A mathematical learning tool can scaffold the learner by performing computations, providing more time for students to test mathematical hypotheses that require reasoning. In the statistics example, learners can focus on why changes to certain parameters affect data–and in what ways, rather than spending all their time calculating measures.
Mathematical tools foster learning at many levels–namely, the learning of facts, procedures, and concepts. Tools can also provide concrete models of abstract ideas, or, when dealing with complex problems, they can enable students to manipulate and think about ideas, thereby making mathematics accessible and more deeply understood.
research suggests that students, in particular girls, may tend to continue to use the same tools because they feel comfort- able with the tools and are afraid to take risks (ambrose, 2002).
The manner in which students learn mathematics influences how well they understand its concepts, principles, and practices. Many researchers have argued that to promote learning with understanding, mathematics educators must consider the tasks, problem-solving situations, and tools used to represent mathematical ideas.
Zero's origins most likely date back to the “fertile crescent” of ancient Mesopotamia. Sumerian scribes used spaces to denote absences in number columns as early as 4,000 years ago, but the first recorded use of a zero-like symbol dates to sometime around the third century B.C. in ancient Babylon.
Mathematics. Cuneiform script, developed by the Sumerians.
The people of Mesopotamia developed mathematics about 5,000 years ago. Early mathematics was essentially a form of counting, and was used to count things like sheep, crops and exchanged goods. Later it was used to solve more sophisticated problems related to irrigation and perhaps architecture.
The first recorded zero appeared in Mesopotamia around 3 B.C. The Mayans invented it independently circa 4 A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth.
The Babylonian numeral system is the first known positional numeral system and it is considered by some as their greatest achievement in mathematics. However, the Babylonians did not have a concept of zero or a digit for it. They instead used a space.
The Babylonians had an advanced number system, in some ways more advanced than our present systems. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today.
Mathematics helps us understand the world and provides an effective way of building mental discipline. Math encourages logical reasoning, critical thinking, creative thinking, abstract or spatial thinking, problem-solving ability, and even effective communication skills.
Mathematics is the science and study of quality, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.
The Mesopotamians were sophisticated mathematicians and were the first to develop the idea of place value based on a number's position in a sequence. Time: The Mesopotamians were the first to divide time units into 60 parts. This concept lead to our 60-second minute and 60-minute hour.
"Zero" is the usual name for the number 0 in English. In British English "nought" is also used. In American English "naught" is used occasionally for zero, but (as with British English) "naught" is more often used as an archaic word for nothing.
0 (zero) is a number, and the numerical digit used to represent that number in numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
mathematician John Wallisinfinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655.
Learning Tools. The manner in which students learn mathematics influences how well they understand its concepts, principles, and practices. Many researchers have argued that to promote learning with understanding, mathematics educators must consider the tasks, problem-solving situations, and tools used to represent mathematical ideas.
Technological Tools. Technological tools are most effective in facilitating students' understanding of complex concepts and principles. Computations and graphs can be produced quickly, giving students more time to consider why a particular result was obtained. This support allows students to think more deeply about the mathematics they are learning. Electronic tools are necessary in mathematics because they support the following processes: (a) conjectures–which provide access to more examples and representational formats than is possible by hand; (b) visual reasoning–which provides access to powerful visual models that students often do not create for themselves; (c) conceptualization and modeling–which provide quick and efficient execution of procedures; and (d) flexible thinking–which support the presentation of multiple perspectives.
Technology enables students to focus on the structure of the data and to think about what the data mean, thereby facilitating an overall understanding of a concept (e.g., function). The graphics calculator supports procedures involving functions and students' ability to translate and understand the relationship between numeric, algebraic, and graphical representations. Transforming graphical information in different ways focuses attention on scale changes and can help students see relationships if the appropriate viewing dimensions are used. Computers may remove the need for overlearning routine procedures since they can perform the task of computing the procedures. It is still debatable whether overlearning of facts helps or hinders deeper understanding and use of mathematics. Technology tools can also be designed to help students link critical steps in procedures with abstract symbols to representations that give them meaning.
Mathematical learning tools can be traditional, technological, or social. The most frequently employed tools are traditional, which include physical objects or manipulatives (e.g., cubes), visualization tools (e.g., function diagrams), and paper-and-pencil tasks (e.g., producing a table of values). Technological tools, such as calculators (i.e., algebraic and graphic) and computers (e.g., computation and multiple-representation software), have gained attention because they can extend learning in different ways. Social tools, such as small-group discussions where students interact with one another to share and challenge ideas, can be considered a third type of learning tool. These three tools can be used independently or conjointly, depending on the type of learning that is intended.
Traditional Tools. Traditional tools are best suited for facilitating students' learning of basic knowledge and skills.
The key characteristic of a learning tool is that it supports learners in some manner. For example, a tool can aid memory, help students to review their problem-solving processes, or allow students to compare their performance with that of others, thereby supporting self-assessment.
Mathematical learning tools should be an important part of students' educational experience. However, a few issues must be addressed before their potential is fully realized. First, use of technological tools is fairly limited in classrooms, despite their potential in changing the nature of mathematical learning. Moreover, software used in schools is often geared towards the practice of computational skills. For example, there may be a potential misuse of the graphing calculator if it is not utilized in the context of sense-making activities. There is a fine line between using a tool for understanding and using it because problems cannot be solved without its use.
To finish, I should note that many other mathematics educators have advocated that the main focus of K-12 mathematics education should be in-depth study of arithmetic, geometry, and a bit of algebra (and little else). For example, Liping Ma, whose approach I wrote about in the latter part of my Devlin’s Angle last October. You may find what she has to say of value. I certainly did.
Of course, it can be beneficial to expose students to various other parts of mathematics. The field of mathematics is one of the great accomplishments of human culture. There is benefit to be gained from showing the next generation some of that intellectual heritage (including calculus, in particular). Partly because it is part of human culture, and a major one at that. But also because it helps them appreciate the enormously wide scope of mathematical applications, providing them a meaningful context and purpose for devoting time to learning the stuff you are asking them to master (which will inescapably require a considerable amount or repetitive practice). But exposure is very different from achieving mastery, and requires a very different teaching approach.
Mathematical thinking takes a long time to develop. The challenge facing today’s math educators is finding the most efficient way to reach that goal. A way that does not fail, and alienate, the majority of our students. There is, I think, good reason to believe this can be done.
If you did not have good mastery of basic arithmetic, you could find yourself disadvantaged in everyday life, and if you did not master arithmetic and some basic algebra (in particular), you could not get off the ground in mathematics. The sooner such mastery could be achieved, the better for all.
The question for mathematics educators is, how best do we develop that way of thinking, and the understanding it depends upon?
Being a mathematician (or a user of mathematics) today is all about using those tools effectively and efficiently. Our mathematics education system needs to produce people with that ability.
The goal is pretty clear. The purpose of education is to prepare the next generation for the lives they will lead. Agreed? So what might that entail for mathematics education? What do our students need to be able to do when they graduate?
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models ... Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful...
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, [or] concrete models...
On the first day of the learning segment, Michelle Makinson engages her learners in a math talk focused on unit fractions, combining into wholes, “parts of,” and the idea of equivalence, using manipulatives to create and explain a visual representation of a contextualized representation / word problem. Her students then share with their partners, explaining their approach. This clip also relates to standard 1 (make sense of problems & persevere in solving them) , standard 3 (construct viable arguments & critique the reasoning of others) and standard 6 (attend to precision).
Fran Dickinson leads a lesson on numeric patterning, helping students to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, Dickinson tells his students that “the first step is to do a pictorial representation… I want you to play around with the tiles, and sketch out what you see happening in those first three patterns, but I want you to pay attention to color-coding. You’re free to use those tiles like I said, or markers if you need them, I can make those available as well." This clip is also indicative of standard 1 (make sense of problems & persevere in solving them).
Patty introduces her lesson by charging students to identify the “big ideas” they should be thinking about when they work with right triangles. Students pair-share their ideas, and Patty notes when they are making reference to available tools and supports, such as anchor charts, around the room. In her commentary, Patty notes that this lesson is intended to develop students’ capacity to engage in modeling mathematical situations. Students identify the Pythagorean Theorem, and Patty prompts them to attend to precision and communicate precisely. In a whole-group sharing, she engages all students to add on to, critique, extend, and clarify each other’s thinking. Students deepen their capacity to make sense of the problem or situation. Patty presents student work from a previous assessment and asks students to critique the person’s strategies and precision, giving advice to each exemplar learner about how to improve their approach. This clip also relates to standard 1 (make sense of problems & persevere in solving them) , standard 3 (construct viable arguments & critique the reasoning of others), and standard 4 (model with mathematics).
Teachers who are developing students' capacity to "use appropriate tools strategically" make clear to students why the use of manipulatives, rulers, compasses, protractors, and other tools will aid their problem solving processes. A middle childhood teacher might have his students select different color tiles to show repetition in a patterning task.
research suggests that students, in particular girls, may tend to continue to use the same tools because they feel comfort- able with the tools and are afraid to take risks (ambrose, 2002). Look for students who tend to use the same tool or strategy every time they work on a task. encourage all stu- dents to take risks and to try new tools and strategies.
They use reasoning and proofto make sense of mathematical tasks and concepts and to develop, justify, and evaluate mathematical arguments and solutions. Students create and use representations(e.g., diagrams, graphs, symbols, and manipulatives) to reason through problems.
Teachers generally agree that teaching for understanding is a good thing.But this statement begs the question: What is understanding? Understanding is being able to think and act flexibly with a topic or concept.It goes beyond knowing; it is more than a collection of in-formation, facts, or data. It is more than being able to follow steps in a procedure. One hallmark of mathematical understanding is a student’s ability to justify why a given mathematical claim or answer is true or why a mathematical rule makes sense (CCSSO, 2010). Although stu-dents might know their multiplication basic facts and be able to give you quick answers to questions about these basic facts, they might not understand multiplication. They might not be able to justify the correct-ness of their answer or provide an example of when it would make sense
A primary goal of teaching for understanding is to help students develop a relational understanding of mathematical ideas. Because relational understanding develops over time and becomes more complex as a person makes more connections between ideas, teaching for this kind of understanding takes time and must be a goal of daily instruction.
They develop their understanding of mathematics because they are at the center of explaining, providing evidence or justification, finding or creating examples, generalizing, analyzing, making predictions, applying concepts, representing ideas in differ - ent ways, and articulating connections or relationships between the given topic and other ideas . For example, in this fourth‐grade classroom, students have already reviewed double‐ digit addition and subtraction computation and have been working on multiplication con- cepts and facts. They mastered most of their multiplication facts by the end of third grade but as part of an extension, the students have used contexts embedded in story problems. They are also illustrating how repeated addition can be related to the number of rows of square tiles within a rectangle. These students’ combined experiences from grades 3 and 4 have resulted in a collection of ideas about tens and ones (from their work with double‐ digit addition and subtraction), an understanding of the meaning of multiplication as re- lated to unitizing (i.e., a row of six as one six), a variety of number strategies for mastering multiplication facts based on the properties of the operation, and a connection between multiplication and arrays and area.
Through reflective thought, people connect existing idea to new information and thus modify their existing schemas or background knowledge to incorporate new ideas. Making these connections can happen in either of two ways—assimilationor accommodation. Assimi- lation occurs when a new concept “fits” with prior knowledge and the new information ex- pands an existing mental network. Accommodation takes place when the new concept does not “fit” with the existing network, thus creating a cognitive conflict or state of confusion that causes what theorists call disequilibrium. As an example, some students assimilate frac- tions into their existing schemas for whole numbers. When they begin to compare fractions, they treat the numerators and denominators separately, as if they represented two whole numbers that have no relationship to each other. Such a student might mentally compare
habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
The practice reflects a long-standing feature of American math education: As early as middle school, students are often split into "tracks" in ways that predetermine who will take advanced classes in high school. The advanced classes are often full of students who are white or Asian and attend suburban schools – while black and Latino students continue to be underrepresented, research shows.
New research suggests that when teachers improve their attitude toward math, it can help to raise student test scores. At Stanford, Boaler and her team designed an online course for teachers featuring research showing anyone can learn math with enough practice, intelligence isn’t fixed and math is connected to all sorts of everyday activities.
Most American high schools teach algebra I in ninth grade, geometry in 10th grade and algebra II in 11th grade – something Boaler calls “the geometry sandwich.”
Georgia mandated high schools teach integrated math starting in 2008. After pushback from teachers and parents, it gave schools the option to go back to the old sequence in 2016. In one large survey, Georgia teachers said they didn’t want to specialize in more than one math area.
The Common Core academic standards, a version of which most states adopted, say high school math can be taught in either format.
Those messages often come from their elementary school teachers, many of whom didn’t like math as students themselves. "Math phobia is real. Math anxiety is real," said DeAnn Huinker, a professor of mathematics education at the University of Wisconsin-Milwaukee who teaches future elementary and middle school teachers.
He said high schools could consider whittling down the most useful elements of geometry and the second year of algebra into a one-year course. Then students would have more room in their schedules for more applicable math classes.