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by Riley Strauss, pre-service teacher
Meet Josie! Josie is a kindergarten student in an urban public school in central Indiana. She enjoys reading and drawing. Right away, it was clear that Josie was an introvert, preferring to work alone, stay quiet, and avoid participating in group activities. However, it was also obvious that she was attentive during instructional time and generally comfortable in the classroom.
– Josie completed 25 problems in her initial fact fluency assessment.
– She had a delayed response on 6 questions.
– She missed one answer.
I was very impressed with Josie's initial assessment. She completed half of the assessment, correctly answering all but one of the questions. It was clear that she had some fact fluency knowledge.
Areas to Grow
Although Josie effectively answered many problems, she was near silent once I started asking how she got her answers. When I asked her how she answered "3 + 2," she shrugged and replied, "I just knew." I gave her multicolored bear figurines to show her thinking, and she placed them in lines that did not match the problem. When I asked her to explain, she simply counted them. Based on this assessment, I decided our goals would revolve around Josie learning to explain her thinking, along with improving in fact fluency.
What Comes Next?
In the assessment, Josie struggled the most with "_+2" problems. To improve Josie's fact fluency skills, I decided that the game "Plus" would be the best option for her. This would help her with adding numbers greater than 1. We would begin with "Plus 2" and gradually incorporate larger numbers.
I decided that the best way to get Josie to feel comfortable sharing her ideas was to put her in a social setting that encouraged dialogue between her and her peers. I figured that her friend Lily would be the best person to help Josie along, creating a comfortable atmosphere for Josie to feel free sharing her mind.
I used index cards to make Josie her own card deck for "Plus." Different than a normal deck of cards, her deck only used the numbers 1 through 5. This was perfect for Josie, because it tested her with the problems she was exposed to and occasionally struggled with during fact fluency. After asking her to solve a problem with this deck of cards, I planned to ask Josie how she came up with the answer. This would reinforce our goals, pushing her towards learning how to explain her thinking.
I made Josie two sets of flashcards. Set "A" dealt with the area Josie seemed to struggle with the most in the assessment (adding 2). It also worked to show her the commutative property, as shown by the bottom two cards. Set "B" dealt with integrating larger numbers and aimed to improve her fact fluency skills. Similar to "Plus," I planned on working through the problems with Josie and helping her to understand how to explain her thought process to me.
After I showed Josie how to play "Plus" during one class, she was eager to play the game with her friend Lily. It helped that Lily was also excited, making the math game desirable and enjoyable for the both of them. As the game progressed, I asked Josie to explain one problem that Lily struggled with. Josie explained it perfectly and with excitement. I began to see growth through social interaction. Josie was gradually opening up to others, feeling able to explain her thinking.
Mr. Strauss: Josie, can you help Lily and me with this problem? We're trying to figure out what "3 + 2" is.
Josie: It's five!
Mr. Strauss: How do you know?
Josie: Because three plus one more is four, and then four plus one more is five!
Whenever I presented Josie with a flashcard or a problem in "Plus," I asked her to explain her thinking. She began to demonstrate with her fingers, counting the first half of a problem on one hand, the second half on the other hand, and then adding them together. As we kept practicing, she started to move away from using her fingers. She began to tell me how to solve her addition problems by solving them in her head. Along with this, she was progressively becoming more efficient at solving the problems I was showing her. We then moved from using our own deck for "Plus" to using an actual deck of cards, using numbers 2 through 9. Josie was excited to take this step forward.
While playing "Plus," the problem becomes "4+2":
Mr. Strauss: How did you get that answer?
Josie: I know that four plus one more is five. Then I did five plus one more, and I know that is six!
Mr. Strauss: Did you need to use your fingers?
– Josie completed 30 problems. That's 5 more than the first assessment.
– She had a delayed response for 5 questions. That's 1 less than the first assessment.
– She got every question right. She had missed one question on the first assessment.
– She did not struggle with "Plus 1" problems on this assessment.
– She improved on her "Plus 2" skills in this assessment.
Josie clearly improved in her fact fluency abilities, but she also improved in her ability to explain her thinking for me. When I asked her to explain the problem "3+2" (the same problem I asked her to explain in the first assessment), she responded with two strategies:
– Three plus one more is four, and four plus one more is five.
– Three plus three more is six. So one less than six is five.
I then asked her to explain "2+3," and she laughed. She explained that it's still five. I asked if it's the same problem as the previous one and she replied, "Yes because it uses the same numbers, and you get five."
Josie became a more effective mathematician this semester. Not only was she able to complete more fact fluency problems, but she also did not need to pause on as many questions. The problems had become familiar because of the games she had played with Lily and me. She also recognized a few from her flashcards. The change between the two Fact Fluency assessments was apparent and shows that Josie has developed in her mathematical skills.
Josie grew immensely this semester. When I first met her, she barely spoke, rarely participated, and preferred to be on her own. When I gave her the initial Fact Fluency assessment, she could not explain her thought process to me. As we went through the semester playing "Plus," using flashcards, and jumping into social settings, I saw her open up and gradually explain her thought process more and more. By the end of the semester, Josie had become a much more vocal student, confidently telling me how she solved problems, sometimes providing me with multiple strategies. Although she's still a quiet learner, Josie has become a much more vocal and courageous one as well.
I firmly believe that Josie's growth in mathematics this semester comes from the work we completed together. While originally Josie would barely talk in class, she grew to explain her thinking thoroughly, and sometimes in multiple ways. She also learned to solve problems without aid (not needing to use her fingers as often). The impact became evident through the second Fact Fluency assessment. Her overall score and the number of problems answered increased while the number of errors decreased. She felt more comfortable and confident in sharing her thinking with me, developing an ability to effectively demonstrate her learning.
I am an extroverted person, which made an opportunity to work with someone who is heavily introverted like Josie unique. This process taught me about understanding learners that are different from myself and providing them with the necessary resources to succeed. For Josie, this meant creating games designed specifically for her, collaborating with one of her peers, and continually asking her to explain her thinking so that it became natural. I'm grateful that I had the chance to work with someone different from myself so that I have the skills to provide introverts with necessary resources in the future. While students like Josie may always be introverted, they also have intelligent thoughts and problem solving strategies to share with others. It's important to provide resources and opportunities that engage all learners.
Riley Strauss is a third-year elementary education major at Butler University. This impact report was originally written for a class project and later adapted for this blog.
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