Math instruction is complex, but it doesn't have to be complicated.
While I have made over 1,000 math videos that are rich in vocabulary and content knowledge, I also understand that these are supplementary tools to enrich students and teachers. They add to but do not replace rigorous initial instruction (sometimes called Tier 1 instruction).
More important than the curriculum teachers use are the instructional strategies they employ when teaching the curriculum (Slavin & Lake, 2008). In other words, the textbook is secondary to solid instructional practices. No amount of worksheets (or even math videos) can ever replace quality initial instruction.
This is a complex issue that cannot be boiled down to a simple blog post. There is, however, an opportunity for all educators to upgrade their instruction regardless of the curriculum they use by employing a simple question and answer feedback loop with students. Click the button below to get a one-page summary of the entire process and keep reading for more detailed information.
Magic 10 questions
In the landmark paper Adding It Up by the National Research Council (2001), which is cited in the Texas Essential Knowledge and Skills (TEKS), five strands of mathematical proficiency are highlighted.
Conceptual understanding
Procedural fluency
Strategic competence
Adaptive reasoning
Productive disposition
Those five strands describe ideal classroom environments in which teachers lead students in robust instruction that requires complex cognition and communication. To kick-start the process, I've adapted 10 question stems that all teachers can use when engaging in a feedback cycle with individual students or during teaching interactions.
What does ___ mean?
What does this remind you of?
What strategy would be best? Why?
What are the steps for ___?
How did you find the answer?
How can you represent your solution?
How do you know you have the right answer?
What other strategies could you have used?
Which parts of the problem were difficult?
Which parts of the problem were easy?
To see how these 10 questions align with the five strands of mathematical proficiency, the process standards in the TEKS, and even Bloom's Revised Taxonomy, click here.
ARF-E cycle
When working with a small group, leading a mini-lesson, or even checking individual student work, starting with a Magic 10 question is a great start. But it's only the beginning.
To maximize initial instruction through enriching mathematical interactions, an entire cycle is needed to stamp the learning. The first step in the Ask/Repeat/Feedback-Encourage/Explore (ARF-E) cycle is to use a Magic 10 question. Once asked, it's the next few steps that will determine how effective the interaction (and learning) will be.
Repeat: Listen to the student's response to the question and repeat it in your own words, including any relevant academic vocabulary
Feedback: Give feedback and/or clarification to the student, ensuring that correct information is provided and reinforced
Encourage or Explore: Based on the student's response, give the child encouragement for his/her work (if correct response is received) or explore with another question to extend the interaction (if incorrect response is received)
Educators are well-versed in the power of questioning yet some fail to seal the deal because they fumble their next steps after the students respond to the question. Learning happens not when teachers share information but when students process it, play with it, and make it their own. These short ARF-E cycles help connect new information to prior knowledge by providing a safe space to wrestle with new concepts.
The ARF-E cycle is aligned with widely-used teacher and instructional evaluation systems, such as CLASS and T-TESS.
To read some sample interactions using the Magic 10 and ARF-E, click here.
Training
If you would like to improve the feedback in your school with the Magic 10 and ARF-E feedback cycles, check out my conference-style and full-day training options.
Support
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References
National Research Council, & Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. National Academies Press.