Video used in teacher learning settings is a powerful tool for engaging participants in understanding children's mathematical thinking. Yet by itself the video, no matter how wonderful, is not sufficient. This example of using video in a preservice course illustrates how teacher educators can engage participants around carefully selected video clips to promote their understanding of children’s thinking and how to assess it.

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Our goals as teacher educators

As teacher educators, we want to help our participants—prospective and practicing teachers—to understand the behavior of children individually. We want participants to observe carefully, to think critically about what they see, and to develop reasoned interpretations to guide their teaching. How can we help participants to accomplish these goals? One method is guided analysis of carefully selected videos.

Imagine that you are in front of your classroom (or in a school-based professional development session for practicing teachers). The topic is *Counting*. You explain to participants that effective mathematics teaching requires insight into children’s thinking and learning. How can a teacher teach well without understanding the child being taught? You tell the class that they will discuss a video illustrating some key aspects of a child's counting skill and knowledge. The participants are tasked to examine the video carefully, develop an interpretation of what the child does and does not understand about counting, and consider how this interpretation can guide teaching.

The video you use meets several essential requirements. First and foremost, the content is suitable for illustrating the child’s thinking and learning of the mathematical topic of interest. Further, the video is intriguing, well made, dramatic, and sometimes even funny. It is not too long. It grabs the viewer's attention and enlivens what they read about for their assignments.

Yet no matter how wonderful the video is, it is not sufficient to simply play it, give your explanation, and then move on. As the instructor, you need to utilize the video effectively if you are to harness its potential for promoting careful observation, analytic thinking, and judicious interpretation, all of which can guide a participant toward productive instructional activities.

This article and its embedded video examples describe how you can do this. The essence of the pedagogy is to:

- Show participants carefully selected segments of the video.
- Help participants observe carefully.
- Repeat a segment or part of it to clarify observations and hypotheses.
- Ask participants for interpretations supported by evidence.
- Use flexible questioning to reveal the thinking behind participants’ interpretations.
- Encourage participants to discuss and challenge interpretations and offer possible alternative hypotheses, and to consider instructional approaches based on what has been learned about the child.
- And finally, help participants to relate the lessons learned from the video to academic papers and ideas concerning children’s mathematical thinking and learning.

Examining videos in this fashion may be more effective than observing children directly. The video allows you to view and review, to go forward and backwards in time, and to engage in deliberate and unhurried contemplation of the evidence. By comparison, direct observation is ephemeral and does not afford the kind of careful study provided by videos of this kind. A video is worth many, many more than a thousand words.

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My Class Interprets Anna Counts

For a year or so my team worked in a preschool in a low-income area of New York. We observed the children and their interactions with their teachers, and we conducted professional development workshops with the teachers and their assistants. We asked the teachers to conduct and video-record an interview or observation of a child in their classroom. One of the resulting videos made by an assistant teacher about Anna counting (see *Anna Counts* for additional commentary about this video), was so informative and interesting that I used it in my class of thirteen Teachers College students, most of them prospective early childhood teachers. I was in turn video-recorded using the video of Anna counting in my class. The following video is my edited combination of those two situations. In summary, the assistant teacher at the New York school recorded the video; I edited it and wrote up an interpretation.

Here is the entire video of Anna counting:

The video begins with the assistant teacher asking Anna, who is about four and a half years of age, to count as high as she can. Almost the entire video just shows Anna counting; the adult says virtually nothing. The original video of Anna counting is just under two minutes long. The classroom interaction video is about 14 minutes in duration.

Here is what happened.

After announcing that we were going to interpret a video involving Anna’s counting, I provided brief background on the child and her school. I then played the first 14 seconds, as shown in the following video clip.

An essential feature of video pedagogy is to help the viewers focus on a small but meaningful slice of behavior. That’s why I asked, “So what do you see so far?” One participant replied that Anna used her fingers. Then, after I asked if she was doing so accurately, a buzz of comments erupted. Someone pointed out that Anna did not count all of the fingers and someone asked me to play the video again, which I did, after launching into a mini-sermon on the need to review videos carefully, as shown in the following video clip. (Please don’t get too disoriented from the initial segment.)

Notice that I stopped the video after Anna reached the number sixteen, and asked what the participants now thought about her use of fingers. One participant proposed that Anna seemed to get confused in going from three to four. Others eagerly chimed in with their interpretation that Anna started to use her fingers only after reaching three, and then was not confused, but needed to “catch up” (that is to synchronize her verbal counting with use of the fingers). I repeated that point, “So she might have to catch up.”

After writing this, I watched the segment again and now think that the need to catch up is not so evident. I now think that she made the transition to finger counting very well and more or less in synchrony with her verbal counting. My second thoughts illustrate an important point: even with the aid of video, observation and interpretation of behavior can be very difficult. Fortunately, in this case, the relation between number words and fingers is not of paramount importance.

Next, I explicitly introduced another key principle. Watch this in the next clip.

After emphasizing the importance of considering alternative interpretations, I went on.

Of course, I was already intimately familiar with the video and knew what points I wanted to make about a child's learning of the counting words. That is the reason I stopped when Anna reached 29, as shown in the following video clip.

I used this clip to illustrate how children use rules to generate the counting numbers. I wanted my participants to think about why Anna stretched out the word *twenty-nine* and used a questioning cadence when she said it. The word *twenty-nine* (better thought of as “two-ten-nine”) indicates to the counter that she needs to say the next tens word, in this case *thirty* (really "three-ten"), and then append to it the unit words in the correct order (resulting in *thirty-one* and so on). Eventually the participants learn that children do not simply memorize the numbers, but use important mathematical rules to generate them.

One lesson for you as teacher educators is that you need to understand the subject matter in some depth in order to help your students understand it as well. For the interpretation of this type of video to be profitable for your participants, you need a deep knowledge of the subject matter, in this case, mathematical understanding of the structure of the counting system, and cognitive understanding of children’s counting.

I then showed the participants the following section of the video, asking them what happened next, and specifically why Anna seemed to make rhythmic movements with her hands, swaying and moving them as she counted from *twenty-five* upwards.

My participants came up with three possible interpretations of Anna's rhythmic hand movements. One cited boredom; another proposed that the rhythm helped Anna to use a pattern in the numbers; and yet another said that the rhythm helped Anna to keep track of whether she said each word in turn. I chose to ignore the boredom hypothesis and also the pattern idea. I could have (and maybe should have) pursued the latter right away, because it was at the heart of what I wanted to teach. Alternatively, I could have tried to engage the participants in a discussion—really an argument—about which of these hypotheses is the most promising.

Instead, I tried to steer the conversation to the deeper idea of pattern, as shown in the following video clip.

I began by asking my participants how they thought memory was involved in Anna’s counting through the twenties. They proposed that Anna memorized the words but did not necessarily understand their meaning (their indication of cardinal value). While this is an important point, at this stage of my lesson I wanted the participants to grapple Anna's ability to generate the counting words after a number ending with the units word *nine*, as in *twenty-nine* or *thirty-nine*.

A bit later, I asked the students to watch the video, and stopped it shortly after Anna said, *thirty-nine*, again with a questioning intonation, as shown in the following video clip.

It was clear that she had verbalized *thirty-nine* in the same way that she said *twenty*-*nine*, stretching out the last syllable.

But what did this mean? The next video clip begins with one student suggesting that Anna was guided by the pattern of the numbers.

By “pattern,” the participant seemed to mean the idea that Anna knew how to append the unit words *one* through *nine* to the tens words. I reminded the participants that they had recently seen the same kind of pattern in a video of a child counting in Turkish. Eventually I summed up their thoughts about the pattern of attaching unit words, but distinguished this from the ability to create the tens numbers. We explore this issue in the following video clip.

This is the most significant and interesting video of all. After saying *thirty-nine* Anna explicitly asks, “What comes after three?” She answers her own question, “I know, four,” and then infers that* forty* must come after *thirty-nine*. At this point, the participants expressed surprise and appeared quite impressed that Anna had developed a rule that relies on the structure of the base-ten system to produce the numbers from the teens all the way through *ninety-nine*. She skipped a few numbers along the way, but that was trivial in comparison to her construction of the rule, which I have never seen another young child do in such an explicit manner. At the end, she recognized the limits of her approach, said that she was not sure what number comes after *ninety-nine*, and that she would have to ask her mother for the answer. She was aware of what she did not know and could not get the desired number by using the rule available to her.

After the video sequence ended at *one hundred*, I asked my participants what they now thought of Anna’s performance.

My class agreed that Anna was was definitely using rules. I then asked my participants to describe the rule carefully, and used framing questions and gentle hints to help them verbalize it. My intention was to help the participants construct their own deep knowledge of Anna’s counting, just as she constructed her deep knowledge of the counting system.

At the very end, a participant had a question about the origins of Anna’s rule, as shown in the next video clip.

One of my participants asked if was Anna taught the construction-of-tens rule in school. I said that I didn’t know. There was no evidence to suggest that Anna learned the rule through a teacher’s intervention or some other form of instruction. So although my class developed a solid and very plausible account of how Anna used a rule to construct the tens numbers, we do not know how the she acquired the rule in the first place. While it is tempting to conclude that Anna discovered the rule herself, the evidence provided by the video simply does not address this issue. The broader lesson is this: be careful to base hypotheses on only the specific evidence available. Don’t propose more than the evidence warrants. Be modest in your claims.

Finally, there were at least two areas in which my lesson fell short. One was in neglecting to ask my participants how Anna’s behavior was related to the assigned reading for that week that discussed the role of rules in counting. Did they see the connections between the reading and their own observations and interpretations? All too often participants feel that what they read for a class is unrelated to their own experience because it is academic knowledge and therefore perceived as not being of any practical value. As teacher educators we need to help our participants understand the academic literature and see how it can be useful.

A second omission was failing to discuss the implications of the video for instruction. I should have, but was worried about time. There is a saying (coined by Elliot Eisner) that the goal of education should not be to *cover* as much material as possible but to *uncover* key ideas. I did not succeed in this goal, but let’s consider what I should have done.

The class had just learned that Anna used a sound rule for generating the counting numbers. One issue I should have raised with the class is what this information implies for teaching Anna. My own answer is that I would help her to practice using her rule, elaborate on it, and consolidate her knowledge. I would certainly *not* limit her counting to small numbers, though the Common Core focuses here on small numbers. Does encouraging using large numbers contradict local standards or the Common Core? That’s a very important issue to discuss. My own view would be in this case to disregard such standards (if they exist), because it is clear that Anna can exceed them. Further, it will not hurt her in any way to expand her knowledge in this area, but if, and *only* if, the instruction is appropriate.

Another issue which would have been beneficial to discuss is what the case of Anna implies for counting instruction in general. I think the implication is that early childhood educators should see the counting words as an exciting gateway to algebra. Of course, children need to learn the numerical (cardinal) *meaning* of these words, which refer in systematic ways to quantity. But at the same time, children can learn about the counting words themselves, apart from the quantities they represent. The counting words embody another kind of mathematical meaning. Studying the counting words can help children to discover the underlying structure of the base-ten system and the resulting rules for generating new numbers. Yes, children must memorize the unit numbers from one to ten or so, but once these are learned the children can develop generalizations about numbers: how to construct the tens and how to append the unit numbers. Although this algebraic learning is abstract, it is not beyond the reach of young children. Given all this, teachers can engage children in verbal counting by ones and tens, in examining the simple table showing the numbers from 1 to 100 arranged in rows of tens, and also in using manipulative activities involving tens and unit sticks or the like.

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Conclusion

As you saw in this article, a well chosen video can attract lively interest from the viewers. A video can provide drama, surprise, vivid examples of abstract ideas, food for thought, and a very different kind of learning experience than can a textbook or journal paper. The video is a kind of intellectual manipulative with which participants can explore children’s thinking and construct a meaningful understanding of it.

But participants need help doing this. The teacher educator needs to guide their participants' construction of knowledge by posing questions in a flexible manner, steering conversation, elaborating on participant ideas, reinforcing key points, challenging interpretations, provoking principled argument, and making explicit the skills of interpretation.

All this (and more) is the pedagogy of the video clip. We need to help our participants to use evidence to support and defend their ideas, to develop critical thinking, and to engage in evidence-based debate. This pedagogy is not dissimilar from how teachers should help children to learn math. The way you teach your participants should set the stage for and resemble how they should teach young children.

The pedagogy of the video clip can also benefit the teacher educator. Engaging in interpretive activities of the kind described in this paper allows you to see participants’ thinking in action, as it relates to specific aspects of children’s learning and teaching. It can help you to learn whether your participants have really understood what they have read or what you have attempted to teach them. Also, frequent use of the pedagogy of the video clip can help you deepen your own understanding of children’s mathematical thinking and learning. I have probably shown my students certain video clips dozens of times. I can honestly say that on many occasions I learned something new about the child’s thinking and the implications for mathematics education over this time. Sometimes I benefitted from the repeated viewing and analysis, and sometimes I learned from astute participant comments.

Video analysis can be a wonderful learning experience for participants. The class tends to come alive when they interpret videos. In fact, many former students have told me that years later, they still remember class videos and the lessons learned from them.

Finally, I find that engaging in video pedagogy is the most exciting, challenging, rewarding, and enjoyable teaching that I do.

Try it.