Communicating & Reasoning as a Tool During Distance Learning

As educators we are prepared for constant change. We are used to the feeling of being comfortable not knowing everything all the time. We rely on our teammates to ask for help about creating a lesson we have never taught before or get advice regarding new strategies to use with a student who is struggling academically or emotionally. School policies change constantly, we adopt new standards, we implement new assessment tools, and we teach multiple grade levels and subjects.

However, none of that prepared us for today’s teaching reality.

We are experiencing a situation that we never really considered. Even after 7 weeks of my school being closed, and being engaged in distance learning, it still feels unreal sometimes.

As an international teacher who has lived abroad for over 15 years, I am used to change. I actually thrive with change.

I love moving to new countries, meeting new people, being immersed in a foreign culture, working at a new school, taking on new positions; but this change has been something else, and I definitely have grown and learned so much from colleagues in these past 6 weeks.

Assessing during distance learning

One topic that I have been hearing distance learning teachers discuss and wonder about is ASSESSMENT.  

How do we assess during distance learning?

How do we know the work belongs to the students and not the parents?

What data will we use to report?

How do we make sure the data is valid?

There is a lot of information out there on this topic, but today I want to share one way you can assess where your kids are and determine what next steps they might need.

AERO Mathematics Assessments Resources, by Erma Anderson

These are the 4 areas to understand if a student is proficient in mathematics.

During distance learning, we wondered if students are doing the work themselves or if they are they getting a lot of support and guidance from their parents. We’re left wondering, are they really proficient? Do they really get it?

We can create virtual learning experiences using different platforms and see if students are solving problems correctly, showing a strategy or model, and of course providing the right answer. Ideally, the problems/tasks we provide allow for problem solving, and not just a simple procedure or memorized fact.

All of this is great, but if we really want to see what kids know, and how they are using all they have been learning to actually solve problems, then we must give them an opportunity to EXPLAIN how they solved.

This can look different across the grade levels. Here are some examples that help students gather their thoughts, make sense of how they problem solved, and use mathematical vocabulary.

3rd grade anchor chart, created by Dani Allie, EAL Specialist
2nd grade poster created by Emerald Garvey, EAL specislist
Kindergarten chart, created by Lori Han, EC Learning Interventionist

During distance learning, we have been providing students with some sentence starters that allow them to elaborate their answers. They show their work and write the correct answer, and they use an audio feature to explain how they solved the problem.

Seesaw lesson in 1st grade with a sentence starter.

Here is a link to an example of a 1st grade student expxlaining how he solved the problem. Skip slide #1 because that is just the teacher’s instructions. Go to slide #3 to listen to the evidence that he is using friendly tens to make nineteen.

Then on slide #4 he is explaining an addition fact showing understanding of tens and ones, showing evidence that a 1 in the tens place is a ten.

We Will Be OK!

I know these are difficult times, AND we will learn from this experience, AND we will become better educators.

School is canceled.  Events are canceled.  And you have no summative assessments in your hand. However, this does not mean your students are not learning! Take the time to develop lessons that allow you to listen to your students.

After this experience of leading instruction from a distance, all of us will be able to appreciate our ability to plan lessons based on their knowledge and skill level, by listening to them, really listening.

If your school has recently closed or is about to, and you want to be prepared and feel succesful, there are many resources for you to choose from, you can go on Twitter and search #virtuallearning or #distancelearning and you will be pleased to see so many resources.

There are several groups on Facebook that also provide a lot of helpful information: “Educator Temporary School Closure for Online Learning” and “Online Teaching for International School Teachers” are just some examples.

Using Apps to Learn Math

Apps could be an amazing tool for learning. Using technology in general is essential in education to prepare our students for their future careers. We need to acknowledge the fact that technology affects the way we work, play, live and learn today.  We can’t pretend smartphones and tablets are not part of our students’ daily life.

As educators, why do we insist on planning learning experiences in our class without the use of technology tools? We need to incorporate them, and use them as the powerful tools they are.

Unfortunately, most schools are not ready for this.

It is not about “allowing” students to use/play technology for a few minutes after students finish their “real” math work.

It is definitely not about using math apps where students have to quickly answer a fact, to then kill a dragon, or an app where they need to beat the clock by answering math facts as fast as possible to win coins and stars.

Fluency is an important part of being proficient in mathematics, but fluency is not just memorizing. Most of the “educational math” apps are focused on drill and practice without any type of focus on the math practices or problem solving skills.

Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meaning and properties of operations. (NCTM 2014, 42) (Paragraph extracted from Developing Numerical Fluency by Steve Leinwand.

The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense. Jo Boaler

If students memorize math facts and procedures without understanding why they work, they will never take an advanced mathematics course because they won’t know how to make connections amongst math concepts.

The creators of these math apps mean well, they believe, I suppose, that they are motivating students to practice facts because the games are engaging. I recently read a review about the best math apps, where every single app focused on using math facts to do tasks such as kill zombies, race on a track, or find secret gates.

The creators of these apps missed the point. They are only focusing on quick answers to math facts.

The idea should be for students to use apps to build their understanding of a math concept via experiences such using manipulatives to “model” their thinking. Ideally, it would encourage and provide a space where students can draw models or visual representations that allow them to make sense of the problem and figure out how to solve it.

Not a List of Skills to Drill

App creators need to look at all aspects of math learning, not only the drill and practice of math facts. We need more apps that help our students “model” how they are thinking about the problem.

What could be the criteria for these new apps? Students need to demonstrate a deep understanding of the math concept or be able to model how they are interpreting the problem with manipulatives. Perhaps the creators of the math apps could mesh their gaming ideas with the tools and strategies found in Braining Camp and Thinking Blocks.

Braining camp is a great bundle of apps with different types of manipulatives. It is easy to use, and the best part is that it does not provide the answers!

Here is an example of how to use base ten blocks to solve a problem using a place value strategy, making sense of what it means to “bundle” 10 ones into a ten, or ten tens into a hundred.

Base Ten Blocks app http://www.brainingcamp.com

Fractions are a perfect opportunity to use apps that allow students to make models! What does it mean to multiply ½ x ½ ? What does that look like?

Dividing fractions does not have to be a set of rules where students flip the fraction! There is no understanding in flipping, just computation.

Look at this app where students can manipulate the fraction tiles to visualize how many one tenths can fit into four fifths:

Fraction Tiles app www.brainingcamp.com

Here is another great math app that is not about quick right answers. It is called Thinking Blocks.

You can see how it gives students opportunities to model their thinking using bar models to represent the problem. Students can then use math facts or repeated addition to figure out the solution.

This app also has a feature where the problem can be read aloud; how great is this for different types of learners?

Another great thing about this app is that there are no time limits to respond. Based on Jo Boaler’s research, we know that timed testing is unnecessary and damaging, as it could be the beginning of math anxiety which blocks the working memory. “Fluency Without Fear” by Jo Boaler.

I am sure there are other math apps out there that allow for problem solving and modeling, so please respond to this blog so we can create and share a list of apps that promote student thinking, and not memorizing.

Until next time!

Essential Component of an Effective Math Lesson

As an elementary math coach, one very important part of my job is to be in the classroom noticing all the small good things students are doing or saying, as well as the different teaching moves that shift students’ thinking about how they approach a math problem.

I usually send an email to the teacher right after my visit, highlighting these positive moments, as we all know that feedback is crucial for students and also for teachers. As teachers we are always learning, and believe in continuous improvement. Feedback helps teachers make their best even better!

I have been in a variety of classrooms from Kindergarten to Grade 5 for the past few weeks noticing and highlighting the great things happening during the SHARE OUT part of the math lesson.

Elementary teachers make a dozen decisions a minute, sometimes the lesson runs longer, or a small group needs more time, or they have a guest speaker coming in, so there are days where the last part of the math lesson has to be very short, or moved to the next day.

However, this is the most essential component of a math lesson because it is where “students listen to other students’ ideas, they come to see a variety of approaches in how problems can be solved and see mathematics as something they can do.” (Teaching Student-Centered Mathematics, John A. Van de Walle)

Student explaining her model and strategy to solve this problem. First I ……. , then I thought ……., lastly I figured out ……..

I saw different teaching moves while facilitating the share out, some teachers ask “Who would like to share how they solved their problem?”  Students like having this choice, however there is the risk that it is always the same students sharing, or the 2 or 3 students sharing the same strategy. You could miss the opportunity of a teaching point right here.

Who Shares?

Something the teacher could try is to choose who will share that day. While students are solving the problem of the day, the teacher is walking around asking questions, and noticing the different strategies students are using.

Then they choose ahead of time the 2 or 3 students that will be sharing. It is good to ask them first “Would you like to share your strategy/ideas at the end?” Students usually say yes.

When the teacher chooses who shares, they can make sure to fit in a teaching point for the day. For example, students could discuss which of the strategies they saw is the most efficient. This is essential in problem solving, as students tend to use a strategy that they are comfortable with, even though sometimes it is not the most efficient. Decomposing for example, is an efficient strategy when adding two digits by 2 digits, but partial sums becomes a more efficient strategy as the students solve with multi-digit numbers.

Ask students to discuss which strategy is more efficient for this problem and why.

Be intentional about the order in which students share their strategies and solutions; for example, select 2 strategies or models that could be connected, ask students to notice the similarities in the 2 strategies/models. This is an essential move to move students from a concrete or pictorial model into a more abstract strategy.

This is a concrete area model next to a pictorial area model. Ask students to make connections between both strategies.

I saw a 5th grade teacher show one student work but covered the name, then he asked “If you look at this work, can you understand how this person solved?” Students had some great feedback about missing labels and how that could help understand what each part of the model represented. One student said “I can see the answer, but no idea how he/she thought about it.” As you can imagine, this was a perfect teaching moment for that teacher to emphasize how to best show how you solve.

When students are used to share their work and explain how they solve, they make sure to have clear models that help their audience understand their work.

How Much to Tell and Not to Tell

When students are sharing their strategies, pay special attention to not take over their thinking. Even if they are struggling to explain, it is important to only ask questions, giving them time to think and elaborate their sentence. Having some sentence starters could help.

Taking over students thinking by talking for them, sends the idea that you do not believe they can explain it themselves, and that can inhibit the discourse you are trying to encourage.

Students show their work and explain how they figured out a solution.

Telling too much eliminates that productive struggle that is key to conceptual understanding. Telling too little can leave students confused.

If an efficient strategy did not emerge from the students, this is the chance to share “another way” to solve, not the only way to solve, but a different way.

Let’s always remember that the person doing the talking, is the person doing the learning.

Differentiating: Using parallel tasks in your math lesson

At the start of this school year in our elementary school, we decided to focus on differentiation strategies all teachers can use in their classroom. To be more specific, we invited all teachers to explore strategies they can use during Tier 1 instruction (RTI), not necessarily differentiating for students with moderate or severe disabilities, but strategies that can make the learning accessible for all students in the classroom.

Students that require Tier 2 or Tier 3 instruction benefit rom support from the Math Learning Support Specialists AND benefit from these Tier 1 interventions and learning experiences in their classroom.

During our first team meetings, all teams from kindergarten to 5th grade explored a variety of strategies we could learn about and put into practice, and were invited to develop a professional team or individualized goal as part of our annual professional growth and evaluation model.

“Parallel tasks” was a popular strategy the teachers identified. I’d like to share some simple strategies so you can give it a try.

1st grade team working and learning together at Shanghai American School

Mathematically Gifted

Let’s not forget about our students who are able to show proficiency during the pre-assessment! We also need to intentionally plan for them. I am not referring to students who get answers quick or have memorized algorithms that give the impression of understanding mathematics; I’m talking about students who can problem solve by choosing the most efficient strategy to solve a problem, show their thinking with a pictorial or more abstract model, and can explain how they solved a problem and why they chose a particular strategy. Do you have some of those students? We do. There are not many, but there are some.

There are many strategies for them, however in this blog I will only explore parallel tasks. However, I do want to mention some strategies to AVOID with these students:

  • MOTS: More Of The Same work. This is the least appropriate way to respond. Students might start to hide their abilities.
  • FREE TIME: Students might find this rewarding, but it does not maximize their intellectual growth. Students will hurry to finish without giving it their best effort. Other students that require more thinking time, will feel they are “not good at math”.
  • HELPERS: This does not stimulate their intellectual growth, and will put students in socially uncomfortable and undesirable situations.
  • PULL-OUT: This practice tends to be unrelated to the regular math classroom, and it does not allow students to go deeper in their understanding of the math content they are learning in class. Unfortunately, many of us did this for years!
  • COMPUTER TIME: Although there are great apps to practice math skills, it does not engage students in their conceptual understanding of math, increase their problem solving ability, nor increase their reasoning and communicating skills needed to justify their thinking.

Parallel Tasks

Parallel tasks are 2 or 3 tasks that focus on the same learning task/learning objective but offer different levels of difficulty. All students should be able to participate in the “Share-Out” at the end of the lesson which is the most important part of the lesson because this is where the teacher learns how students are tackling the problem, what strategies are they using, and what misconceptions they might have.

You can assign students to a particular task or you can give them options. I prefer to provide different options as this gives students more ownership of their learning pathway. If they choose a task that is too difficult, they can move to another one.

Example

Learning target: Represent and solve problems involving addition and subtraction

Lesson objective: Use place value knowledge to subtract within 100

Standards: 2.OA.1, 2.NBT.5, 2.NBT.7

There are many great things to notice about these two slides.

First of all, the teacher will use the “slow release” function, to release each sentence one by one allowing students to notice and wonder about the context of the problem. Using Numberless Word problems is essential when creating parallel tasks. (To learn more about Numberless Word problems click here: https://bstockus.wordpress.com/numberless-word-problems/ by Brian Bushart)

For example, she/he might ask students, how many kinds of apples are there? Or how many apples could there be? (Estimation180 is a fantastic website that can support this type of thinking and dialogue in your class).

After the “release” of how many apples each child has, students might wonder about the question, and will then come up with the best questions. Give them a chance to discuss with a partner first before you ask a student. This gives them the chance to practice their question, and/or craft a question with their partner.

Next the teacher will show the 2nd slide; this is when students have the choice of numbers they can choose to solve the problem.

Notice how ALL students will be showing the strategies they used to solve the problem, explaining how they solved it, and making sure that their argument supports their work (Claim-Evidence-Reasoning).

If students need to use manipulatives to solve, the first set of numbers allows them to use any type of manipulatives such as discs, unifix cubes, or single cubes. The second set of numbers will require ones and tens.

If you want students to move from using base ten blocks in order to try other strategies, encourage students to use the third set of numbers as they will have an incentive to think about another strategy.

For problems involving computation, you can add multiple sets of numbers to allow for ways to vary the difficulty level. You could also use parallel tasks to vary the level of problem solving and reasoning.

Open Middle

“Open Middle” is a great resource created by Robert Kaplinsky www.openmiddle.com that provides parallel tasks for the same standard, however the lesson objective might vary because students will show proficiency of it using deeper problem solving skills, than only computational skills. Take a look.


“Open middle problems generally require a higher Depth of Knowledge than most problems that assess procedural and conceptual understanding.   They support the Common Core State Standards and  provide students with opportunities for discussing their thinking.” Robert Kaplinsky.

During the active engagement of your lesson, walk around to observe the strategies and different approaches that students are using to tackle the problem. This is where the teacher plans which students will be sharing their ideas making sure that a variety of strategies are shared. The teacher uses this time to look for opportunities to help students make connections between the different ideas shared.

Last week, I was in a 1st grade classroom when the teacher asked a student to share how he persevered when solving a tangram puzzle. It was a fantastic share-out moment, and we all learned from this 6 year-old talking about not giving up!

Next steps

When you are thinking about creating parallel tasks for your lesson, start by identifying the big idea you want to focus on and think about what your students might need. You could start by using different sets of numbers, the number of operations they can use, or making them open to allow for deeper problem solving, etc.

Start with a task from your original lesson then create parallel tasks in order to allow students to choose the task they will work on while making sure that sometimes the most difficult task is the first one. This will ensure that students consider all options before choosing their task.

If you are interested in reading more about parallel tasks, refer to “Teaching Student-Centered Mathematics” by John A. Van de Walle.

Other samples of parallel tasks:

1st grade sample. More resources at http://www.openmiddle.com
2nd grade sample
3rd grade sample. Based on #tapediagramtuesday by Duane Habecker
5th grade sample

Are You Asking the Right Questions?

How do you know you are pushing your students’ thinking? The questions you ask in class can either push their thinking, and let them show you how they thought about solving a problem; or they can shy away if they feel they don’t have the right answers.

“Problem solving is the major theme of doing mathematics and ‘teaching students to think’ is of primary importance. ‘How to think’ is the theme that underlies genuine inquiry and problem solving in mathematics.” Erma Anderson 2019

Asking questions to our students, gives us the opportunity to see how they think, and gives them the opportunity to create a mathematical argument.

We know that effective teachers of mathematics ask purposeful questions. But what does that mean?

“Guess my thinking” type of question

This type of question is the one where the teacher has a specific answer in his/her head, and is hoping the student guesses it.

It could go like this:

T-  “What is one strategy you could use to solve this problem efficiently?”

S-  “A number line?”

T- “Yeah, it could be that but, what is one strategy you could use to solve this problem efficiently?”

S- “Decomposing??”

T- “Mmmmm. Think about it. What is the one strategy you could use?”

S- “ I dunno.”

This conversation can go on, or it can stop right away. This question does not help the student figure out what strategy is more efficient. Also this type of question does not give the teacher any evidence of what the student thinking process is about, or what misconception might he/she have.

The most dangerous part of using this type of question, is that students shut down immediately, because they realize they are never going to guess what the teacher has in his/her head.

“How are you thinking about it?” type of question

This type of question allows the student to explain how he/she thought about a problem. It gives him/her the chance to pause and think about the steps he/she took, and what conclusion he has gotten to.

Here is where the teacher asks questions to find out what the student is thinking about (as opposed to a guessing game).

How did you think about it?

Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both the curriculum and instruction should begin with problems, dilemmas, and questions for students. – Hibert et al., 1996

Allow students to search for solutions, and resolve incongruities

A few weeks ago I was in a 1st grade class, working with a couple of kids working on the learning target: Represent and solve problems involving addition and subtraction. Ms. Jones had ask me to see if these 3 students needed some extension tasks, as they already were showing proficiency of adding 2 digits. We wanted to make sure they were not using only algorithms, and if they showed understanding of place value.

Ms. Jones and I did a number talk on using numbers flexibly to add and subtract. Then she gave them a few problems to solve with their partners. She gave them some paper with number lines on it, so they could use the decomposing strategy. For example when adding 26 + 25 perhaps they could start on the 26 on the number line, and then take a big jump of 10. Then 10 more, then 5.

After solving two or three problems, one student looked at me and said:

S- “Miss Caty, this is very silly. (He was giggling) I am not going to use the number line for this problem.”

Me- “Tell me more.”

S- “Well, look at the problem. If I am going to add 99 + 14, I don’t need a number line. All I have to do is take a 1 from the 14, and give it to the 99. Now I know I have 100 and 13.”

Me- “That is so true. Write that down.”

A number line is a good strategy, but there are other strategies that could be more efficient for this problem.

Allow students to wonder why things are

How do we know a number is even or odd?

Last week, our students in 2nd grade were exploring even and odd numbers, as part of their learning target: Work with equal groups of objects to gain foundations for multiplication.

One lesson gave them the opportunity to use color tiles, and make arrays with different numbers. They worked in pairs, and during the “discovery” time of the lesson, they came up with conclusions about why numbers 3, 5, 7, etc could not be arranged in arrays. One little tile was always left out.

The objective for this lesson is for students to discover the difference between even and odd numbers.

Notice how the lesson was a DISCOVERY session, not a lesson to memorize what is even or odd.

Students made rectangles with all even numbers.

Another lesson had students discover how even numbers could be split into 2 equal groups.

I was in 2 different classrooms that week, were teachers were working on the same lessons. We were very pleased to hear students make their own conclusions about why numbers were even or odd. In both classrooms, students needed the concrete manipulative to make sense of what even numbers look like.

However, in the next couple of days, when we sat down to look at student work, and analyze their “communicating & reasoning” in their notebook, we saw that several students were not using what they learnt in class to explain how they knew if the number was even or odd.

Look at this example here, where the student says that once you see the number in the ones place is 5, then you do not need to look at anything else!

Then he continues and explains that if you want to find out a number is even, then the number needs to have a partner. This second statement shows what he learned in class. The first statement is something he had memorized.

Another student, said that he knew a number is even when “apparently” you can count like this 2, 4, 6, 8. He also says a lot of other things, but the word apparently is on the 5th line. Take a look.

One student wrote that if you think about the word EVEN it has 4 letters, and if it is ODD   it has 3 letters, so there you go. Think no more!

So we decided to ask more questions to see why they were using that rationale, after they had discovered in class how an even number can be split equally in 2 groups. Or also it could be even if every part of the number had a pair.

We finally realized that a lot of our students have memorized that “a number is even if it ends in 2, 4, 6, 8, 0” And a number is odd if “it ends in 1, 3, 5, 7, 9”

A lot of our students go to tutoring after school, and their tutors have them memorize all these concepts, without understanding why the rule works.

We asked questions like:

“What do you mean because 2, 4, 6, 8?”

“What makes you think 5 is odd?”

“What did you discover when you use the color tiles to make arrays with numbers 0 to 20?”

The good thing about asking purposeful questions, is that if you ask to find out what they are thinking about, or why they says what they say, you can find out the misconception, and with a few more questions, students can make connections between those thoughts and their learning experiences. You will hear a lot of “ahhhh” and “ohhhh”, when students make connections, or when they finally understand why that rule they memorized, makes sense. Memorizing in not fluency.

We want our students to be fluent in mathematics.  “To be fluent means not just being able to do something, or memorize something, but being able to reliably do it accurately and efficiently. Fluency is an outcome of a progression of learning and sufficient thoughtful practice provided at each grade.” Erma Anderson, 2019

Do you want to find out what your students know about even and odd?

 Use “How are you thinking about it?” type of questions, to find out!

Student work

These are some samples of student work, after we conferred with each of those students, and they had one more chance to show what they learned, not what they had previously memorized.

Convince me!

Communicating & Reasoning in Mathematics

1st grade class ready to use sentence starters to explain how they solve problems.

A few years ago, a good friend of mine, who was teaching kindergarten, told me that she had this conversation with one of her 5 year old students:

Student: “Miss Koshika, why don’t you trust me?”

Miss Koshika: “What makes you say I don’t trust you?”

Student: “Well, every time I tell you my answer, you say convince me, prove it!”

Miss Koshika laughed out loud. She was happy to share that with me, because she knew she had to push her students to explain how they solved a problem, even when the problem was only 5 + 6.  She was proud to share that her students were getting better at explaining how they knew the answer was 11.

To this day, she is one of my favorite teachers in the world, and I always share this story when I am presenting at conferences or workshops. It makes my audience laugh, and I make my point at the same time.

It has been about 5 years since I started trying to make sense of what communicating and reasoning looks like, and sounds like, for each grade level from kindergarten to 5th grade. I started to focus on the reasons why students should explain their thinking, how their strategy works, or how they solve a problem.  This explanation might sound different at the different grade levels.

I also have been exploring some of Jo Boaler’s research, she is a professor of Mathematics Education at the Stanford Graduate School of Education. These studies found that students achieved at higher levels when they could make sense of numbers, when they could explain how to use numbers flexibly, and when they used mathematical vocabulary to make sense of the problem.

“Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive. Teachers can create such mathematical excitement in classrooms with any mathematics question, by asking students for the different ways they see and can solve the problems, and by encouraging discussion of different ways of seeing problems.” Jo Boaler, Visual Math Improves Math Performance.

Number Talk with 2nd grade students at Shanghai American School.

Providing opportunities for students to explain and reason about how to solve 46 – 13, goes above and beyond a simple correct answer. There are multiple ways to solve it, and just as important, students should get opportunities to explain how they solved, and share the steps their brain took, to arrive to a solution.

In the book Principles to Action, published by the National Council of Teachers of Mathematics (2014), they share these 8 Effective Mathematic Teaching Practices:

“Principles to Action” writing team: Steve Leinwand, Daniel J. Brahier, DeAnn Huinker, Robert Q. Berry III, Frederick L. Dillon, Mathew R. Larson, Miriam A. Leiva, W. Gary Martin, and Margaret S. Smith

The 2nd teaching practice: Implement tasks that promote reasoning and problem solving, refers to engaging students in solving and discussing tasks that promote mathematical reasoning and problem solving, and allowing multiple entry points, and varied solution strategies.

 The 4th teaching practice: Facilitate meaningful mathematical discourse, requires teachers to facilitate discourse among students to build shared understanding of mathematical ideas, by analyzing and comparing student approaches and arguments.

There are still many parents and teachers that belive they should teach the exact same way they were taught, which is, using formulas and memorizing facts. But thanks to the research from Jo Boaler, and the National Research Council, amongst others, we know that students learn best when teachers plan for interactions and discourse in the classroom, that helps students to make sense of mathematics.

How can we facilitate meaningful mathematical discourse among our students?

Here are 3 ideas:

  1. USE NUMBER TALKS BEFORE YOUR LESSON

Using Number Talks (Sherry Parrish), is a great way to provide opportunities for students to explain their thinking, to convince you and other students about their solution.

Students see problems in different ways, but they not always have the right vocabulary, and confidence to convince others. Sometimes they do not even know how they solved!

Look at this example of a Number Talk in 4th grade:

The teacher asked students to solve in their head. No paper or pencil. Lots of students had their heads tilted to one side, others had hands in the air, you could see them thinking.

The teacher then asked them to share with a partner, and explain how they solved. This went on for a few minutes. The room is loud. Thinking about mathematics is fun and the room can be loud.

Joshua shared how he solved, he explained that he decomposed 24 into 20 and 4, and he multiplied 20 x 6 because that is an easy number for him. He got 120. Then he continued to multiply 4 x 6 and he knows it is 24. Then he said he added 120 + 20, and he got 140. Lastly he said, he added the remaining 4. His answer is 144.

Some students made a signal that means “I agree”. They also thought about it like that.

“Who saw it differently?” asked the teacher.

Another student explained that she multiplied 24 x 5 because that is easy for her. She knows it is 120 because 20 x 5 is 100, and 4 x5 is 20. Then she just added 24 more. So 120 + 24. She also got 144.

“Who did saw it differently?” A student said: “I did it very similar to Joshua but I drew a picture of an area model in my head.” He explained how he pictured a rectangle cut into tens and ones with the 20 and the 4. Then the 6 on the left side. Now he could multiply, just like Joshua 20 x 4, and 6 x 4.

(To learn more about Number Talks, read my blog “Introducing Number Talks.” Or, better yet, get the book Number Talks by Sherry Parrish.)

When we allow time for students to share how they solve a problem, not only are we providing opportunities for those students to practice how to explain their reasoning; we are also providing a variety of strategies to all students to solve problems. They now have new strategies to try. They also were able to make a connection between Joshua’s strategy, and the visual representation of an area model.

2. HAVE STUDENTS SHARE HOW THEY SOLVED, LET THEM DO THE TALKING, THEN WRITE IT DOWN

Here is another example of what we can do in our class to promote reasoning and meaningful discourse. This is how an explanation could sound like in 2nd grade.

This is not a Number Talk, this is part of a lesson. Here, students are able to use manipulatives, or visual representations to solve the problem. Then they come up to the board, show their solution, and explain how they thought about this problem.

Notice how the student is not describing the answer, or even mentioning the answer. The goal is to explain how he/she thought about the problem, and how he took each step.

Focus in Grade 2: Teaching with Curriculum Focal Points, figure 2.22, page 88 – 89. NCTM, 2011

For this same problem of 346 – 159, there are other strategies to use. Using base ten block visuals, and connecting that with an algorithm with re-grouping, is just one way to solve this problem, but it is not the only way.

Another student might solve like this:

Solved using 2 strategies, first compensation, then subtracting by parts.

You will be surprised to see how many different ways students come up with. The key is to facilitate meaningful discourse. Once students have their strategies, it is the job of the teacher, to be a facilitator, and invite students to explain how they solved, while other students listen.

Then they can use sentences like:

“I agree with _______, and will like to add this other step: _____________.”

“I disagree with ________ because __________________________.”

“I saw it a different way, this is how I saw it: _________________.”

“My strategy is similar to __________________, but I did it like this: _________.”

“First I used __________ to figure out that _________. Then I decided to ____________ and I found that ___________. Finally I discovered that __________________ was the answer.”

3. USE SENTENCE STARTERS OR ROLE PLAY

If your students are young and do not know how to start to explain how they solved, you could provide them with some sentence starters. This is also very helpful when they are just starting to write.

Mrs. Martin, a 1st grade teacher, wanted to give her students some ideas to start thinking about how they solved the problem, besides only mentioning which strategy did they use. She found that a lot of her students were only using “I used a number line”, or “I used a number bond” when she asked them to explain how they solved.

So she added these few sentences to the students recording sheet. She told them she needed to know how their brain worked, to see how they got their answer.

This is a numberless word problem. Notice the sentence starters to guide students to explain their thinking.

Another 1st grade teacher, Mrs. Tustin-Park did this poster with her students. She then asked one of her students to explain how he solved a problem, she listened carefully, and she followed the sentence starters. She wrote it down on the poster. Each strategy has the student’s name. Students love using strategies that their friends “created!”

Poster made with 1st grade students.

A week after she did this poster with her students, they were all using the starters on their own, and explaining their reasoning. She told me it was magical. She said “I didn’t do anything else. It just happened. Having that visual, just made it!”

The same week that Mrs. Martin and Mrs. Tustin-Park used these strategies in their 1st grade classrooms, another teacher in 5th grade asked if I could come and co-teach a lesson with her. She said she needed help having her students be more clear about how they explained their reasoning. Bingo! My mind was already spinning.

I was not going to use sentence starters with these 5th graders, so I had to come up with another strategy.

Ms. Tornstebo and I planned the lesson together. We chose a number talk, and for the lesson we chose a rich problem that involved using visual models to multiply fractions, and a lot of reasoning. Students were engaged, working in pairs. Lots of conversations were going on.

This was the problem:

After they solved the problem, we gathered them back on the carpet for the share-out part of the lesson, and we asked for volunteers to share their work. We had already seen a few people using different models, it was going to be a great share-out.

Students came up and shared with detail how they had first used a model to show a model withthe 24 pieces. Then they found what 1/6 of that was, finally 2/3 of it. The model showed the answer. There were no algorithms, really, it was so good to see them using models to figure out what the fraction meant.

When they showed their recording sheet on the document camera, we saw students didn’t even fill on the part of explain how you solved.

One of the students, Siri, had solved the problem beautifully, but this is what she explained:

“I know the answer is 12 because 16 – 4 =12.”

So, we told the students we had some great strategies to solve the problem, and we recognized their perseverance to solve it using models. However we wanted to work on the “explain your answer” piece. We told them we were now going to role play, and I was going to pretend to be Siri, and I was going to write how “I, Siri” solved this problem.

Siri agreed, so she started explaining step by step how she thought about the question, and I moved my lips pretending I was her, and at the same time I summarize the steps “I” used to solve the problem. I made sure to not include the answer. I just wrote how “I, Siri” had seen the problem, and how “I” went about solving it.

That was it. They were laughing at my silly role play, and Siri felt like a rockstar!

Notice how she solved this problem without multiplying numerators by numerators, etc. She solved by making sense of the problem with a picture. Other students decided to use fraction of a set, so they drew 24 small circles that represented each piece of art. That is how powerful modeling in mathematics is. But that is for another blog.

Class was over, so they didn’t have time to work on their explanation, so we agreed they would do it in tomorrow’s class.

I didn’t work with Ms. Tornstebo the next day, but she sent me their recording sheets, and it was beautiful. They now knew how easy it was to explain the different steps they took, to solve the problem.

It was a great week!

Here are a few pictures of students explaining thow they solve problems:

Kindergarten student explaining how he knows that 5 and 3, is the same as 4 and 4. In Kindergarten they use Seesaw to record their explanation. Or they can tell their teacher and she records.
Another kindergarten student explaining what he is wondering.
This 3rd grade student has 4 strategies to solve 8 x 7, he can choose which strategy to explain to the class.
Becky is a 4th grade student. She solved this problem using a place value chart. She has a strong sense of place value. She really understands that dividing is making equal groups.
Steve is also in 4th grade, he chose to use partial quotients to solve the same problem. He will explain his thinking bubble and how using it, is very helpful for him.
2nd grade class during a number talk. They are explaining how many lightning bugs are there, and how they got the answer.
This 1st grade student decided to make a 10 to add these numbers easily.
He also solved by making a 10, but he used the 5 and 5 to make it. This is how his brain saw it.
This 1st grade student is using unifix cubes to convince her teacher of her solution.
In kindergarten, students explain how many dots they count, and they can prove their answer by using manipulatives.
This is me during a number talk in 1st grade. I love listening to students explain their way of thinking.

I hope you are ready to use one or more of these ideas in your class. Let me know how it went!

Introducing Number Talks

What is a Number Talk?

  • A Number Talk is a short, powerful tool for helping students develop computational fluency and number sense.
  • Number Talks are not necessarily directly related to the math curriculum. They are not intended to replace current curriculum or take up the majority of the time spent on mathematics (5-10 minutes).
  • Number Talks allow students to make connections and find relationships and patterns.
  • Number Talks also allow students to use the language of mathematics.

What is the focus of the Number Talk?

  • The conversation is the focus of the Number Talks, and the teacher takes on the role of facilitator.
  • Children develop computational fluency and number sense while thinking and reasoning like mathematicians.
  • When they share their strategies with others, they learn to clarify and express their thinking, thereby developing mathematical language.
  • Students share their mathematical thinking and develop efficient, flexible, and accurate computation strategies that build upon the key foundational ideas of mathematics such as composition and decomposition of numbers, our system of tens, and the application of properties.

For example, if a 1st grade student can only solve 13 -7 by using an algorithm, it means he/she has not been exposed to any other strategies to solve this kind of problem. During a number talk, one student might solve this by thinking of adding 3 to the 7 to make a 10, then adding 3 more to get to 13. Yes, this problem can be solved by adding 3 and 3 more.

Students explain how they solved the problem, and the teacher makes a visual representation of these explanations.

Another student might solve this problem by decomposing the 7 into 3 and 4, so take away 3 from the 13 is 10, and then take away 4 more, it is 6. This strategy uses friendly tens as well.

When students are exposed to different ways to solve problems, and specially different ways to use numbers flexibly, they start making sense of numbers.

Algorithms are not always the most efficient strategy, and number talks provide students with a variety of strategies where other students use numbers flexibly to solve problems mentally.

How does Number Talks allow opportunities to make sense and persevere?

  • Students look for number relationships to plan their strategies and seek alternate ways to verify their reasoning.
  • Students develop flexibility in looking at problems from multiple perspectives!
  • As students share their answers and strategies, they must evaluate other ideas and approaches, which further develops this mathematical disposition.

Do Number Talks help build fluency of basic facts?

  • Yes, mental computation is a key component of number talks because it encourages students to build on number relationships to solve problems instead of relying on memorized procedures.
  • Yes, mental computation causes them to be efficient with the numbers to avoid holding numerous quantities in their heads.
  • Yes, repeated experiences with reasoning strategies are effective in committing facts to memory; memorizing is not.
Multiplication problems can be solved by using numbers flexibly. All of these strategies are based on place value.

USING SIGNALS!

*Tell your students that you are going to be doing a Number Talk. They are to be thinking in their heads, and trying to figure out the answer.

*Tell them that they should be ready to share how they figured out the answer.

*You can use these signals for your Number Talks, or you can make your own signals with your class.

NUMBER TALK MOVES for teachers and students!

*You can introduce these Number Talk moves in one day, or a few each day.

WHAT DOES THE TEACHER SAY?

*After providing some wait time, ask students to share with their partner. When students explain to their peer first, they gain vocabulary and confidence to share with the class.

WHAT DOES THE STUDENT SAY?

*In order to help students use their mathematical language, show them these examples to share their thinking.

Your students can also come up with their own math language!

READY?

All number talks follow a basic six-step format.

  1. Teacher presents the problem: Problems are presented in many different ways: a word problem, number problem, ten frames, dot cards, models. You can show problems on a document camera or write on the board. Present today’s problem on the board: “How many legs are there on 5 horses and 2 roosters?”
  2.  Students figure out the answer. Give time to figure out the answer. To make sure the students have the time they need, ask them to give a “thumbs-up in front of chest” when they have determined their answer.
  3. Students share their answers. Teacher collects different answers on the board. “Does anybody have a different answer?” Only answers on this step.
  4. Then, students share their thinking. Have students share with a partner before they share out with class. This helps them be prepared to share. Have three or four students explain their thinking to the class. “Did anybody do it differently?” This is the most important part of the number talk, because it is here where students can visualize how other students solved the same problem.
  5. The class agrees on the “real” answer for the problem. There might be different answers (Which one doesn’t belong?). Models and explanations are very helpful for students to see where their thinking went wrong, help them identify a step they left out, or clarify a point of confusion.
  6. The steps are repeated for additional problems. Thank the students for their participation in the Number Talk.

Different examples of Number Talks

*Subitizing: How many do you see? How do you know?

How many do you see? How did you count?
How many? How do you know?

*Different strategies to add, subtract, multiply and divide:

  • Doubles/near doubles: 298 + 297
  • Making Landmark or Friendly Numbers: 98 + 52
  • Adding up in Chunks: 57 + 36
  • Compensating: 37 – 18
  • Partial Products: 14 x 16
  • Doubling and Halving: 15 x 16

Repeated Subtraction: 100 ÷ 25

*Different types of images that promote discourse

wodb.ca
Can you make this number?
mashupmath.com
estimation180.com

This blog was created using a variety of resources such as:

“Number talks” by Sherry Parrish

“The First 10 Days of school” from Erma Anderson

Article “Math Perspectives” by the Teacher Development Center

“Teaching Student Centered Mathematics” by John A. Van de Walle

There are Number Talk videos from Sherry Parish on youtube, so you can watch her teaching moves during a number talk.

How many fourths are in one half?

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.

Then the question is “How many 1/8 fit in 3/5?” Students need to realize that 1/8 in their model is equivalent to 5/40.
Area model is an efficient model for fractions.

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.”

Numberless Word Problem using a Student-Centered-Approach

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.

Ethan’s strategy

 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!

This type of number talk allows students to use numbers flexibly, and allows students to communicate their reasoning.

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.

Start the lesson only showing this part of the slide.

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.

Finally reveal the number.

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.

This is the strip of paper that goes onthe student’s notebook.

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

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.

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。