growth mindset language will you be sure to include?


6.  Routines & Structure: What daily mathematical routines will you put into place, and how will they help build mathematical discourse in your classroom? How will your math block be structured?  What management routines will you have in place to help your classroom run smoothly?

7.  Opportunities:  How will you regularly opportunities for all students to interact with rich math tasks? In what ways will students grow in the math practice standards as well as the mathematical content standards?  What types of explorations and problem-solving will you use in your classroom?  Will they promote perseverance in your students?  How will students be encouraged to find and explore their own mathematical questions? 


8.  Modeling:  How will you model creativity and risk-taking?  In what ways will you provide examples of collaborative talk and respectful debate? How will students know that this is a safe classroom to take risks? What will you do to share your own wonderings and questions with your students? How can you be intentional about modeling perseverance?

How can you take risks this year in order to grow as a math teacher?  What resources can you use to help you to learn more about best practices in mathematical instruction? Using these 8 ideas as a starting point should help you on your way!






Monday, July 23, 2018

Looking for Explorations/Investigations to do with your class?



Explorations and investigations help your students to take ownership of their own learning and are a great way to get students excited about math!  They encourage critical and creative thinking skills.  They build a sense of math community, and make us all better mathematicians.

If we get started with predetermined explorations, and our students become comfortable with the format, they may begin to come up with their own explorations that you can embed into your instruction!  

I have posted about a number of explorations either that I have done with students or have seen others do. However, I wanted to remind you of some good places that you can go to find an exploration that works for you and your students!

WIM:  The weeks of inspirational math from Youcubed are all set up and ready to go for you.  I have blogged about them before, and I can't say enough about how they not only encourage a growth mindset, but also that they are a lot of fun!

Math Solutions:  This location is full of exploration options for you.  Many have a great connection to literature.

Math for Love:  The free lessons on this site typically involve investigations.

100 Numbers to get students talking: This task has step by step directions and examples of how to use it to build your students' group work abilities this year.  

Finally, I have blogged about some different explorations that you could try in your room.  You might find one that will be a great review or introduction for your students this year.  To find the blog posts, look over at the right side of my blog at the labels.  Click on the explorations label, and it will show you all of my posts about explorations.


If you have other great explorations to share, please post in the comments below.

What a great way to start your year of math learning!  Beginning with some explorations will give you plenty of time to get to know your students, build your classroom culture, and develop your routines.  Let me know if I can help in any way!

Monday, July 2, 2018

Own your mistakes. Strive to be better.

I recently had this conversation with my 16 year old son as I investigated what had happened to our mailbox. I didn't even need to see the ding in our new second car's hood to have a pretty good idea what had happened...

My frustration in this incident was not that he broke the mailbox,but rather that it took him so long to admit that he had.  After an admission, we discussed what had caused the accident, and ways we could work to help it not to happen again. 

Owning our mistakes helps us to grow.  As I have transitioned from the classroom to my current coaching role, I have had the opportunity to put all of my focus into math educational research and resources.  I learn something new most days, but one of the things I have learned that I must do is own my own mistakes as a teacher.

I have read dozens of math books in the last two years, but three have really had an impact on me.


These books have helped me to better form in my mind the necessary shifts that need to happen in our classrooms to help all of our students truly grow as mathematicians. As I read each book, I recognized myself--not always in their examples of best practices.  I saw all of the mistakes I made as a classroom teacher.  Over and over again. It hit me hard.  

Thankfully, I also saw myself in some of the good practices, but that did little to make me feel better about the ways that I felt I had failed my students.  After owning these mistakes, I identified the causes (fixed mindset, poor models, little time to keep up with best practices...) and now I am working to help it not continue to happen.  

I talk often with teachers about the shifts in math instruction that need to happen, but I find it hard for most of them to understand and visualize what this should look like. I continue to share these books with teachers all of the time, but I know that not all of them have the time or desire to read them (especially those who don't see themselves as mathematicians).  I get it.  

So--I have set some goals. My first goal is to put the concepts from these books in front of teachers through the PD we offer.  Not just workshops and presentations--but also through preparing PLC modules that will highlight some of these concepts in ways that teachers can try in their own classrooms and then discuss with their colleagues.  

My second goal is to continue to grow my own understanding through practice, practice, practice! (and not practice of 25 problems on a page...) As I go into classrooms to help or lead a lesson, I am going to do all I can to model the ideas I have read about in these books, discussed on Twitter, and witnessed in classrooms and webinars. I know that I will continue to make mistakes, but I also know that these mistakes will only help me to grow stronger in my practices.

Finally, I know I still have plenty to learn!  If I want to make changes that benefit our students and teachers, I need to continue to push myself in my own learning. Through reading new books, participating in Twitter chats, and attending math conferences, I believe I can continue to learn more and grow as a teacher and mathematician.







Monday, June 25, 2018

Using Counting Collections in the Classroom

Have you tried counting collections



Counting Collections: Kindergarten - a common core classroom friendly exercise from Luna Productions on Vimeo

This activity is a great opportunity for our primary students to gain a better understanding of counting and number, and with some modifications, I think it could be used, at some level, in the upper elementary classroom as well.

You will need to begin by creating some collections.  In the link above, they give some examples of objects you can gather for counting, but I'm sure you can find other items around your home or classroom that will work as well.  You will probably want them to be smaller in size so that storing them doesn't become much of an issue.  Hopefully, you can find other teachers in your classroom to join you, and then you can find a common space to share your different collections. This handout will also be able to help guide you as you plan for your collections and fine-tune the activity.

I think that these would be a great way to kick off your math habits to start the year.  You would be able to learn a lot about your students by interviewing and talking to them as they work on organizing and counting their collections.  

For older grades, I have thought that you could have them count objects in multiples or fractions to get a total.  You could also have them count by sets.  Packs of items...Can they count by 24 or 36? What if you offer them decks of cards? Can they count by 52s?  While they may not be fluent as they move through these unusual multiples, it will certainly aid them in developing mental math strategies.  Introducing fresh concepts through counting offers all students an access point, so students can practice new concepts using the math routine of counting.

Here is an example of a Counting Collection in a 3rd grade classroom:
Counting Collections: Third Grade - a common core classroom friendly exercise from Luna Productions on Vimeo.


Counting collections can adjust as your students develop their number sense. It is a routine that will allow students to think about better ways to organize, more efficient ways to count, and concepts of number. It can be used all year long.

What do you think about this?  How can you make it work in your classroom?  Please share your ideas!

I'd love to join in on the fun as your class does a counting collection!  If you are okay with that, let me know when you think you would like to do one, and I will see if I can join you.


Monday, June 18, 2018

Rethinking Homework


As we rethink the role that homework is playing in our students' learning, we should look at new ways to have our students practice their learning rather than just a set of practice problems each night.  

One idea to consider is a math reflection question for the students to respond to.   Building their metacognition through these written responses has proven to build better mathematical thinkers.  

You might decide to have a menu of questions for students to answer, or you might begin by having them all answer the same question--whatever works best for your students.

Click below to a link of some potential questions.  Reflecting on these ideas each day will help to build the types of thinkers we hope to cultivate.




Friday, June 8, 2018

Building Understanding and Developing Culture


I have done this problem with first graders for the past couple years.  The first time I offered it to students, I was surprised by how difficult they found it (due to the majority of my experiences being with older students).  When I did it subsequent times, I offered a lot of "up-front" guidance to help them think about it, and I highly encouraged them to use manipulatives and images to help them make sense. However, they still struggled...

I have decided that if  I work in a first grade classroom next year, we will again do this problem. I have been pondering some different ways to approach it.  

I think it will make a great numberless word problem that we can do with a slow reveal in hopes that students work to make sense of it rather than just trying to solve it. Using the slow reveal will offer a great way to directly instruct the students in the process of notice/wonder and making sense of the problem before trying to solve it.  This use of numberless problems should help them as we continue to build understanding throughout the year.

Doing a general overview as a class, offering the students manipulatives and encouraging them to work with a partner to solve it.  Once they think they have solved it, they must find another pair who has a solution, and each group must convince the other that their solution is correct.  I like this idea, too, as it should help to build mathematical culture int the classroom--collaboration, sharing methods, convincing others of our solution... I think that the deliberate teaching of convincing others with math talk will need to come before this lesson, so that both sets of students don't just say the answer they got and move on.  

I could provide a picture of two spotless ladybugs and then have the students work to put the spots on to match the words in the problem. This one seems very direct, but it does emphasize the importance of using visuals to help us make sense of the problem. Again, I think partner work is a great way for them to approach this.

I think the sharing out of this problem is important and recognizing the different ways that students went about solving it. Determining as a class what we think the correct answer is and acknowledging how we worked with our partners to solve it.  Maybe a gallery walk of our thinking?  

A possible follow-up activity could include students writing their own kind of problems like this and then switching problems with others.

This problem is written with 1st grade in mind.  Certainly, this type of problem could be adapted for different grade levels:
* I have six pieces of candy in my purse.  There are four more pieces of gum than mints. How many pieces of gum do I have, and how many mints do I have?
* We have 27 students in our class.  There are 5 more girls than boys in our class.  How many boys and girls are in our class?
* There are 114 vehicles in the parking lot.  There are 72 more vehicles with 4 wheels than vehicles with 2 wheels.  How many vehicles have two wheels?  How many vehicles have four wheels?
* In my closet, there are 18 shoes on the floor. However, there are 2 more right shoes than left shoes.  How many complete pairs of shoes do I have in my closet?
* I found thirty-five coins in my car.  There were four times as many pennies as all of the other coins combined?  How many pennies did I find in my car?

What do you think?  How might you approach a problem like this with your students? Do any of the ideas I shared seem to be better than the others?  I'd love to hear your thoughts.