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Last week, I had a great time teaching & learning division of fractions with one of my teachers. I have not stopped thinking about how models really provide a visual representation of what dividing fractions really mean. Our students use concrete models to gain understanding of what fractions are, how to put them together, take them apart, and take parts "of" it. It is true that when we come to dividing fractions, it gets tricky. This is why models play such an important part.

Here is a picture of a concrete model that shows how to divide ½ by ¼ . Using a concrete manipulative, such as these fraction tiles, allow students to identify the ½ in a whole, and the ¼ in a whole. Then they will think "How many fourths FIT in a half?" or "How many groups of 1 fourth FIT in a half?"

Using fractions tiles is essential when teaching & learning fractions. The great thing about these manipulatives, is that the whole is already identified, and students can use the little fraction tiles to figure out how to divide fractions. These concrete tool gives students different visual representations so they can picture what a half is, or what a fourth is, etc.

THERE IS NO NEED TO INVERT FRACTIONS!

There is no need to teach students strategies such as "flip" the fraction to solve division of fractions. When we provide opportunities for students to visualize what fractions are, they build conceptual understanding, therefore, they can make sense of what dividing fractions is all about.

In this next problem, students solve 4/5 ÷ 1/10 , they understand that this equation means "how many one tenths fit into four fifths?"

Students used a great app called "fraction manipulatives" created by Braining camp, and they used a pictorial representation of what 4/5 looks like, then they made sense of how many 1/10 fit in it. They came to the conclusion that there are eight 1/10 in there. Isn't this great? No tricks, no rules, no algorithms! Students can see what the answer is, by trusting the model!

To build conceptual understanding, students should move from CONCRETE manipulatives, to PICTORIAL representations, to finally move into more ABSTRACT strategies.

Here is one more example: 3/4 ÷ 3/12 or how many 3/12 fit into 3/4?

Then we moved into another problem that brought some productive struggle. Just like we planned in our lesson.

Here is what happened:

The fraction tiles work when the fractions are alike, but when working with fifths and eights, it can get messy, my favorite part of a math lesson, when it gets messy!

"There is a remainder" they said. Students need to understand what to do with this part that is left.

It is not easy to manipulate eights and fifths in the same whole, but a pictorial model can do that.

After they use a pictorial representation, they can start making groups of 1/8 in their model. Students will be able to visualize and understand that 1/8 of the rectangle is equivalent to 5/40. How many groups of 5/40 are in 3/5?

They will count 4 groups and some leftover. The left over is 4/5. Because 5 of those will make 1 more group, since they do not have 5, that becomes its own fraction.

Concrete manipulatives and pictorial representations (models) are made to build conceptual understanding. That is why we need to use problems that make sense, such as ½ divided by ¼, or 4/5 divided by 1/10. To provide our students with opportunities to visualize and build understanding of what dividing fractions really mean.

In grade 6, when we get to fractions like 3/5 divided by 1/8, students can start making connections, and with the conceptual understanding they built in grade 5, start using other strategies that are more efficient for that grade level. They will be able to answer questions like: Why? How do you know? How does that work? How does that look like?
Models are used to build conceptual understanding. To understand WHY a strategy works and HOW it works.
This reflection made me remember a student in grade 5 last year, who could multiply and divide fractions by multiplying numerator by numerator, and denominator by denominator, without making any sense, just getting the answer quickly. She could divide by inverting the fraction.

I asked her if she wanted to play with fractions tiles, and she said yes. I gave her a problem with fourths and twelves. She started making a whole with fourths, then she made a whole with twelves. Then she solved 3/4 by 4/12. She moved the pieces, put the twelves inside the fourths, and she gasped! She got very quiet. I asked "what is happening?" She looked at me, and said: "Miss Caty, is this why 3/12 is equivalent to 1/4? Because 3 of these fit in 1 fourth? Is this how dividing fractions was invented?"

I said: "Yes. Someone had so much fun playing and understanding fractions with models, that they created a shortcut. The short cut is to invert fractions."

Let's explore a lesson using an upside-down approach to teach math. It is basically the opposite from the traditional I-do-you-do approach, where the teacher lectures the class with a specific way to solve a problem depending on the content, and students repeat the same steps the teacher just showed, but using different numbers. In this upside-down approach, or Student-Centered approach, the students do all the thinking, all the problem solving, all the modeling, and all the reasoning to solve a problem.

Some teachers have been finding elements of this upside down approach really effective, for example naming the strategies after the student that first came up with it, is really effective.

It might not sound like a big deal, but students get so excited about trying "their peer's strategies" so much more than using the decomposing strategy (even though it is the same strategy.)

However, there are some other elements of this same approach, that could be more challenging, for example how to make sure that the lesson was effective if the work in students' notebook is unfinished or extremely messy, or if students chose to solve problems with a strategy that is not the most efficient.

Let's explore a 2nd grade lesson on solving problems of addition and subtraction with re-grouping with my friend and colleague Richard Ziegler.

Number Talk

Mr. Ziegler's first move is to start with a quick number talk, to warm up his students' brains. Sometimes, he uses an equation such as 26 + 36 and students solve it in their head. Only mentally. Then he asks students to share with each other how they solved.

After that, Mr. Ziegler asks a few students to explain how they solved. They explain step by step, and he uses numbers and pictures to represent the strategy that the students are describing. It is an art!

Sometimes Mr. Ziegler uses a math talk, it could be an image, and the questions could be: What do you notice? What are you wondering? How do you know? Convince your partner. The process is the same, he asks students to think about it on their own, mentally, then share with a partner, and finally, to share with the class.

Check out the books "How Many?" and "Which One Doesn't Belong" by Christopher Danielson. They are great resources for math talks.

I will write another blog on number talks soon. Stay tuned!

Mini-lesson

The easiest way to use numberless problems is to use the slow-release function in a power point presentation. This will allow you to release information about the problem little by little.

If you do not want to use a computer, you could create some strips of paper and use them little by little, just as the slow-release function works on the power point slides.

The mini-lesson starts here. What are dragonflies? Have you seen them? What are toads? There are pictures on the slides to make the context very clear to students. There are no numbers yet. Students are not trying to guess answers quickly. They are wondering about this real situation.

Release one more part of the problem.

Mr. Ziegler then asks: What can the numbers be? What numbers could not be? Students here explore estimation. It is about building number sense.

Students start talking about many possibilities. They make sense about the fact that the snake ate more than the toad. How many more? "We do not know yet" they say.

Students are engaged, they want to know more, how many? How many? Please tell us Mr. Ziegler!
"How do you make students engage in math class? Give them a problem to care about!" Steve Leinwand at MSIS 2017.
The next slide reveals that the snake ate 26 more dragonflies than the toad ate. How many did the toad eat? The toad ate 39.

Mr. Ziegler pauses. "Hmmm. What can the question be?"

Multiple hands go up!

This is an essential teaching move, as students learn that there are many answers that can be found with this information, they begin to understand that there is not only one answer to a problem, it depends on what we are figuring out. There are many questions out there, and also many answers.

Students come up with all sort of questions:

*How many dragonflies did the snake eat?

*How many dragonflies did they eat altogether?

*Why do they eat the dragonflies? Why don't they eat grass?

Some questions don't make sense. Get ready for that. Ha!

Mr. Ziegler then reveals the question!

"Ahh, I knew it" says one student.

This was the mini-lesson. The goal is to spend 8-10 minutes talking about the context of the problem. Having some student to student discourse about what the problem is about, and what are we going to figure out. The teacher is a facilitator.

At this point the teacher has not asked students to use any specific strategy. The questions are about the context, and having a clear idea about what is it that we want to find out.

Students can solve this problem in many different ways. They can use manipulatives, or draw models, or use numbers flexibly. Students already have ideas about how to solve problems, let's hear them out.

This problem has 3 different combinations of numbers. Students can choose to start with the first set of numbers, and move on to another set. Or they can solve just the first set. It is not about quantity of addition facts, but quality of problem solving strategies. Students are encouraged to use different strategies to solve the different sets. This problem allows for differentiation.

Active Engagement﻿

"Is everyone clear about what we need to do now?"

"Off you go," says Mr. Ziegler.

Students now go to their tables and work in pairs. They can use concrete manipulatives, drawings, or numbers.

The role of the teacher now is to walk around and identify who needs support, or who has some clarifying questions. It is here, where the teacher can pull a small group, and perhaps review a strategy that was shared the day before and that this group might need to review again.

The teacher can also observe if there are students that need an extra challenge, maybe there are students that solved all 3 sets of numbers.They can come up with another question using the same context. There are opportunities of differentiation when using numberless problems, and also by extending the questions.

The teacher could be talking to different groups or, individual students and checking for understanding. Also this is the opportunity to individually ask students to try new strategies. For instance, if a student has been using the same strategy for the past few lessons, the teacher would suggest he/she tries a new strategy as he/she seems to have mastered the previous one.

Mr. Ziegler chooses to sit with a group of students and is asking them questions about the strategy they are going to use. I hear him ask a student if he could also use a number line to solve this problem. His voice is very melow and friendly, it sounds like an invitation, not a challenge. The student says "yes, of course I can!" Mr. Ziegler is such a talented teacher.

This student solved using the decomposing strategy, and then took a risk and solved with a different strategy.

I sat with a student that used the number line to solve all three sets of numbers. I asked her in a few different ways if she would be willing to try a new strategy, but she said no. She really liked using the number line, she said. Later in the day, during our debrief session, I made sure to let Mr. Ziegler know about this, so he could use his amazing and inviting voice next time so he can encourage this girl to try a new strategy.

This student chose to use the same strategy for all three sets of numbers.

Share-Out

"It is time to come back to the carpet" says Mr. Ziegler. "Now we are all going to share some of the strategies we use in 2-RZ."

This is the most essential part of the lesson because it is here that students will be exposed to a variety of strategies, and the teacher might introduce a new strategy.

During the active engagement part of the lesson, the teacher is walking around and talking to different groups or students to check for understanding, and also choosing the strategies that will be shared today in order to hit the learning target.

Students come up to the front and show their work under the document camera or any other device that helps students have a good visual. It is recommended to have 3 or 4 students share their strategies, this gives the rest of the group a variety of different ways they can try the next day.

If a student comes up with a strategy for the first time, the teacher might ask the class if they should call this strategy "Laura's strategy" or the name of the student that came up with that particular strategy. Believe me, naming the strategies with students' names makes the strategy suddenly very, very polular!

If most students used a similar strategy or not a very efficient strategy, it is here where the teacher can share an efficient strategy that he/she would have used. This strategy then, is one more option, but not the only option. Making an emphasis on how efficient this strategy might be, and/or giving the new strategy a special name, will make your students really want to try it.

Mr. Ziegler had 2 students share their strategy. The first student shared how she solved using pictorial base ten blocks. She explained how she added the tens, and made 5 tens. Then she explained how she added the ones and got 15 ones, so she made another ten and added it to the 5 tens she previously had. She explained how she ended up with 6 tens and 5 ones, which is 65 dragonflies.

Student #1 used a pictorial base ten blocks strategy.

The second student came to the front and showed how he had solved the problem by using the decomposing strategy. He explained step by step, how very similar to her classmate, he had added the tens first, then the ones, and added a new ten. He also found that 65 dragonflies was the solution.

Student #2 solved by using the decomposing strategy.

Mr. Ziegler then came and did his magic! He showed to the class how these two strategies are connected! You can see on the picture, how he is showing both strategies at the same time, and allowing students to make the conenction, because both strategies are based on place value. They just had different visual representations, that is all. Magic.

Mr. Ziegler then guided students to see the connection between both place value strategies.

If you run out of time, or you don't have enough time in your lessons, you can start you lesson with the "share out" part from the day before. If you have enough time during your lesson (some teachers have 60 minutes for each lesson) you could try to start with a slide for "Some of thework from yesterday...." and you could have a quick teaching point about a very efficient strategy before starting your lesson for today. Then do all the other parts of the lesson. This is ideal as you get 3 teaching points throughout your lesson.
This could be the overall structure of a Student-Centered Lesson:
Part 1- Some of the work from yesterday...... (teaching point) 4-5 min

Part 2- Mini lesson (what is the problem about/context) 8-10 min

Part 3-Active engagement (teaching points to individual students or small groups) 15-20 min

Part 4-Share out (learning happens here! Multiple teaching points) 10-15 min

Something to consider....

Before you try this approach to teaching math, you need to be comfortable with students trying the different strategies at different times.

One of the biggest concerns is that sometimes you want your students to try a specific strategy right away, that day, so you can assess their work and see if they get it. But with this approach, this might take a few days. So if you are patient, your students will start making sense of what are the most efficient strategies to use, and will be able to explain it themselves, and apply it.

This approach allows them to own their learning. It ensures a balance between content standards and mathematical practices, which makes a balanced math approach to learning.

Connection to Mathematical Practices

Which math practices are used on a lesson like this?

They use problem solving skills, by choosing the most efficient strategy on their own. Making sense of what the problem is about to choose how to go about it. (The opposite of following a set of rules.)

They use modeling to solve problems, by showing how their strategy works with a visual representation. This representation helps them illustrate the problem and find the answer.

They use reasoning & communicating skills, by convincing their classmates about how their strategy works.

Give it a try and let me know how it went!
If you want to dive deep into this approach, read the book "Teaching Student Centered Mathematics" by John A. Van de Walle

As teachers, we continue to learn new approaches to explore mathematics and are faced with new opportunities to provide the best learning experiences for our students. We are constantly participating in different professional learning to keep our teaching practices in line with the most up-to-date research on how students best learn. We collaborate constantly with fellow teachers, administrators, and instructional coaches to learn from each other, and to make sure our classrooms are true learning environments.

作为老师，我们一直在学习对数学进行探索的新方法，同时还面临着向我们的学生提供最佳学习体验的机会。我们不断参加各种职业培训，以在给学生提供最好的学习体验这件事上让我们的教学实践跟上最新的研究成果。我们不断与其他老师、行政管理人员以及指导教练合作，相互学习，来保证我们的课堂是真正适合学习的环境。

I am also a parent of twins in elementary. As parents, we want to be informed and involved in our child's education, so we can help at home. We engage with our kids asking questions about what they learned at school at every dinner conversation. We want to be a positive influence on our kid's emotional and academic life.

正是因为这些原因，作为上海美国学校浦西校区的小学部数学指导教练，作为正在读四年级的双胞胎的家长，我开始参与到我们的家长社区中来。在过去两年的时间里，我们举办了多场"数学交流会"，介绍了我们在上海美国学校教授和学习数学的方法。自第一场活动吸引了90位家长以来，这个活动的反响非常好。

These first meetings were about sharing with our parent community that recalling facts, procedures, and formulas is merely arithmetic, and while it seems to "work" for elementary and middle school, it does not prepare students to solve real mathematical problems, which is the main goal of mathematics: to solve "messy" or "unfamiliar" complex problems. Mathematics is about making connections and solving problems by making sense of numbers, by using numbers flexibly, by using concrete and visual representations to build understanding. When students truly understand a concept and they can visualize it, use a model or representation, and can explain it, then they are ready to make a connection to a new math concept where they can apply their knowledge accordingly to solve the new "messy" problem.

在这些活动中，我们让学生家长认识到，记住数学事实、程序和公式仅仅是停留在算术层面，尽管这似乎对于小学和中学来说"奏效"，但这并不能让学生解决真正的数学问题，解决"棘手的"或"不熟悉的"复杂问题才是数学教学的主要目的。数学就是要对数字进行理解，以灵活的方式使用数字，使用有形和直观的方法帮助理解，从而建立联系，解决问题。当学生真正理解一个概念，能够对其进行可视化呈现，使用模型或表征，并对其进行解释的时候，他们就能够与全新的数学概念建立联系，运用他们学到的知识解决新出现的"棘手"问题。

Parents continued to want to learn more about this conceptual approach to learning mathematics, so we invited them to participate in four sessions called "Math in March."

家长们希望进一步了解数学学习的概念化方法，于是我们邀请他们参加了四场名为"三月数学"的活动。

We started by showing the progression of addition and subtraction strategies our students use to build understanding in solving problems. Parents were engaged using a variety of CONCRETE manipulatives, to make sense of place value when adding and subtracting without the need to use any rules or tricks such as "carry" or "borrow". They were able to use place value strategies to understand how to make tens, or how to "unbundle" a ten when subtracting. They continued to use PICTORIAL representations such as tape diagrams or number lines to model the problems. They understood the difference between modeling and fake modeling (which is just drawing an answer found by an algorithm). Using visual representations is how we really MODEL a problem, then we can choose other strategies to help us solve.

首先我们向家长们展示学生们用来解决问题的加减策略。家长们使用一系列"有形"教具在加减运算过程中对位值进行理解，无需使用"进位"、"借位"等任何规则或技巧。他们能够使用位值策略理解如何加到十，或者是在进行减法运算时如何对十进行"拆分"。随后，他们使用尺图、数轴等"图形"表示对问题进行建模。他们理解了真建模和假建模（只是用算法求出答案）之间的差别。我们使用直观的方法对问题进行建模，随后我们可以选择其他策略帮助我们解决问题。

We moved onto ABSTRACT strategies and then expanded into multiplication and division, and in later sessions into fractions. (Fractions were a hit with the parent community. They were so excited to use fraction tiles, unifix cubes, number lines, area models, and tape diagrams to understand how to add, subtract, multiply and divide fractions conceptually!)

接着我们介绍了"抽象"的方法以及乘除法和分数运算。（分数深受家长喜爱。他们非常兴奋地使用分数瓦片、方块积木、数轴、面积模型和尺图理解如何从概念上对分数进行加减乘除！）

The constant sounds of "ahhhhh" and "ohhhhh" were just fabulous when parents were shown a visual representation of what ½ x ¼ looks like! Parents were able to see past the formula and tricks and able to see in a new, clearer way what one half of one fourth really looks like, and why the product is one eighth. The room was loud, Chinese and English language, iPhone flashes, unifix cubes everywhere! Our parents were doing mathematics!

当看到 ½ × ¼ 的直观表示时，家长们发出了一阵阵的惊叹声。家长们能够透过公式和技巧以更加清晰的全新方式看到四分之一的二分之一到底长什么样以及乘积为什么是八分之一。家长们议论纷纷，有说中文的，也有说英文的。大家纷纷拿出手机拍照，到处都是方块积木。看哪，家长们在学数学！

Annie Barnard, mother of 5th grade and kindergarten students, said of the experience, "Most of us were impressed with simple questions we had been taught in the past and knew how to do, but never got to understand why. You taught us to visualize the math. Seeing the answer make sense in front of our eyes really gave us a good vibe for the first time, and got us wanting to learn math again. We're now helping each other understand how to solve math problems without worrying about not being able to remember the right way - because there isn't one way to solve math, there are many ways."

作为五年级学生和幼儿园学生的妈妈，Annie Barnard表示："让我们大多数人印象深刻的是那些我们过去学过并且知道如何解答但从不知道其中缘由的简单问题。我们在用直观的方法学习数学。亲眼看到答案呈现在眼前真可谓让我们眼前一亮，也让大家升起了再次学习数学渴望。我们现在能够相互帮助，理解如何解答数学问题，而不用担心记不住正确方法，因为解答数学问题并不只有一种方法，而是有很多方法。"

What can parents do at home to support their child to feel successful in mathematics?

家长们在家里能够做些什么，从而让他们的孩子在数学上获得成就感呢？

*If you find a mistake in their homework, instead of saying, "That is the wrong answer, do it again." Try saying, "guide me through this problem, how did you get that answer?" Most likely, when explaining their strategy, they will catch their own mistake.

*如果你在他们的家庭作业中发现了错误，不要说"这个答案不对，重做一遍"，而是要说"给我讲讲这个问题，你是如何得到这个答案的"。大多数情况下，他们在解释做题策略时，就会发现自己的错误。

*When doing homework, ask: "Can you think of a different way to solve that?"

*在做家庭作业时，你可以说："你能不能想出解这道题的另一种方法？"

*Play with numbers, for example: find all the different ways to make 24, or all the different ways to solve 16 x 9.

*玩数字游戏，例如：找到得到24的所有方法或解答16 × 9的所有方法。

*Solve math puzzles.

*解答数学难题。

*Play math games to practice fluency, fun games, not flashcards!

*通过玩数学游戏做到熟练掌握，要玩有趣的游戏，而不是数字卡片！

Across both campuses, at SAS we use a balanced approach to teaching and learning mathematics. Teachers and students not only focus on concepts and procedures, they focus on the use of visual representations to model real-world situations, they problem solve, they communicate and reason in their thinking, and make connections to their future learning.

在上海美国学校的两个校区，我们使用一种两者兼顾的方法开展数学教学。老师和学生不仅专注于概念和程序，还专注于使用直观方法来对现实生活情形进行建模，解决问题，沟通并推理他们的想法，与未来学习建立联系。

If you are introducing fractions in your class, allow students to use unifix cubes to make sense of what fractions really are and look like.

You can start with fractions that have the same denominators so students understand that the whole is made of a number of parts. A fraction is part of a whole. You can have 1 whole divided in 4 parts, and you can have 1 whole divided in 8 parts. The most important thing is for students to understand that the wholes have to be the same size. For instance 1/2 small pizza and 1/2 large pizza do not make 1 whole pizza because the wholes are differen sizes.

After students have had a lot of opportunities to explore adding a lot of fractions with the same denominators, you could start posing problems with "alike" denominators. For instance start with a problem like 1/4 + 1/8. Give students unifix cubes and let them figure out how they could represent these 2 fractions using the same whole as a reference.

After they represent these 2 fractions with the same whole, they can add the 2 fractions. When students work in pairs, they can help each other to make sense of how to represent eights and fourth using the same whole. It might take some time, but they will get there.

Step 1- Ask students: using unifix cubes, how can you make 1 whole that can be divided in "eights" and also in "fourths"?

Ask students to SHOW how their whole can be divided in fourths and eights. Some students might make a whole with 4 unifix cubes only, and when they try and divide that in eights, they will realize it doesn't work. They will try again. Do not solve for them.

Have students explain their thinking, and share with the class how did they get to that conclusion.

After they have their fractions represented, they can identify 1/8 and 1/4.

Step 2- Ask students to show 1/8 of the first whole, and 1/4 of the second whole. Label each fraction. Students will get a visual representations of what both fractions look like related to one another. It has to make sense to them.

Step 3- Ask students to put their fractions together and find an answer. If students answer 3/4, ask them to go back to what they identified as 1/4, ask: "Does that make sense?" They can discuss and reflect on it, and come to understand that each of those 3 unifix cubes look just like the 1/8 from step 2.

This whole time, students do not need to use any paper or pencil, and defnitely no need to learn a set of rules to find the common denominator. That will come later. Students will make sense of it on their own. Just give it time.

Concrete & Pictorial Representations

Here is one more example to use in your class, to allow students to make sense of adding fractions with alike denominators without using memorized steps. These strategies allow students to understand why if you are adding thirds and sixths, the answer can't be ninths! Because they have never seen ninths in the representations they are creating.

Step 1 - Pose the problem on the board. 4/6 + 1/3. Ask them to use unifix cubes to represent sixths and thirds using the same size whole. Let them struggle a bit. They can work in pairs, and talk to their partners. They will start with thirds and eventually will make sense of using 6 unifix cubes to also represent thirds. Have students explain to each other how they came up to that conclusion. Explaining their thinking is for their own learning, some students finish making sense of what they learned, when they hear themselves explain, when they organize their thoughts, and make sense of it in words. Learning is fascinating!

Step 2 - Ask students to label their fractions.

Step 3- What is the total fraction now? Have students explain why their answers is 1 whole. Have them show how these 2 fractions make 1 whole. This is a perfect moment to ask students if they could come up with other models to represent this same problem. Perhaps they can use fraction tiles, or fraction circles. They can also draw the fractions if you do not have other manipulatives in your class.

There are many concrete manipulatives to allow students to build conceptual understanding of fractions.

Finally, it is time to use pictorial representations. The bar model or tape diagram, is a recomended model to represent fractions. It is easier for students to use a rectangular model than a circular model to be more accurate with the equal number of parts.

Most students will start using 2 tape diagrams, and that is ok, as long as the 2 wholes are the same size.

Eventually they should start partitioning the same whole in both sixths and thirds. They can use 2 different color markers to see their work. This is a good strategy for when they start sbtracting fractions.

Use a variety of good problems to help students see where we use fractions in the real world. Here are some ideas:

If Billy has 35 cookies and shares 1/5 of them, what fraction does he have left?

If I read ¾ of my book, and I have 80 pages left, how many pages is my book?

I am running a 3 km race. I am almost there. I have 1/6 left. How much do I have to run?

Is the sum of 4/5 and 7/8 under or over one? Explain how you know.

Enjoy playing fractions with your students!

I have been teaching for about 29 years. Most of those years in elementary. Math has always been my most favorite class to teach, but as most teachers did in the past, I used the I-do-you-do approach. I did my best at explaining how to solve problems using manipulatives and pictures, and quickly moved into abstract strategies using tricks, and helping my students memorize steps. I remember how subtracting with re-grouping was always a "hot topic" in my classes.

I thought I was doing justice to my job by taking extra time to review the steps to re-group when solving subtraction. I was patient and used small group instruction to review the steps. I offered extra sessions during recess to review steps for subtracting using the algorithm efficiently. I thought this was best practice. This was back in the early 2000's.

It wasn't until several years later, when I started using a student-centered-approach to teaching and learning math. That was one of the best years in my long journey as a teacher. That was the year when my students dicovered on their own, what re-grouping really means, and why we use it. That was the first year, that re-grouping made sense!

It is ideal to teach addition and subtraction strategies at the same time. Math problems can be solved by adding or subtracting. Use concrete manipulatives for students to see what re-grouping looks like.

When students can use concrete manipulatives, like these base ten blocks, they can make sense of what "take-away" looks like. For instance, when taking away 128 from 274, students will start by taking away the hundreds, to then take away the tens.

Then they have 154 left. When they want to take away 8 from the ones, they will realize they do no thave enough tens, so they organically will "exchange" a ten for 10 ones, and take the 8 ones from there. At this point, there are no rules or tricks. Students know that they have enough to take away 8, they can SEE that they do not have enough ones but they have enough tens.

It is important to allow students to work in pairs, so they discuss this, so if one of them is not thinking of a strategy, the other student will probably come up with it, and all of a sudden, no one knows who came up with it, it was their work combined!

After that students just see what they have left. They have 1 hundred, 4 tens, and 6 ones.

It is important to allow students to use manipulatives for a few days, so they really learn to trust the process of using these base ten blocks to find the answer. A lot of students do not trust this process and quickly use an algorithm to find the solution, an dthen just represent the answer with the base ten blocks. That is fake modeling. That is not using manipulatives to build understanding.

After students get use to using manipulatives, and understanding how to make sense of place value, they can explore other strategies like "subtracting by parts" where they decompose the number, and subtract part by part.

When students develop strong number sense, and use numbers flexibly, they could come up with other strategies like "subtracting by place." It is ideal to allow this strategy to organically emerge from one of the students, instead of teaching it. When they use their own words to explain a new strategy, it suddenly makes sense.

I have come to realize that when your main focus on the lesson the problem, with engaging context, students really care about the context, and feel free to explore different strategies. They are not focused on using one strategy efficiently, they care about the math that is involved to figure out that good problem.

Today, I do not have to teach the algorithm strategy, nor do I teach any tricks and steps to solve a subtraction problem, because when students build understanding of using place value strategies with base ten blocks, and using decomposing strategies, they make sense of how the algorithm works. They figure out that algorithm tricks like "borrow from the "neighbor", is nothing but realizing that if you don't have enough ones, well, then you take it from the tens!

Do you have students that solve ALL problems using only the traditional algorithm? Do they stop and think about the strategy that could make the most sense for that particular problem?

Here are some strategies that support students in building understanding of what adding really means, before using "carry-on" as a set of rules to be followed to solve any problem.

It is essential for students to move from CONCRETE manipulative, to PICTORIAL strategies, to end in more ABSTRACT strategies. This progression will bring understanding and fluency, as students will have a variety of mental strategies to solve problems.

It is important to allow students to explore concrete manipulative to understand how to bundle ones into tens, and tens into hundreds when it is necessary.

When solving a problem with manipulative, students tend to start putting together the hundreds, then the tens, and the ones.

When students get more than 10 ones, they immediately re-group them and put them with the other place value that it belongs.

Then they count how many of each and find an answer. There are no rules or procedures, the concrete manipulative allows them to make sense of what they have in all.

After using these concrete experiences to make sense of addition with regrouping and making sense of place value, students can move into pictorial strategies. One strategy could be using the place value chart. Students can use real place value discs, or draw the place value discs.

Then they make a connection with the previous concrete strategy, and will know what to do with more than 10 ones.

Use the term "bundle" for students to understand that 1 ten is a bundle of ten ones.

Students can make a connection between using the manipulative and using this place value chart. It is not a new strategy, it is a connection. Same strategy different visual representation.

Now it is time to move into more abstract strategies but still based on place value, not based on a memorized set of rules. For example this expanded form strategy is another visual representation of the exact same thing they did using the place value chart.

Again, students will likely start by adding from left to right (unlike the algorithm), and will make sense of what to do with more than 10 tens, and more than 10 ones. They had the previous experience of "exchanging" the manipulative, and "bundling" the place value discs. They do not know what re-grouping even means, but they understand what the value of these numbers are in all.

Finally, students can make a connection with another visual representation. This abstract strategy is usually called partial sums.

Students can see the same values than before, they will start by adding 300 and 110, and can mentally solve using the visual representation they have built previously, and come up with 410, then mentally again add 13 by breaking it down into 10 and 3.

When students become fluent with these strategies, they will eventually make a connection with the traditional algorithm, and they will finally understand that "carry" is nothing more than "bundling", but bundling makes sense.

Even if the math program you use in your school encourages students to solve 15+ addition problems using the standard algorithm, you could make a decision and create an environment of exploring the beauty of mathematics, and allow your students to solve 3 or 4 addition problems using CONCRETE-PICTORIAL-ABSTRACT strategies to really understand how math works, and how strategies based on place value are all connected!
"As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research and this has meant that unproductive and counter-productive classroom practices continue." Jo Boaler, Fluency Without Fear.

By inviting children to solve problems in their own ways, we are initiating them into the community of mathematicians who engage in structuring and modeling their "lived worlds" mathematically. Fosnot and Jacob, 2007
Teaching mathematics through problem solving requires you to think about the types of tasks you pose to students, how you facilitate discourse in your classroom, and how you support students use of a variety of representations as tools for problem solving, reasoning, and communication.

This is a different approach from "do-as-I-show-you" approach where the teacher shows all the mathematics, demonstrates strategies to solve a problem, and then students just have to practice that exact same skill/strategy, perhaps using a similar problem.

Teaching mathematics through problem solving means that students solve problems to learn new mathematics through real contexts, problems, situations, and strategies and models that allow them to build concept and make connections on their own.

The main difference between the traditional approach "I-do-you-do" and teaching through problem solving, is that the problem is presented at the beginning of the lesson, and the skills, strategies and ideas emerge when students are working on the problem. The teacher listens to students' responses and examine their work, determining the moment to extend students' thinking and providing targeted feedback.

Here are the 4 essential moves in a math lesson using a student-centered approach or problem-solving approach:

Number Talk (5-7 min) (Connection)

The mini-lesson starts with a Number Talk. The main purpose of a Number Talk is:

*to build number sense, and

*to provide opportunities for students to explain their thinking and respond to the mathematical thinking of others.

Please refer to the document Int§roducing Number Talks. Or watch this video with Sherry Parrish to gain understanding about how Number Talks can build fluency with your students.

Here are some videos of Number Talks so you can observe some of the main teaching moves.

The role of the teacher during a number talk is crucial. He/she needs to listen carefully to the way student is explaining his/her reasoning, then use a visual representation of what the student said. Other students also share their strategies, and the teacher represents those strategies as well. Students then can visualize a variety of strategies to solve a problem. They learn how to use numbers flexibly, there is not just one way to solve a problem. When students have a variety if strategies in their math tool box, they can solve any problem, they can make connections with mathematical concepts.

There could be multiple answers for these kind of math talks. Students make sense of their answer and convince the class using their reasoning and math vocabulary.

There are a variety of resources that can be used for Math Talks. Note: the main difference between Number Talks and Math Talks, is that one allows students to use numbers flexibly leading them to fluency, develop number sense, and opportunities to communicate and reason with mathematics; the other allows for communicating and reasoning, building arguments to critique the reasoning of others, the use of logical thinking, and the ability to recognize different attributes to shapes and other figures and make sense of the mathematics involved.

2. Using problems to teach (5-7 min)

Problems that can serve as effective tasks or activities for students to solve have common features. Use the following points as a guide to assess if the problem/task has the potential to be a genuine problem:

*Problem should be appropriate to their current understanding, and yet still find it challenging and interesting.

*The main focus of the problem should allow students to do the mathematics they need to learn, the emphasis should be on making sense of the problem, and developing the understanding of the mathematics. Any context should not overshadow the mathematics to be learned.

*Problems must require justification, students explain why their solution makes sense. It is not enough when the teacher tells them their answer is correct.

*Ideally, a problem/task should have multiple entries. For example "find 3 factors whose product is 108", instead of just "multiplying 3 numbers. "

The most important part of the mini-lesson is to avoid teaching tricks or shortcuts, or plain algorithms. Our goal is always to help guide students to understand why the math works (conceptual understanding). And most importantly how different mathematical concepts/ideas are connected! "Math is a connected subject" Jo Boaler's video

"Students can learn mathematics through exploring and solving contextual and mathematical problems vs. students can learn to apply mathematics only after they have mastered the basic skills." By Steve Leinwand author of Principles to Action.

3. Active Involvement (20 min)

This is the opportunity for students to work with partners or independently on the problem, making connections of what they know, and trying to use the strategy that makes sense to them. Always making sure to represent the problem with a visual representation. It can be any model that helps student understand what the problem is about.

The job of the teacher during this time, is to walk around asking questions to students to guide them in the right direction, but without telling too much. Allowing students to come up with their own solutions and justifications.

Teacher can clarify any questions around the problem, not the solution.

Teacher emphasizes reasoning to make sense of the problem/task.

Teacher encourages student-student dialogue to help build a sense of self.

Some lessons will include a rich task, or a project based learning, or a number problem (find 3 numbers whose product is 108). There are a variety of learning target tasks to choose from, for each grade level on the Assessment Live Binders website created by Erma Anderson and Project AERO.

Again, keep in mind that some lessons will follow a different structure depending on the learning target for that day. Regardless of instructional design, the teacher should not be doing the thinking, reasoning, and connection building; it must be the students who are engaged in these activities

4. Share (10-12 min) (Link)

The most crucial part of the lesson is here. This is where the teaching/learning happens, not only learning from teacher, but learning from peers reaching their unique "zone of proximal development" (Vygotsky, 1978).

We bring back our students to share how they solved their problem. Sometimes they share with a partner first, to make sure they are using the right vocabulary, and to make sure they make sense of their answer. Then a few of them can share with the rest of the class. But sharing with a partner first is helpful so everyone has the opportunity to share.

"Talk to each other and the teacher about ideas - Why did I choose this method? Does it work in other cases? How is the method similar or different to methods other people used?" Jo Boaler's article "How Students Should Be Taught Mathematics."

Students make sense of their solution. The teacher listens and makes connections between different strategies that students are sharing. Teacher paraphrases the strategy student described, perhaps linking it with an efficient strategy.
"It is a misperception that student-centered classrooms don't include any lecturing. At times it's essential the teacher share his or her expertise with the larger group. Students could drive the discussion and the teacher guides and facilitates the learning." Trevor MacKenzie
If the target for today's lesson was to introduce the use a number line, for example, this is where the teacher will share that strategy as another possible way to solve today's problem!

This could also be a good time for any formative assessment, using See Saw, using exit slips, or any kind of evidence of what they learned today.
References.
"Teaching Student-Centered Mathematics" Table 2.1 page 26, Van de Walle, Karp, Lovin, Bay-Williams

"Number Talks", Sherry Parrish

"How Students Should be Taught Mathematics: Reflections from Research and Practice" Jo Boaler

"Erma Anderson, Project AERO Assessments live binders?

"Principles to Action", Steve Leinwand

" Turning Teaching Upside Down ", by Cathy Seeley

"Four Inquiry Qualities At The Heart of Student-Centered Teaching"

By Trevor MacKenzie

"The Zone of Proximal Development" Vygotsky, 1978

*** Here is a link to my favorite places to plan Math at SAS padlet, you will find a variety of resources, videos, articles, etc. By Caty Romero

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Last week, I had a great time teaching & learning division of fractions with one of my teachers. I have not stopped thinking about how models really provide a visual representation of what dividing fractions really mean. Our students use concrete models to gain understanding of what fractions are, how to put them together, take them apart, and take parts "of" it.

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Shanghai American School Disciplinary Transfer Goals

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Turning Teaching Math Upside Down

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Video: Modeling Algebraic Thinking CCSS from Grade 2- IM3 at SAS

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Video: Modeling Algebraic Thinking CCSS from Grade 2- IM3 at SAS

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Modeling in mathematics CCSS algebraic progression

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Video: Flexible Thinking and Conceptual Understanding at SAS

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Video: Flexible Thinking and Conceptual Understanding at SAS

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Flexible Thinking & Conceptual Understanding

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Video: Progression of addition & subtraction

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Video: Progression of addition & subtraction

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Learning targets

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Learning targets

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Learning targets

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Learning targets

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Learning targets

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With clarifications

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PLC

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Paul Lockharts

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K-5

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by Cathy Seeley

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Article: Never Say Anything A Kid Can Say

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Article: Never Say Anything A Kid Can Say

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by Steven C. Reinhart

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Article: Memorizers are the lowest achievers

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Article: Memorizers are the lowest achievers

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by Jo Boaler

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It's time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences.

All children are different in their thinking, strength and interests. Mathematics classes of the past decade have valued one type of math learner, one who can memorize well and calculate fast.

Yet data from the 13 million students who took PISA tests showed that the lowest achieving students worldwide were those who used a memorization strategy - those who thought of math as a set of methods to remember and who approached math by trying to memorize steps. The highest achieving students were those who thought of math as a set of connected, big ideas.

The U.S. has more memorizers than most other countries in the world. Perhaps not surprisingly as math teachers, driven by narrow state standards and tests, have valued those students over all others, communicating to many other students along the way - often girls - that they do not belong in math class.

The fact that we have valued one type of learner and given others the idea they cannot do math is part of the reason for the widespread math failure and dislike in the U.S.
Related: Common Core math experts say teachers need to stop using shortcuts and math tricks
Brain science tells us that the students who are better memorizers do not have more math "ability" or potential but we continue to value the faster memorizers over those who think slowly, deeply and creatively - the students we need for our scientific and technological future. The past decade has produced a generation of students who are procedurally competent but cannot think their way out of a box. This is a problem.

We don't need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.

Mathematics is a broad and multidimensional subject. Real mathematics is about inquiry, communication, connections, and visual ideas. We don't need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.
Related: Should we stop making kids memorize times tables?
Technology leaders are publically arguing that calculation is not math, and that math is a much broader subject. Conrad Wolfram, one of the leaders of one of the world's most important mathematics companies, Wolfram-Alpha, urges schools to stop emphasizing calculating and focus instead on problem solving, modeling, thinking, and reasoning as these are the mathematical abilities that students need in the workplace and their high tech lives. This broad, multidimensional mathematics is the math that engages many more learners and puts them on a pathway to life long success.
Related: Testing gives third-graders upset stomachs, tears, even fevers
Part of the problem in the U.S. is the desperation of many parents to advance their children in math, pushing them to higher levels of math faster and sooner, somehow believing that a resume packed with advanced math courses will guarantee their future. Bill Jacob, a mathematics professor at the University of California, Santa Barbara, speaks openly about the dangers of students being pushed to higher levels of mathematics too soon. "I know it is hard to persuade parents that their students shouldn't race to get calculus, but I really wish they wouldn't. Too much content and depth is left out when they do." said Jacob, who is not alone in saying that he would rather have students in his university mathematics courses that have breadth in their mathematical experiences than any additional Advanced Placement courses. Experts in England are giving the same advice to parents of high achieving students. Geoff Smith, chairman of the British and International Math Olympiads warns that accelerating children through the system is a "disaster" and a "mistake". He, like me, recommends that high achieving students explore the mathematics they are learning in depth, instead of rushing forward.

Mathematics is not a subject that requires fast thinking. Award winning mathematicians talk about their slow, deep thinking in math. Fields Medal winning mathematician Laurent Schwartz wrote in his autobiography that he felt stupid in school because he was one of the slowest thinkers in math. Eventually he realized that speed was not important - "What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant."

Some school districts, such as San Francisco Unified, are trying to slow down the math experience, requiring that advanced students go deeper rather than faster. Students still reach calculus but the pathway to calculus consists of deep understanding rather than procedures and memorization. This is an important move. There is no harm in students being introduced to higher-level mathematics earlier, as long as the mathematics is enjoyable and ideas can be explored deeply. Third graders can be fascinated by the notion of infinity, or the fourth dimension, but they do not need a race through procedural presentations of mathematics.
Jo Boaler is a professor of mathematics education at Stanford University, co-founder of This story was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Read more about Common Core. www.youcubed.org, and author of What's Math Got To Do With It: How Teachers and Parents can Transform Mathematics Learning and Inspire Success. (Penguin, 2015)
New brain science tells us that no one is born with a math gift or a math brain and that all students can achieve in math with the right teaching and messages. The classrooms that produce high achieving students are those in which students work on deep, rich mathematics through tasks that they can take to any level they want. No one is told what level they can reach and no one is held back by narrow questions that limit students' mathematical development and creativity.

Many people across the U.S. have gross misconceptions about mathematics learning, thinking that mathematics is a narrow subject of memorization and speed and that people who do not calculate fast or memorize well are not 'math people'. We need to change the conversations about mathematics, communicating to all children that they can learn. We also need to change the way math is taught, valuing the different ways of thinking that are so important to the subject. Mathematics, itself, needs this and although change is hard, it should be embraced.

We need to broaden mathematics and open the doors of mathematics to all students. When we do this we will see many more creative, energized young people equipped to think quantitatively about our ever-changing world. We all need this.

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Memorizers are the lowest achievers and other Common Core math surprises

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It's time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences. All children are different in their thinking, strength and interests.

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Building Fluency

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Multiplication, division, fractions.

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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Assessment resources

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You have been invited to view a private LiveBinder named: “AERO Mathematics Assessment Resources”. Please enter the access key that was given to you with this link. If you have not received an access key, please contact the author of this binder.

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By Caty Romero

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Modeling with manipulatives in Middle School

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Modeling with manipulatives in Middle School

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Shanghai American School sample