IFL Recommends 6/13/23

Jun 13, 2023 | IFL Recommends

This week’s recommendation comes from:

Joe Dostilio posing in front of trees

Joe Dostilio  

IFL Mathematics Fellow

Joe says, “If you’ve read some of my past IFL Recommends, you might be noticing a pattern that I am often reading and listening to things about music and math and sometimes about the connections between them. As my kids approach the time in their lives when they may start to learn to play musical instruments, I thought about maybe starting to play an instrument again too … like, you know, something to fill up all the free time I have! I ran across this video and found the explanation of how math has been used to create different tunings for instruments and thought I’d share it. Enjoy!”

The Mathematical Problem with Music, and How to Solve It 

Yuval Nov

“There is a serious mathematical problem with the tuning of musical instruments.”- Yuval Nov

The serious problem is one that mathematicians like Galileo, Newton, and Euler tried to solve. Watch the video to hear a very detailed and interesting explanation of how ratios and scaling ratios have been used to create different tunings. The video includes several audio demonstrations, so try to listen with headphones to try to hear the differences in tunings!   

Planning for Charting In and Across Lessons 

Jun 6, 2023 | Mathematics Teaching and Learning, Spring 2023

By Kristin Klingensmith 

Charting can be a meaningful learner-centered routine that “serves as a public record of the intellectual efforts of and meaningful contributions from the learning community that can be referenced later or in future discussions” (Klingensmith, Speranzo, Dostilio, Bill, 2020). In our previous charting article, we shared our thinking about what charting is and isn’t, as well as examples of charts from math classrooms in some of our partnering districts. We also shared some information about how charting, when done intentionally to leverage the learning community’s knowledge, leads to more equitable instruction. Charting student responses gives students a voice in their own learning. They can see the impact of their contributions and understand that their ideas and perspectives are valued. This empowerment fosters a positive classroom culture and encourages students to actively engage in learning. 

In our work with teachers over the years, the role of charting in and across math lessons has been a common topic. We have spent time talking about how charting can help students see patterns and variations in thinking and connect different types of representations. We have discussed how charting could be used as a touchstone for discussion because students’ ideas may be charted before, during, and after the discussion. The act of charting allows in-the-moment thinking to be documented so that it can be revisited, revised, and refined over the course of a lesson and across lessons.  

Here are some guiding questions that can help think through how charting may be used to actively document students’ collective understanding at the moment and overtime. There are a lot of questions, and we are not suggesting that each represents an idea that needs to be charted. Instead, we recommend using the set of questions to think through the charting process to surface information critical to supporting your students’ collaborative construction of new knowledge. 

Why Consider the Relationship Between Existing and New Knowledge? 

A starting point for thinking through charting is to consider how a chart or charts can be used to publicly mark students’ ideas, claims, and reasoning related to the mathematical concept or relationship being explored. Students may draw on existing knowledge or make conjectures, which are important to consider in the planning stages.  

When anticipating the knowledge with which students are entering the classroom, consider their diverse backgrounds, experiences, and perspectives. Acknowledge and value the different ways students may approach the mathematical concept or relationship and how they may have encountered it in the world outside of school. Intentionally including diverse perspectives in your planning creates opportunities for each student to contribute and be represented in the charting process. 

The following questions can help guide your planning and ensure that the public record, the chart, of student thinking aligns with the mathematical ideas–concepts and relationships–being studied.  

    • What mathematical idea will be explored and solidified across the lessons? 
    • What prior knowledge do students have that can serve as a foundation from which the new knowledge will be constructed?
    • What are some examples of the mathematical idea(s) in students’ lives outside the classroom? 
    • How is student understanding of the mathematical idea(s) anticipated to grow from lesson to lesson? 
    • What mathematical ideas underly the strategies and/or procedures students are expected to use? 

Classroom Connection 

What prior knowledge do students have that can serve as a foundation from which the new knowledge will be constructed? 

This is a great question to utilize when thinking about the knowledge assets students are bringing into the classroom. To confirm and make public students’ existing knowledge, one third-grade teacher we worked with engaged students in a charting activity at the beginning of a unit of study about understanding fractions as numbers. The teacher started by asking the students to think about the word “half.” She then asked them to draw a picture of half and share with their neighbor how they knew it was half. Then she asked students to share with the class other fractions and anything else they knew about fractions. A few students wrote fractions on the chart, a few others drew diagrams, and the teacher recorded ideas they shared. 

charting work by students

Charting what students know about fractions and where they see or use fractions in their life grounded their upcoming study in meaningful ways and gave them a chance to hear about their peer’s knowledge and experience. Through this charting activity the prior knowledge that the teacher anticipated was confirmed, and she gained insight into students’ thinking. 

Why Consider Ways to Make Sense of and Show an Understanding of the New Knowledge? 

Another critical aspect of thinking through the charting process is identifying the types of representations that could be included. Brainstorming the representations that may surface during the lessons is important because there are several ways to represent mathematical ideas, and there is no “best” way.  

Students come into the classroom with different preferences and strengths when it comes to expressing their thinking and may move through representations in different orders. While some students begin by creating pictures or manipulative models (visual and physical representations) before recording expressions or equations (symbolic representations), other students may think symbolically and then make a visual or physical representation to aid in calculation. Regardless of the types and order of representations, students benefit from translating the math across various representations, so making connections between and among the representations that they and others create is critical. 

These questions can help guide your planning and ensure that the representations created by the learners in your classroom also aligns with the mathematical ideas–concepts and relationships–being studied. 

    • Which real-world and/or mathematical contexts represent the mathematical ideas? 
    • What visual representations illustrate these mathematical ideas? 
    • What symbolic representations show these mathematical ideas?
    • How might students talk and write about these mathematical ideas using their own words and language?
    • What new words or vocabulary related to the mathematical ideas may be needed? 
    • How do the representations, including language, change over the series of lessons? 
    • What connections between and among the representations within and across lessons can be made? 
    • Which of the representations are most likely to come from students? 
    • Which of the representations are new and may need to be co-constructed with students? 

Classroom Connection 

How might students talk and write about these mathematical ideas using their own words and language? 

This question spurred the use of a combination of learner centered routines during the comparing fractions unit in one third grade classroom. The teacher used a combination of quick writes and turn and talks to generate contributions that were then used to create a chart for comparing fractions with like numerators. 

Student charting

Why Consider the Impasses/Barriers Students Might Encounter? 

Part of the charting process should involve identifying the possible impasses/barriers students may encounter as they work to construct new knowledge and link it to existing knowledge. Flawed reasoning, inaccurate conjectures, overgeneralizations, and misconceptions are a natural part of looking for and reasoning about mathematical structures and applying repeated reasoning (Mathematical Practices 7 and 8). Some approaches work until they don’t. Some rules expire.   

Taking time during planning to anticipate the sticky points allows for greater intentionality in addressing them when they surface. The goal isn’t to avoid the impasses and barriers but rather to be ready to address them in supportive, timely, and public ways as a natural part of the learning process.  

These questions can help determine impasses that may signal critical charting opportunities. 

    • Where might there be a disconnect between the ideas, strategies, and representations used in these lessons and those used in earlier lessons or grade levels? 
    • Where might there be disconnects between the language students use and the formal mathematics vocabulary and/or the language of instruction?
    • What procedure is expected, and what do students understand or need to understand about the conceptual roots of that procedure?
    • What are the common misconceptions related to the new knowledge in these lessons, and what is the root of the misconception?
    • What overgeneralizations might students bring into or make during these, what about the structure is leading to the overgeneralization, and how is this structure different?  

Classroom Connection 

Where might there be disconnects between the language students use and the formal mathematics vocabulary and/or the language of instruction?  

What are the common misconceptions related to the new knowledge in these lessons, and what is the root of the misconception? 

These two questions lead to a third-grade teacher engaging students in a charting activity that involved revising the claim about comparing fractions with like numerators (shown below). The first of these two questions surfaced a common disconnect between everyday language and the language of mathematics. Every student is learning mathematical language, and public charting can be used to refine language in collaborative, non-threatening, and public ways. The second question highlighted a lingering misconception about the relative size of the pieces in the whole based on the denominator.  

students charting

During this charting activity, students   

    • revised “bottom number” to “denominator;”
    • revised “top number” to “numerator;” 
    • corrected the statement “the bigger the bottom number the bigger the piece” to say “the fraction with the denominator 6 has more pieces than the fraction with the denominator 4” and “when there are more pieces in the whole, the pieces are smaller;” 
    • discussed two different meanings of “bigger” and decided that it was appropriate when describing comparing the size of the pieces like in “smaller pieces” and “bigger pieces” but should be revised when making the mathematical statement “3/4 is greater than 3/6;”
    • added the symbolic statement “ 3/4 > 3/6 ” to the claim. 
child at board

If you have been using charting as a learner-centered routine in your classroom, we would love to hear your story. We invite you to share any questions that resonate with you and other questions you think need to be added to this list. You can tell us here. 

For more information about the use of learner-centered routines in math classrooms, check out these two articles. 

IFL Recommends 6/6/23

Jun 6, 2023 | IFL Recommends

This week’s recommendation comes from:

Kristin Klingensmith

Kristin Klingensmith  

IFL Mathematics Fellow

Kristin says, “I recently came across this article originally published in 2021 about using pictures to spark math conversations. Even though it’s been two years since it was published, it still offers great how’s and why’s of using pictures combined with a little curiosity to engage students in mathematically interesting and accessible conversations. The article provides ideas for pictures that illustrate a range of mathematical concepts and links to resources where you can access pictures for use in your classroom, even if it might not be until next year! As I read the article, I saw immediate connections to two of the mathematical practices. When engaging in the conversations described in the article, students are likely to make use of mathematical practice #2, Reason abstractly and quantitatively, and #7, Look for and make use of structure. Through this conversation, teachers have the opportunity to elicit student thinking and prompt students to make connections between and among representations.”

How to Spark Engagement in K-8 Math With Picture 

Kristen Acosta 

“We can show students that math isn’t about just algorithms and shortcuts—it’s about seeing patterns and applying what we know to what is seen.”  – Kristen Acosta, math educator 

Using pictures is a good way to help students make sense of math problems. Read on to learn some strategies and examples for various grade levels and different mathematical concepts.  

IFL Recommends 5/30/23

May 30, 2023 | IFL Recommends

This week’s recommendation comes from:

Laurie Speranzo in front of the Cathedral of Learning

Laurie Speranzo

IFL Mathematics Fellow

Laurie says, “Recently I was back on the University of Pittsburgh’s campus for IFL meetings, and I was again awestruck by the grandeur of the Cathedral of Learning. It is a building at Pitt that houses classrooms but also rooms representing nationalities all over the world. The rooms are beautiful and educative. They also honor the students who leave their homelands, their families, their language, their communities to come to Pittsburgh for a university education. Many of the students in our partner districts also live this experience as their families immigrate and make their homes in the United States. They bring to us their identities and their lived backgrounds, and for some, that means being able to share with their classmates ways of doing math that is fostered in other countries.”    

Cathedral of Learning 

University of Pittsburgh 

“The Nationality Rooms are both at the heart of the University and at the center of community life of the city of Pittsburgh. The story of the Rooms is the story of the communities that are the heartbeat of Pittsburgh.”

Built in 1921, the Cathedral of Learning is a 42-story building that functions as the centerpiece of the University of Pittsburgh’s main campus in the Oakland neighborhood of Pittsburgh. Take a virtual tour of the nationality rooms at the Cathedral of Learning and learn more about this beautiful building here: Welcome to the Nationality Rooms | Nationality Rooms (pitt.edu). Explore this comparison of mathematical notation to see how some of our Latin students note their math versus what they see in the United States: Microsoft Word – OPERATION DESCRIPTION2.doc (todos-math.org). 

The Power of Public Charting in Math Classrooms to Engage Every Student in Accountable Talk® Discussions

May 23, 2023 | Mathematics Teaching and Learning, Spring 2023

By Laurie Speranzo and Joe Dostilio

Having students put their work on chart paper can serve the class’s discussion of math. Gallery walks have their place. But effective charting is a meaningful routine to use as it “serves as a public record of the intellectual efforts of and meaningful contributions from the learning community that can be referenced later or in future discussions.” (Klingensmith, Speranzo, Dostilio, Bill, 2020).

In an previous article we unpacked four learner-centered routines that afford math students greater access to classroom discourse. We examined those same four routines through the lens of equitable instruction in a follow-up article. Now, it is time to take on the fifth learner-centered routine: charting!

Learner-centered routines work to

        • spur discussion,
        • increase student ownership of understanding, and
        • create space for student voice

So, what is charting?

Charting IS 

Charting IS NOT 
  • A public record of class thinking 
  • A work in progress throughout the class period or the unit of study 
  • Focused on why the strategy or the deep math works 
  • Mirrors the language that students use (including the first language of students who are English learners) 
  • Includes different types of representations (pictures, diagrams, tables, graphs, expressions/equations) 
  • Possibly messy  
  • A single students’ work (without additional info/annotation) 
  • A procedure or list of the “right” steps students follow without conceptual understanding 
  • A list of definitions of mathematical terms  
  • Decoration for the classroom 

 

 

 

But how do we routinely create useful, authentic charts that support students as a means of preparing to and engaging in Accountable Talk® discussions?

♦ Supporting Accountability to the Learning Community

The learning community is a group that actively listens and reacts to each other’s thinking. Often in a discussion that is accountable to the learning community students will be asked to agree or disagree and explain why. Students may be adding on to each other’s thinking.

By keeping track of the growing thinking publicly, students have a chance to not only hear, but see the progression. Students need to hear ideas multiple times, process them, and see them. Charting aids in being able to see the mathematics—in all its representations! It is very hard to follow the verbal description of a graph but having it displayed allows a deeper level of access.

Charts are Created from Student Thinking and Student Work

To that end, it is crucial that the students’ ideas be the starting point of the charts. There may be a strong hankering to design the perfect chart and laminate it for coming years! 🎵 “Let it go!” 🎵 We all attend to things in our lives where we have an investment; this is true of charts as well! If students contribute to the chart, they are more likely to refer to it throughout (and even after) the lesson or unit. Using a piece of student-generated work or idea to co-construct a chart not only honors students as authors of mathematics, it promotes students engaging in being mathematicians, working together over time to figure things out and moving toward precision through their shared process of learning.

Charts are Publicly Visible and Don’t Disappear

Large interactive screens/touch-sensitive boards in the classroom are amazing! However, the work disappears when the next image or task goes up on the screen. The same with student whiteboards that get erased regularly. Creating a class record of thinking or long-term tool requires a public display that can be referenced as students engage in small group and whole group discussions, and after. That means you may need to use chart paper sparingly, but strategically choose what you want to showcase for the unit of study.

NOTE: Using interactive white boards allows for collaboratively created and edited work to be sent to interactive notebooks. Students can refer to the cumulative version of the class thinking later when it has landed in their personal accessible space. It can also be brought back up on the interactive white board as needed. Technology can be impactfully used as long as it is available when the students want to/need to/ask to access it and not only when the teacher wants to show it.

♦ Supporting Accountability to Content Knowledge

Being accountable to the students’ content knowledge means that there needs to be a focus on helping the students develop conceptual understanding. Use of representations—physical, visual, contextual, verbal and symbolic—helps students construct knowledge and see the structure of the mathematics. Public charting of the representations that students are using or studying helps students attach verbal language to the models. And the representations get cleaner and more precise over time.

Charts Become More Precise as Students Add to Their Thinking Throughout a Lesson or Unit 

In this classroom at the Environmental Charter School in Pittsburgh, PA, the teacher started with a student-generated definition of proportional relationships. They added to and refined the definition as they learned more throughout the unit. You can see the use of student sticky notes to indicate needed changes and teacher-recorded ideas that came out of the whole-class discussions.

Initial Charting  charting in action
Revised Charting  charting in action
Continued Revision of Charting  charting in action

This teacher in Fabens Independent School District outside of El Paso, TX, asked students to record their thinking about the dimensions and volume of cones. Students started with what they knew and added to the chart over two days of exploration. When students were working on problems related to conic volume, the teacher regularly asked students where they could go to find helpful information and this chart became a visual go-to for students as they explored additional 3-D figures.

Class Period 1
Class Period 2 charting in action 

♦ Supporting Accountability to Rigorous Thinking

In math, there are ways to press on rigorous thinking so that students move forward in their conceptual understanding. Two ways we can do that are to:

          • Connect mathematical representations and
          • Press for mathematical reasoning.

Connecting mathematical representations means being able to study at least two – maybe there are two representations you know and feel comfortable with or maybe you are looking at one you have in your toolbox and one crafted by another person. Having public access to what is being connected is crucial.

Charts Capture Students’ Conceptual Understanding

Unpacking mathematical reasoning takes time as students figure out not just the “what” but the “why” of the math. In the scenario below the students are both

          • connecting the symbolic tables and equations to the visual representation of the graph, and
          • figuring out the meaning of the key features as they related to the solution of the systems of linear equations (point of intersection, slope and y-intercept of each equation, solution determined via substitution):
Beginning of Whole Group Discussion Charting in action
End of Whole Group Discussion Charting in action

As students worked on this problem and discussed different solution methods, student language was not mathematically precise from the beginning. As the teacher co-constructed the chart with students, the teacher added mathematical vocabulary as students made connections between the graph and the context of the situation

Student 1:            The lines cross when it’s 7.

Teacher:              When what is 7?

Student 2:            Wait, they’re both 30 at 7.

Teacher:              So, the point of intersection is (7, 30). What does that mean in the context of the problem?

The precise mathematical vocabulary, “point of intersection”, was added to the chart only after students discussed connections between representations and collectively made sense of what it means to be a “point of intersection” during the whole-class discussion.

How does charting support equitable teaching and learning?

When only a few students, like the ones who raise their hands first and often, are the only ones who contribute to a discussion, the classroom is not equitable. Consider these questions.

      • From whom are teachers hearing?
      • Whose ideas are being charted?
      • What assumptions are being made about students’ ability to contribute to the class knowledge?
      • Why is it important to get multiple student perspectives?

By charting collectively, the entire student community can feel that their thinking is important, even if it is messy or imprecise or even off the mark at first. Consistently and publicly revisiting and adding to/revising a chart as a class provides space and opportunities for all voice and sends a clear message that learning is a collaborative endeavor that happens over time.

 

So, what might you try or do more of? Consider:

      • How can you use a student’s or student group’s work to launch a classroom chart?
      • How might you have students add to someone’s thinking to ensure all of the big ideas are made public?
      • How can you use student work and student language to make for a stronger reference tool?

Thanks to

      • Maria Apodaca, Fabens Middle School, Fabens Independent School District, TX
      • Trisha Doerr and Robert Reffert, Environmental Charter School, Pittsburgh, PA

References

Klingensmith, K., Speranzo, L., Dostilio, J., & Bill, V. (2020). Accountable Talk Mathematics Discussions: Teacher’s Guide. Pittsburgh. PA: University of Pittsburgh.

® Accountable Talk is a registered trademark of the University of Pittsburgh.