How are proof and proving conceptualized in mathematics curriculum documents in the USA and Japan?

Only a few international comparative studies have reported on proof and proving in curriculum documents. This report proposes a method of comparing the meaning of proof-related words in two specific countries’ curriculum documents (the USA and Japan) through quantitative and interpretative analyses. Using a text mining approach to explore text data, we found that the co-occurrence network of the words “proof” and “prove” in curriculum documents from the two countries is quite different. In the USA, the word “proof” is concerned with justification and “prove” is used as a general process, while in Japan “proof” is more related to discovery and “prove” is more associated with specific theorems.This research was supported by JSPS KAKENHI (JP20KK0053, JP20K14013)

Identifying, counting, and characterizing superfine activated-carbon particles remaining after coagulation, sedimentation, and sand filtration

Superfine powdered activated carbon (SPAC; particle diameter ∼1 μm) has greater adsorptivity for organic molecules than conventionally sized powdered activated carbon (PAC). Although SPAC is currently used in the pretreatment to membrane filtration at drinking water purification plants, it is not used in conventional water treatment consisting of coagulation–flocculation, sedimentation, and rapid sand filtration (CSF), because it is unclear whether CSF can adequately remove SPAC from the water. In this study, we therefore investigated the residual SPAC particles in water after CSF treatment. First, we developed a method to detect and quantify trace concentration of carbon particles in the sand filtrate. This method consisted of 1) sampling particles with a membrane filter and then 2) using image analysis software to manipulate a photomicrograph of the filter so that black spots with a diameter >0.2 μm (considered to be carbon particles) could be visualized. Use of this method revealed that CSF removed a very high percentage of SPAC: approximately 5-log in terms of particle number concentrations and approximately 6-log in terms of particle volume concentrations. When waters containing 7.5-mg/L SPAC and 30-mg/L PAC, concentrations that achieved the same adsorption performance, were treated, the removal rate of SPAC was somewhat superior to that of PAC, and the residual particle number concentrations for SPAC and PAC were at the same low level (100–200 particles/mL). Together, these results suggest that SPAC can be used in place of PAC in CSF treatment without compromising the quality of the filtered water in terms of particulate matter contamination. However, it should be noted that the activated carbon particles after sand filtration were smaller in terms of particle size and were charge-neutralized to a lesser extent than the activated carbon particles before sand filtration. Therefore, the tendency of small particles to escape in the filtrate would appear to be related to the fact that their small size leads to a low destabilization rate during the coagulation process and a low collision rate during the flocculation and filtration processes