The effects of teaching a mathematics problem solving strategy on the problem solving ability of grade nine students

Forty grade 9 students were selected from a small rural board in southern Ontario. The students were in two classes and were treated as two groups. The treatment group received instruction in the Logical Numerical Problem Solving Strategy every day for 37 minutes over a 6 week period. The control group received instruction in problem solving without this strategy over the same time period. Then the control group received the treat~ent and the treatment group received the instruction without the strategy. Quite a large variance was found in the problem solving ability of students in grade 9. It was also found that the growth of the problem solving ability achievement of students could be measured using growth strands based upon the results of the pilot study. The analysis of the results of the study using t-tests and a MANOVA demonstrated that the teaching of the strategy did not significaritly (at p s 0.05) increase the problem solving achievement of the students. However, there was an encouraging trend seen in the data.

The impact of integrated programming on student attitude and achievement in grade 9 academic mathematics and science

This research studioo the effect of integrated instruction in mathematics and~ science on student achievement in and attitude towards both mathematics and science. A group of grade 9 academic students received instruction in both science and mathematics in an integrated program specifically developed for the purposes of the research. This group was compared to a control group that had received science and mathematics instruction in a traditional, nonintegrated program. The findings showed that in all measures of attitude, there was no significant difference between the students who participated in the integrated science and mathematics program and those who participated in a traditional science and mathematics program. The findings also revealed that integration did improve achievement on some of the measures used. The performance on mathematics open-ended problem-solving tasks improved after participation in the integrated program, suggesting that the integrated students were better able to apply their understanding of mathematics in a real-life context. The performance on the final science exam was also improved for the integrated group. Improvement was not noted on the other measures, which included EQAO scores and laboratory practical tasks. These results raise the issue of the suitability of the instruments used to gauge both achievement and attitude. The accuracy and suitability of traditional measures of achievement are considered. It is argued that they should not necessarily be used as the measure of the value of integrated instruction in a science and mathematics classroom.

Teachers' professional learning within communities of mathematics-for-teaching practice

Ontario bansho is an emergent mathematics instructional strategy used by teachers working within communities of practice that has been deemed to have a transformational effect on teachers' professional learning of mathematics. This study sought to answer the following question: How does teachers' implementation of Ontario bansho within their communities of practice inform their professional learning process concerning mathematics-for-teaching? Two other key questions also guided the study: What processes support teachers' professional learning of content-for-teaching? What conditions support teachers' professional learning of content-for-teaching? The study followed an interpretive phenomenological approach to collect data using a purposive sampling of teachers as participants. The researcher conducted interviews and followed an interpretive approach to data analysis to investigate how teachers construct meaning and create interpretations through their social interactions. The study developed a model of professional learning made up of 3 processes, informing with resources, engaging with students, and visualizing and schematizing in which the participants engaged and 2 conditions, ownership and community that supported the 3 processes. The 3 processes occur in ways that are complex, recursive, nonpredictable, and contextual. This model provides a framework for facilitators and leaders to plan for effective, content-relevant professional learning by placing teachers, students, and their learning at the heart of professional learning.

Student mathematical proofs by Edward Stephen (?) Wigglesworth, 1788

Sheet with two handwritten mathematical proofs signed "Wigglesworth, 1788," likely referring Harvard student Edward Stephen Wigglesworth. The first proof, titled "Problem 1st," examines a prompt beginning, "Given the distance between the Centers of the Sun and Planet, and their quantities of matter; to find a place where a body will be attracted to neither of them." The second proof, titled "Problem 2d," begins "A & B having returned from a journey, had riden [sic] so far that if the square of the number of miles..." and asks "how many miles did each of them travel?"

Notebook of Obadiah Ayer, 1708-1716

The leather-bound notebook contains academic texts copied by Obadiah Ayer while he was a student at Harvard, and after his graduation in 1710. There is a general index to the included texts at the end of the volume.

Student notebook of Samuel Dunbar, 1721-1722

This leather-bound volume contains substantial transcriptions copied by Samuel Dunbar from textbooks while he was a student at Harvard in 1721 and 1722. There is a general index to texts at the end of the volume. Dunbar's notebook provides a window into the state of higher education in the eighteenth century and offers a firsthand account of academic life at Harvard College. Notably, he often indicated the number of days spent copying texts into his book.

Student mathematical textbook of James Freeman, 1774

Manuscript volume containing portions of text copied from Nicholas Saunderson’s Elements of algebra, Nicholas Hammond’s The elements of algebra, and John Ward’s The young mathematician’s guide. The volume is divided into two main parts: the first is titled Concerning the parts of Arithmetick (p. 1-98) and the second, The elements of Algebra, extracted from Hammond, Ward & Saunderson (p. 99-259).

Student mathematical notebook of Thomas Noyes, 1793

This sewn volume contains Noyes’ mathematical exercises in geometry; trigonometry; surveying; measurement of heights and distances; plain, oblique, parallel, middle latitude, and mercator sailing; and dialing. Many of the exercises are illustrated by carefully hand-drawn diagrams, including a mariners’ compass and moon dials.

Student mathematical notebook of Ebenezer Hill, 1785

This mathematical notebook of Ebenezer Hill was kept in 1795 while he was a student at Harvard College. The volume contains rules, definitions, problems, drawings, and tables on arithmetic, geometry, trigonometry, surveying, calculating distances, and dialing. Some of the exercises are illustrated by hand-drawn diagrams, including some of buildings and trees.

Student mathematical notebook of Ephraim Eliot, 1779

Handwritten mathematical notebook of Ephraim Eliot, kept in 1779 while he was a student at Harvard College. The volume contains rules, definitions, problems, drawings, and tables on arithmetic, geometry, trigonometry, surveying, calculating distances, and dialing. Some of the exercises are illustrated by unrefined hand-drawn diagrams, as well as a sketch of a mariner’s compass. The sections on navigation, mensuration of heights, and spherical geometry are titled but not completed. The ink of the later text, beginning with Trigonometry, is faded.

Commonplace book of Benjamin Penhallow, 1721-1724

This leather-bound volume contains excerpts copied by Benjamin Penhallow from books he read while he was a student at Harvard in the 1720s. The volume contains extracts from two texts: Johanis Henrici Alstedii's (John Henry Alsted / Johann Heinrich Alsted) Geometria Domini, and the anonymous text "The Legacy of a dying Father; bequeath'd to his Beloved Children, or Sundry Directions in Order unto a well Regulated Conversation," from 1724 (originally published in 1693-4). The last page of text in the volume contains the hymn "The Sacred Content of Praise" first published in 1734, and added after Penhallow's death.

Student notebook of James Varney, 1721-1725

The bound notebook contains academic texts copied by Harvard student James Varney in the early 1720s. The texts are written tête-bêche (where both ends of the volume are used to begin writing). The front paste-down endpaper reads 'James Varney his book 1724,' and the rear paste-down endpaper reads 'Joseph Lovett' [AB 1728].

Student mathematical notebook of William Emerson Faulkner, 1795

Leather hardcover notebook with unruled pages containing the handwritten mathematical exercises of William Emerson Faulkner, begun in 1795 while he was an undergraduate at Harvard College. The volume contains rules, definitions, problems, drawings, and tables on geometry, trigonometry, surveying, calculating distances, sailing, and dialing. Some of the exercises are illustrated by unrefined hand-drawn diagrams, including some of buildings and trees.

Student mathematical notebook of William Tudor, 1795

Notebook containing the handwritten mathematical exercises of William Tudor, kept in 1795 while he was an undergraduate at Harvard College. The volume contains rules, definitions, problems, drawings, and tables on geometry, trigonometry, surveying, calculating distances, sailing, and dialing. Some of the exercises are illustrated with hand-drawn diagrams. The Menusration of Heights and Distances section contains color drawings of buildings and trees, and some have been altered with notes in different hands and with humorous additions. For instance, a drawing of a tower was drawn into a figure titled “Egyptian Mummy.” Some of the images are identified: “A rude sketch of the Middlesex canal,” Genl Warren’s monument on Bunker Hill,” “Noddles Island,” “the fields of Elysium,” and the “Roxbury Canal.” The annotations and additional drawings are unattributed.

Mathematical notebook attributed to Thaddeus Mason Harris, ca. 1787

Hardcover notebook containing handwritten transcriptions of rules, cases, and examples from 18th century mathematical texts. The author and purpose of the volume is unclear, though it has been connected with Thaddeus Mason Harris (Harvard AB 1787). Most of the entries include questions and related answers, suggesting the notebook was used as a manuscript textbook and workbook. The extracts appear to be copied from John Dean's " Practical arithmetic" (published in 1756 and 1761), Daniel Fenning's "The young algebraist's companion" (published in multiple editions beginning in 1750), and Martin Clare's "Youth's introduction to trade and business" (extracts first included in 1748 edition).