NSF Awards: 1350068

The goal of this project is to identify Algebraic Knowledge for Teaching (AKT) that is useful to develop students’ algebraic thinking. Through an exploration of U.S. and Chinese elementary expert teachers’ video-taped lessons, this project has discerned a type of AKT named *Teaching through Example-based Problem Solving* (TEPS), which is documented in a recent book with Routledge (Ding, 2021). This approach emphasizes engaging students in the process of working out an example task through pertinent representation uses and deep questioning. With regard to representation uses, a teacher may situate a worked example in a real-world context (e.g., word problem), which can be modeled through “concreteness fading.” For deep questioning, a teacher may ask concept-specific questions to promote meaning-making and comparison questions to promote connection-making. TEPS has been illustrated through selected Chinese and U.S. lesson episodes (with 25 annotated video clips) on two fundamental mathematical ideas, inverse relations and the basic properties of operations.

## Meixia Ding

Lead PresenterAssociate Professor

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

## Ana Stephens

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?

## Meixia Ding

Lead PresenterAssociate Professor

Thank you Ana! The worked example in a textbook lesson would be the sample task presented before the practices (either guided or independent). In our project, we studied how a teacher unpacks an example task before asking students to practice more tasks. There are different styles. Some teachers will just show and tell the example while others will engage students in the process of working out the example.

## David Kung

FacilitatorDirector of Policy

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?

## Meixia Ding

Lead PresenterAssociate Professor

Hi David! Great to know that you lived in China and your kid went to school there.

Thank you for your great questions! Here are my responses:

1) It is very interesting to think about the language factor. In my project, I did observe this influence (but I did not study it in depth). For instance, one of the topics is the associative property of addition. This property is implicitly used in a strategy called "make ten to add", which was taught in both US and China. Say, 9+4=9+(1+3)=(9+1)+3=10+3. (There is a special way to document this decomposing process in China; the US classrooms have varied ways to record this). Before this lesson, Chinese students have learned the idea of "10 + something = ten something." I am sure that you know "11" is called "ten one" in China, and "12" is called "ten two" and so on (place value). Chinese teachers would highlight "10 + something = ten something" which shows HOW EASY when one adds a number to 10! With this prior knowledge, students are motivated to learn lessons like "9+ something" by figuring out how to make a 10 to add. In short, yes, Chinese children do benefit from the language factor in some sense.

2) Yes, I definitely see how my study is related to Liping Ma (1999). I think it makes one step further as Liping only interviewed teachers in both the US and China and did not observe classroom teaching. With the NSF support, I had the privilege to collect and analyze actual classroom lessons focusing on parallel early algebra lessons, which provided a lot of insights for gleaning useful knowledge for the field. In addition, China has undergone math education reform since 2001. The classroom teaching in my study is more aligned with the spirit of Chinese math education reform.

3) Yes, I agree the "conventional" approaches in the US may not be conventions elsewhere. My intention in using the word "conventional" was more in a decontextualized way. For instance, I consider teaching keywords as "conventional," which could occur in different countries. With that in mind, I was hoping to identify useful knowledge that can be used to develop children's algebraic knowledge in a more meaningful way. So, that's the goal of my project - to identify "integrated" insights from both US and Chinese expert teachers' classrooms.

4) So, the identified AKT was documented in this video that I shared here. It is a teaching approach called TEPS: Teaching through example-based problem-solving. Briefly, the example of a lesson was situated in a real-world context (e.g., a word problem), which was unpacked through the process of engaging students to solve the problem (not just to present a worked-out solution to students). During this process, representations and deep questions were used to help students see the "big ideas" (or fundamental concepts) behind the worked example.

Thank you again for so many insightful questions!

## Myriam Steinback

FacilitatorConsultant

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.

## Meixia Ding

Lead PresenterAssociate Professor

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.

## Myriam Steinback

FacilitatorConsultant

Thank you for your response. Very interesting - I believe as you do - that teachers need to work on the problems themselves. Showing is what we want to get away from; we want students (and teachers) to have opportunities for thinking and problem-solving.

## Meixia Ding

Lead PresenterAssociate Professor

Thank you Myriam! It is interesting to hear your comment. I have been teaching a math methods course. Many of my preservice teachers bring back from the field a popular lesson structure named "I do, we do, and you do." The "I do" section refers to the new lesson (worked example). I always argue back that a good math lesson should not contain a PURE "I do" section during which the teacher shows and tells everything.

## Myriam Steinback

FacilitatorConsultant

Hurray for you!!!! That’s how it should be. Allowing thinking is the best.

## Meixia Ding

Lead PresenterAssociate Professor

Thank you for your encouragement, Myriam!

## Noelani Ogasawara Morris

FacilitatorDemonstration Teacher

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?

Myriam Steinback

## Meixia Ding

Lead PresenterAssociate Professor

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:

Thank you for your great question!

## Noelani Ogasawara Morris

FacilitatorDemonstration Teacher

Thank you for clarifying as that does make a lot of sense to me as I reflect on my own teaching process as well. I often worry that as a CGI teacher I am 'over teaching' strategies but recognize your research touches on how teachers can support the students to work towards building on to the student's thinking but adding a layer of efficiency through the introduction of new tools to use.

## Meixia Ding

Lead PresenterAssociate Professor

Thank you so much for your very thoughtful question and discussion! It helps me think deeper. And, I cannot agree with you more :)

## Ann Podleski

I am interested in ideas that can help older students who haven't had the benefit of early education in math that promoted engaging in fundamental ideas of inverse relationships and basic properties of operations. (I was struck by the goal of stressing how a teacher can shift student's focus through their instructional approach and thinking about some ideas for college teaching. I think the idea of having the student answer questions about why two problems are different is something I could try.

## Meixia Ding

Lead PresenterAssociate Professor

Thank you, Ann! I also just watched your video. What a cool project your team has been doing with the bigger students!

## Abigail Helsinger

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!

## Meixia Ding

Lead PresenterAssociate Professor

Thank you, Abigail! It is great that we share the same view about the important role that "real" world situation plays. I just watched your video and noticed this connection between our projects!