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.
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- 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 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.
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- 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.
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.
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- 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.
During this charting activity, students
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- 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.
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.