There are two things I love in math – activities which build number sense and puzzles. There is a great need to promote number sense in our classrooms – especially at the elementary level.

British researchers studied low, middle, and high achieving math students solving problems. What was the difference between the low and high achieving students? High achieving students used number sense to understand and solve problems.

Kwame Sarfo-Mensah shared three great numeracy activities in a recent Edutopia post which are great.

The Hundred Challenge – Students use four given numbers (i.e. 1, 4, 6, 8) to create equations which equal each of the numbers from 0 to 100.

Do Now: Equation or Expression of the Day – Students solve an equation which represents that day’s date.

Cryptarithmetic Puzzles – Digits are replaced by letters, and students have to solve the letter/number puzzle.

I’m excited to try some of these with my class. They might need to be adjusted a bit for elementary students, but they’re too good not to try.

Kids love problem solving. I really believe this. They like to figure things out. If given the right environment, students like to challenge their thinking and struggle with concepts.

I was reminded of this last weekend when I checked my “Shared With Me” file on Google Drive. I had a number of files from students titled “Math Puzzles.” A few of my students created puzzles to challenge their classmates.

Everyday we start our math lesson with a math puzzle and some mental math problems to build number sense and improve our problem solving. My students love completing these tasks, so some of them they made their own puzzles.

This got me wondering if students really don’t like math, or do they just not like how math is typically presented. I had six students who spent their own time creating puzzles. I didn’t tell them they had to do it or assign it for work. They enjoyed the challenge and wanted to show their creativity. It doesn’t get much better than that – creativity, engagement, and problem solving. Now I have to keep that excitement going.

Teaching problem solving is a tough task. Students have to read a word problem, understand what to do, complete the computation, give an answer, and then determine if the answer is reasonable. So how do teachers best support students in that endeavor?

There is no magic bullet to help students become better problem solvers. They need to be exposed to different problem types. They need to see, hear, and learn how other mathematicians around them are solving problems. Some students also need a structure to help them solve problems and explain their thinking.

One strategy I’ve use is called “I know… So I… To find… Therefore…” This helps students work through a problem and explain their thinking. Here’s a breakdown of each of those parts.

I know… What important information was given in the problem?

So I… What did you do to solve the problem?

To find… What answer did you get?

Therefore… What is the answer with the correct label?

Let’s look at the following problem to walk through the process.

David started his coin collection with 14 coins. He added 3 coins to his collection at the end of each month for 5 months. How many coins are in David’s collection after 5 months?

Students would answer the question with: I know David has 14 coins in his collection. He adds 3 coins each month, and this happens for 5 months. So I added 14 + 3 + 3 + 3 + 3 + 3 because he gets 3 coins after each of the 5 months to find 29. Therefore, David has 29 coins in collection after 5 months. (Students could also say they added 14 + (3 x 5) instead of using repeated addition.)

This takes some modeling at the beginning of the year, and I always supply a graphic organizer with the “I know.. So I… To find… Therefore…” components already listed. I created a copy of the graphic organizer on Google Docs.

Students pick up on the process pretty quickly. Using the graphic organizer slows their thinking, which improves focus during the problem solving process. It also helps improve a student’s explanation by giving them a structure to follow.

Last week, my students were working on problem involving fractions. The problem was: A quarter is 1/4 dollar. Noah has 20 quarters. How much money does he have? Explain.

I gave my students graphic organizers. (We’re not quite ready to take the training wheels off yet.) Here are two examples of how an average student completed the graphic organizer.

I have adjusted this approach over the years. I’m sure I’ll continue to make slight changes in the future to try and improve the supports I’m providing for my students. What strategies and supports do you use with your students to help them be better problem solvers?

Saturdays are my time to reflect on what I read, heard, and discussed throughout the week. It is my “exit ticket” for the last seven days. So here are three ideas that made me think this week. (They are not in any particular order.)

1 – Did you know there are three meanings of multiplication? I didn’t know them until a couple years ago. A blog post by Jeff Lisciandrello reminded me that the meanings are not widely known. They are: equal groups, rates, multiplicative comparison, rectangular array, and Cartesian products. Jeff has a great description of each meaning in his blog post.

2 – Michael Pollan is a food writer and the author of books like “The Omnivore’s Dilemma,” “The Botany of Desire,” “In Defense of Food” and “How to Change Your Mind.” His new book, which is only available as an audiobook on Audible, tackles our complex relationship with caffeine. Pollan explores the effects of caffeine in our society and how it impacts our bodies. I have not listened to the book, but read an interesting preview in the Washington Post. In his book, Pollan says, “Something like 90 percent of humans ingest caffeine regularly, making it the most widely used psychoactive drug in the world and the only one we routinely give to children, commonly in the form of soda. It’s so pervasive that it’s easy to overlook the fact that to be caffeinated is not baseline consciousness but, in fact, is an altered state.” That’s a pretty sobering thought.

3 – The third slot is again reserved for something which will put a smile on your face. I’ve never really believed in the whole idea of a spirit animal. Then I saw this video of a dog sledding down a hill. I now complete embrace the idea of a spirit animal and believe I’ve found mine.

Every once in a while there’s a lesson that makes me take a step back and think about how hard it is to truly learn and understand something. For example, fractions are not an easy concept to grasp conceptually. It’s one thing to learn that 1/3 is greater than 1/4 because the smaller denominator is the bigger fraction. It’s another thing entirely to understand conceptually why 1/3 is greater – each fraction has one part but the parts of 1/3 are larger than the parts of 1/4.

Fractions are even a difficult concept for adults to conceptually understand. In the 1980s, the A&W company tried to compete with McDonald’s quarter pounder. They created a third pounder which was cheaper and bigger than the quarter pounder. It beat McDonald’s burger in taste tests and was accompanied by a marketing campaign. So a company produces a bigger, less expensive burger, and the public likes the taste. Seems like a slam dunk, right? No. The public didn’t go for it. Why? During focus groups it was determined that customers thought they were getting less burger for their money because they mistakingly thought 1/3 is smaller than 1/4. A&W’s problem wasn’t a quality of product problem. A&W had a math problem – literally.

This is another reminder of how difficult it is to really understand some concepts. I can manipulate the numbers in my head and explain why 1/3 is greater than 1/4. It’s easy for me to forget that it takes time, modeling, and repetition to truly understand how to compare fractions with like numerators.

This struggle with fractions reminded me of a concept I first heard in the book Make It Stick. The idea is called the curse of knowledge. “The curse of knowledge is our tendency to underestimate how long it will take another person to learn something new or perform a task that we have already mastered.”

It is hard to remember how difficult it is to first learn something. Take driving for instance. An experienced driver doesn’t think about all the decisions he makes behind the wheel of the car. He has built a knowledge base over years that allows the process of driving to become almost second nature. Now put a 16-year-old behind the wheel and you’re reminded of how much a person needs to learn to drive a car. You’re also reminded that this knowledge is not obtained overnight.

I try to go into lessons with the curse of knowledge on my mind. It is important to remember that the 9- and 10-year-olds sitting in my classroom haven’t had the years of math exposure I have. It’s why thinking through the lens of our students is so important.

In an attempt to combat this curse of knowledge, I try to anticipate prior to a lesson which concepts will give students difficulty and why. My goal is to overcome the curse of knowledge as a teacher. In the process, hopefully, my students will conceptually understand difficult concepts like fractions and know the mathematical difference between a third pounder and a quarter pounder.

I read an article a few years ago that really changed the way I talk to my students in math class. The article is called 13 Rules That Expire. It was published inNCTM’s August 2014 issue of Teaching Children Mathematics.

The gist of the article is that there are math “rules” used in classrooms. These are often presented as rules that always work, but in the case of these 13 rules, there is an expiration date. The article goes into greater detail about each of the rules, including the expiration date (or expiration grade). My goal is not to thoroughly discuss all 13 rules in this post, but wanted to at least list them.

When you multiply a number by a ten, just add a zero to the end of the number.

Use keywords to solve word problems.

You cannot take a bigger number from a small number.

Addition and multiplication make numbers bigger.

Subtraction and division make numbers smaller.

You always divide the larger number by the smaller number.

Two negatives make a positive.

Multiply everything inside the parentheses by the number outside the parentheses.

Improper fractions should always be written as a mixed number.

The number you say first in counting is always less than the number that comes next.

The longer the number, the larger the number.

Please Excuse My Dear Aunt Sally.

The equal sign means Find the answer or Write the answer.

There is one rule I wanted to discuss a little further. Before reading this article, I was a big fan of teaching students to solve problems with the help of keywords. This often included making an anchor chart with the keyword on one side and the meaning on the other side (i.e. in all means add, how many more means subtract).

One common keyword strategy to teach students is that the word altogether means you should add. While that does work most of the time, it doesn’t always work. Take for instance the following problem:

Jimmy read 11 books in fourth grade, and Brian read 8 books. Jimmy, Brian, and Elyse, read 37 books altogether. How many books did Elyse read?

If students attack word problem by simply looking for keywords to help them solve problems, they will most likely see altogether and add 11 + 8 + 37. If students are thinking about the context of problem, and not just the keywords, they will see that they have to subtract 19 (11 + 8) from 37. This requires students to understand what is happening in the problem and not simply use the keyword as their only problem solving tool.

Teaching problem solving to our students is not an easy endeavor. It takes many tools in the mathematician’s toolbox. It takes constant repetition and exposure to many problem types. Teaching problem solving is not easy, but it is one of the most important skills we teach in any subject area.

It’s always great to show students the wonder of math and numbers in the real world. Today, February 2, 2020 is one of those opportunities to have fun with numbers. Not only is today Super Bowl Sunday and Groundhog Day, but it is also an eight-digit palindrome. A palindrome is a word or phrase that reads the same forward and backward – Anna, nurses run, or my daughter’s favorite book Taco Cat.

Today is 02/02/2020 or 02022020. It is symmetry at its finest. How rare is a global eight-digit palindrome? The last time it happened was 909 years ago – November 11, 1111 or 11/11/1111. The good news is you won’t have to wait another 909 years. There will be another in a mere 101 years on December 12, 2121… 12/12/2121.

But wait, the math fun isn’t done yet. Today is the 33 day of the year and there are 333 days left in 2020. Thank you leap year for that. With all this math fun, it’s like Christmas in February!

Saturdays are my time to reflect on what I read, heard, and discussed throughout the week. It is my “exit ticket” for the last seven days. So here are three ideas that made me think this week. (They are not in any particular order.)

1- If you’ve read more than a couple posts on my blog, you probably know that one of my greatest passions is math education. I am a total nerd when it comes to better understanding how to teach math, especially at the elementary level. I am also a huge podcast fan. So, when I heard Marilyn Burns on a podcast talking about math education, it was like Charlie finding Willy Wonka’s Golden Ticket. “I think that most teachers begin teaching the way we were taught,” Burns said on the podcast. As her teaching career progressed she started to change her thought process. “I got really curious about how do I get kids to be the stars in the classroom rather than me being the star in the classroom. So everything shifted for me from ‘How do I make myself as the most important person in the room?’ to making the students the most important people in the room.” Isn’t that what we should be striving for in every classroom?

2 – A teacher in Nebraska had his favorite pair of shoes stolen from his classroom. His students all chipped in to buy him a new pair. His reaction to the gift shows the impact a great teacher can have on his students. It’s impossible to not see how much he and his students care about each other. On a side note, when I watched this video, somebody must left some chopped onions near me.

3 – I generally save #3 for something light hearted. This story from CBS’s Steve Hartman fits that mold but also has an important message. The arts are so important to our children. We should do everything we can to protect music and art education in our communities. Seeing the look on this nine-year-old boy’s face when he sees the Michigan Marching Band proves that the arts are critically important.

A few years ago I was doing a lot of one-on-one work with a student who struggled in math. We often met before class to do some mental math and try to improve his number sense. After school, we would revisit the skill we discussed in class that day. He made a lot of progress and was starting to feel much more confident about math in general and problem solving in particular. Then we introduced fractions.

Shortly after we started our unit on fractions, he put his head down in frustration. I asked him what was wrong, and he replied, “I was just starting to get math and then… then fractions happened.” It was as if he had contracted a deadly virus named fractions. We talked about how good problem solving doesn’t change just because we moved from whole numbers to fractions. Over the next month or so, we continued to work and his confidence with fractions improved as well.

My approach with that student would be slightly different today. Instead of moving him right into word problems with fractions, I would use numberless word problems to help ease some of the anxiety. The numberless word problems would be a sort of scaffold to the problems with fractions. This would help him see that we’re still using the same problem solving strategies we used with whole numbers.

Numberless word problems are just that – word problems without numbers. Let’s look at the following problem.

Brian eats 1/8 of the pizza and Julie eats 4/8 of the same pizza. How much of the pizza did Brian and Julie both eat?

This is a pretty basic word problem involving fractions. Some students might struggle with this problem simply because they are intimidated by the fractions. So, why not take the fractions out all together?

Brian eats some pizza and Julie eats some pizza. How much of the pizza did Brian and Julie both eat?

Now we can have a conversation about what is happening in the problem. We can talk about how one person has some amount of pizza and another person has another amount. We want to know how much they have together. That hopefully leads students to see that this is an addition problem. Once they understand the context of the problem and have determined the operations they need to use (addition), then they can start looking at specific numbers.

Depending on the students and the problem, I might put some whole numbers in the problem before moving to fractions. This will give the student another opportunity to see the context of the problem before the fractions are introduced. I did this today with my class.

This is a wonderful strategy because it helps students focus on the context of the problem. The focus on the narrative portion of the problem before worrying about the numbers. They can’t worry about the numbers, because they aren’t there yet.

We just wrapped up a unit on fractions which covered equivalent fractions, common denominators, simplest form, comparing fractions, and ordering fractions. Comparing and ordering fractions gave me an opportunity to use a strategy which I absolutely love: clothesline math.

As the name suggests, I put a clothesline up in my room (really just a piece of yarn) and students use it to create a number line. It’s a great strategy because it works with whole numbers, fractions, decimals, algebra, and on and on.

I fold a notecard in half and write the fraction on one half. This allows the fold of the card to hold the number on the clothesline. Since I’m writing the numbers myself, I can make the numbers fit whatever lesson or standard I need. For example, in Pennsylvania fourth graders are only comparing fractions with denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100. So, I create fractions cards which fit those parameters.

There are times when I will place cards on a table and let students choose their own number. Other times, I want to be a little more strategic with who gets each card, so I will distribute them to students. This allows me to ensure that students are getting fractions that match their math abilities. I give students who have a better grasp of the concepts fractions which are in the sixths, while struggling students might only get fourths. If a student has to draw a model of a fraction, it is easier to draw fourths as a visual representation than sixths.

Once all students have the cards, we discuss which numbers should be placed on the clothesline first. Yesterday, my students determined that 0 and 1 should be the first two numbers. Then they decided that 1/2 should go next because it is the benchmark we used in our lessons on comparing and ordering fractions.

Once we have 0, 1/2, and 1 placed, we start to have a discussion about the other fraction cards. I allow students to talk with a partner as we move through the numbers. When students go to place their fraction, they have to share three things:

Say the fraction

Share which two numbers their fraction goes between

Explain, using math vocabulary why you placed it there (common denominators, common numerators, relative location to the benchmark, etc.)

Once we get started, my role is to simply facilitate the conversation. I don’t tell students whether they are right or wrong when they place their fraction. Yesterday, three of the fractions were placed in the wrong spot. Once every student placed their card, I asked if there were any changes we needed to make. Then the conversation moved into a bit of error analysis. After some discussion and debate, the cards were placed in the correct order. Once again, I let my students talk about what they saw and what needed to be changed.

Students were drawing models on whiteboards, comparing fractions using common denominators, and one student even asked to use the fraction tiles we have in our room to model his fraction.

This is such a wonderful activity because it checks so many boxes. In addition to comparing and ordering, students are experiencing many of the Standards for Mathematical Practice. Students are constructing viable arguments and critiquing the reasoning of others, modeling with math, attending to precision, and looking for and making use of structure. I am a stickler about precision in this activity. One student said, “The number on the top in 4/8 is smaller than the number on the top in 5/8.” I will ask them what we call the number on the top and ask them to restate their reasoning. “The numerator in 4/8 is smaller than the numerator in 5/8.” If the students stops there, I will prompt them. “What does the numerator represent?” I will prompt the students until they explain that the numerator in 4/8 represents 4 parts, and the numerator in 5/8 represents 5 parts. The parts are the same size since the denominators are the same, so 5/8 is greater than 4/8.”

I also make sure there are some equivalent fractions included in the cards. When students have an equivalent fraction, they use a paperclip to stack the numbers to show they fall on the same point on the number line, or clothesline.

Each year I introduce this as a whole-group lesson. This allows students to experience those rich conversations with students of all math abilities. Then I use it as a small group activity. Students work in groups of 3-4 and deal cards to one another. They take turns placing their cards and have a similar discussion. As an exit slip, I’ll have students choose three fractions and write how they ordered them. This can also be done by taking a picture and using an app or other digital sharing tool.

This is a great activity that can be used in so many applications throughout the grade levels. The conversations are fantastic, and students are engaged in the activity. Hopefully you find it as valuable as I do.