In this session we will explore ways in which making the math explicit, digging deeper for generalizations, and using probing questions supports math learning in the classroom. You will do math, watch students doing math, and notice what teachers do to make the math explicit.
The work of elementary math classrooms focuses on solving problems, gaining computational fluency, seeing and doing geometry and measurement, and understanding data. The foundations of algebra arise naturally throughout students’ work with numbers, operations, and patterns and by using familiar and accessible contexts to investigate how one set of values changes in relation to another. This work anchors students’ concepts of the operations and underlies greater computational flexibility.
Work in early algebra in the elementary classroom has the potential of enhancing the learning of all students. The teachers with whom the Investigations team collaborated during the development of the curriculum commented on this potential in their classrooms. Teacher collaborators reported that students who tend to have difficulty in mathematics become stronger mathematical thinkers through this work.
As one teacher wrote,
When I began to work on generalizations with my students, I noticed a shift in my less capable learners. Things seemed more accessible to them.” When the generalizations are made explicit—through language and representations used to justify them—they become accessible to more students and can become the foundation for greater computational fluency. Furthermore, the disposition to create a representation when a mathematical question arises supports students in reasoning through their confusions.
At the same time, students who generally outperform their peers in mathematics find this content challenging and stimulating. The study of numbers and operations extends beyond efficient computation to the excitement of making and proving conjectures about mathematical relationships that apply to an infinite class of numbers. A teacher explained, “Students develop a habit of mind of looking beyond the activity to search for something more, some broader mathematical context to fit the experience into.”
The Common Core State Standards say this about Understanding Mathematics:
The Standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.”
There are many opportunities for students to justify their mathematical understandings as they notice patterns, regularities, and early algebraic generalizations. As you work through this session consider how early algebra experiences enhance the learning for all learners. How do representations, contexts and teacher questions support students’ learning?
Solve the following problem.
8 x 12 = 4 x 24
Describe what you notice about the factors and how they behave in this problem in your notebook.
Does this happen for other multiplication problems?
How do you know?
What is your conjecture? Will it always work?
What Math Practice do you see enacted here?
See how students in one classroom shared their thinking.
My students all worked individually in their journals and many of them solved both sides of the equation to justify that it was true. Some of them saw the connection between the first factor on the left side being cut in half on the right side, and the second factor being doubled. I asked them to decide if this could be proven without solving each side to show that 96 = 96. This is the array they used to represent what they were saying.”
I followed it up with,
Who thinks they can convince us that this will always work?
To move to a more generalized representation, I said,
I have taken this number (8) and cut it in half. What have I done to the other (12) number?
The students used ‘whatever’ as a placeholder for any number, when they stated:
If you take half of whatever number and double the other number, the product will be the same (and we fine tuned ‘whatever number’ and ‘the other number’ to ‘factors’): If you halve one of the factors in a multiplication problem and double the other factor, the product is the same.
The authors of this article share how explicitly stating generalizations and justifying reasoning with examples has made the math more accessible to all learners across the grades. You will read classroom vignettes focused on two different students.
Use your notebook to write your thoughts about how each student in this article benefited from making the generalizations explicit as they justified their reasoning through spatial representations. How does a classroom culture where looking for regularities and making conjectures benefit the range of learners in all classrooms? Include a quote from the article that you want to remember.
Work in early algebra is fundamental to the experience of young students…
Watch and listen to Investigations author Susan Jo Russell talk about early Algebraic Reasoning
In this activity you will solve a Crazy Cake puzzle, watch videos at different grade levels and consider from both the roles of teacher and students in each of the learning environments.
Watch at least three of the following video clips. Think about the role of the students and the role of the teacher. Respond to the following questions in your notebook for at least three of the videos that you viewed.
In this clip the students discuss their attendance for the day –that’s represented with clothespins. They discuss what the notice about the chart and then double check to make sure they are correct.
Watch Kindergarteners Taking Attendance
In this clip the teacher reminds her students about the work they did the previous day with adding two single digit numbers. You will watch her set up the new task for students; and then watch the class as they share their thinking about adding three single digit numbers.
Watch - Video Coming Soon - Please check back on Thursday
This 3rd grade class has been discussing how many hundreds are in 82 + 392. Watch how they mentally solve the problem and share their thinking with a partner.
Watch 82 + 392
Solve the Crazy Cakes puzzles on page 1.
Watch as students share their strategies for three different Crazy Cakes.
Reflect on all the videos you viewed. Describe the characteristic the different learning environments had in common- the physical setting, the teacher role, the students’ role in your notebook.
In order for differentiation to impact student learning, we need first make sure that our classrooms are places where making sense of mathematics is at the center of the work for both students and teachers and, that we believe that all students, are capable of doing important mathematics.”
In this activity you will solve problems, consider how students reason algebraically as they solve multiplication problems, read short vignettes of students’ strategies, write an expression to match their work and write a general claim that describes how the strategy could work for all multiplication problems.
Elementary age students can and do think algebraically as they develop strategies to solve story problems. Activity 2 and 3 will focus on multiplication generalizations and how they engage and support the range of learners.
Students develop a sense of how multiplication works by using a variety of representations, such as skip counting, story contexts, arrays and drawings of multiplicative situations. In the context of performing calculations and learning factor combinations, students implicitly use the properties by focusing on ways to count groups of objects. This work lays the foundation of three properties of multiplication – associative, distributive and commutative.
Note: Students verbalize and represent the actions of the properties of operations such as multiplication in their own words. The focus is on representing, proving and stating them, not on using formal names for the properties. Having noticed a pattern, students understand the behavior of the operations with whole numbers, and use them when solving problems.
Complete the following Related Story Problems. Follow the directions. Remember to justify your solutions. You may want to print the problems to record your thinking on paper.
When students are given multiplication combinations they do not know, they are encouraged to build up to the answer by beginning with a part of the problem they already know.
Think about how each student solved the problem.
In your notebook:
In this activity you will watch a class engage in solving the related problems you experienced in Activity 2. You will watch students as they interact in a whole class setting as well as with their partners, and consider how the teacher, Ms. F, plans and facilitates the lesson.
Note: Parts of this session have been adapted from an article by Judy Storeygard and Nikki Faria-Mitchell, publication pending.
Keep these benchmarks in mind as you watch the students in following video.
Do the students:
Ms. F begins the lesson by making connections to previous experiences and setting expectations for working on the new problem. She tells the class that she has planned a partner turn and talk during the lesson. She is explicit about the ‘big idea’ she wants the students to think about as they work with their partner.
Students Working in Pairs
Ms. F. is intentional about how she groups students and how she intervenes to keep their discussions on track. Next you will watch two pairs of students working independently. After you watch each pair of students, record what you learned in your notebook.
Ms F. shares…
I think carefully about how to pair students, taking into consideration their learning and behavioral strengths and needs, as well as their social relationships. It was particularly challenging for me to find partners for David and Jeremy, who have attention and focusing difficulties. David has great math ideas, can grasp the content fairly quickly, and make connections within our curriculum.
However, he has a difficult time sitting still and expects to be called on whenever he raises his hand; he needs the confirmation that he is thinking correctly. If he is not sharing, he tends to lose focus. Talking with another student during turn and talk is one way to keep him engaged without him taking over the whole class discussion. If he shares first with another student, he also is able to express his ideas more clearly for the whole class.
I decided to pair him with Tasha. Tasha is not as strong a math student as David, but she can help him slow down so he can complete his work. My goal for her was to become more fluent with multiplication facts, solidify her understanding of equal groups, and begin to see the relationship between the two problems.
Watch Tasha and David as they work together.
Record what you learned in your notebook.
How did Tasha and David work as partners?
Ms F. shares…
Jeremy is quiet and tends to do what he is told, but often is not paying attention and needs a push to think about what he is doing. Nikeesha is a solid math student, but I find that working with Jeremy gives her confidence. In their case, the interaction is not as much back and forth as the typical turn and talk. Nikeesha is able to give him some guidance, which also solidifies her understanding.
I spend a lot of time modeling what it means to help another student, not to tell the answer but to offer resources and examples and let the other student “do” the mathematics. The questions Nikeesha asks Jeremy helps him think about what he needs to do in order to solve the problem. This is thinking he would not have done if left to work on these problems independently. I want him to see the connection of the skip counting between 8 x 3 and 8 x 6, to know that he can use a resource such as our skip counting books, and to gain confidence in explaining his work.
Turning and talking with a partner who could help him focus and actually do some mathematics helped Jeremy begin to make sense of the mathematical relationships in the problem. Nikeesha’s ability to help Joshua and to see the relationship between 8x3 and 8x6 increased her confidence.
Watch Nikeesha and Jeremy work as partners.
Record what you learned in your notebook.
How did Nikeesha and Jeremy work as partners?
Watch Ms. F facilitate the discussion.
Ms F. shares…
Planning and Implementing a Class Discussion…
As I listen, and converse with the pairs, I plan how to structure the whole group discussion. I choose students to share who will bring up some of the key mathematical points. I also think about students’ needs and strengths. In this case, I made sure to call on Jeremy because I wanted to see if he could sustain his focus and communicate the mathematics he had done with Nikeesha.
I began the discussion by making a quick sketch so that students would have a common reference point. I asked Jeremy to show what he had done.
I called on Carlos to share last because I knew he could express the rule we had been working on. Although I knew that not all of the students had reached this understanding, I wanted them to hear one of the students articulate the rule and remind them of where we would begin next time.
Checking in after the group discussion
David, who had done good mathematical work during Turn and Talk was unable to sit during the group. He and I have a signal he gives me when he knows he needs to remove himself. He usually sits close enough that he can follow what is going on, but sitting for a discussion remains challenging for him. Turn and talk is valuable for him because it is a structure that allows him to focus and learn mathematics. I made a mental note to check in with him about his work after the whole group discussion.
Turn and Talk
Ms. F uses Turn and Talk experiences to make the math explicit in a lesson and to keep all students actively learning. Read how she plans and sets expectations.
When my 3rd grade students turn and talk, I try to be explicit about the goals so that the students know the expectations. Students stay focused longer because they work independently, receive immediate feedback as they talk with a partner, and then resume work in their small groups. This prevents students from working independently for too long, a situation that tempts them to engage in conversations unrelated to math. It gives them permission to talk, but for a specific purpose. It also allows them the opportunity to clarify confusions.
I typically use this strategy between one and three times during any given math lesson. Turn and talks may take place during a mini-lesson, independent or group work, or during group share time. Although I plan ahead for when I might use Turn and Talks, I also may incorporate one on the spur of the moment, depending on how the students are behaving and understanding. For example, if I become aware that many students are not grasping something, I ask the class to discuss the concept with a partner. I might come across a student’s strategy that would be helpful to others, and ask students to talk about why this strategy may be useful.
During this lesson on related multiplication problems, I anticipated the students would need time to turn and talk. The students were in different places in their understanding.
Before I asked the children to turn and talk I gave the students instructions to ensure that they had a focused discussion. The mathematical goal was for students to develop an understanding of the strategy that doubling (or halving) one factor in a multiplication expression doubles (or halves) the product. I also wanted them to consider the question at hand: How can what I learned from the first problem help me solve the second problem?
Turn and talks are an integral part of our classroom math culture. By using this routine regularly, I feel that all students know what is expected. They are required to be part of a productive math conversation. Sometimes I give a verbal warning to my students who have a difficult time staying on task. I will let them know ahead of time that they will be talking with a partner about a given task. This heads-up allows them to prepare for the upcoming conversation.
In the beginning of the year, we practice turn and talks using nonacademic topics, e.g. their favorite music, meals. I know they need to practice these guidelines before they can focus on mathematics.
After each turn and talk, we come together and I ask them to evaluate how they did: Did you take turns talking and listening? Did you stay on the topic? Did you ask clarifying questions?
Turn and Talk
Turn and Talk is a powerful strategy that can help students engage with mathematics at different levels of understanding and with different learning and behavioral needs. This flexible strategy takes time and practice to be effective, how it helps build classroom community and includes students who would struggle to focus on their own or in a whole group for an extended period of time.
Rigorous math experiences for ALL Students…
Although it is challenging for teachers, all students need to be included in rigorous mathematics content and practices as advocated in the Common Core. Turn and Talk is one strategy that can actively engage students in mathematics. Above all it is important to be explicit about the mathematical goals for the Turn and Talk and the questions students need to address in their conversations, carefully plan how to pair students, and establish a safe learning environment so that the teacher and students can acknowledge their strengths and learning needs and work together to maximize opportunities to learn mathematics.
Consider how these types of questions will support you in supporting all students so they do more than solve problems.
Describe how making math explicit through questions that promote algebraic reasoning benefit your Sam and Charlie. How will it support one and stretch the other? Record your response in your Sam and Charlie Journal.
...noticing generalizations about addition, subtraction, multiplication, and division are events that occur frequently in a classroom when lessons are structured to elicit students’ ideas.... investigating such generalizations takes students to the heart of their study of number and operations.”
Describe how a classroom culture where looking for regularities about operations benefits the range of learners in all classrooms? Include a quote from the readings or video in this session.
Complete the Session 5 Notebook page using the indicated prompts.
In the final field of your notebook, reflect on the key take-aways from this session for your own learning and record ideas that you will implement to support math learning.
Making the math explicit…
Please contact ETLO to report any broken links or other problems with this page.