What’s critical in planning a mathematics lesson for English Learners?
Often, when we work with groups of teachers planning lessons, we ask them about lesson goals and objectives. Frequently, the answer is a specific mathematics standard, a topic, or a particular procedure they want students to carry out. While this procedure is part of the goal, it is not sufficient on its own.
A powerful goal for a lesson combines conceptual understandings, mathematical practices, and uses of language to explore both ideas and practices. Conceptual understanding lies in the many connections between procedures, representations, ideas, and scenarios. When completing Collaborative Lesson Inquiry Cycles, this point is where the journey begins.
Haiwen: From Compound Probability to Independent Events
One example comes from our work with high school mathematics teachers in the Jurupa Unified School District. We were working on a data science lesson, and the initial topic was compound probabilities:
If you know the probability that a card is red and the probability that a card is a face card, what is the probability that it is both red and a face card?
There are well-defined procedures (computing probabilities) and representations (two-way tables) for this topic, but as a group we decided we needed a deeper idea.
We settled on the concept of independent events:
Does one condition make another more or less likely?
This question has familiar applications within a deck of cards: being red and being a face card are independent—they do not make each other more likely. Being a heart and being red, however, are highly dependent on each other. If a card is red, it has a 50–50 chance of being a heart, compared to 25% in general. Conversely, a heart is 100% guaranteed to be red.
To plan the lesson, we decided to ground this concept in students’ own opinions about engaging current controversies, which we provide in the article published in the February 2025 issue of Mathematics Teacher: Learning and Teaching PreK–12.
As for mathematical practices, several are salient:
Reason abstractly and quantitatively
Attend to precision
Look for and make use of structure
In terms of language, students further engage in the following practices requiring language:
Expressing conditionality and contingency (e.g., “The probability that someone passes the class given they have attended tutoring is…”)
Linking and comparing probabilities (e.g., “The probability of…is the same as the probability of…given…”)
Advancing contextualized rationales (e.g., “We think…and…are independent because…”)
Monique: From Listing Values to Relating Domains and Range
A second example comes from our work with high school mathematics teachers in East Side Union High School District. We were working on a Math 1 lesson where the topic was finding domain and range from a table or a discrete graph. This procedure is very clearly defined, but we wanted to offer students opportunities to make more connections, especially because domain and range are important across all of high school algebra and beyond. We settled on a more substantive idea, that is:
If two relations have the same domain and range, are they identical?
For example, the domain could be a single group of students. These students are related to grades in English and in Math. The overall range, or set, of grades in both classes is probably the same. A particular student, however, may have a B in English and an A in math. Simply put, grades in English are not the same as grades in math.
Student | English Grade | Math Grade |
Alberto | A | B |
Brenda | B | B |
Carl | B | A |
Donna | A | A |
To plan the lesson, teachers collaborated to design a task in which students would individually find the domain and range of their tables and discrete graphs, then partner up and share their domain and range to find if their partner’s matching domains and ranges led to the same relation. By comparing their data they would “bridge the gap” between their two graphs through talk.
The salient mathematical practices included:
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Look for and make use of structure
In terms of language, students engaged in the following practices requiring language:
Describing representations (e.g., “My graph/table has a domain of… Do you have something that matches my domain?”)
Relating domains to corresponding ranges as sets (e.g., “The range that corresponds to my domain is…”)
Checking if domain and range match (e.g., “I think mine and yours are (similar, different)…because…”)