Table 1 Theoretical blended Math-Sci sensemaking framework
From: Cognitive framework for blended mathematical sensemaking in science
0 | Students recognize mathematical relationships in a provided formula | |
1 Qualitative | Description | Students use observations to identify which measurable quantities (variables) contribute to the phenomenon Example: force and mass make a difference in the speed of a car |
Pattern | Students recognize patterns among the variables identified using observations and can explain qualitatively how the change in one variable affects other variables, and how these changes relate to the scientific phenomenon in question Example: the smaller car speeds up more than the big car when the same force is exerted on both | |
Mechanism | Students demonstrate qualitative understanding of the underlying causal mechanism (cause-effect relationships) behind the phenomenon based on the observations but can’t define the exact mathematical relationship Example: it is easier to move lighter objects than heavy objects, so exerting the same force on a lighter car as on a heavy car will cause the lighter car to speed up faster | |
2 Quantitative | Description | Students recognize that the variables identified using the observations provide measures of scientific characteristics and can explain quantitatively how the change in one variable affects other variables (but not recognizing the quantitative patterns yet), and how this change relates to the phenomenon. Students are not yet able to express the phenomenon as an equation Example: recognizing that when variable A has value of X, variable B has value of Y |
Pattern | Students recognize quantitative patterns among variables and explain quantitatively how the change in one parameter affects other parameters, and how these changes relate to the phenomenon in question. Students not yet able to relate the observed patterns to the operations in a mathematical equation and can’t develop the exact and accurate mathematical relationship yet Example: recognizing mathematical relationships such is direct linear and inverse linear among others | |
Mechanism | Students can explain quantitatively (express relationship as an equation)(express relationship as an equation) how the change in one variable affects other variables based on the quantitative patterns derived from observations. Students include the relevant variables that are not obvious or directly observable. Students are not yet able to explain conceptually why each variable should be in the equation beyond noting that the specific numerical values of variables and observed quantities match with this equation. Students cannot explain how the mathematical operations used in the equation relate to the phenomenon, and why a certain mathematical operation was used. Students can provide qualitative causal account for the phenomenon Example: In Fnet = ma, multiplication makes sense because when applied force on the mass of 50 kg increases from 10 to 20 N, acceleration increases by 2. That only makes sense for a multiplication operation | |
3 Conceptual | Description | Students can describe the observed phenomenon in terms of an equation, and they can explain why all variables or constants (including unobservable or not directly obvious ones) should be included in the equation. Students are not yet able to explain how the mathematical operations used in the formula relate to the phenomenon Example: In F = ma, the F is always less than applied force by a specific number, so there must be another variable subtracted from Fapplied to make the equation work. The variable involves the properties of the surface. So, the equation should be modified: Fapplied-(variable) = ma |
Pattern | Students can describe the observed phenomenon in terms of an equation, and they can explain why all variables or constants (including unobservable or not directly obvious ones) should be included in the equation. Students are not yet able to provide a causal explanation of the equation structure Example: In Fnet = ma, multiplication makes sense because as applied force on the same mass increases, acceleration increases linearly, which suggests multiplication | |
Mechanism | Students can describe the observed phenomenon in terms of an equation, and they can explain why all variables or constants (including unobservable or not directly obvious ones) should be included in the equation. Students can fully explain how the mathematical operations used in the equation relate to the phenomenon in questions and therefore provide causal explanation of the equation structure, that is how the equation (the variables and the mathematical operations among the variables) is describing the causal mechanism of the scientific phenomenon Example: Since greater acceleration is caused by applying a larger net force to a given mass, this shows a positive linear relationship between a and Fnet, which implies multiplication between m and a in the equation, or Fnet = ma | |