The Teachers’ Multiplicative Reasoning Research Group studies the ecology of teachers’ cognition around interrelated multiplicative topics––including multiplication, division, fractions, proportional relationships, and linear functions. We are particularly interested in ways that teachers’ reasoning about multiplication, division, and fractions support their reasoning about ratios and proportional relationships and in how number lines and strip (tape) diagrams can support this reasoning. Our theoretical framework is based on two complementary perspectives on proportional relationships, one of which we call multiple batches and one of which we call variable parts. The multiple batches perspective has been investigated in past research primarily with students (see Figure 1). The variable parts perspective has been largely overlooked by past research but holds promise for making applications of proportional relationships to linear equations and other topics more accessible (see Figure 2).
Our methods coordinate detailed case studies with recent advances measuring teachers’ multiplicative reasoning. The psychometric work is done in collaboration with Dr. Laine Bradshaw and is based on determining profiles of multiplicative reasoning. The profiles are determined using Diagnostic Classification Models and summarize strengths and weaknesses with four core components of reasoning about fraction arithmetic in terms of quantities. The components are referent unit, partitioning and iterating units, recognizing multiplication and division situations, and forming multiplicative comparisons. Figure 3 shows the distribution of teacher profiles in a national sample of 990 in-service U.S. middle grades teachers.
Our project seeks to shed light on how future teachers develop understanding of central middle and secondary grades mathematics. We expect our project to inform improved teacher education content courses and to lay a foundation for studies of middle grades students’ reasoning about ratio and proportional relationships.
Walimu 'Multiplicative Sababu ya Utafiti Group masomo ya ikolojia ya walimu 'utambuzi kuzunguka interrelated multiplicative mada ikiwa ni pamoja na kuzidisha, mgawanyiko, vipande, mahusiano ya uwiano, na kazi linear. Kwa mfano, tunaweza kuona jinsi walimu wanavyoweza kuunga mkono hoja zao kuhusu kuzidisha, kugawanya, na vipande, na jinsi mistari ya nambari na michoro ya mkanda inavyoweza kuunga mkono hoja hizo. "Mfumo wetu wa nadharia unategemea mitazamo miwili ya ziada juu ya mahusiano ya uwiano, moja ambayo tunaita ""mzigo wa wingi"" na nyingine ambayo tunaita ""sehemu ya kutofautiana.""" Mtazamo wa mizunguko mingi umechunguzwa katika utafiti wa zamani hasa na wanafunzi (tazama Picha ya 1). Sehemu ya kutofautiana inaonekana kuwa ni muhimu kwa utafiti wa zamani, lakini ina ahadi ya kufanya matumizi ya mahusiano ya uwiano kwa equations linear na mada nyingine zaidi kupatikana. Mbinu zetu kuratibu masomo ya kesi ya kina na maendeleo ya hivi karibuni kupima walimu'multipliative hoja. Kazi ya psychometric ni kufanyika katika ushirikiano na Dk Laine Bradshaw na ni msingi wa kuamua maelezo ya sababu multiplicative. Profiles ni kuamua kwa kutumia Diagnostic Classification Models na muhtasari nguvu na udhaifu na nne msingi vipengele vya kufikiri kuhusu sehemu arithmetic katika suala la kiasi. Vipengele ni kitengo cha kumbukumbu, kugawanya na iterating vitengo, kutambua kuzidisha na mgawanyiko hali, na kuunda kulinganisha multiplicative. Picha ya 3 inaonyesha usambazaji wa profiles ya walimu katika sampuli ya kitaifa ya walimu wa darasa la kati wa 990 nchini Marekani. Mradi wetu unatafuta kutoa mwanga juu ya jinsi walimu wa baadaye wanavyoendeleza uelewa wa msingi wa kati na sekondari darasa la hisabati. "Tunatarajia mradi wetu kuelimisha kuboresha kozi za maudhui ya elimu ya walimu na kuweka msingi wa masomo ya wanafunzi wa darasa la kati ""kufikiria juu ya uwiano na mahusiano ya uwiano."""