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Concept or Fluency for the Win?

March 15, 2017

This year my students have been having a blast learning their multiplication facts. They compete against themselves on mad minute tests, graph their progress, and celebrate moving from one set of facts to the next. They work through a binder full of challenging multiplication puzzles and games. They’re having fun NOW, but before they could do that, we worked for weeks on developing conceptual understanding of multiplication. This class has me wondering: when it comes to basic math facts, which is more important: conceptual understanding or fluency?

Concept for the Win

One thing I know for sure this year: my students have a solid understanding of multiplication. They can solve multiplication word problems, draw pictures to show their understanding, use different problem-solving strategies, and have great discussions with their peers about multiplication concepts. I am proud of them for the work they have done, and I love listening to them explain their thinking. I am especially happy that they seem to like math when they’re working on multiplication! Since I know their understanding on state testing will be measured through word problems and not basic algorithms, I am a happy teacher. This is why concept wins over fluency.

Fluency for the Win

But we are teachers, not testers. As much as we want them to do well on that state test, our students need to memorize their multiplication facts, and the only way to do that is with practice. Outside the classroom, their math skills will not be put to the test through complex word problems. They can’t draw a picture to solve every real-world mathematical dilemma. Their employers will not wait for them to decompose bigger numbers in order to make it easier to solve problems. This goes for basic addition and subtraction as well. When I see students using their fingers to count and solve 7 + 8, I cry a little inside. Real world application: this is why fluency wins over concept.

And The Winner Is…

Conceptual understanding and basic fact fluency are equally important! Even taking standardized testing out of the equation, students need to have an understanding of multiplication. It’s important for them to know that 5 x 7 represents five groups of seven items. I also believe that, once they understand the concept, it makes them even more motivated to memorize the fact tables. They can connect the facts to prior learning and real-world application. Fluency and understanding have to lean on each other to create the strongest knowledge base for our students.


The Calculator Controversy

March 3, 2017

Recently I reached out to one of my best friends who happens to be a professor of Mathematics at a university in Iowa. Even though we teach at different levels, we have been able to use each other as valuable resources in the past. To some extent, a student is a student, and we can always learn from each other. This time, I reached out to her to ask what trends and challenges she is noticing with her math students. She forwarded my question on to some other instructors in her department, and I was excited to read their responses. One that came up: college math professors are finding that students are “too dependent” on calculators, and are unable to complete basic computations by hand. This got me thinking about my 3rd graders…

The Easy Button?

Here in Tennessee, students are able to use a basic 4-function calculator on a portion of their state test. This wasn’t the case when I was an elementary student, but the test and the curriculum have changed since then. The test is more complex and it is also timed. I am sure that having students complete all test items by hand would take more time, and the last thing we want is for students to run out of time to show what they know! Are calculators the answer? Maybe. As teachers we want students to develop a conceptual understanding, but we also know that students are growing up in a technology-heavy world. We can’t fight technology- and we shouldn’t- we just need to find its place in our math instruction.

Balancing Act

So, how do we find the balance? Maybe splitting the test into 2 parts, calculator and non-calculator, is the way to go. I think the key is teaching students to know when they need to use a calculator and when they should be able to use other strategies to solve problems. Taking the time to ensure students have the conceptual understanding of the skill is important before showing them how to solve problems using a calculator. In my classroom, I do not pull out the calculators until we are doing test review. We look at problems first and decide if we need calculators or if we can solve the problem by hand. We even play games like Beat the Calculator to practice proper use of the tool. The novelty of the calculators has worn off, and most of my students are able to determine if they need one or not. I always encourage them to use the calculators to check their paper-pencil work.

Do you use calculators regularly in your math classes? I would love to hear about how teachers tie them into their math lessons, and what pros and cons you’re seeing as a result!

Featured Math Organization

Math Toolboxes

February 20, 2017

Do you use math toolboxes for your students? If so, you know how wonderful this resource is for students of all ages! I have used math toolboxes in each grade I have taught, from Kindergarten to 3rd grade. These are great to have in class, and you can even help parents and students create math toolboxes for home!

Here are some things I love about math toolboxes:


I love, love, love anything that helps to keep my classroom looking neat and organized! The small pieces can be kept in Ziplock bags and stored in cheap dollar store containers, then stacked neatly when not being used. You could have students keep them in their desks, but my students have so much stuff in their desks already, and we don’t use the toolboxes on a daily basis. Plus, storing them in a cabinet or cubby eliminates the chance of lost or damaged materials…not that anything ever disappears inside a student desk, right? ☺

Student Choice

I love that students can choose whatever manipulatives help them make sense of problems. This gives all students the freedom to solve problems in ways that work for them. One of my favorite things about letting children explore math tools is watching them use different tools in ways that I would have never considered! It is so fun watching them work out strategies and find new ways to use the various math manipulatives.

Home/School Connection

Your students can easily build a math toolbox to keep at home to help them when they are working on homework or extra practice. Simple household items like egg cartons, dried beans, toothpicks and paper plates are great for home toolboxes. If you’re fortunate enough to be able to provide 2 sets of things such as base ten blocks, play money, and measurement tools for school AND home- that’s wonderful! But you don’t need fancy or expensive manipulatives to make an effective toolbox!

So, what goes in a math toolbox?

It really depends on your grade level and your standards. I would always include items for place value practice no matter your grade level- this is a skill that students are always going to use. Here’s a list of some things that can go in your math toolbox:

  • Base ten blocks
  • Small cups (great for working on equal groups in younger grades)
  • Judy clock (or a paper plate clock with a bobby pin or paperclip for hands)
  • Dried beans
  • Popsicle sticks or toothpicks
  • Unifix cubes or any 1” cube/tile
  • Egg carton
  • Pattern blocks
  • Play money
  • Number line
  • String
  • Ruler
  • Grid paper- I have used small grid paper strips that are laminated to use with dry erase markers
  • Calculator- these are important to practice with if your students are allowed to use calculators on standardized tests

The key is to introduce each item gradually and let students have an opportunity to explore and practice with them before adding them to the toolbox. This is especially important if they’re creating a home toolbox so they will be able to effectively use materials without your supervision.

Do you have other materials you’ve added to your math toolboxes? I’m always looking for new resources- especially things that the students make themselves! Giving them ownership of their materials by letting them create their own tools helps keep them invested in the process and gives them a feeling of responsibility.

Featured Math

“I’m Not a Math Person”

February 4, 2017

Have you ever said those words, or heard another teacher say them? How about students? I have to admit I used to say it ALL THE TIME before I became a teacher. I never felt confident in math. In college, I took my math classes in the summer to ensure a passing grade. Now I no longer believe that “math people” and “non-math people” exist. Math is problem-solving and problem-solving is an essential skill that all students can master. We have to make sure that, as math teachers, we’re bringing the best attitude possible to our students.

The Mindset Began

I know exactly when I realized that I “wasn’t good” at math. I was a 3rd grader. I can remember spending hours at our kitchen table, tears flowing as my mom tried to help me understand the math concepts in my homework. I wasn’t blessed with a teacher who took the time to work with me when I was obviously struggling. Fortunately, I went on to have some wonderful teachers, but I never fully regained my confidence in math. In college, however, I was able to shift my focus from being a hesitant math student to being a confident and enthusiastic math teacher.

It Only Takes One Teacher

We know how one teacher can change our entire view of a subject. During my time at the University of Tennessee, I took a required course on teaching elementary mathematics. Over the semester I fell in love with teaching math, thanks to a professor who showed us what amazing things we can do with math instruction. We have to make math relevant to our students, and we have to motivate them by making it interesting. You can do so much with cross-curricular tasks, too! I love incorporating math into my social studies lessons. Calculate distance when studying geography, incorporate elapsed time when looking at timelines/sequence of historical events, and practice measurement and data skills when your students are learning about landmarks. Reach out to your Special Area teachers- find out what they’re working on with your class, and see how you can tie in math skills!

Attitude Is Everything

Most importantly- get excited about math in your classroom! Even if you don’t feel it, and even if you don’t consider yourself a “math person.” Your students will feel your excitement and immediately engage in your lessons. Be a learner with your students- help them explore math tasks and learn from them as you guide their discovery. You will be amazed at how a positive attitude toward math will change your lessons, and you may just discover that you are a math person after all.


The Place Value Break-Down

January 19, 2017

The more I dig into my students’ math work this year, the more I am realizing that many of them lack a solid understanding of place value and its role in mathematical problem solving. This is particularly evident in my Tier 2 and 3 math students- those students who are not working on grade level and yet are being expected to master grade level standards.

The Shocking Realization

At the beginning of the year, I was able to give my students a diagnostic assessment to pinpoint where they were individually and where I should begin their instruction in small groups. The thing that was most apparent: a lack of place value understanding in 3 digit numbers. During a counting question, I observed students counting “107, 108, 109, 200…207, 208, 209, 300..” I was shocked that so many third grade students were not able to correctly count through 3 digit numbers. Suddenly, my frustrations from trying to teach rounding and operations with 3 digit numbers made sense. I realized I could not expect students to understand rounding with 3 digit numbers if they did not understand the values of the digits in a number.

The Break-Down

After identifying the pattern of lacking understanding in place value, I looked back through prerequisite standards in K-2. Kindergarten is the only grade that has Counting and Cardinality as a standard domain. Within that domain, there is the standard K.CC.A.1: Count to 100 by tens and ones. I thought back on my four years as a Kindergarten teacher and remembered assessing this standard. I simply asked my students to count. I stopped them when they got to 100. I placed a check mark by that standard and moved on. They could count to 100- great! That’s as far as I needed to assess them. Fast forward to third grade, and we assume students come to us with a knowledge of counting, and that, if they can count to 100, they can count to 1000. If I want my students to develop a deeper understanding of mathematical processes, this is not an assumption I can afford to make.

My Observations

This year I have been fortunate enough to observe some amazing math teachers in grades above me. I saw some incredible teaching and learning happening in these classrooms. As a result, I learned some great ideas to take back to my third graders. The one thing that stood out to me above all else: teachers are constantly teaching and reinforcing place value, year after year. As the mathematical processes get more difficult in the upper grades, we can’t forget that some of our students may not have a concrete understanding of place value. Without solidifying that, they are going to continue to struggle more and more each year as the math gets “harder”. We have to keep that in the front of our minds as we teach math, and make a conscious effort to bridge the place value gap for our struggling students.

How can we help our students develop a deeper understanding of place value?

This year I have used traditional base ten blocks and mats, as well as resources from the National Library of Virtual Manipulatives. I let my students manipulate these virtual base ten blocks so they can see the break-apart as they decompose numbers. Students need practice decomposing 3 digit numbers by tens, not only traditional expanded form. They need to see that 486 is 48 groups of 10 with 6 ones remaining. In addition, they need to see the number as 4 hundreds, 8 tens, and 6 ones.

These practices can also be used to strengthen addition and subtraction skills. Prompt students to represent numbers in various forms to show their understanding of place value and its role in mathematical problem solving. This is a skill students will need to develop math reasoning skills beyond the surface-level memorization and recall.

Featured Math Reading

Math Is More Than Just Numbers, It’s Letters, Too.

January 9, 2017

Hello, magical math teachers! I am so happy you have stopped by to check out Tenspire! I am so thankful to have this space to share ideas, triumphs, challenges, and the latest Math-related topics with you!

Each school year brings a fresh sense of excitement, a new group of sweet faces, and new challenges. With the path of education constantly finding new directions, it’s no surprise that teachers are faced with changing curriculum and instructional strategies. For me and my wonderful third graders, one of the biggest shifts has been the level of reading that is now involved in math instruction. We are seeing math skills presented differently than the way we learned them. Students are being asked to solve math problems in the context of stories rather than worksheets full of basic algorithms.

Out With The Old

For teachers, this means retraining our brains to prepare, plan, and execute math lessons differently than we have before. We have to think of our math block as an extension of our reading block. We need to spend time helping students learn how to interpret mathematical text and use stamina to take apart word problems using reading comprehension strategies. In addition to understanding the four operations, they’re searching for clue words, applying important information, and discerning the process they need to use without it being stated for them. This can provide amazing learning opportunities as well as challenges for both teachers and students. At first, you may need to do some vocabulary front-loading and reinforcement, but once your students develop the habit of searching for the “meaning in the math”, you will be amazed at the conversations you hear as they solve problems on their own!

Discussions and Problem-Solving

Those of us who consider ourselves “math people” may look at the necessary changes as fighting an uphill battle. Many of our students come to us with low reading skills, making the task even more difficult. We are focused on making sure they understand the basic math- computation and procedure. However, if we want our students to develop a conceptual understanding, we need to promote discussions about problem-solving in our classrooms. Word problems are a great way to do that! Even simple mathematical text can provide layers of information for students to pull apart and analyze. This also provides a great opportunity for partner and group work- something we are always looking to incorporate into our classes. The extra reading practice that students get- without even realizing it- is an added bonus!

Are you noticing this reading-heavy trend in your math classes? What can you do, as the teacher, to help strengthen your students’ reading skills while teaching your math standards? Instead of looking at the downsides and challenges, try viewing it as an opportunity to promote math conversations and help your students put their ELA skills to work!