# Scenario: Holly teaching probability

## Context –curricular, professional, other

As is the case in the United States, North Carolina has state mandated goals and objectives for mathematics instruction at each grade level.  The North Carolina Standard Course of Study (NCSCOS) indicates that in grade 4 students are to “maintain” skills in permutation and combinations from Grade 3, which state “the learner will understand and use simple probability concepts,” which are further explained as determining the number of permutations and combinations of up to three items and solving probability problems using permutations and combinations. (NCSOS).

This school used the EnVisions curriculum, as mandated by the district (local authority).  Students in this school were not grouped by ability level for math instruction, and therefore the class was a heterogeneous mix of abilities.  The lesson was taught by Holly, an undergraduate pre-service teacher, and the lesson took place in a school-based placement in early March of her final semester (which ended in mid-April) in the teacher education program.

## Scenario

Holly taught a lesson on probability, which she explained to the class was “the likelihood that an event will happen.” Students were seated at their desks and looking at the interactive whiteboard during this portion of the lesson.  Since this was the second lesson on the topic, Holly reviewed what the students had learned the previous day about probability and the use of a tree diagram to determine the total number of possible outcomes.  After verbalizing an example to determine the number of outcomes when selecting different color marbles from a bag, Holly showed the class a slide which displayed 2 orange, 5 yellow and1 blue t-shirts.  She then asked the class, “What is the total number of outcomes of this problem?”  After discussing this for a few minutes, Holly said to the class, “Let’s look at another example, using the same problem. What is the probability of picking a blue shirt?”  Students raised their hands, Holly called on one, the answer was given, and usually Holly repeated the answer and then provided a brief explanation.  This process was repeated for each of the following questions:

1. What is the total number of outcomes for this problem?

2. What is the probability of picking a blue shirt?

3. What is the probability of picking an orange shirt?

4. What is the probability of picking a blue or an orange shirt?

5. What if I ask you what is the probability of picking a yellow or blue shirt?

Holly then displayed the next slide she made on the interactive whiteboard, which showed 2 squares, 2 lines, 4 circles, 5 arrows, 1 oval and 3 half-circles.  Holly explained, “I would like you in your notebook to answer individually the probability of selecting each shape.” This meant students needed to answer the following questions:

1. What is the total number of outcomes for this problem?

2. What is the probability of picking a square?

3. What is the probability of picking a line?

4. What is the probability of picking a circle?

5. What is the probability of picking an arrow?

6. What is the probability of picking an oval?

7. What is the probability of picking a half-circle?

After a short amount of individual work time, Holly called one student at a time up to the interactive whiteboard to write the answer next to the corresponding question.  After each of the seven questions had been answered, Holly asked the class:

Holly: If I were to add up all of the numerators, what should I get?

Students: 17

Holly: Why should that be the case? If I add the squares, lines, circles, arrows, ovals, and half-circles, I should get 17. Why? (Pause) There are six different shapes. Why should I get 17 for my numerator?

Student: Because there are 17 shapes all together.

Holly: We know that our total number of shapes is 17, so if we add all of the numerators, we should have

17 because I asked you to find the probability for each of the shapes listed.

Holly then went on to have students describe events as “certain,” “likely,” “unlikely,” or “impossible,” based on their probability.

# Knowledge Quartet Coding Commentary

## Scenario: Holly teaching probability

Curricular materials, such as textbooks, serve as a starting point and a guide for teaching, but rigid adherence to given materials may indicate a lack of specialized content knowledge on the part of the teacher (Stein, Remillard & Smith, 2007).  One aspect of this specialized knowledge is the ability to critically use curriculum, which is described by analyzing given or mandated curricular materials and modifying them in order to meet the needs of students (Rowland et al., 2009; Ball, Thames & Phelps, 2008).

This episode was selected as an example of ATB because the teacher critically used the curriculum rather than strictly adhering to the given curriculum.  Critical use entails careful selection of tasks rather than doing every problem provided and/or following the sequence exactly as written in the teacher’s manual.  It may also mean supplementing with additional resources and teaching strategies to improve on what was provided.

After the lesson, Holly was interviewed and asked if and how she used the Teacher’s Guide in her lesson planning.  To this she replied:

“I went through the book, and I looked some stuff up on-line. The only question I pulled from the book

was the first one…the one with the different colored shirts. Because I thought it was a good visual, and I

could show the shirts so that those kids, like our ESL kids (students whose first language is not English), could actually count and look at the different

colored shirts. So I pulled that one, but even in the book, it only had one or two questions associated with

it, but I wanted to pull more out of it.  (So) I asked a question for each color because I wanted them to see

then that the total number of colors should equal a whole fraction…And then the other question, the one

with all different shapes, I just made that one up. They had one kind of like it in the book with like

crescent moons and stars or something, but it was smaller, and I thought, “They’re going to get this. I’m

going to give one a little more challenging,” so that’s why I (gave them the second problem with the ‘

shapes), to see what they did. And, they got it.”

Holly’s explanation about her planning indicated that she carefully selected, adapted, and extended problems in the Teacher’s Guide to best match her students’ instructional needs.  She did this by considering her ESL students (who were currently learning English) and using pictures to present the problems, writing more questions for the first problem, and increasing the difficulty of a problem in the Teacher’s Guide to generate the second problem.

In summary, in this example the teacher taught a lesson that met the same goals and objectives as delineated in the curriculum and critically used the text book by carefully selecting tasks from the existing text book and supplementing with her own.

This is a positive example of a negative use of ATB.

We felt that in this scenario it is a good thing that she didn’t stick to the text book, but in other scenarios the adherence to the textbook would have been more helpful.  Delores’ example is a negative example of ATB because Eoin didn’t stick to the textbook but it would have been helpful if he had.

## References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics teaching. Thousand Oaks, California: Sage.

Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319-369). United States of America: Information age publishing.