Student Understanding of the Sign of Negative Definite Integrals in Mathematics and Physics

Abstract

In a study of student understanding of negative definite integrals at two institutions, we administered a written survey and follow-up clinical interviews at one institution and found that “backward integrals”, where the integral was taken from right to left on the x-axis, were the most difficult for students to interpret. We then conducted additional interviews focused on backward integrals at a second institution. Our analysis uses the concept image framework and a recent categorical framework for mathematical sense making. We found that students were most successful using the Fundamental Theorem of Calculus to determine the sign of an integral when a symbolic expression was provided. When considering a definite integral in a graphical context, students often had difficulty if they viewed the integral as a spatial area, stating that area must always be positive. Some students were able to conceptualize $$\Delta x$$ Δ x or $$dx$$ dx as a difference or change and thus a signed quantity; when these students were able to view the area as a sum of smaller pieces, they were more successful in justifying a negative backward integral. Students who used amounts or similar images often had difficulty making sense of the negative sign. Even more progress was made when students either invoked or were asked explicitly about a specific physical context to be represented by the backward integral, other than spatial area. The context provided a meaning to the difference represented by $$\Delta x$$ Δ x or $$dx$$ dx and thus to the sign of that difference and the definite integral.

Three Basic Skills of Rectilinear Motion Chart Interpretation; What Do Students Already Have and How Should They Be?

“…Tercermin pada indikator keterampilan memaknai luasan di bawah kurva yang masih menjadi tren kesulitan yang paling banyak dibandingkan kedua indikator keterampilan dasar lainnya. Temuan ini sejalan dengan penelitian Bajracharya yang menyatakan banyak siswa yang kesulitan memaknai luasan di bawah kurva yang seharusnya menjadi "shortcut" ketika menentukan jarak tempuh/perpindahan melalui grafik kecepatan-waktu 𝑣(𝑡) dan/atau menentukan kecepatan melalui grafik percepatan-waktu 𝑎(𝑡) (Bajracharya et al, 2023). Apalagi ketika siswa diminta mentransformasi grafik yang membutuhkan kombinasi keterampilan memaknai luasan di bawah kurva dengan keterampilan membaca grafik menjadikan hal tersebut sangat sulit dipecahkan siswa.…”

“…Artinya, nilai kecepatan pada interval waktu 4s sampai 5s lebih besar dibandingkan dengan interval waktu 0s sampai 2s. Penelitian sebelumnya juga mendapati bahwa pemaknaan curam/landainya kemiringan garis masih menjadi tren kesulitan bagi siswa sehingga perlu mendapat perhatian khusus bagi guru (Ayop & Ismail, 2019;Bajracharya et al, 2023;Susac et al, 2018).…”

“…Namun, mereka gagal dalam menentukan perubahan posisi dalam interval waktu 4s hingga 6s, sehingga mereka merepresentasikannya pada grafik posisi dengan perubahan posisi 3m. Pemahaman yang benar tetapi tidak konsisten juga sering menjebak siswa untuk memilih jawaban yang salah (Bajracharya et al, 2023;Skrabankova et al, 2020). Berdasarkan soal tersebut, hanya 27% siswa yang dapat menjawab dengan benar pada pilihan (B).…”

“…Siswa seringkali salah ketika menentukan kecepatan berdasarkan grafik 𝑥 − 𝑡, begitupula menentukan percepatan pada grafik 𝑣 − 𝑡 (Susac et al, 2018;Taqwa & Rivaldo, 2018). Luasan di bawah kurva juga seringkali tidak diperhatikan oleh siswa (Bajracharya et al, 2023). Padahal, interpretasi luasan di bawah kurva yang identik dengan menghitung integral ∫ 𝑦𝑑𝑥 merupakan "shortcut" dalam menentukan perpindahan pada grafik 𝑣 − 𝑡 serta menentukan kecepatan pada grafik 𝑎 − 𝑡 (Susac et al, 2018).…”

Students as Prospective Teachers’ Understanding of Integral Based on the APOS Theory in Terms of Gender Difference

“…Maharaj (2014) found that students had difficulty applying rules for integration. In addition, several studies reveal that students also have difficulty using integral concepts in the course of calculus (Bajracharya & Thompson, 2014;Bezuidenhout & Olivier, 2000;Cui & Bennett, 2006). This shows that students' understanding of integral needs serious attention, especially for prospective teachers in order to ensure no mistakes when teaching the concept of integral later.…”