Put Math in their Hands

I often talk to teachers about the importance of concrete representations for our students.  The use of concrete tools helps all students to gain a better understanding of the function at hand.  It also creates a visual representation that will be a reference for students to use as they advance into more advanced mathematical ideas.

In this post, I want to share some simple concrete representations that teachers can help students develop as they work to make sense of a new concept.

Doubles/Doubles +1: Both the rekenrek and ten frames help students to make more sense of what a double is and how we build from that to doubles + 1.

Doubles on ten-frame:  3 + 3 =6

Doubles + 1:  3 + 4 = (3 + 3) +1 =7

Doubles on rekenrek: 6 + 6 = 12

Doubles +1:  6 + 7 = (6 + 6) + 1 = 13

The rekenrek is a great tool for doubles facts between 10 and 20.  Student can see that all of the green beads have been pulled over, which is 10.  They only need to add on the white beads.


Addition/Subtraction within 100:  For this skill to take hold, I find beaded number lines and base ten blocks to be good concrete tools to use because they carry over so nicely into an open number line representation.  Open number lines are important for students to use to build number sense.

Addition on Beaded Number Line:  17 + 35 = 52

Addition with Base Ten:  17 + 35 = 52

Subtraction on Beaded Number Line:  44 - 18 = 26

Open number lines are important, but they are often too abstract for many of our students.  Combining the use of the beaded number line or base ten blocks with the creation of an open number line helps for it all to make more sense for the students.

Multiplication:  For learning multiplication facts, creating arrays is a nice way to start.  This should help students to see how repeated addition is connected to multiplication.  Any small object will work to help students create arrays.  Using the number card templates will allow them to create a visual more quickly at times when they need a visual but don't have time to build an array.

Multiplication with Base Ten Blocks:  22 x 23 = 506

The 22 and 23 were created at the top of this image and on the left side of the problem using base ten blocks.  Multiplying a blue ten and anothe blue ten gives you an orange 100.  The same continues until we end up with 4 hundreds, 10 tens, and 6 ones for a total of 506.

For multi-digit multiplication, building the array using base-ten blocks works well to connect the arrays that student made for facts under 100 with the longer problems that they are ready for now.  Besides--the base-ten blocks transition nicely into graph paper representations and finally into the area model of multiplication for multi-digit numbers.

Decimals:  The beaded number line can be used for addition and subtraction of decimals less than 1, and it can also be used to round and compare decimals.

Rounding on the beaded number line:  0.86 is closer to 0.9 than 0.8

Adding decimals on a beaded numer line:  0.6 + o.14 = 0.74

I hope that you find some of these ideas helpful and that you find ways to incorporate them into your explanation of numbers with your students.  There are many ways that can be used for different problems.  If you have a concrete tool that works well for teaching these skills, don't think you have to switch to one of these--do what works best for you and your students.  The key is to get the concrete math in their hands!  This way it will stick in their heads.  😊

What do I do with all of these ideas?

I recently completed providing some professional development to teachers in our district.  At the end of the final session, I asked teachers to come up with some goals of things they would like to try.  Hopefully, they will get a chance to try it now before the end of this school year instead of waiting until next fall.  Here were some of their takeaways:

Goals are important as they help us and our growth mindset. It is good practice to after a training to take one big idea back with you to help you grow as an educator! 

Of course, while participants found resources and ideas to put into their classrooms, questions remained.  These are some that were left on my question board at the end of the session:

How do I fit all of these ideas into my workshop along with my activities?
  With everything that a teacher has to fit in during the day, I see where this can be a concern.  I understand how difficult it might be to get in your lesson each day.  Some of the activities/resources that I have shared with you might be good opening activities in lieu of or in addition to the mini-lesson.  In addition, once you have taught the routine, the activity might be an activity that students can complete independently or with a partner.

  In all, we need to be intentional as we plan our lessons to determine what pieces of our text we want to use--maybe doing all of the math journal problems won't be necessary?  Maybe students can practice a few of the problems instead of the whole page, and that will leave you extra time for some of the extra resources you are getting.  I might also recommend that a technology station that students use, like IXL, is not necessary every day.  The rich problem solving activities and thinking routines will provide them with far more realistic practice than the rote practice of problems that some online sites might offer.  Mix it up!  Every day doesn't have to look the same, and building our students' conceptual understanding and problem solving skills is worth finding the time for.

  If you continue to struggle finding ways to make it work, consider a coaching cycle or meeting to help you wrap your brain around all of your options.

How do I use the number rack and beaded number line whole class?  How can I teach them when and how to use that tool?
  The rekenrek and beaded number line are great tools for our students to use to make sense of problems.  We should encourage their use at all elementary grade levels.  The more that these tools are used as part of our instruction, the more likely that the students might be to use the rekenrek independently.

   There are many online resources to help you better understand the use of the rekenrek and beaded number line.  
Here are a few:  Rekenrek Activities 
                              The rekenrek as a visual model  
                              Top 5 Rekenrek Activities
                             Beaded number line for 3rd grade
                             Beaded number line for 4th grade
                             Beaded number line for 5th grade

How do I set a growth mindset in math? I have kids that give up before even attempting a problem.
   Great question! I believe that setting this tone from the beginning of the year is key.  Jo Boaler's youcubed website  has some great resources about this topic.  If you look through her Weeks of Inspirational Math activities, growth mindset is prevalent, and it will help guide your classroom learning.  She also has other resources available that focus on the growth mindset.

   This topic has many resources available online with just a little searching.  Certainly, praising effort over intelligence is a key change that can have a big impact!

If you look at the word cloud on the right side of the screen, you may notice that I have written about these topics in the past as well.  Maybe following one of those links will give you other ideas or resources, too.

I enjoy providing PD for our district, and participant feedback is important to me.  I want to be sure that I am meeting the needs of our teachers.  I hope some of these resources will help!

Dr. Nicki and Beaded Number Lines

If you have worked with me in the last year, you know that I have a great love of the beaded number line.  This tool was introduced to me last summer by Dr. Nicki Newton, and I have found them to be a good concrete manipulative for students to use to gain a better understanding of math concepts.

 There are ways to use these number lines beyond just counting, and Dr. Nicki has created some videos pinpointing some uses at various grade levels.  (She had one for 3rd grade, too, but it has been taken down right now.  If it (or any others) comes back, I will add it to this post!) I would highly recommend watching her ideas in the videos below as well as making some for your classroom!

General Info about Beaded Number Lines:


4th grade:




5th grade:




These are not difficult to make, but they can be a bit tedious.  If you want a class set, you might consider asking some parent volunteers to split the job.  If you teach students in 2-5, it might be a good beginning of the year activity.  It might be a test of patience, but it might also teach you something about your kiddos, too!  

Let me know if I can be of any help!

Concretely, Pictorially, Symbolically

I had the greatest time seeing Dr. Nicki this summer! She really is a wealth of knowledge, and I got a bajillion great ideas and resources from her.  If you are unfamiliar with her, here is a link to her blog.  She has written lots of books about teaching math, and she just really is ALL THAT.

Anyway, one of the big ideas she left us with at this conference was the fact that students need to learn their math (we focused on facts) in three stages: concretely, pictorially, and symbolically. We often spend so much time on the symbolically part that we forget about the other two stages!  However, these stages are key for ALL grade levels! All kids won't stay in the same stage for the same amount of time, but it is our duty to be sure that they know how to do it and what it means.

For example 2 + 4:
CONCRETELY:

PICTORIALLY:

SYMBOLICALLY/NUMERICALLY:

2 + 4 = 6

I know that I was guilty of glossing over or skipping some of these steps sometimes--esp. with my higher math kiddos. However, research shows that students need to be able to understand each of these steps to gain a deeper understanding of what is happening,  Some students may need to stay in the concrete stage longer than others, but all of our students should be able to explain what is happening in the concrete and pictorial stages.  They may move more quickly to the symbolic stage, but they need to be able to explain how it all works,  Students who just have math fact recall are like students who are good decoders in reading.  They can say what you want to hear, but they have no comprehension of what is going on...

Thankfully EM builds these stages in regularly for our students.  We are so lucky to have such a great resource.  

Dr. Nikki shared a myriad of tools to help us build the concrete part into our lessons.  One thing she had us make was a beaded number line.  It would be a good thing for you to put into your first weeks of school.  Many grade levels would be able to make it independently.  If you don't think  your kiddos could, you could ask for parent volunteers to make them or to help the students make them.  This could be used for all grade levels. Two colors of pony beads and a strong piece of string for each child is all you need. :)

This video has some good ideas for you on ways to use it.

Using a Beaded Number Line from Karen Richardson on Vimeo.


Here also is a virtual beaded number line that you could use along with the kids' own numberlines.  It has some great features, too.

http://mathsframe.co.uk/en/resources/resource/69/itp_beadstring

I'd love to hear how you use these number lines in your room, or let me know if I can help you with ways to put the beaded number line into your lessons.