This morning, I stumbled upon a video clip of two students double counting. These students were immersed in an authentic problem – how do we read two counts? I loved hearing their two voices and watching their gestures as they navigated two counts, uninterrupted by adult conversation.
This video reminded me of a question that I have been mulling over for months now: How do students construct a mathematical identity that is rooted in the context of who they are and who they want to become outside the classroom? In particular, I wonder, how does the way we teach students to construct viable arguments and critique the reasoning of others nurture or neglect their identity as a whole person?
The predominant ways that I have seen students taught to engage in this math practice have been:
sentence stems like “I disagree because…”.
orchestrated turn taking
I think these structures and norms are used with the intention of giving students equal opportunity to participate and feel like their ideas were heard. But do they work? Does each student feel like the opportunity to argue for something they believe to be true has been genuinely presented to them? Does each student feel like their true voice is heard? When I think about the word “argue”, I think about passion, frustration, and joy. I also think about family and community. What does a productive argument look like in the lived experiences of each of our students? Are there sentence stems, raised hands, and orchestrated turns taken?
I am not saying that we should abandon structures and norms in our classrooms. Structures and norms are intended to ensure that arguments are productive. I wonder if we need to reconsider how the structure and norms come to be. Who benefits from the structures and norms?
When I watched the video of these students double counting, I think I saw passion, frustration, and joy, but I don’t want to project my thoughts and feelings onto the students. I saw back and forth conversation, ideas being exchanged, and thinking being revised, all while an adult observed.
This video clip was a spring board -a push- for me to re-examine classroom norms and routines that I always used and accepted as “good practice” for all students. I am going to sit with the questions:
Where did our classroom norms and routines come from and who do they really serve?
Whose culture have I preserved when I cultivated and enacted classroom norms and routines?
I haven’t been able to write much on my blog for the last two years because I have been busy writing the 5th grade course for the Illustrative Mathematics K-5 math curriculum. I am humbled and honored to share some of the story my colleagues and I are writing.
From the start of the year, we want students to know they are capable of engaging in grade level mathematics. In the Opportunity Myth (2018), data has shown there is an opportunity gap for historically marginalized students between the grade level expectations laid out in standards and students’ opportunities to engage with this content in their math classes. Oftentimes, grade level content is withheld from students because they are perceived as being not “ready” and, therefore, they are restricted to only engage in below grade level work. If we want to close the opportunity gap, we have to do many things differently than we have done in the past, one of which is re-examine the sequence of the content we put in front of students. The Illustrative Mathematics K-5 story of fifth grade was intentionally written so all students have access to grade level content.
Typically, many 5th grade math curricula start the year with units that focus on place value concepts and whole number multiplication and division algorithms. When considering whether to start the year this way, we wondered:
What message does the mathematical content in the early units of the year send to students about what it means to ‘do math’ and what is valued in math class?
What assumptions might educators make about students based on their successes or challenges with multiplication and division algorithms at the beginning of the year?
How might these assumptions impact students’ access to grade level content throughout the course of the year?
Extend an Invitation
From the start of the year, we want to send the message that mathematics is an opportunity to be curious, collaborative, and playful. To do this, we begin by introducing students to the concept of volume. Being a new concept to fifth graders, it naturally invites students to be curious and creative, while at the same time offers teachers the opportunity to notice and build on what students do know and can do. It also diminishes, if not eliminates, the assumptions both teachers and students might make about what it looks like to be ‘good at math’.
Students begin their study of volume with many opportunities to build with unit cubes and use familiar words and phrases to explain the layered structure of rectangular prisms. Teachers listen for and display the words students use, such as “slices”, “groups”, or “layers”, to describe how they counted the cubes to determine the volume. Students connect their less formal language to multiplication expressions that represent the volume of rectangular prisms before they are introduced to more formal math vocabulary and procedures.
How do the expressions 5 x 24 and 6 x 20 represent the volume of the rectangular prism? Explain or show your reasoning.
Throughout unit 1, the measure of prism side lengths were chosen to encourage students to make sense of concepts of volume while they continue to strengthen their fluency with multiplication. Over time, students connect their informal strategies for finding the volume of rectangular prisms to generalized formulas. Later in the year, after learning the standard algorithm for multiplication in unit 4, students solve problems involving volume with larger multi-digit numbers as side length measures. This progression increases the opportunities for students to be successful with multiplication and division algorithms and decreases the likelihood that they will be perceived as not ready for new grade level topics because of their proficiency with computation. Therefore, we believe these choices will decrease the opportunity gap and increase student access to grade level content.
Put Fractions in the Forefront
Beyond Unit 1, we want to continue to maximize students’ opportunities to access grade level content. In Units 2 and 3, we introduce grade level content while providing ongoing opportunities to reinforce prior grade level understandings and fluencies, all within the work of fractions. While many curriculums organize whole number computation before fractions, this sequence has implications for extending an opportunity gap. If students are perceived as not being able to use whole number multiplication and division algorithms correctly, they may be confined to remediation cycles and denied the opportunity to engage with grade level fraction concepts and procedures. We know that understanding fraction operations is a major indicator of success with topics in high school mathematics so we want to ensure that each student has the opportunity to make sense of grade level fraction understandings and operations early in the year ( Siegler et. al., 2012, p.695).
Continue to Build on Prior Grade Level Content
In units 2 and 3, students revisit the meaning of fractions, multiplication, and division. They connect what they know about these topics to new learning.
Students begin unit 2 by interpreting a fraction as division of the numerator by the denominator. Through this work, they continually deepen their conceptual understanding of fractions and division. In the example below, students connect what they know about unit fractions and division.
Complete the table.
number of people sharing a sandwich
amount of sandwich for each person
Choose one row of the table and draw a diagram to show your reasoning for that row.
What patterns do you notice?
As the unit progresses, students extend their understanding of multiplication as they multiply a whole number by a fraction. In the example below, students leverage what they know about whole number multiplication to solve problems involving multiplication of a whole number and a unit fraction.
Find the area of the shaded region. Explain or show your reasoning.
In unit 3, students continue to use area diagrams to make sense of fraction multiplication. They build on the new understanding of multiplying a whole number by a fraction while applying prior grade understanding of area. They recognize, when multiplying fractions, the unit is the size of the piece being tiled within the unit square. They count those unit tiles by multiplying the number of tiles in each shaded row by the number of shaded rows.
For example when asked to determine the area of the shaded region below, students know that the unit square is partitioned into a array, so the size of each tile is . They then multiply to find how many tiles are shaded and determine that the area of the shaded region is. Students later apply these understandings to multiply any two fractions.
Students are offered another opportunity to revisit the major work of prior grades in the context of the major work of 5th grade when they are introduced to division of a unit fraction by a whole number and a whole number by a unit fraction. As shown in the example below, the sequence we chose offers students the opportunity to be curious and creative while deepening and extending their understanding of the meaning of division.
Sequencing fraction multiplication and division before whole number algorithms invites teachers to notice and celebrate what students do know about multiplication and division and minimizes the chances of teachers and students making assumptions about what students don’t know about fraction operations.
Make Algorithms Accessible
By the time students encounter whole number multiplication and division algorithms in unit 4, they have had significant experience building the necessary fluencies and understandings to be successful. Teachers have had significant opportunities to notice what each student knows and can do in regards to applying concepts of multiplication and division.
In telling the story of 5th grade, we chose to begin with an invitation to engage in grade level mathematics, introduce fractions early in the year, and postpone multi-digit multiplication and division algorithms until later in the year. We think these choices will:
send the message that each student is creative, curious, and capable of doing grade level math in the beginning of the year.
support teachers to notice and build on what students know and can do.
increase opportunities for students to access grade level content.
We hope these choices lead to a world where more historically marginalized students have the opportunity to know, use, and enjoy mathematics.