Communicating & Reasoning in Mathematics
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.
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:
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:
- 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.
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:
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.
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!”
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:
I hope you are ready to use one or more of these ideas in your class. Let me know how it went!