Knowing and Teaching Middle School Mathematics: A Professional Development Course for In‐Service Teachers

This article describes a professional development course intended to improve the content understanding of middle school mathematics teachers. The design of the course included three professional learning strategies: problem solving, examination of student thinking, and discussion of research. The concepts studied in the course included multi‐digit subtraction, multi‐digit multiplication, operations with fractions, and concepts of area and perimeter. Results from pre‐ and post‐tests administered to the nineteen participants indicate a significant increase in the mean score for each concept and document growth in the teachers' content understanding. In particular, their solutions moved from primarily procedural to more conceptual. Responses to an open‐ended survey indicate other important aspects of the professional development. Examples of teachers' work and comments are included.     

Selective cleavage of acetals with ZnBr2 in dichloromethane

A selective cleavage of acetals of 1,2- and 1,3-diols has been achieved under mild conditions using ZnBr₂ in dichloromethane at room temperature. Acetal types cleavable by this procedure include benzylidene, isopropylidene and cyclohexylidene acetals. This method is compatible with several other types of hydroxyl protecting groups such as Bn, Bz, TBDPS, TIPS and TBDMS.     

Synthesis of Enantiopure Primary Amines by Stereoselective Ring Opening of Chiral Octahydro-1,3-benzoxazines by Grignard and Organoaluminum Reagents

Chiral 1,3-perhydrobenzoxazines 1, 2, and 9-14, prepared by condensation of 8-(benzylamino)menthol with different aldehydes, react with alkylmagnesium bromides and trimethylaluminum leading to the open amino alcohols 3a-d, 4a-d, and 15-20 in excellent chemical yields and good to excellent diastereomeric excess. The sequential elimination of the menthol appendage by heating with P(2)O(5) and the benzyl group by hydrogenolysis lead to primary amines 7a-d, 8a-d, and 27-30 in excellent chemical yields and ee. The addition of the alkyl group from the Grignard derivatives and the methyl group from the trimethylaluminum occurs from opposite sides of the heterocycle, yielding the final primary amines with the same stereochemistry.     

A novel synthesis of enantiopure octahydropyrrolo[3,4-b]pyrroles by intramolecular [3 + 2] dipolar cycloaddition on chiral perhydro-1,3-benzoxazines

[reaction: see text] Condensation of N-substituted glycines with chiral 3-allyl-2-formyl perhydro-1,3-benzoxazines forms an azomethine ylide that cyclizes to give octahydropyrrolo[3,4-b]pyrrole derivatives. The [3 + 2] dipolar cycloadditions are stereoespecific leading to a single diastereoisomer. The chemical yields are dependent on the reaction temperature and the presence or absence of a base.     

The reaction of β-aminoenones with substituted acetonitriles. Regiospecific synthesis of 2(1H)-pyridones

Unsymmetrically substituted β-aminoenones react with malononitrile, cyanomethylphenylsulfone, benzoyl-acetonitrile and ethyl cyanoacetate, in very mild conditions, to yield regiospecifically 3-functionalized 2(1H)-pyridones in high yields.     

Volumetric Properties of Binary Mixtures of Isomeric Butanols and C8 Solvents at 298.15 K

Excess molar volumes, V _m ^E, over the whole composition range for binary mixtures of 1-butanol, 2-butanol, and 2-methyl-2-propanol + 1-octanol, or 2-octanol, or di-n-butyl ether, or n-hexylacetate were determined at 298.15 K from density measurements carried out with a vibrating-tube densimeter. Small V _m ^E values, both positive and negative, are displayed by mixtures containing 1- or 2-octanol, whereas positive and larger values are always found for mixtures containing dibutyl ether and hexylacetate. These results can be justified in terms of H-bonding interactions and/or steric hindrance due to the branched alkyl chains. Partial molar volumes at infinite dilution of the isomeric butanols in the C8 compounds were also calculated from the apparent molar volumes in dilute solution. The solute-solvent interactions and the effects of the local organisation of the solvent around the butanol molecules were discussed using the void and cavity volumes as different estimates of the intrinsic volume of the molecules. The volumetric behavior of butanols seems to be determined by the solute-solvent interactions rather than packaging effects.     

On the [2,3] sigmatropic rearrangement of allylic nitro compounds

Allylic nitro compounds undergo relatively clean [2,3] sigmatropic rearrangement upon heating in refluxing 1,2-dichlorobenzene in the presence of DABCO to give the corresponding allylic alcohols via the intermediate allylic nitrite. [reaction: see text]     

Reductive Ring Opening of Ephedrine and Pseudo-Ephedrine-Derived Oxazolidinium Methiodides by Sodium Borohydride. A Direct Access to New Chiral Amine-Borane Adducts

Sodiun borohydride reacts with oxazolidinium methiodides derived from ephedrine and pseudoephedrine, in ethereal solvents, cleaving regioselectively the C-N bond of the ring and leading, in good yields, to chiral amine-borane adducts of know absolute configuration. The ¹HNMR data allow to predict the most stable conformation.     

Highly Homogeneous Stereocontrolled Construction of Quaternary Hydroxyesters by Addition of Dimethylzinc to α‐Ketoesters Promoted by Chiral Perhydrobenzoxazines and B(OEt)3

A highly efficient enantioselective addition of Me₂Zn to α‐ketoesters, assisted by a chiral perhydro‐1,3‐benzoxazine ligand, is described. This novel catalytic system offers homogeneous elevated enantioselectivities in the preparation of α‐hydroxyesters that bear a quaternary stereocenter, with a minor dependence on electronic and steric effects when aromatic, heteroaromatic, or aliphatic α‐ketoesters are employed. The catalyst can be recovered and reused without loss of activity.