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Mathematics Online resources Problem Solving

Prodigy Math

Humans are naturally motivated to solve problems, especially problems that they can relate to. Unfortunately, most classrooms limit problem solving activities to word problems in textbooks, which students often cannot identify with. Prodigy is a Mathematics App/Website that develops problem solving skills by tapping into the intrinsic attraction that many children have to games involving quests, wizards, and battles. In the game, students have to answer progressively more difficult word problems to defeat wizards, unlock health and potions, and ultimately defeat a dragon. While many of the problems in the game are similar to ones in textbooks, the ability to the program to adapt difficulty based on previous answers, and rich context in the game still makes this game worth considering for elementary school children.

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Mathematics Fractions Online resources Virtual Manipulative

Fraction Strips

Students who are not yet “fully operational” or able to reason using abstract concepts, can still learn form mathematical concepts with support. They typically need to develop these abstract representations of concepts in three stages, using: 1) concrete representations (like a geo-board) ; 2) semi-concrete representations (like a drawing); and 3) abstract representations (like a formula). While concrete “manipulatives,” like base-ten blocks, are widely used and effective in introducing concepts like place-value, evidence suggests that “virtual manipulatives” can be similarly effective.

Toytheater has an excellent selection of “virtual manipulatives” that students can use online, including a realistic version on fraction strips or fraction blocks. The image below will take you to their website (in a new tab) where you can explore their fraction strip app, along with other virtual manipulatives.

Click on the image to go to toy theater.

Comments and suggestions are always welcome. Thanks for visiting!

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Mathematics Fractions Sheltered English Texts

Text: Introduction to Fractions

Fractions present students in early to middle elementary school with a big challenge. For the first time, they are faced with numbers that do not represent whole numbers, and do not follow the same operational rules as whole numbers. For that reason, in some countries – like Japan – operations with fractions are not taught until students reach fifth grade. In the U.S., the common core standards introduce fractions in third grade, but emphasize developing an understanding of fractions as numbers, and developing a strong intuitive understanding of whole number fractions before diving into addition, subtraction, multiplication, and division with mixed fractions.

The set of texts below are designed to support 2-3 lessons introducing third graders to fractions. The texts focus on connecting fractions to students’ everyday experiences, providing students with concrete experiences working with fractions, and using concrete and semi-concrete experiences to introduce the symbols used to represent fractions.

“Develop understanding of fractions as numbers for fractions with denominators 2, 3, 4, 6, and 8.” (p. 40)…“Understand a fraction 1 ∕b as the quantity formed by 1 part when a whole (a single unit) is partitioned into b equal parts; understand a fraction a ∕b as the quantity formed by a parts of size 1 ∕b”

Related MA Common Core Standards

Why three texts?

Many states require students who are not proficient in English to learn all subjects – e.g. Math, Science, Art, and Social Studies – in English along with native speakers. For teachers, many of whom already struggle to differentiate lesson plans for students with different academic backgrounds, learning styles, and interests, the requirement to teach equivalent content to students at varying levels of English proficiency poses yet another challenge.

The texts below are designed to provide equivalent content instruction (along with content-specific academic language) to students at three levels of English proficiency – early intermediate (“WIDA 2-3”) intermediate (“WIDA 4-5”) and advanced to native speaking (“WIDA 6” and native speakers). The basic template is also designed to be further differentiated in terms of complexity – for example, by using more complex assessments or examples.

In any case, check out the texts below – hope you like them, and don’t hesitate to leave comments or suggestions! [note, links will open a new tab outside of this website]

Categories
Mathematics Methods

A learning video analysis

In Mathematics instruction, the mental processes that are taking place in students’ minds are often as (or more) important than the output that they produce. But, instructions to “show your work” aside, it can often be difficult to evaluate whether a student has fully grasped a concept like a slope-intercept formula, long division, or place value. One way to gain a better understanding of the mental models that students approach problems with is to record or observe students as they attempt to solve (relatively) open ended problems. According to Goldsmith and Seago (2013), this system for gaining insight involves three steps:

  1. Select representations for particular purposes – find out which representations will support your students’ access to the content
  2. Recognize the math involved in a particular representation – for example, students need to know that there are 100 cents in a dollar in order understand  how to use 100s blocks to solve coin problems
  3. Link representations to underlying ideas and other representations – for example, students learn to skip count because they have been asked to memorize the multiples of certain numbers. So, often, students don’t know that skip counting is a form of multiplication

The analysis below examines this video using the three steps listed above. (Note: this exercise was carried out for EDUC7520.01 – Math and Technology in Teaching – at Boston College)