The Lost Art of Counting

There have been many negative effects on education that stem from standards and standardized testing. These have been discussed and rehashed over and over through the years since standards and testing began. I have been looking specifically into the direct effects on math instruction.

One of the biggest negative effects of standards based education is the narrowing of the scope of what our children are learning. Due to the very specific way that the standards are written and tested, most districts and curricula have narrowed their focus to the specific skills that are listed in the standards. I live and teach in the state of Virginia and I will be using specific examples from the VA SOL’s, but my research shows that Common Core states have similar issues and I am guessing that other states with their own standards would prove the same.

When I started interviewing students in grades 3-5 who were struggling with math, I noticed some big similarities among them. The most noticeable is that they struggled to count numbers higher than 100. When I started looking into this, I realized that in Virginia, there are no specific standards for counting past 120 in second grade. Because of this, most teachers don’t do any counting work past that goal.

The counting standards in Virginia include counting forward and backwards to and from 120 and skip counting by 10, 5, and 2 from any multiple of that number up to 120. Almost all of our students are successful with these skills, so we didn’t really think past them. I will admit that when I was a third grade teacher I was just as guilty.

It used to confuse me why, if my students know place value, how can they struggle so much with things like rounding, comparing, and ordering???  What I have discovered is that students struggle with these skills because they can’t count the numbers they are expected to work with. In VA we expect 3^rd grade students to work with whole numbers up to 6 digits and 9 digits for grades 4 and 5, but there are no specific standards that require counting those numbers.

My belief is that if any numbers we expect students to work with, we should also make sure they can count them. We should be able to pick a random number and students should be able to count forwards and backwards from that number by 1’s, 10’s, 100’s etc… Counting directly relates to the skills of comparing, ordering, rounding and mental computation. If our students were comfortable with counting these numbers, then teachers wouldn’t have to rely on shortcuts, poems, procedures and tricks to teach these skills.

When I was in school, I can vividly remember counting chorally well into middle school.  Choral counting and counting circles are a great way to bring counting back into classrooms.  Choral Counting and Number Talks are the 2 routines that have had the biggest positive impact with the struggling students I work with.

First of all, here is a link to a book that will help you start Choral Counting in your own classroom.  It is called “Choral Counting & Counting Collections” by Megan L. Franke, Elham Kazemi, and Angela Chan Turrou. If you don’t have the means to get the book, just google choral counting and you will find a wealth of information and videos that you can use.

Here is a great video that shows choral counting in an upper elementary classroom. https://vimeo.com/44837082

Remember, if you are starting from scratch, use easy numbers to help students learn the basics of choral counting. I usually start with 3 digit numbers and skip count by 5’s, 10’s, or 100’s. Eventually you want to be able to start at any number and skip count by a variety of numbers that cover the numbers that your class is responsible working with. (In VA, that would be whole numbers up to 6 digits in third grade and 9 digits in fourth and fifth grades.) The most powerful part of the activity is looking back and finding the patterns and discussing what they notice and wonder about.

Don’t forget that you can also do coral counting for fractions and decimals. Here is a video for a fraction example. https://www.engageny.org/resource/nti-november-2012-rigor-breakdown-skip-counting-by-fractions

And here is a wonderful decimal example. Listen to the students discuss the patterns. https://www.youtube.com/watch?v=Qf_VBRFCnuc

Counting Circles are another great routine that takes no preplanning. Anytime you have a couple minutes to kill you give the kids a number and the amount you are counting by and go “around the circle” counting. https://www.youtube.com/watch?v=Yk0_sa-wo_U

Remember that the more fluency students have with counting numbers the more those numbers will make sense to them.

Bring back the Art of Counting!!

There are no Butterflies in Math!

I truly believe that the math shortcut or “trick” that annoys me the most is using a butterfly to compare fractions. It annoys me so much because teaching students the underlying conceptual knowledge for this trick is so simple. We should be teaching students the deep knowledge of math and not shortcuts and tricks that rob them of important mathematical understanding.

I was having a conversation with a fifth grade teacher about fractions and I asked her what she liked most about teaching fractions and what scared her about teaching fractions. She stated that teaching fraction computation was her worst nightmare and I told her that we would tackle that hurdle together. She then said something that surprised me. She said that her favorite thing to teach during the fraction unit was comparing and ordering fractions. When I asked her why, she said that the students loved butterfly math and they got it easily and rarely had any trouble with it.  (Cue eye twitch)

I then asked her if she taught her students why the butterfly trick worked. She looked at me for a few moments and asked, “What do you mean?”  At that point I figured that this was a trick that she had been taught but she never knew why it worked herself; therefore, she could not pass that knowledge on to her students.

After I asked a few more teachers about this, only two of the twelve teachers knew why the butterfly worked even though ten of them had taught it to students. It seems as if we all complain about student’s lack of number sense, but then we use procedures that further corrode the number sense that they do have.

I start teaching equivalent fractions using colored tiles. We make 1/3 by using 1 green and 2 blue tiles. We then add another identical row and discuss that 2/6 is still 1/3 because 1/3 rows are green. Students then make several more fractions that are equivalent to 1/3 by adding more rows. I then give them time to explore some other fractions and their equivalents.

When we start comparing fractions we make boxes on grid paper that are similar to the models of equivalent fractions we had been building.  As students explore making fractions on grid paper they will make some generalizations and discoveries. Such as, we can use both denominators to create a box that can be used to represent both fractions. If we were going to compare 3/5 and 2/3 we would draw a box that was 5 blocks by 3 blocks.

We then make two of the same size box and shade in the two fractions. Students then have to discuss with their group how we can use this diagram to answer the question, which fraction is greater? It may take a while, but students will start to discover that you can count the number of shaded squares to compare the fractions. The box with the most shaded squares is the greater fraction.

Because our students are allowed to use grid paper on their assessments, I do not push them beyond this discovery.  Some students will make more discoveries and generalizations on their own as they explore this concept.  Some years students realize that the number of squares shaded in can be determined by cross multiplying, but some years it isn’t discussed.

Don’t skip the exploration and discovery in order to make a short cut. Math is discovery! Don’t cheat our students out of that process.

Down with Math Rules!!

Rules

I was never a person who did well with rules that I didn’t understand. My parents can attest to the fact that I was never an enthusiastic follower of rules. If I didn’t have a legitimate reason for why there was a rule, I didn’t think it was important that I followed it. I drove my parents crazy.

Math, as we all know, is just filled with rules. Back when I was in school we were given a math rule to follow with little to no explanation and then given a sheet of what seemed like a million practice problems to do. We could get them right if we just followed the rules. But, as I previously stated, I was not one to do this. I was that kid who was continuously asking, “Why?” To my all too often disappointment, I was told over and over that it was just the way math was.

I refused to accept that math rules were just arbitrary things written by some ancient math dictator. They had to make sense. I needed for them to make sense. I started doing my own experiments with the rules to see what would happen if I didn’t follow them. Although I discovered that the rules had mathematical reasoning behind them, I felt as if I had learned more than my peers because I knew why the rule was written. If there was a rule that I couldn’t figure out on my own, my teachers were usually not much help. They knew how to teach the rule, but didn’t usually understand it any more than I did. This frustrated me to no end.

Even in college I heard, “Just do it the way I showed you.”  I swore as an education major that I would never ask my students to do something that I couldn’t explain to them. I was determined that my students would know the reasoning behind all the rules.

During my first several years as an elementary school teacher, I struggled with balancing the curriculum that the district handed me with the desire for my students to understand math and not just memorize rules. I dove into the works of Marilyn Burns and John Van de Walle. I read everything I could find that they wrote and replaced curriculum lessons with activities from their books. Since that time, my district has decided to write their own curriculum that stresses conceptual knowledge over procedural memorization.  I love it!!

If you don’t understand the difference, consider this example. When you were little, I’m sure your mother told you not to touch the stove because it was “hot.” She didn’t spend time explaining what that meant, but expected you to obey. Think about which is a more powerful lesson, listening to your mom or touching the stove and learning what “hot” meant for yourself.

Math is the same. Math rules are just generalizations that mathematicians have discovered through the years. They aren’t wrong, but memorizing them doesn’t give students a true understanding of the math they are trying to do.

For example, the rounding poem is the way many teachers use to teach their students to round numbers. If you are not familiar with this poem here it is.

Find the digit

Look right next door.

Five or more

Raise the score.

Four or less

Let it rest.

I’m sure there are several other versions out there, but they all say basically the same thing. It is just a procedure written in a rhyming format. The problem comes in if you ask a student why 38 rounds to 40. You get an answer along the lines of “8 is more than 5” or they will recite the poem to you. They don’t understand that 38 rounds to 40 because 38 is closer to 40 than 30.

This is why it is more powerful to have students use a number line to round. They actually see that 38 is closer to 40 on the line. If they get enough time to explore rounding on the number line, they will discover those generalizations from the poem themselves. They will see over and over that when the number in the next place is 5 or higher you round up to the next place and when it’s 4 or less you stay where you are.

I cannot effectively describe the excitement in the room when they make these generalizations for themselves. It is so much more powerful than just memorizing a poem. They truly understand the math concept and are more likely to retain that information long term.

I have heard from teachers, “I show them the number line and explain to them why the poem works, so they understand it.” While I admit that they may have some understanding of the generalization, it is nowhere near as powerful as personal exploration and discovery.

Over the summer I will be blogging about several other common rules and tricks that should be buried in the mathematical graveyard.