The post Flipping the I Do, We Do, You Do Script appeared first on Math Coach's Corner.

]]>Students taught through passive approaches follow and memorize methods instead of learning to inquire, ask questions, and solve problems. Jo Boaler, *What’s Math Got to Do With It?*

*This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.*

The past 15-20 years have seen huge shifts in math instruction. New standards deemphasize rote memorization of procedures and outline an approach to teaching math that balances procedural fluency, conceptual understanding, and problem-solving. That is in large part due to the fact that the way we *use* mathematics has changed in the past 50 years. Our mobile phones now have the power to do calculations that previously were either done manually or using room-sized computers. That’s not to say that procedural fluency is no longer important, it’s just not the only thing that’s important.

Now think about the traditional model for teaching math; *I Do, We Do, You Do*. This model is also called the* gradual release of responsibility. *Look back at Jo Boaler’s quote at the beginning of this post, and do you see that this traditional approach is very passive? Students are basically just memorizing and mimicking a procedure modeled by the teacher. Yet this is still the predominant instructional model in math classrooms. The lessons in most math textbooks are structured based on the model. Unfortunately, this model does little to address problem solving or conceptual understanding.

But what if we flipped the script? Now the model becomes *You Do, We Do, I do. *

So now we have a more student-centered model that incorporates problem-solving. It’s much less scripted. Let me illustrate with an example.

Say, for example, that I want to introduce all the combinations for a number, in this case 5, to a Kindergarten class. I could present a very scripted lesson, such as the one shown below. Recognize that this is very passive learning for students. They are merely mimicking the teacher’s actions. And the teacher is doing all the explaining.

Contrast that with this approach. This time, I’m going to give pairs of students two plates and 5 counters. Then I’m going to pose the following problem: *Stephan has 5 cookies he wants to put on two plates. Use your counters and paper plates to show how he might place the cookies. Can you think of more than one way?*

After giving students a few minutes to work on the problem *(you do),* I’ll solicit the combinations they discovered and record them on the board or chart paper. I can question them about their process and we can discuss any patterns we see *(we do)*. To extend the activity, I might ask, *Huh, I wonder if we found all the ways to make 5? *Then we could discuss different ways to organize our data to help us determine if we found all the combinations. Finally, I’ll consolidate the learning *(I do)*, asking students to reflect on what they learned and stating my teaching point: There are many different ways to make numbers and we call them *combinations**. *

Notice that with this lesson, students are actively involved. They are working on the problem with a partner, which encourages great mathematical conversations. They are making their own discoveries, and that helps them take ownership of the learning. And then the lesson is tied up neatly with a bow!

Change can be hard! The *I Do, We Do, You Do* model has been around for a very long time. Let’s break that mold and put students in charge of their learning!

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]]>The post You Write the Story: Student-Generated Math Word Problems appeared first on Math Coach's Corner.

]]>Part of the problem is that over the years very well-intentioned teachers have tried to turn the process of solving word problems into an algorithm—if you follow these steps, then you can solve math word problems. As a result, we see widespread use of “strategies” like CUBES. Unfortunately, CUBES only works with the simplest of problems. Consider this problem:

Not a terribly complicated problem, right? It’s a fairly straightforward multiplication problem with a little extra information thrown in. So let’s apply the CUBES strategy. To start with, nowhere in the CUBES strategy does it actually tell students to *read the problem!* And if you’ve ever watched students use CUBES to try to solve a problem, you’ll quickly see that they don’t. They scan the text, circling the numbers as they go. We’ve basically given them permission not to read. Next, notice that keywords don’t help **at all** in this problem. The only keyword is *increase, *which indicates addition. Finally, since students don’t really understand what the numbers in the problem represent, they can’t identify that the $8 increase is extra information. Hence, they end up adding the three numbers, not even realizing they are adding together money and memberships. For an alternate method that actually helps students solve math word problems based on **understanding**, check out this post on the Three Reads Protocol.

So how can we actually help students become more accomplished in solving word problems? Well, we emphasize *quality* over quantity. Instead of solving pages of word problems, we let students write their own! If students can write word problems, trust me, they can solve them. One of my favorite activities is called *You Write the Story. *It’s super effective and super low-prep—an awesome combination! Students are given an expression, and their task is to write a story problem, draw a model showing the problem, and then solve it. You can see from the index cards below how easy it is to differentiate the task.

Of course, writing word problems is a skill we need to teach. Don’t rush this! Do plenty of guided writing before you assign it as an independent task. Here’s a process I find to work well. First, I’m going to ask students what they notice about the expression. This helps them focus on the magnitude of the numbers and the operation (addition). Next, model what it sounds like to develop the idea for a story problem. It’s a story, so we need to think about the characters, the setting, and the action taking place.

Here’s what it might sound like:

*For some reason, I’m thinking about milking cows! Silly, right? So, the story is about Farmer Jon and his son. The story takes place in Farmer Jon’s barn. What’s happening is that Farmer Jon and his son are milking cows. And the numbers in the story represent the cows that Farmer Jon and his son milked. Now I’m ready to put it together and write my story.*

Farmer Jon and his son were milking cows in the barn. Farmer Jon milked 34 cows and his son milked 18 cows. How many cows did Farmer Jon and his son milk?

We also want to model the process for drawing a representation. A simple part/whole diagram is perfect for helping students visualize what the numbers in a problem stand for and what number they are looking for. Remember, it’s important for students to label their diagrams so they can make a connection to the meaning of the numbers in the problem and their relationships.

You Write the Story can easily be adapted. Students need to be able to solve problems with the unknown in any position. So how about a card that looks like this?

And, you can even provide students with a model and ask them to write a story, provide the equation, and solve the problem. Here are a couple of examples.

So, there you have it! Are you ready to give You Write the Story a try? If you do, I’d love for you to drop a comment and share your experience.

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]]>The post Long Division Choice Board appeared first on Math Coach's Corner.

]]>*This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.*

I’m currently rereading Jo Boaler’s book, *Mathematical Mindsets*, which was recently released in its second edition. Chapter 3 is titled *The Creativity and Beauty in Mathematics.* Does that strike you as odd? Do you think your students would describe math in that way? The main point of this particular chapter is that there is a huge disconnect between the way we teach math in our schools and the way math is actually used in the real world. Our instruction largely centers on performance—performing calculations or answering questions. Yet, Boaler refers to Conrad Wolfram’s four stages of math:

- Posing a question
- Going from the real world to a mathematical model
- Performing a calculation
- Going from the model back to the real world, to see if the original question was answered

Unfortunately, Wolfram states that 80% of the instructional time in our mathematics classrooms is focused on Step 3, performing a calculation.

Something that Boaler goes on to discuss is the importance of having students explain their thinking:

Many parents ask me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.

So how can we shift the emphasis away from calculations? It will require a hard and honest look at the types of tasks we give students to do. Consider long division. How do students benefit from a page with 30 long division problems? Either they know how to do it, and they’ll get them all right, or they don’t know how to do it, and they’ll get them all wrong. We need to move toward more thinking and reasoning-type tasks.

Consider this long division choice board.

Yes, there are problems that require only calculation, but there are more thinking/reasoning-type problems. It’s used like a tic-tac-toe board, with students choosing the three problems they want to solve. I would suggest telling them that the only option they *can’t* choose is diagonal from the top left to the bottom right because that would result in them only performing calculations. Every other option presents them with one calculation problem and two thinking/reasoning problems.

Click __here__ to grab a copy of the long division choice board.

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]]>The post Number of the Day with Ten Frames appeared first on Math Coach's Corner.

]]>The daily routine begins by showing the number of the day (which is the number of days in school) using ten frames. Each day a student gets to draw another dot on the ten-frame. You could have students add a dot sticker, but it was kind of cool to see the kid-drawn dots.

What happens next is the magic of this routine. The teacher had put adhesive magnet squares on the back of the ten frames (which you see on the board), so the kiddos could move them around. When I walked in, the students were showing different ways to make 83 using two addends by sliding the ten frames into two columns. So what you see in the picture below is a student showing 83 as 50 + 33. Brilliant, right?

As students came up to show different ways to decompose the number, the teacher recorded their expressions on a mini whiteboard.

With each new way to decompose the number of the day, the class counted aloud each addend (practicing counting by tens) and confirmed that the expression was correct.

THEN, the class moved on to three addends! Just awesome.

Best question of the day: “So, class, no matter how we move the numbers around, what number will we always have?” *EIGHTY-THREE!*

Do you need a ten-frame template? Just click __here__ to grab one that prints three to a page.

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]]>The post Composing and Decomposing Numbers appeared first on Math Coach's Corner.

]]>So what does that have to do with math you might ask. I’d say a lot. Think about the ultimate goal of math instruction—students with strong skills and flexibility with numbers. One of the key factors in reaching that goal is helping our students learn to compose and decompose numbers. It’s a skill that starts in Kindergarten and follows students throughout their mathematical career, and it’s something that should be present in your classroom on a daily basis.

Composing and decomposing numbers became part of our instructional vocabulary after the Common Core State Standards for Mathematics (CCSSM) were adopted in 2010. Consider the following examples:

A Kindergarten student uses __number bracelets__ to find all the combinations for a given number. This is actually the foundation for learning basic facts, but it goes beyond that. Say the student is now presented larger numbers, like 8 + 5. They know that a combination for 5 is 2 and 3. They also know that 8 and 2 makes 10. They can now solve it by decomposing the 5 into 2 and 3, adding the 2 to the 8 to make 10, and then adding the remaining 3 to get 13. Now *that’s* mental gymnastics. And trust me, the time it takes a child to do this is a lot less than the time it takes me to explain it!

In a 1st-grade class, a student uses their knowledge of number combinations to solve 8 = ? + 3, modeling the problem with counters.

A 2nd-grade class is working on place value and writing numbers in expanded form (135 = 100 + 30 + 5). We don’t typically refer to it as decomposing the number, but that’s really what it is. Check out this __post__ I wrote about the importance of showing students that numbers can be decomposed in more than one way and grab a little freebie. Students need to understand that 135 absolutely can be represented as 100 + 30 + 5, but it can also be decomposed into 130 + 5 or 100 + 20 + 15. That’s the basis for understanding subtraction with regrouping.

A 3rd-grade class is working on multiplication facts. A student is trying to master the fact 6 x 7. They realize they can decompose the 6 into 3 + 3 and then knows that 6 x 7 is just twice 3 x 7 (a fact they happen to know).

In the 4th grade classroom down the hall, students use an area model to solve 12 x 13, leading to a deeper understanding of the multiplication process.

And in yet another classroom, a 5th-grade student decomposes 11/7 into 7/7 and 4/7 to change it to the mixed number 1 4/7.

Strong, flexible mathematicians. Go for the gold!

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]]>*This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.*

Although most clocks are now digital, the standards require students to read both digital and analog clocks. When you think about it, reading an analog clock is a **VERY abstract concept**. We say the numeral the hour hand is pointing to, but we count by 5s for the minute hand. Not to mention the fact that they have to remember which is the minute hand and which is the hour hand. While digital clocks are easier to read, they don’t provide a visual referent for times. For example, there is nothing to show that 2:55 is almost 3:00. Students must know there are sixty minutes in an hour, so 55 minutes past 2:00 is almost to the next hour.

Whenever students encounter a new skill or concept, it’s important for them to have concrete experiences. In the case of time, students should work extensively with **geared student clocks** to experience the relationship between the minute hand and the hour hand.

Here are some suggestions for a progression of teaching time to the hour and half hour in First Grade:

- Have students practice showing and writing times to the hour using geared student clocks. Provide visuals, such as an anchor chart, to help students remember that the hour hand is the shorter hand and the minute hand is the longer hand. Tip: There are fewer letters in the word
*hour*, and the hour hand is shorter. When writing time, explain that the 00 following the hour in a time like 3:00 indicates that the time is zero minutes after 3 o’clock, or 3 o’clock and zero minutes. - Allow students to explore what happens to the minute hand when the hour hand moves from one hour to the next. Ask students to show the time 3:00. Then ask them to move the hour hand slowly until it reaches the next hour. Repeat this with additional times to the hour. Ask students what they notice and wonder about the minute hand and the hour hand.
- After students are secure with times to the hour, provide them a frame of reference for times to the half hour by asking them to estimate where the minute hand would be halfway between two hours, for example 3:00 and 4:00, and show you on their geared clocks. Remember to have students defend their answers. Because students in First Grade aren’t required to understand 5-minute increments, it’s not really necessary to show them that the minutes are counted by fives. It’s enough for them to know that an hour has 60 minutes, so times to the half hour show a time halfway from one hour to the next and are written with 30 for the minutes.
- Have students show time a time to the hour and then move to the half hour. For example, have students show you 3:00 and then move the hands on their clocks to 3:30. Again, it’s not necessary for them to count by fives. What you do want them to recognize is that the minute hand points to the 6 and the hour hand is halfway between the 3 and the 4 on the clock. Alternately referring to the half hour times as
*half past*provides students with another frame of reference. Repeat this activity to provide students with lots of practice. - Once students have separately practiced times to the hour and half hour, give them mixed practice by alternating asking students to show times to the hour and half hour, so now they have to think a little more about the process. Again, stick with those geared clocks!
- When students move toward writing times to the half hour, I always have them decide on and write the hour first. I have them circle the two hours the hour hand is between, and then write the hour it has already gone past.

I hope these tips for teaching students to tell time to the hour and half hour are helpful to you! And now I have a FREE resource for your kiddos to practice reading time to the hour and half hour.

Use **this link** to grab your freebie.

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]]>The post Brain-Boosting Equations appeared first on Math Coach's Corner.

]]>Brain-Boosting Equations is an activity for 1 to 4 students. It can be played as a competitive game or a collaborative task. The overall goal of the activity is to choose three cards and create equations, using any of the four operations, equaling the numbers 0 through 10. The deck of cards also includes a Wild Card that can be used for any value.

To play as a competitive game, the three cards are chosen and all the players work individually to write equations that equal 0 through 10 using the numbers on the cards. The player that creates all eleven equations first is the winner. If time is short, then the player with the most equations when time expires is the winner. To use as a collaborative task, the students work together to create the equations.

Think of all the math your kiddos are doing as they try to come up with equations! Wouldn’t this also make a great homework assignment?

Grab your free copy of Brain-Boosting Equations: An Elementary Equations Game using **this link**.

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]]>The post The 3 Reads Protocol for Solving Word Problems appeared first on Math Coach's Corner.

]]>Over the years, very well-intentioned teachers have developed strategies designed to help students **solve word problems**. Two such strategies that are still quite prevalent are “problem-solving” models and the use of keywords. The idea is that if you follow these steps and look for these keywords, you will be able to solve any word problem. Unfortunately, it’s just not that simple, and despite their widespread use, these strategies are not very effective.

If you look at the CUBES problem-solving model, reading the problem is not even one of the steps! And if you’re thinking, *Well, of course students know to read the problem!* you might want to watch this model in action. I have more often than not seen students just literally start circling numbers (and not even the labels that go with the numbers…) without ever having read the problem. And keywords are not reliable either. Some word problems have no keywords, and keywords in multi-step problems end up confusing students because of the mixed messages they send.

So can we just agree that something else is needed and put these “strategies” to rest? Students fail at solving word problems for one reason—they don’t understand what the problem is asking them to do. It’s a comprehension problem, so students need reading comprehension skills.

Let me first say that if you search the Internet for *3 Reads Protocol*, you’ll find that there are slightly differing versions. What I’m about to describe is the version that I find to be particularly effective. Regardless of the version, we are reading the problem three different times and each reading has a different focus.

The 3 Reads Protocol is a guided learning experience. Students are presented with the problem in stages, and with each read the teacher asks probing questions. Looking at an example is probably the easiest way to understand the protocol, so let’s dive in.

To begin the 3 Reads Protocol, the teacher presents the students with a problem, and the class reads the problem together. Probably the easiest way to do this is with a PowerPoint or Google Slides file. Notice that with the first read, there are no numbers and no question. We just want the students to understand what the story is about and make a mental picture. Without numbers, students have to focus on the meaning of the words! After reading the problem together, the teacher asks what the story is about and calls on students for responses. Don’t be surprised if the responses are very general at first (*girls, flowers, *etc.). Ask for additional details, if necessary. Ideally, for this problem, you’d like the students to offer the names of the girls and the types of flowers.

For the second read, the problem is again presented to the students, but this time it includes the numbers. Read the problem again whole class. The questions you will ask now are all related to the numbers in the story. Our goal is for the students to understand that it’s not just 10, it’s 10 *daisies*. Students might also offer relationships—e.g., Natassja picked more daisies than Ayriale.

Finally, with the third read, students are asked to generate questions that could be answered using the information in the problem. Even though the problem looks just like it did for the second read, don’t skip the reading part! Some problems won’t lend themselves to very many different questions. I like to use this problem as an example because many different questions can be generated. Why? Because there are lots of different numbers in the problem. Here’s a sampling of questions that could be asked. I’m sure you can think of many others.

- How many flowers did Ayriale pick?
- How many flowers did Ayriale and Natassja pick?
- Which girl picked more flowers? How many more?
- How many daisies did the girls pick?

That’s the protocol in a nutshell! Once a question or questions have been generated, you can have students go on to solve the problem.

Frequently asked questions

**1. When students are solving word problems independently, do I ask them to ignore the numbers and the question?**

No! That would be pretty much impossible for them to do. By routinely solving problems using the 3 Reads Protocol with either the whole class or in small groups, you are helping students develop good reading habits that will transfer to their independent work. When they are working independently, the idea is that they will automatically think about the context, identify what the numbers mean within that context, and better understand what the question is asking them to find.

**2. Where do I find problems for the 3 Reads Protocol?**

I’m sure you can find some that have already been prepared, but it’s super easy to make your own! Just set up a PowerPoint or Google Slides file and format it however you like. Maybe you want a colorful border or a particular font. Use problems that you already have from your resources—textbooks, supplemental books, etc. You’ll need two slides for each problem. On the first slide, type the problem from your resource, leaving out the numbers and the question. On the second slide, add in the numbers. Use a nice big font so students can easily read the problem when projected on your interactive whiteboard. That’s all there is to it!

If you have other questions, add them in the comments below, and I’ll add them to the FAQ. I’d also love to hear how the 3 Reads Protocol is working out in your classroom!

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]]>The post What Great Teachers Do Differently appeared first on Math Coach's Corner.

]]>The title of his presentation was ** What Great Teachers Do Differently**. It was fast-paced and hilarious! I mean this guy could seriously be a stand-up comic. In between all the jokes, though, he was very thought-provoking. Some of the points I really liked:

*If kids come to us lacking something (social skills, behavior management, etc.), we can either whine about it or fix it.**In a great teacher’s classroom, nothing happens randomly. Everything is intentional.**Great teachers have the ability to ignore.*Ignore*does not mean*empathize*,*criticize*, or*roll your eyes*.**Great teachers accept responsibility.*

But probably my favorite thought of the day was his 10-Days-Out-of-10 rule, which basically states that every student deserves to be treated with dignity and respect every day. Period. Along with that, he listed 3 things that have absolutely no place in a school environment:

- arguing
- yelling
- sarcasm

I purchased his book right after the conference and devoured it! He published **the 3rd Edition** in 2020, and it’s on my reading list for this summer. I’ll call it my 10 Year Refresher Course!

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]]>The post Rebranding “Show Your Work” appeared first on Math Coach's Corner.

]]>In math classrooms across the world, students are told on a regular basis to “show their work”. I wish I had a nickel for every time those words came out of my mouth during my educational career. It is certainly done with good intentions—it is critical that students are able to communicate mathematically. Both *communication* and *reasoning and proof* are process standards that should be incorporated into our math instruction.

And analyzing the “work” a student has shown provides the teacher with valuable feedback.

My problem is not with the process, it’s with the words, so I have been experimenting with rebranding “show your work”.

As I see it, there are two major problems with asking students to show their work. First, the words hold a very negative connotation in the minds of students. It’s something they *have* to do. Furthermore, the words are often delivered in a way that is not conducive to cooperation. *“John, if I’ve told you once I’ve told you a million times, you’ve GOT to show your work.” “Valerie, I’m not taking this paper until you show your work!”*

Second, many students don’t show their work because they don’t know what the heck it means! My favorite is the student who circles the multiple-choice answer they think is correct and then x’s out the other choices. If you ask, they are “showing their work”.

For my suggestions, I will address the second problem first. Students have to be specifically taught what it means to show mathematical thinking (see how I’m rebranding it?). This happens through a great deal of modeling and coaching, and it starts by communicating your expectations at the beginning of the year. Students need to understand *why* it’s so important for them to communicate their process. Then you need to teach students *how* to document their thinking. You have to be consistent with your expectations throughout the year because it needs to become a habit for students. Something they just do without thinking about it. Keep in mind, however, that if you’re going to convince students that you are committed to this process, the assignments you give them must be consistent with your message. Which of the two assignments below sends the message that the process is as important as the solution?

Now, to overcome the negative connotation of the words “show your work”, we have to stop using them. Think about it, when you are solving a problem do you think to yourself, “Hmmm, I’ve got to show my work.” I don’t think so. What I DO do, is make notes to myself as I interact with the problem. Those are now my go-to words when working with the students—I **document my mathematical thinking** by **making notes** as I i**nteract with the problem**.

Using this approach, students are more open to the process and, through coaching, they learn how to take notes on their own and determine important information. It’s a thinking process, not a rote procedure.

I’d love to hear your comments about this!

You can grab the Mathematically Speaking problem **using this link**.

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