# That Day We Added It All Up 那一天，我们聚在一起学数学

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.

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.

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.

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.

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!)

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!

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.”

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

*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.

# Making Sense of Adding Fractions

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.

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!

# What does re-grouping really mean?

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.

# Teaching Mathematics through Problem Solving- An Upside-Down Approach

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:

1. 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 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

“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

# When it makes sense!

The look…

I get it now!

This is the look that makes my job worth it, when a student finally gets it!

I have the fortune of working with very dedicated students at Shanghai American School. Most of them have been working on mathematics since the young age of two. They start counting with their grandmas at the park. They start adding numbers. As soon as possible they start working with tutors. However, there are plenty ocassions when they do not know why something works, or how it works. That is where the magic happens. When we are able to listen to our students, to find out where they are in their understanding of mathematics. Some of them know the answer, they just do not know how they got it, they do not know how that looks like, they have trouble explaining their reasoning, and most importantly they struggle with making connections of different mathematical concepts.

When teachers are listening to their students, they can ask questions and guide their student to discover HOW math works, and how all mathematics are connected!

Let’s provide all our students with plenty of opportunities in the classroom to make sense of mathematics, by allowing them to experience some productive struggle while solving an authentic problem, a messy problem some call it, a problem that allows them to make connections, to solve in different ways, where students can focus on MAKING SENSE of what the problem is about.