Let’s explore a lesson using an upside-down approach to teach math. It is basically the opposite from the traditional I-do-you-do approach, where the teacher lectures the class with a specific way to solve a problem depending on the content, and students repeat the same steps the teacher just showed, but using different numbers. In this upside-down approach, or Student-Centered approach, the students do all the thinking, all the problem solving, all the modeling, and all the reasoning to solve a problem.
Some teachers have been finding elements of this upside down approach really effective, for example naming the strategies after the student that first came up with it, is really effective.
It might not sound like a big deal, but students get so excited about trying “their peer’s strategies” so much more than using the decomposing strategy (even though it is the same strategy.)
However, there are some other elements of this same approach, that could be more challenging, for example how to make sure that the lesson was effective if the work in students’ notebook is unfinished or extremely messy, or if students chose to solve problems with a strategy that is not the most efficient.
Let’s explore a 2nd grade lesson on solving problems of addition and subtraction with re-grouping with my friend and colleague Richard Ziegler.
Number Talk
Mr. Ziegler’s first move is to start with a quick number talk, to warm up his students’ brains. Sometimes, he uses an equation such as 26 + 36 and students solve it in their head. Only mentally. Then he asks students to share with each other how they solved.
After that, Mr. Ziegler asks a few students to explain how they solved. They explain step by step, and he uses numbers and pictures to represent the strategy that the students are describing. It is an art!
Sometimes Mr. Ziegler uses a math talk, it could be an image, and the questions could be: What do you notice? What are you wondering? How do you know? Convince your partner. The process is the same, he asks students to think about it on their own, mentally, then share with a partner, and finally, to share with the class.
Check out the books “How Many?” and “Which One Doesn’t Belong” by Christopher Danielson. They are great resources for math talks.
I will write another blog on number talks soon. Stay tuned!
Mini-lesson
The easiest way to use numberless problems is to use the slow-release function in a power point presentation. This will allow you to release information about the problem little by little.
If you do not want to use a computer, you could create some strips of paper and use them little by little, just as the slow-release function works on the power point slides.
The mini-lesson starts here. What are dragonflies? Have you seen them? What are toads? There are pictures on the slides to make the context very clear to students. There are no numbers yet. Students are not trying to guess answers quickly. They are wondering about this real situation.
Mr. Ziegler then asks: What can the numbers be? What numbers could not be? Students here explore estimation. It is about building number sense.
Students start talking about many possibilities. They make sense about the fact that the snake ate more than the toad. How many more? “We do not know yet” they say.
Students are engaged, they want to know more, how many? How many? Please tell us Mr. Ziegler!
“How do you make students engage in math class? Give them a problem to care about!” Steve Leinwand at MSIS 2017.
The next slide reveals that the snake ate 26 more dragonflies than the toad ate. How many did the toad eat? The toad ate 39.
Mr. Ziegler pauses. “Hmmm. What can the question be?”
Multiple hands go up!
This is an essential teaching move, as students learn that there are many answers that can be found with this information, they begin to understand that there is not only one answer to a problem, it depends on what we are figuring out. There are many questions out there, and also many answers.
Students come up with all sort of questions:
*How many dragonflies did the snake eat?
*How many dragonflies did they eat altogether?
*Why do they eat the dragonflies? Why don’t they eat grass?
Some questions don’t make sense. Get ready for that. Ha!
Mr. Ziegler then reveals the question!
“Ahh, I knew it” says one student.
This was the mini-lesson. The goal is to spend 8-10 minutes talking about the context of the problem. Having some student to student discourse about what the problem is about, and what are we going to figure out. The teacher is a facilitator.
At this point the teacher has not asked students to use any specific strategy. The questions are about the context, and having a clear idea about what is it that we want to find out.
Students can solve this problem in many different ways. They can use manipulatives, or draw models, or use numbers flexibly. Students already have ideas about how to solve problems, let’s hear them out.
This problem has 3 different combinations of numbers. Students can choose to start with the first set of numbers, and move on to another set. Or they can solve just the first set. It is not about quantity of addition facts, but quality of problem solving strategies. Students are encouraged to use different strategies to solve the different sets. This problem allows for differentiation.
Active Engagement
“Is everyone clear about what we need to do now?”
“Off you go,” says Mr. Ziegler.
Students now go to their tables and work in pairs. They can use concrete manipulatives, drawings, or numbers.
The role of the teacher now is to walk around and identify who needs support, or who has some clarifying questions. It is here, where the teacher can pull a small group, and perhaps review a strategy that was shared the day before and that this group might need to review again.
The teacher can also observe if there are students that need an extra challenge, maybe there are students that solved all 3 sets of numbers.They can come up with another question using the same context. There are opportunities of differentiation when using numberless problems, and also by extending the questions.
The teacher could be talking to different groups or, individual students and checking for understanding. Also this is the opportunity to individually ask students to try new strategies. For instance, if a student has been using the same strategy for the past few lessons, the teacher would suggest he/she tries a new strategy as he/she seems to have mastered the previous one.
Mr. Ziegler chooses to sit with a group of students and is asking them questions about the strategy they are going to use. I hear him ask a student if he could also use a number line to solve this problem. His voice is very melow and friendly, it sounds like an invitation, not a challenge. The student says “yes, of course I can!” Mr. Ziegler is such a talented teacher.
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.
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.
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.
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.
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
Making Sense of Mathematics 
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