THIS is what I keep trying to say...!

I have continual discussions with teachers and administrators about our much-needed shift in our mathematics classrooms.  However, it seems to me that, too often, people get caught up in the shift in content from grade to grade rather than the real and necessary shift in our instructional models and methods.  Matthew Oldridge says this so much more eloquently than I ever could in a recent blog post. (well worth the 6 minutes to read!)

I especially like when he says, "Assume your students are all capable of high levels of mathematical thought."  It reminds me of the saying we see on Pinterest and Facebook all of the time:  "MY TEACHER THOUGHT I WAS SMARTER THAN I WAS, SO I WAS."

This statement of his also resonated with me:  "If you treat mathematics like broccoli, kids will probably try and avoid it. Treat it like the world’s most delicious cake. Stop pretending it’s just some medicine that must be taken. Make it all interesting. Yes, you can."

Yes, you can, indeed!

Another puzzle website!

I have posted before about the algebraic thinking mobile puzzles from Solve Me Puzzles.  They are great for increasing our students' understanding of equality, and they give even our youngest students some exposure to algebra concepts.

Solve Me Puzzles has another great puzzle called Who Am I?  These place value puzzles ask students to figure out the value of the number from the clues.  There is some similarity between these puzzles and our Everyday Math place value puzzles.

On both sites, the puzzles progress in difficulty.  I think both puzzle sites could be used in some way with students in 1-5.

There is a third puzzle available on the Solve Me Puzzles menu.  It is called Mystery Grid.  You might have students who enjoy it, too.  It has some similarities to Sudoku puzzles and KenKen puzzles.  (I don't find it quite as much fun as KenKens...)

These puzzles might work well as an opening or closing activity during your day, a partner activity, or a station during math workshop.  They will definitely help build our students reasoning, thinking, and mathematical skills!

Yes, he does know your number, but not your age!

Giving students opportunities to explore and work things out with others needs to be happening in our classrooms.  The best way to do this is for students to notice and wonder and explore their own ideas, but sometimes we can provide opportunities that allow them to take a risk at trying it themselves.  I have done this activity with students before and found it to be one that we all enjoyed!

Is it really mind-reading? Have your students participate in this sequence of steps by Rick Lax. Encourage them to write down the steps that he has the participant do.  

Give them cubes and ten frames and whatever manipulatives they are comfortable with...Can they figure out why he is able to figure out everyone's number?  Let them collaborate with others as they explore the sequence of this exploration.

A next step might be for them to see if they can write their own "Read your mind" puzzle.

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!

Prove it!

I have really been working with my Firsties groups to teach them how to "prove" their answer.  They need to prove it to themselves first through either manipulatives or a representation, and then be prepared to share their steps so that they can prove it to someone else.

Today's task was a pretty basic one, but I find that those are important for them to do as they grow their ability to explain to each other what they did and how they know.  Today I asked them to find the sum of 6 + 8.

The variety and levels of their strategies was, as always, quite interesting...

This little one counted out 6 green and 8 orange.  She needed to think about what to do with them, but she then realized she needed to put them together.  She then proceeded to count all to get to the sum of 14.

This guy counted out 6 and 8.  He organized them into a line and then counted all.  His counting was a bit easier because of the line and the differentiation between the 6 and 8, but he still counted all of the cubes to get 14.

A similar approach with the counters...this little guy organized his 6 yellow and his 8 red.  He began with the 6 at the top and then counted on the next 8 to get 14.

Here was one who grouped his 8 and 6. He then started with the 8 and counted on the next 6 to get 14.  He decided to build the plus sign while he was waiting for the others.

There was some organization in this guy's work.  He grouped his 8 and 6 and then counted on from 8 to get his 14.  While he was proving it to the group, one of the others,  who had used a much more mottled strategy, blurted out, "It's a ten frame!" We took the time to explore how it was like a ten frame, but not quite, so we moved two yellow to the end of the black blocks to make our ten.  They immediately saw the 10 +4 and called out, "14!"

Now it was my turn.  I started with a stack of 6 and a stack of 8.  This time, however, we broke two off of the 8 to identify that each of the stacks was now worth 6, and the students called out, "6 + 6 =12." They then knew they needed to add the two back to get the 14.

This little activity doesn't take us long, but it shows me so much about the students' thinking and levels.  They have all been taught in the classroom to "start big and add on."  However, when working independently, they don't all do that.  When we look at each other's strategies, we identify how it works for each person.  We identify what looks the same and what looks different.  I model and discuss the way to organize our blocks so that they are easier to total.  In the end, however, we have a discussion about the fact that there are many ways to get to the same place.  We all need to use what makes sense to us.  

I was glad to see when I was in one of these student's classrooms today leading a 3 act task that one of the students that I see regularly was organized in her representation of the problem.   We were able to share her representation with the class and discuss how the organization that she used helped to make finding the sum a little easier.  

They are 6 and working to make sense of it all.  😃

Odd and Even: A Visual Exploration

Sometimes we need reminders about how important some things are.  I know our math should be visual, but sometimes, when you see someone teach something visually, it hits you upside the head!  How had I not seen that before?

I recently participated in a great webinar with @MonicaNeagoy entitled "Planting the Seeds of Algebra."  It was a very interesting and informative webinar.  If you would be interested in listening to her ideas on ways we can make our math come alive for our students and prepare them for increased math connections as they progress through math curriculum, you can access the webinar here.

Early in the webinar, she discussed an exploration she does with students where she discusses odd and even numbers.  She had the students create snap cubes models to match the story of double-decker buses and car trailers she was talking about.

6-A double decker bus with 3 windows on each level

3-A car trailer with the cab and two cars on back

After the models are created, students sequence them

And a pattern begins to emerge

Students then looked at the models, which were visually clear in the difference between odd and even numbers.  Students were then encouraged to continue their thinking about odd and even numbers by exploring the sums of these numbers, and hopefully finding and testing some common patterns in these additions.  Again, Ms. Neagoy goes into much more detail in her webinar, if you are interested.  (It is worth your time!)

EVEN + EVEN:  4 + 8 = 12

EVEN + ODD: 2 + 9 = 11

ODD + ODD:  5 + 11 = 16

Well--this discussion and model got me thinking about a similar exploration for older students.  Using the same odd/even models, what happens when you multiply by an even number an even number of times or an odd number an odd number times?  What can our students determine through this exploration?  How will it change as the numbers increase, or will it?  Can students determine and odd/even rule that accompanies the multiplication of whole numbers?  

EVEN x ODD:  4 x 3 = 12

EVEN x EVEN: 4 x 6 = 24

ODD x EVEN: 7 x 4 = 28

ODD x ODD: 7 x 7 = 49

What a worthwhile way to spend a class period!  The primary version of the odd/even exploration and the older student exploration may be subjects that you discuss in your classroom already, but do you make a point to do it in such a VISUAL way?  And are your students figuring this out on their own, or are you telling them the patterns that exist?  

The shifts in math instruction show us that students will learn and understand better if the math is visual and open.  Taking these basic ideas of instruction and creating a visual tool for our students to use to figure concepts out on their own are steps that we want to be taking as teachers

Let me know if you try this with your students and how it goes. I would also love to hear about other odd/even explorations that you have found for elementary school students!