Concrete Learners: Hands On and Real Life…Every Day

In teaching division this year, I’ve never before used so many counters for so many consecutive days in a row.  I’ve got a core group of students who feel really great about division, some of them have even been memorizing multiplication at a very fast rate, which allows them to make better sense of division.  But I also have the exact opposite end of the spectrum as well.  As soon as they see that division symbol, their eyes glaze over and they become fearful of the problem. They worry about what to do and they think they cannot divide (even through they’ve been dividing all of their lives, they just haven’t seen the number sentence for it).

To help struggling learners, I’ve been trying to make it more concrete. Our youngest learners often need to see visual representations of numbers so that the concept is not so abstract.

In this set of problems, fruit was being divided equally into bags. I decided to lay out paper bags for this student.  The problem was 14 divided by 2, and he could easily solve it. He was both proud and excited to write his answer.  Win!

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Another student though, had a little more trouble. The problem was 8 divided by 4.  Because she had done 10 divided by 2 right before this one, she forgot to put away the 2 counters to start. She hit a wall very quickly and gave up. Instantly, she had her hand up for more help.

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To help her, we re-calibrated a bit by checking the problem again.  She very quickly realized that her counters started out wrong, and she was able to fix it. That check back to the problem is what I want her to do in the first place, great mathematicians do that without any prompting.  It was clear to see that she wasn’t connecting the number sentence with the manipulatives at all. It was a quick 1 minute conference on the importance of paying attention to detail/being precise, a math practice standard that many students struggle with. I told her that the next time she should try that strategy before asking for help.

Giving these quick teaching tips while conferencing with students makes WAY more sense when those tools are right there in front of them.  I used to try to draw on student’s papers, help them extend patterns etc… but the most struggling students need those tools in their hands to actually act out and see the problem. That is when I’ve noticed teaching tips given to them have become ultra powerful!



How Much is Enough? How Much is TOO Much?

I must ask the question to myself over and over again, almost daily. How much practice is enough practice for my students? Today I struggled as a math teacher. Writing this post hurts my heart a little.  I am wrestling with a new math series, trying to give it a try to have it be at fidelity while still balancing the needs of my classroom.

We all have learners that fall on the spectrum of different levels of understanding regarding the learning target. I am prepared for that daily.  What sometimes gets me though, is the way that our new math series assigns a number of problems to a student.  In one math lesson, the students are to solve 6 review problems, a large hands on problem, 30-35 independent practice problems from the book and a 20 problem homework page.

This is absurd in my mind for two reasons:

1.  The students who understood the learning target, certainly don’t need to do it 50+ times.

2.  The students who didn’t understand the learning target, absolutely CAN’T do it 50+ times.

This is what the book page looked like (only the bottom half):

Maybe no more worksheets?


We are only beginning division. Asking students to learn the concept of division, and then 3 days later have them divide by 8 and 9 is just crazy.  I know that as their math teacher, I can step in and give modified assignments, lessen the number of problems and the level of difficulty. I know that is why I am the professional in front of them, giving them what their brain needs.  But this isn’t just this particular lesson this particular day, it is just about every lesson, just about every day. Having to modify everything can be exhausting, and it makes those struggling students feel sad that they can’t do it all.

Why do math textbooks have this almost mindless repetition for students who get it, and then induce pure panic in those that don’t?  Watching students who struggle try to answer even one problem is heartbreaking, leaving them feeling broken when they see that there are 29 more to go.

This is why I’ve gone away from the traditional model of math, in favor of deeper problem solving and a more project based feel.  Watching student’s faces drain of the love of math is totally heartbreaking and I really can’t do it anymore. Math is an amazing, beautiful subject and I know I can help my students grow.  I can prove it with data, and I can prove that I don’t need to do it with endless worksheets.

Even with a new math series that led us to believe that things would feel different, I am realizing that being an effective teacher and lifelong math learner is truly the only way to help students grow. There is no magic textbook, no perfect program.  Instead, we must search and find the best tools, and tailor our instruction to every student at every level.

I hope you can share the best ways that you know how to do this with me (all of us!) as well.


When Number Sense is Missing: Extend Patterns

Sometimes math is like a stinking onion.  You encounter a student who is having a problem, so you peel back a layer to discover more misconceptions which leads to more and more problems until you have a big stinking mess and lots of tears. The good thing though, is a stinking onion can be cooked in a way that tastes delicious.

We are deep into our division unit, and many students are struggling with strategies to divide by 8 and 9.  These are much larger numbers that we are working with, so the most fragile students are even more sensitive as they calculate their answers.  Red faces, frustrated brows, and a lack of perseverance sets in quickly.

A student that was trying to divide 48 by 8 was really struggling.  She had already tried to solve it by making equal groups, she had tried repeated subtraction, she had tried to figure out the related multiplication fact, but she kept getting lost in the process. We decided to find the related multiplication fact together.

I asked her to try thinking of a friendly eight fact.  8 x 2 felt good, she was able to solve that.  She then jumped to 8 x 5.  She counted by fives until she had 40. I waited to see if she’d make the connection that 8 x 6 is just one more eight added to the group.  I waited, and waited.  Nothing. So I wrote it out for her…8 x 6.  I asked her, “What is 40 and 8 more?”


I mean, there was complete and utter silence.

I was actually speechless that she made it this far and didn’t know how to add 40 and 8! In a calm and non-judgmental way, I wrote out the following sequence on a dry erase board:

0 + 8

10 + 8

20 + 8

30 + 8

40 + 8

After she solved the first two, she immediately saw the pattern and we both breathed a sigh of relief.  I didn’t leave the school day feeling very good though.  This concern has been in the front of my mind all night. How is this child supposed to learn to multiply and divide when her number sense is so fragmented?

So many of us (I am guilty of this myself) have stressed memorization of math facts without any strategy or number sense behind it.  This moment today has convinced me that this type of practice MUST happen regularly for students, and those strategies need to be shared out loud often. Extending patterns is essential for students to become strong mathematicians.  It is part of the math practice standards and truly is an important skill.

Promote Number Sense in the Classroom: Extend Patterns

How do you promote number sense (or extend patterns) in your classroom?


The Half Game: Teaching Division Concepts Early

I wish I had more time to post about my 20 minutes of math play per day quest. If I had the time I would post what we’ve been doing each day, mostly so that I don’t forget what I’ve done!

Anyway, here is another one that happened kind of informally. The other day, my 5 year old daughter was staring at our window in the kitchen.

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She looked at me and said, “Mama, what is half of 12?”

Always in awe of her thought process, I answered, “Six!”

She immediately shouted, “You’re right!”

When I asked her how she knew this, she pointed at the window and said that there were six squares in the top half.  How she saw this when I only see a window, I’ll never know. Tiny minds are so fascinating!

So I pulled out some playing cards and asked her to figure out half of the pile I gave her. We started small.

Teaching division at a young age.

This one was way too easy for her. So I gave her a bigger pile.  Suddenly, once the pile was bigger, she was all mixed up about how to split it up. I gave her two cups, and told her to give the same to both of the cups.

Teaching division concepts early

Her little mind started turning and this is what she ended up with in the end!

Teaching Division Concepts Early

At this point we counted both sides to be sure they were the same.  Then I showed her how to skip count, so that we could use that shortcut next time.

Math play with a deck of cards is totally endless! We’ve been trying out all sorts of different things like:

  • sorting by number and counting them
  • sorting by color and counting them
  • sorting by shape and counting them
  • putting two cards together to find the total number of shapes
  • playing war to learn about greater numbers

I forget the power of playing games sometimes, and how it can help with number sense so much. I want to think of ways I can extend this to my third graders to help them with division concepts. We started division this week and it feels like a slap in the face to many of them. I plan to share some real life division with them before even showing any symbols and numbers.

Do you have any card games you play in class? I would love to hear them!

Here are some more awesome math teachers you can connect with to learn more about real life math:

Real Life Math

A monthly REAL WORLD math blog link-up hosted by 4mulaFunFourth Grade StudioTeaching to Inspire in 5th, and MissMathDork.