Hello! We love to hear your feedback about the TEPS approach and your advice on future research directions. Thank you in advance!

Thanks for sharing your video. This looks like a great project, and I would love to check out your book! I was wondering if you could give an example of what you mean by a worked example?

Such interesting work! Having lived in China (with a kid attending 4th grade there) I find comparisons between US and Chinese classrooms fascinating. 

Are there cultural or language differences that affect how kids approach early algebra topics? I'm thinking here of the research on numbers – how in Asian languages, the ways we express numbers more closely matches base ten thinking, and thus makes number sense easier. 

How do you see this work as related to Liping Ma's seminal work comparing classrooms in the same two countries? 

I was a little confused by the use of the word "conventional" - which seems very culturally-based. What's conventional in the US might not be conventional elsewhere. I'd love to know how you're thinking about this word.

I'm also interested in the Algebraic Knowledge for Teaching described as the goal of the study. Can you briefly describe what you learned on that front? 

What an interesting approach and comparison. I remember years ago watching videos comparing the way that math was taught in Japan and in the US - so illuminating! What specific differences have you found that have informed the development of your program? What have you determined to be factors for AKT? I would like to hear more about the worked examples you mention. Also, say more about concreteness fading. 

Dear Myriam,

Thank you so much for your encouragement!

Before conducting this NSF program, I conducted a series of elementary textbook comparisons focusing on early algebra concepts between US and China (e.g., the equal sign, the distributive property, and the inverse relations). The main difference is that the big ideas in the US textbooks were often presented without sense-making or used as an implicit computational strategy. I also observed many US  teachers' struggles with teaching these early algebra concepts. These have inspired me to pursue comparative classroom research between US and China: How will EXPERT teachers in both countries approach these topics? Here is my project website that listed the relevant prior studies: https://sites.temple.edu/nsfcareerakt/publicati...

The pedagogical dimensions of AKT were based on three factors: worked examples, representations, and deep questions, which were based on the IES recommendations (Pashler et al. 2007). My project findings have enriched these dimensions as I found that worked examples should not be simply shown to students in classrooms. Representations and deep questions play a key role in unpacking the example. Eventually, through learning the worked example, students should be prompted to see the undergirding big idea/concepts.

Worked example in my project extends the literature as it is somewhat different from the way used in laboratory studies (showing example solutions to students). Although the textbook would present the full solution of a worked example, a teacher is expected to take time to engage students in the process of solving the example task. 

Concreteness fading has drawn increasing attention (Fyfe, McNeil, Son, & Goldstone, 2014; Fyfe & Nathan, 2009).  It is a representational sequence starting from the concrete of an object and then fading into semi-concrete and eventually to abstract. In my project, we define that a real-world situation is concrete, manipulatives/diagrams that represent that situation are semi-concrete, and the corresponding number sentences are abstract. This sequence was found by prior studies most supportive for learning and transfer. This is the most frequent sequence we observed in Chinese lessons when teaching the worked examples. 

It's so exciting to see this approach that encourages using real world examples when problem solving and growing mathematical understanding.  I was particularly excited to hear you say that your practice encourages more the process over the final answer.  I have also read and used the ideas from "Thinking Mathematically" in my teaching practice.  At my school and classroom, we follow the CGI (Cognitively Guided Instruction) philosophy of teaching mathematics.  The philosophies of thinking about students as mathematicians seem very similar.  How is the TEPS approach different from CGI?

Hi Noelani, Similar to your experiences, I also have benefited a lot from reading these two books: CGI and Thinking Mathematically! My early algebra project partially originated from these books. As a result, there are clear connections.

For your question, how is TEPS different from CGI? I think there are a few aspects - let me know if this makes sense:

1. In the CGI book, different types of word problems were introduced, and children demonstrated their own ways to solve them. The CGI book mainly documented children's thinking processes (often alternative ways), which were not necessarily linked to formal solutions (e.g., numerical equations). In TEPS, children's ways are encouraged. However, the teacher would ask further questions to guide students to model quantitative relationships (e.g., using tape diagrams) and solve the word problem using equations ("concreteness fading" here). The mathematical emphasis is more in understanding the quantitative relationships and why certain operations were used. 
2. In CGI, all types of word problems were discussed, which also served as the basis of my project. However, the inverse relationships among related word problems (e.g., "inverse word problems") were not discussed. In other words, solving word problems appeared to be a goal in CGI. However, TEPS used related word problems as a context to help make sense of the big ideas (e.g., inverse relations). So, solving a word problem is not the eventual goal. Rather, teachers are expected to engage students in sense-making of fundamental ideas (e.g., inverse relations, basic properties of operations) either implicitly or explicitly through problem solving.
3. Related to the above aspects, I think CGI is more about children's thinking while TEPS is more about teachers' instructional approach (or knowledge for teaching). The latter stresses how a teacher can shift children's thinking from informal to formal understanding of the big ideas.

Thank you for your great question!

This is great! I love hearing about these early introductions to algebra. I use algebra everyday in the "real" world. It's so important kids learn the skills early and then understand how critical the skills are in everyday life. Thanks for your work!