Language in Mathematical Thinking
You just can’t think mathematically without language. (Try it!) You may have internal thought bubbles, talk to yourself aloud, or collaborate with others. However, if you are thinking mathematically, you are calling upon language. And, working through ideas verbally supports and develops mathematical thinking.
As Phil Daro, mathematics educator who co-directed the development of the Common Core State Standards for mathematics, described, “there is a new relationship between language and content that is quite explicit in the Standards” and “knowledge, cognition, [and] language, these are all threads of a single fabric of learning.”
So, as we work to develop mathematical thinking and reasoning in each and every student, it is critical that we simultaneously develop and support students’ mathematical language. One essential strategy that serves to support these interwoven goals is to provide students with sentence frames and starters.
Sentence Frames and Starters as a support for mathematical thinking
Sentence frames and starters are skeletons of sentences that help students organize and communicate their thinking.
Purposeful sentence frames and starters play a dual role – they focus student attention on the mathematical thinking at hand and support them to articulate that thinking by providing a starting point or frame for the thought. When students implement sentence frames and starters, they increase their language production, collaboration, and mathematical thinking.
In order to implement sentence frames to develop mathematical thinking, the frame itself needs to focus on the thinking, rather than answers or outcomes. Consider the following scenario. Students are working to notice and use mathematical structure as they connect a story (problem context) to a graph that represents the story/context.
If we provide students with a sentence starter like, ‘____ matches ____’, we are clearly positioning them to focus on the answer and bypass the articulation of their thinking that led to the answer. Responses that integrate that sentence frame may sound like, ‘A matches 2’, and end there, failing to highlight the underlying thinking.
If we provide students with a sentence frame like, “I noticed…so I looked for…’, we draw them back to what they first noticed about the story or the graph, i.e. elements of mathematical structure. Students then continue to share what they did with their noticings that helped them make the connections. Responses that integrate a ‘thinking-focused’ sentence frame like this one may sound like, ‘I noticed the flat part of the graph, so I knew that was when the distance wasn’t changing, so I looked for a story that talked about when the runner stopped.’ This response raises critical mathematical ideas and provides a lot of fodder for classmates to consider and discuss.
When are sentence frames and starters most helpful?
Sentence frames and starters provide organization of language and thoughts for students. As such, whenever students are organizing their thinking and the language necessary to describe their thinking, sentence frames are super helpful. In particular, we find sentence frames and starters to significantly increase student language production and advance student thinking if they are provided when students:
- begin to describe their ideas with a partner
- share in the full group
- reflect on what they learned – in writing or orally
Sentence Frames and Starters remotely
Sentence frames and starters can provide the same support for students to develop mathematical thinking and language while learning remotely. And, with careful planning we can overcome challenges as we implement sentence frames and starters and ensure that students use them – even from a distance!
Teachers are working in many different contexts – synchronously, asynchronously, in hybrid models, etc. In each of these contexts, sentence frames and starters can continue to provide their critical support. Some ideas to include sentence frames and starters in remote instruction include:
- Include them on common workspaces for students to respond to during partner work or individual work – Jamboard, Google slides, etc.
- Include them on print materials and provide space for students to respond using them
- Prompt sentence frames and starters and provide time for students to use them in synchronous sessions and screencasts or pre-recorded video
- Provide them to parents/family (especially if you have frequently used starters or frames!)
Would you like to learn more?
Webinar: Building Math Discourse and Multiple Representations into Your Remote Teaching, November 18th, 7 pm – 8:30 EST Register here.
Read more about Connecting Representations in Chapter 4 of Routines for Reasoning
