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CAREER: Algebraic Knowledge for Teaching: Evidence from Inverse Relationships

DRL-1350068

The goal of this five-year CAREER project is to identify, from a cross-cultural perspective, essential Algebraic Knowledge for Teaching (AKT) that will enable elementary teachers to better develop students’ algebraic thinking. This video reports the main findings of our year 1 data. Based on US and Chinese expert teachers’ integrated instructional insights of inverse relations, we have identified three AKT components: unpacking one worked example sufficiently, using concrete representations to model quantity relationships, and asking comparison questions to promote connection-making.

## Meixia Ding

PresenterHi Everyone, We are eager to hear about your constructive feedback and questions to improve our study!

## Miriam Gates

These are really interesting findings. I’m curious if you can briefly discuss your methods for identifying these components.

## Meixia Ding

PresenterDear Miriam, Thanks for your question! Let’s see if I have addressed your question. If not, I will re-answer :)

The three components/dimensions are identified from the IES recommendations for “organizing instruction and study to improving student learning” (Pashler et al., 2007).(http://ies.ed.gov/ncee/wwc/practiceguide.aspx?s...). Overall, there are several recommendations. All of them have been drawn from high-quality studies in fields of cognitive science, experimental psychology, and education.

In my study, the first dimension of “worked examples” (Interweaving worked examples with practice problems) is aligned with the IES recommendation #2. The second dimension of “Representations” (Connecting concrete and abstract representations) is a combination of the IES recommendations #3 and #4. And, the third dimension of “deep questions” (Asking deep questions to elicit students’ self-explanations) is based on the IES recommendation #7. These selected recommendations seem to most directly relate to classroom instruction. These dimensions are also consistent with the key aspects of Mathematical Knowledge for Teaching, a notion developed by Deborah Ball and associates. In fact, this three-dimension framework has been used in my previous studies on teachers’ lesson planning and comparative textbook analysis, which has been found to be fruitful.

The above IES recommendations provide general directions for organizing instruction to improve learning. However, without explicit guidance, it was found that many teachers could not successfully implement these instructional principles in lesson planning (Ding & Carlson, 2013), let alone complex classroom teaching. For example, even if a textbook provides a worked example using a word problem context to teach inverse relations, instead of simply asking students to study the example, how can a teacher unpack it to help students make sense of inverse relations? How can this teacher guide students to purposefully represent and solve this word problem to illustrate inverse relations? What types of deep questions can this teacher ask to make inverse relations explicit to students? These questions call for an exploration, along with these dimensions, of detailed and transferable Algebraic Knowledge for Teaching (AKT) that promotes students’ algebraic readiness. My study aims to explore useful AKT based on these dimensions.

## Meixia Ding

PresenterHi Miriam, Not sure if you want me to explain how we used the three-dimension IES recommendations to identify the findings based on our videotaped lessons :) – Meixia

## Miriam Gates

This is really helpful background! Thanks so much for sharing it. In my (unclear) question, I was interested in your video viewing methods. That is, how did you come to the conclusions that you discussed in your video (summarized around minute 2:59). In particular, I’m wondering if you have a observation instrument for classroom practices aligned to high AKT in the classroom or plans to produce one.

## Meixia Ding

PresenterHi Miriam. Yes, we used a coding framework that is aligned with the AKT-components (worked examples, representations, and deep questions). This framework was modified based on my previous study on teachers’ lesson planning published in Elementary School Journal; but it was refined several times based on our initial video analysis.

## Miriam Gates

Will you be writing about that coding framework soon? I would love to see how you have operationalized these challenging constructs.

## Meixia Ding

PresenterYes, we are in the process of refining our conference papers into journal manuscripts. Hopefully, they can appear in good journals soon. To have a sense about the coding framework we used, you may be interested in checking our our Elementary School Journal article. As I mentioned, the coding framework used for video analysis was modified from this existing one.

Ding, M., & Carlson, M. A. (2013). Elementary teachers’ learning to construct high quality mathematics lesson plans: A use of IES recommendations. The Elementary School Journal, 113(3), 359–385

## E Paul Goldenberg

Your video listed three teaching practices that you took as particularly important to the teaching you’d like to see more in US elementary schools. One that stood out was your emphasis on unpacking: both the nature of it (e.g., whether it was deeply about one worked example or shallowly about a few) and how much class time was spent on unpacking (of any kind?). A 3 minute video is naturally too short to articulate exactly what you mean by unpacking and how it differs from other classroom discourse. Could you comment on specific features that you believe are the elements of this kind of discussion that make the most important difference to learning?

## Meixia Ding

PresenterDear Paul, Thanks for great questions! Although there are cross-cultural differences, we found that expert teachers in both countries share similarities in unpacking worked examples through representation uses and asking deep questions, which may shed light on the to-be-explored AKT in this study. Below are elaborations of some specific features.

(1) Worked examples. Our findings suggest that expert teachers in both countries spent sufficient time unpacking worked examples. Some Chinese teachers spent about 40% of class time unpacking just ONE worked example. Given that worked examples are examples of mathematical principles, it makes sense for teachers to sufficiently unpack a worked example to help students understand the targeted idea. This finding shed light on ways to implement the IES recommendation on worked examples in classrooms. While it may be effective to interleave worked examples and practice problems in laboratory or in a homework setting, it seems to be unrealistic and ineffective if a teacher presents multiple worked examples in a short class period. As observed from some US classrooms where 3-4 repetitive examples were discussed, each worked example was discussed in a quick pace with very little depth. Our video data suggests that it is of upmost importance for teachers to engage students into the process of unpacking 1-2 typical worked examples.

(2) Representations. To unpack worked examples, many of the expert teachers situated the teaching of inverse relations in word problem contexts such as shopping, which helped students make sense of this relation. This is different from the traditional focus on number manipulations. Uniquely, the Chinese classroom discussions focused more on inverse quantitative relationships, which were often explicitly written on the board (e.g., original – taken away = left over; left over + taken away = original). We also noticed that expert teachers used representations to model quantitative relationships. Which is different from the common use of representations as reported in the literature, that is, to find answers (e.g., counting cubes to figure out 4 × 6 = 24). In some of our U.S. videos, we did observe this type of representation use. One exciting observation with the U.S. classrooms was that some teachers introduced tape diagrams to their students, which was presented in their textbooks (e.g., Go Math). This type of schematic diagram is highly recommended by the common core and widely used by East Asian classrooms. However, due to the tape diagrams being new to U.S. teachers, we noticed that teachers tended to directly present full diagrams to students, which was in contrast to Chinese teachers’ engaging students into the process of co-constructing the diagram.

(3) Deep questions. To unpack worked examples, expert teachers ask many deep questions. These questions often asked students to compare types of representations and various solutions, which appear to elicit students’ understanding of deep structures. Typical comparison questions are “what are the similarities between these two solutions? What are the differences?” “Compare this group of number sentences, what patterns do you find?” “Can you change this multiplication word problem to its reverse problem (a division problem)?” These comparison questions may facilitate students’ connection-making, which are the key to develop structural understanding. In fact, comparison techniques have drawn renewed interests in the literature. Prior studies have found that many US teachers did use comparisons but often without efficiency. Our cross-cultural videos indicate that Chinese teachers prompted students to compare multiple solutions resulting from multiple ways of thinking while some US teachers tended to focus on computational strategies that led to the answer. Our next-step is to explore in a great detail about the condition and purpose of Chinese and U.S. expert teachers’ use of comparisons. In short, while the literature defines “deep questions” as questions that may elicit explanations of causal relationships, our video data contributed new insights about deep questions to be asked in elementary classrooms.

## E Paul Goldenberg

Thanks, Meixia.

## Courtney Arthur

I am curious if there were findings that surprised you, and why?

## Meixia Ding

PresenterHi Courtney, Yes, there are some surprising findings. Even though all expert teachers were selected based on certain criteria (e.g., 10+ year teaching experience, have obtained some teaching awards or have good teaching reputation-recommended by the principal, and obtain good score of our AKT-based written interview), the quality of U.S. teachers’ classroom instruction demonstrated great differences, which may related to teachers’ different levels of expertise in teaching mathematics. The differences among Chinese teachers are much smaller. This is likely due to the reality that all Chinese teachers are specialized in teaching mathematics; but this is not the case for U.S. elementary teachers. Therefore, even though we have selected U.S. expert teachers (e.g., national board certified teachers), they may be experts of other areas but not necessarily mathematics. One thing needs to mention is that, students in all of the US expert teachers’ classrooms behave nicely (very rare classroom behaviors are observed).

## Gerald Kulm

Hi Meixia. Nice project! I am happy that you are following up on your work with MKT. Looking at and comparing expert teacher practices offers a great way to unpack effective instruction. How do you plan to apply the findings in your work with preservice or inservice teachers? For the teachers in your study, how does Math Knowledge play a role in these characteristics of MKT?

## Meixia Ding

PresenterHi Dr. Kulm, It is so nice to hear from you! Thanks for encouragement. We will conduct intensive data analysis of our years 1 and 2 data (year 1: inverse relations; year 2: the basic properties of operations) in this upcoming year (year 3). Findings about the use of worked examples, representations, and deep questions will be stored in a booklet as teaching materials (typical video clips will be provided). For preservice teachers, I will integrate findings into my methods course to support their lesson planning and teaching practice. For inservice teachers, we plan to provide summer workshops (end of year 3) to both U.S. and Chinese expert teachers who will re-teach their lessons based on our findings in year 4. This part of data will provide insights for refining the booklet. To make broader impact, we will provide workshops introducing project findings to novice teachers or teachers in needy schools in Philadelphia. For the question about teacher knowledge, we did not directly measure their math content knowledge. However, as I responded to Courtney’s question, I do observe differences of teachers’ classroom instruction (especially within the US teacher group), which may be related to their math knowledge.

## Joni Falk

A really interesting video on approaches to Algebra in China and US. Thanks for submitting this! I think you might also be interested in visiting Davida Fischman’s video on professional development in Algebra… which you can find at:

http://videohall.com/p/784

Hope the two of you get to connect.

## Meixia Ding

PresenterThank you very much, Joni! I will definitely check it out!

## Jeffrey Barrett

It is good to have this analysis to show expert strategies for addressing ideas deeply in elementary teaching situations. Please comment on the cultural traits or characteristic patterns that you find relevant to these differences in approaches to teaching mathematics across the two countries. Also, what were your criteria for finding expert teachers?

## Meixia Ding

PresenterHi Jeffrey,

Thanks for your comments. The goal of this study was to identify AKT based on teachers’ insights from both countries. However, we do find cross-cultural differences in classroom teaching. Broad differences include (a) Chinese lessons are all 40 minutes and US lessons range from 30 to 90minutes; (b) Chinese students usually stay in their seats (rows and columns) for the whole lesson. Small group work is often integrated throughout the lesson. The format includes 2 students in the same desk discussing with each other or 2 students turning around to talk people behind them. In the US classrooms, students usually come to the front sitting on the carpet to take the new lesson. Students then go back to their seats (or just find a spot) to complete assigned problems. In addition to these general differences, we identified some specific ones in alignment with the AKT components.

1) Worked examples. Chinese teachers spend more sufficient time on unpacking one worked example (In fact, most lessons just discuss one worked example). Most US teachers discuss a couple of worked examples that did not necessarily show variations. Moreover, Chinese worked examples are always situated in story problem contexts. This is not necessarily the case of US classrooms.

2) Representation. The sequence of Chinese teachers’ representation uses is generally from concrete to abstract. The concrete representations are mainly story situations, pictures, and diagrams. Manipulatives may be occasionally used by students. The tools are mainly “sticks” or some cards pre-made by students. All these concrete representations are used as a tool to model quantitative relationships with abstract number sentences always being used for formal solution. In contrast, US classrooms contain more diverse concrete representations (fingers, cubes, dominos, chips etc.). These tools may or may not be used to model quantitative relationships. For instance, many teachers allow students to use the concrete representations as solutions (e.g., drawing out the answer) or use them to find computation answers (e.g., counting cubes to find the answer). Finally, as I have explained in a previous post, tape diagrams have been used in several US teachers’ classrooms; but it is widely used by Chinese teachers. While US teachers tend to fully present the diagram to students, Chinese teachers are more skillful in engaging students in the process of co-constructing these diagrams.

3) Deep questions. Chinese teachers tend to ask students to compare different types of solutions. Some US expert teachers do so as well. However, more US teachers invite students to contribute different strategies. Yet, the comparisons among these strategies are rarely made or the comparisons remain at surface levels.

Regardless of the above differences, we found that expert teachers in both countries do share similarities in using worked examples, representations, and deep questions.

## Elizabeth McEneaney

Very nice project! It’s interesting that many years after the TIMSS video project and related analysis that US teachers still are rushing through problems and passing up opportunities for richer and more varied representations of the mathematical ideas. Your comparative approach with these case studies is helpful for showing that this tendency still exists.

## Meixia Ding

PresenterThank you, Elizabeth! It is great to link back to the TIMSS video analysis.

## Andrew Izsak

Hi Meixia,

Nice project. Two questions.

1. Why did you decide to focus on inverses? Will you be examining other aspects of algebra in future years?

2. You frame your work in terms of knowledge but, as Paul pointed out, your results are about instructional practices. How do you understand relations between the two?

Andrew

## Meixia Ding

PresenterDear Andrew, Thanks for your comments and questions!

1. This project focuses on two early algebra ideas (1) inverse relations-year 1 topic, and (2) the basic properties of operations (e.g., the commutative, associative, and distributive properties) – year 2 topic. These topics are fundamental mathematical ideas that were suggested by Carpenter et al. (2013) Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School. My initial study on early algebra topic was about the equal sign. My later studies extended the topics to the inverse relations and the basic properties. I suspect my future studies will examine other aspects of early algebra, probably in a deeper way (e.g., taking a deeper look at the modeling & problem-solving perspectives).

2. This study aims to glean Algebraic knowledge for teaching (AKT), which is aligned with Ball’s mathematical knowledge for teaching (MKT). This is a type of knowledge that I wish to glean from expert teachers’ instructional practices, which will be documented in forms that will be useful for others.

Further posting is closed as the showcase has ended.