IFL Recommends 6/27/23

This week’s recommendation comes from:

Aaron Anthony smiling for the camera

IFL Staff Blurb 

We thought we would end the 2022-2023 school year IFL Recommends with this Dr. Rochelle Gutiérrez podcast on rehumanizing mathematics. The IFL was fortunate enough to have Rochelle speak to us about the dimensions in equity back in 2019. We are delighted to share some more of Rochelle’s words of wisdom related to humanizing mathematics in this podcast. 

Dr. Rochelle Gutiérrez on Rehumanizing Mathematics

Podcast with Alec Patto

“I have an abuela rule in my classes and in my work, and the abuela rule is this. We need to be able to communicate with each other in ways that are understandable to our grandmothers. I am not saying that we need to water down or distill or somehow dumb down the ideas that we have, but our abuelas carry such wisdom in this world. We do not want to waste their time with jargon when we can just say things simply.”  

Dr. Rochelle Gutiérrez, professor of curriculum and instruction at the University of Illinois 

In the podcast, Dr. Gutiérrez talks about reconnecting math to our lived experience, and sharing the world’s rich history of different approaches to mathematics with kids. 

IFL Recommends 6/20/23

This week’s recommendation comes from:

Aaron Anthony smiling for the camera

Aaron Anthony  

IFL Director of Operations  

Aaron says, I recommend the Chartr newsletter. I love data and data visuals and have found this daily newsletter to be a quick and fun way to take in a hodgepodge of topical information. I also like to use it for inspiration on ways we at the IFL can use charts to quickly and clearly share our own data.

chartr

“Bite-sized, visually appealing and factual content will always have a place in journalism. If you’re bored of stock images and fake news, we think you’ve found the right place.”  chartr 

If you are a visual thinker who enjoys data-driven insights, check out the chartr newsletter. Published several times a week, these five-minute reads look into data in business, tech, entertainment, and society. 

Productive Talk + Purposeful Play = Learning in Action 

By Michelle Cianciosi-Rimbey and Kristin Klingensmith with Michael Telek 

Calls for ensuring that purposeful play is a critical component of early childhood classrooms have been made by researchers and educators (Hadley & Newman, 2022; Allee-Herndon & Roberts, 2021; NAEYC, 2022). Zosh and her colleagues (2022) present purposeful play on a spectrum, ranging from self-directed play, guided play, playful instruction, and direct instruction. These types of play mainly highlight the degree of adult guidance, child autonomy, and the presence of learning goals. What we consider here is a focus on a high degree of guidance from teachers with clear learning goals in the areas of math and literacy instruction.

A critical component of purposeful play is engaging children in talk. The amount of time we engage in talking with young children matters. Research shows that the number and quality of the conversations that adults have with young children directly affects how they learn to talk (Shonkoff & Phillips, 2000). The number of total words, and the variety of words that parents and adults use with the child on a daily basis, the number of conversations and discussions, and the number of positive affirmations are related to language development (Hart & Risley, 1999).

In addition, oral language development is considered an unconstrained skill. In other words, it is a skill that can never be fully mastered because there is always more to learn.

IFL fellows share four strategies for embedding productive talk into purposeful play. 

    • Expect a Talking Community
    • Find Out Their What and Why
    • Show and Discuss
    • Go with Their Flow

These strategies can be used to spark, strengthen, and enhance language skills that positively impact learning for each and every student.

Expect a Talking Community

An expectation that talk is used to explore and explain ideas is essential in supporting young children in language development, as well as conceptual understanding. Teachers can provide opportunities for children to listen to and respond to one another by repeating back what was said, putting what was said in their own words, and adding on to what was said. Teachers can also provide opportunities for students to make and support claims and offer counter ideas.

Students should be welcome to use the language that they use at home while at school to communicate their ideas. Whether a student is acquiring English as their only language or as an additional language, hearing, saying back, saying it another way, helps build and strength their oral language.

Approaching learning through talk allows children the much-needed time to process new ideas and information and integrate them into their existing thinking. Often, children need time to verbally make sense of and articulate their thinking with the community of learners who are also engaged in the same experience and journey.

Dramatizing Events in a Read-Aloud

The first example of incorporating purposeful play involves a read-aloud of the picture book, The Girl Who Wore Too Much: A Folktale from Thailand. In the book, a girl who is getting ready to go to a dance cannot decide which of the many dresses and jewels she should wear to a dance. She decides to wear them all and learns a valuable lesson. After the teacher read the story, she placed costume jewelry and clothing in the dramatic play area in the classroom. She encouraged children to visit the area later in the day to recreate the events in the story which generated excitement and interest among her learners. This activity encourages students to retell the story through dramatic play, as a way to support their comprehension of the text and encourage verbal interactions among their peers.

Book the girl who wore too much

Find Our Their What and Why 

An expectation that talk is used to explore and explain ideas is essential in supporting young children in language development, as well as conceptual understanding. Teachers can provide opportunities for children to listen to and respond to one another by repeating back what was said, putting what was said in their own words, and adding on to what was said. Teachers can also provide opportunities for students to make and support claims and offer counter ideas.

Students should be welcome to use the language that they use at home while at school to communicate their ideas. Whether a student is acquiring English as their only language or as an additional language, hearing, saying back, saying it another way, helps build and strength their oral language.

Approaching learning through talk allows children the much-needed time to process new ideas and information and integrate them into their existing thinking. Often, children need time to verbally make sense of and articulate their thinking with the community of learners who are also engaged in the same experience and journey.

Conservation

In this video, students were playing at the water table and the teacher joins them and begins engaging them in a conversation about how the 1 cup of water looks different but doesn’t change in volume when it is placed in different containers.

The video focuses on conservation of volume an idea related to the conservation of quantity and linear measurement. Through multiple cognitively engaging experiences, students had an opportunity to develop, revise, and refine their thinking and reasoning related to conservation. Through these experiences, students also had several opportunities to further develop their use of verbal-language when sharing their thinking and reasoning. The combination of experiences and discussion lay the foundation for a fully developed understanding of conservation which can take years to establish.

 

Show and Discuss

Pictures, images, charts, and manipulatives all serve as scaffolding tools for understanding concepts and language. This strategy allows the visual representation, in this example manipulatives, to serve as a scaffolding tool that helps guide and shape understanding through discussion, rather than as a product of the child’s understanding.

Playful Interactions around Oral Language Development

During our discussion of robust vocabulary instruction with coaches, they described how teachers incorporate playful interactions with sophisticated language for their youngest learners. One story that shared involved the word “exhausted.”

After children learned the meaning of the word, exhausted, through a read-aloud the teacher incorporated playful interactions with the word throughout the school day. An example of one of these interactions included the teacher taking students to the playground and asking them to run as fast as possible. When the students returned to the classroom, she asked them how they felt as they collapsed on the floor. This playful interaction cemented students’ understanding of the word, exhausted, not only because they experienced how it felt to be exhausted, but because the conversation that took place following their rigorous run around the playground, provided opportunities to extend their knowledge about the word while building their oral language development. The teacher posed questions such as, “How does this experience connected to the word, exhausted? How do you feel when you’re exhausted?” These thoughtfully planned interactions with words across a variety of contexts supported children’s learning.

Playful Interactions Around Oral Language Development Graphic

Go with Their Flow

Staying on topic is an incredibly valuable skill, and one that teachers can use to support children’s language development and engagement in discussions. When teachers take language turns with young children and stay on the same topic, children are significantly more likely to reply than if the teacher responds but changes the topic (Dunham & Dunham, 1996). This strategy makes use of a child’s prior knowledge and allows the child to elaborate on topics that interest them in the moment because the teacher’s semantic elaboration supports the likelihood of the child to stay on topic and engage in more language related to the topic. This strategy is especially useful in less structured learning opportunities but can also be utilized in more formalized learning situations because it honors the child’s thinking pathway.

Sorting and Categorizing

The students in this video had been exploring sorting and categorizing in mathematics and had engaged in activities focused on matching by single attributes, then sorting by those attributes, and finally sorting and categorizing by two or more attributes. They had experience sort, resorting, and sub-sorting by color, size, and shape.

In this video, the students and their teacher were in a play center with a bin of balls. The teacher asks them what they can do with the balls and one of the students responds with “sort them.” The video shows the student sorting the bin of balls by color. At one point in the video, the teacher turns to another adult in the room and asks for the word “blue” in Bangla and then uses it to engage a Bengali student in the group. The students also decide to resort the balls according to their size. Throughout the activity students, including those whose primary language is not that of instruction, have opportunities to make claims, give justifications, and debate where to put a multi-colored ball.

References

Allee-Herndon, K. A., Roberts, S. K., Hu, B., Clark, M. H., & Stewart, M. L. (2022). Let’s talk play! Exploring the possible benefits of play-based pedagogy on language and literacy learning in two Title I kindergarten classrooms. Early Childhood Education Journal, 50(1), 119–132. https://doi.org/10.1007/s10643-020-01141-6

Allee-Herndon, K. A., & Roberts, S. K. (2021). The power of purposeful play in primary grades: Adjusting pedagogy for children’s needs and academic gains. Journal of Education201(1), 54-63. https://doi.org/10.1177/0022057420903272.

Dunham, P. & Dunham, F. (1996). The semantically reciprocating robot: Adult influences on children’s early conversational skills. Social Development, 5(3), 261-274. DOI: 10.1111/J.1467-9507.1996.Tb00085.X

Hadley, E. & Newman, K. (2022). Prioritizing purposeful and playful language learning in Pre-K. The Reading Teacher, 76(4), 470-477. https://ila.onlinelibrary.wiley.com/doi/10.1002/trtr.2161

Hadley, E., Newman, K. & Mock, J. (2020). Setting the stage for TALK: Strategies for Encouraging Language-Building Conversations. The Reading Teacher, 74(1), 39-48. https://ila.onlinelibrary.wiley.com/doi/abs/10.1002/trtr.1900

Hart, B. and Risley, T. R. (1995). Meaningful differences in the early experiences of young American children, Baltimore, MD: Paul H. Brooks Publishing.

Hart, B. and Risley, T. R. (1995). The social world of children learning to talk, Baltimore, MD: Paul H. Brooks Publishing.

Shonkoff, J. P., & Phillips, D. A. (Eds.). (2000). From neurons to neighborhoods: The science of early childhood development. National Academy Press.

Zosh, J. M., Gaudreau, C., Golinkoff, R. M., & Hirsh-Pasek, K. (2022). The power of playful learning in the early childhood setting. In NAEYC (ed.), Developmentally Appropriate Practice in Early Childhood Programs Serving Children from Birth Through Age 8, 4th ed., 81–107. Washington, DC: NAEYC. Retrieved from https://www.naeyc.org/resources/pubs/yc/fall2022/teachers-questioning-math-learning

IFL Recommends 6/13/23

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 

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.