# Mathematics of Grace: Limitations and Perturbation (Productive Struggle)

In my last blog I wrote about one of the first things I noticed about the mathematics of Grace. She used her fingers to solve addition situations like 7+4 by constructing more advanced finger patterns, where one finger could mean one or eleven and six fingers could mean six or sixteen. This allowed her to find the answer to many addition situations, but it had limitations. Her fingers only helped her when the sum of the addition situation was less than twenty.

Given that fact that Grace could use three fingers to mean both three and thirteen, the research team found evidence that she could work with some imagined (figurative) material. We wondered how she could learn to solve more advanced problems by developing her ability to work with more advanced mental material.

I have adopted Von Glasersfeld’s definition that learning takes place when instead of a child experiencing an expected and satisfying result of their scheme, they are a bit perturbed (perplexed), which leads to them accommodating scheme through the tools they use or the operations they use to once again allow them to experience that expected and satisfying result. (von Glasersfeld 1995 – Radical Constructivism) In the following clip you will see how Grace answered a problem after multiple attempts. This is an example of what I would call productive struggle (Common Core language) or perturbation (radical constructivist language). Her struggle produced a new way for her to solve problems. In the video clip below Grace experiences some perturbation because she is given a situation where the sum of the addends exceeded her finger patterns to twenty. Watch how she resolves this perturbation. When the video begins she had already counted seventeen counters into a cup. Let’s see what happens next.

You can see from the video that Grace has to work the problem multiple times. She just isn’t very satisfied with her answer. This behavior is typical of a student in the beginning stages of constructing. They aren’t really sure what to do, but they are figuring it out. As the teacher, this is the time for me to watch and continue to give her more situations like this one (sums just beyond twenty). I don’t want to jump in too quickly with a strategy.

Did you notice that when she is satisfied with her answer and finally explains her thinking, she describes it by lifting seventeen sequential fingers and then continues to count with four more sequential fingers? This strategy is definitely an adaptation to her adding scheme. I believe that she now can mentally imagine seventeen fingers and four fingers and join them in her mind. Finally, she uses movement to count the collection of fingers (the seventeen and four fingers) for which she doesn’t know the amount.

We met with her the next day and she was able to answer situations in which she used her new strategy to find the totals 18 & 7 and 14 & 9. This was an exciting moment for me because, in giving her the space to learn, I watched her construct and be able to answer problems that used to limit her. Knowing the research about the mathematics of students empowered me to know the type of situation to present to Grace to encourage that learning. I love watching a student gain mathematical power. Seeing that twinkle in their eye is priceless.

In my next blog I will write a little bit more about Grace and how my experience with her is a great example of how knowing the research about the mathematics of students can empower a teacher.