Math, a Journey of Understanding

Jackie Walts

Middle school special education teacher

Syracuse City School District and the IFL have been partners in education since 2014 with the goal of improving student learning. The goal has been to increase student learning and the approach has been two-fold, grow teacher conceptual content knowledge while refining instructional practices. In support of the district’s goals, our eclectic team of SOAR^* certified trainers focuses on the training and implementation of the IFL’s SOAR Mathematics survey tools and activities during tiered instruction (intervention). We work with teachers to explore learning progressions for mathematical concepts and analyze student responses to identify knowledge assets and areas of unfinished learning.  During these endeavors, we have seen firsthand the role that having a deep conceptual understanding of the content plays in supporting student learning, especially for those students with significant areas of unfinished learning.

Teachers with a conceptual understanding of mathematics and pedagogical content knowledge have the ability to support student learning without taking over the thinking for them.  With a blended understanding of content and pedagogy, teachers are able to leverage representations and students’ initial sense making to support their entry into more sophisticated thinking and reasoning about mathematics.  Let’s see an example of how a teacher uses her understanding of multiplying with one fractional factor and the importance of making connections between representations to support a group of fourth grade students.  The students are solving the Eating Apples task which involves one-third of an apple being eaten every day for five days.

The teacher circulates during the explore phase of the lesson while the students work collaboratively in small groups. Read the exchange that teacher has with one of the small groups of students who have arrived at an inaccurate solution.

T: What do you know about the Eating Apples task so far?

S1: Well, we know we can multiply.

T: Say more. What and why are you multiplying?

S2: Josiah likes thirds so for five days we can multiply five times one-third.

S1: We wrote 5 x 1/3.

S3: There are 5 days and he eats 1/3 of an apple each day.

T: Ok, so how many apples has Josiah eaten after 5 days?

S1: 5/15 of an apple!

T: How did you figure out 5/15of an apple?

S3: We multiplied five times one and then whatever we do to the top (all students chime in to chorus) we have to do to the bottom.

T: Work as a team to show the situation with a model.  Be ready to talk about the model you create and how it shows the amount of apple Josiah eats over five days. I’ll be back.

The teacher recognizes that the students have over generalized and multiplied the numerator and denominator by 5.  The teacher does not tell them that they made an error, but rather, presses the students to create a model of the situation.  By asking student to model the situation and be ready to talk about how their representations were related, she set up an opportunity for them to advance their thinking. The next transcript shows their exchange when she returns.

T: Tell me about the model you created.

S2: We drew a circle and colored 1/3 of it for each apple.  We did that five times.

S1: Each circle is one day.

S2: No, each circle is one apple.

T: So is the circle an apple or a day?

S2: Each circle shows 1/3 of an apple that he ate each day. We needed five for the five days, to show the 5 in 5 x 1/3.

T: Before, you said Josiah ate 5/15 of an apple in five days. Where do you see 5/15  in your model?

S1: Well, we changed our answer. We figured out he ate 1/3.

S2: 5/3 of an apple.

T: How did you determine that he ate 5/3 of an apple?

S3: We have thirds and we counted one, two, three, four, five thirds.

T: So how many apples is that? What does 5/3 mean?

Students: (No response.)

T: (Points to students’ diagram) If we put all of these thirds together, what would we have? Talk about it and figure out a another way to record 5/3.

The teacher asks the students about their model and then presses for clarification about the meaning of the factors and how the model represents their original solution.  The students share that they discovered their original solution was not accurate and were able to provide reasoning for why they revised their solution.

The teacher’s instructional choices in this example were intentional.  The teacher’s conceptual and pedagogical knowledge are the basis for the instructional decisions she made.  She was able to orchestrate this learning opportunity, meeting students where they were in their understanding, because she asked questions to assess their thinking and then chose a path along which to advance their understanding.

In contrast, a teacher who has not had yet developed a conceptual understanding of the mathematics or grown pedagogical practices that support conceptual development is likely to focus on automatized procedures and computational fluency, relying on standard algorithms and rote procedures. Imagine how different the learning experience would have been if the teacher had said, “The rule of what you do to the top, you do to the bottom, does not work here. Multiply the whole number and the numerator.” Students who are taught to perform a series of steps, may arrive at a correct answer but not understand why and how it is correct.