Category: Research Paper

Welcome to the "Mathematics Research" page! This is specifically devoted to making you a better mathematician and researcher. Please complete the following steps:

- Read the description.
- Examine the wonderful possibilities and implications of math research by watching the Math in Action video or reading the Andrew Wiles interview.
- To get some ideas about what you may wish to research, visit some of the websites in the resources tab.
- Review the methodology section to get ideas on how to go about planning and conducting your research or proof.
- Visit the resources section for miscellaneous information on gathering and reviewing information and some interesting or wacky math research facts.

Mathematics is the study of patterns, numbers, quantities, shapes, and space using logical processes, rules, and symbols. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematicians investigate patterns, formulate new conjectures, and determine truth by drawing conclusions from axioms and definitions. A mathematician can be an artist, scientist, engineer, inventor or straightforwardly, an independent thinker. He/she is commonly more than one of these at once.

"Ethno-mathematician" Ron Eglash is the author of *African Fractals ,* a book that examines the fractal patterns underpinning architecture, art and design in many parts of Africa. By looking at aerial-view photos -- and then following up with detailed research on the ground -- Eglash discovered that *many African villages are purposely laid out to form perfect fractals*. with self-similar shapes repeated in the rooms of the house, and the house itself, and the clusters of houses in the village, in mathematically predictable patterns.

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The above picture shows a fractal pattern. A fractal is a shape made from repeating one pattern on different scales. This repetition often makes the shape appear irregular.

As he puts it: "When Europeans first came to Africa, they considered the architecture very disorganized and thus primitive. It never occurred to them that the *Africans might have been using a form of mathematics that they hadn't even discovered yet* ."

His other areas of study are equally fascinating, including research into African and Native American cybernetics, teaching kids math through culturally specific design tools (such as the Virtual Breakdancer applet, which explores rotation and sine functions), and *race and ethnicity issues in science and technology*. Eglash teaches in the Department of Science and Technology Studies at Rensselaer Polytechnic Institute in New York, and he recently co-edited the book *Appropriating Technology* . about how we reinvent consumer tech for our own uses

*Ron Eglash: Mathematician* Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns.

'I am a mathematician, and I would like to stand on your roof.' That is how Ron Eglash greeted many African families he met while researching the fractal patterns he’d noticed in villages across the continent.

Typically, the study of math is divided into two major categories: pure mathematics or applied mathematics. Pure mathematical research involves significant mathematical exploration and the creation of original mathematics. Pure mathematics seeks to develop mathematical knowledge for its own sake rather than for any immediate practical use. Applied mathematics seeks to expand mathematical techniques for use in science and other fields or to use techniques in other fields to make contributions to the field of mathematics. Boundaries between pure mathematics and applied mathematics do not always exist.

What do the mathematicians have to say about math?

Click the image above to find out!

*Biographies of Mathematics Researchers*

*Google timeline of mathematicians*and their discoveries from 300 BC to the present- Pythagoras (approximately 569B.C.)
- Biography
- Biography and contributions

- Euclid (approximately 330 B.C.) Wrote comprehensive compilation of geographical knowledge used for over 2000 years.
- Biography
- Biography and discussion of elements

- Da Vinci (1452) Fifteenth century inventor/innovator who utilized math to detail and design inventions
- Work and biography
- Museum exhibition
- Biography

- Galileo (1564) Regarded as mathematic and scientific genius who was an innovator of the telescope that provided proof of Copernicus’ theory that earth revolves about the sun.
- Biography, timeline, articles, and interactive experiments (falling objects, projectiles, inclined planes, and pendulums)
- Video - Looks at Galileo’s contribution to the technology of the telescope
- Article - Looks at Galileo’s contribution to the technology of the telescope

- John Wheeler (1764) astrophysicist best known for coining the term, “black holes”
- Video interview
- Oral history transcript

- John August Roebling (1806) Pioneering architect of suspension bridges. Designer of the Brooklyn Bridge, NY, NY.
- Biography
- Images

- Dr. Robert Goddard (1882) pioneering work with rockets is responsible for the NASA space program
- Biography - timeline
- Biography

- Le Corbusier (1887) architect who developed the concept of modular human beings to design buildings based on anatomical geometric proportions
- Biography

- David Smith (1906) sculptor of geometric solids constructed from counterbalanced stainless steel shapes
- Biography
- Biography

- Sylvia Earle (1935) invented “JIM”, a deep sea diving suit, requiring advance knowledge of engineering.
- Video

- Diana Eng (1983) fashion designer who integrates her knowledge of mathematics, science, technology and fashion to create a collection of “magical” clothing.
- Work samples
- Biography

Overlap also exists in research in the field of mathematics. However, for our purposes, we will subdivide the fields of study by their primary purposes. In the chart below, the following purposes are outlined by Jeff Suzuki in

*But How Do I Do Mathematical Research ?*Suzuki, Jeff. "But How Do I Do Mathematical Research?”." Mathematical Association of America, 2010. Web. 26 July 2010. <http://www.maa.org/features/112404howdoido.html >.

This research seeks to justify a conjecture using logical reasoning. This category also includes finding alternative justifications for previously-proven theorems.

This research seeks to expand current mathematical concepts.

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Math instruction is generally broken down into five math strands: numbers and operations; algebra; geometry; measurement; and data analysis and probability. Though concepts are more advanced than others, even the youngest children can learn basic math strategies that will prepare them for future learning. Students also learn problem solving as a way to think critically about and integrate math strategies. Teachers should use a variety of instructional methods to keep lesson interesting and fun for all students, and to ensure that their lessons are reaching students of all learning styles. Assessment and evaluation can be done through tests and quizzes as well as one-on-one conferences and journal writing

Keywords Algebra; Curriculum; Data Analysis; Formative Assessment; Geometry; Journal Writing; Learning; Lesson Planning; Manipulatives; Math; Measurement; Numbers; Probability; Problem Solving; Structured Learning; Word wall

Teaching math is an important job for instructors who work with learners of any age. The goal of the math teacher shouldn't just be for the student to understand the concept or strategy being taught, but also for the students to be interested in the learning process. Ideally, students should find mathematics both intriguing and enjoyable.

Even the youngest children seem to be hard-wired to do math and be interested in numbers. From their earliest days, babies seem to have a basic understanding of mathematical concepts like adding and subtracting. Watch two objects move on a screen in front of him or her, a baby's face will often register surprise when another object is introduced, indicating a simple understanding of addition. Children may be ready to learn math at a very early age, and, when they are given opportunities, will usually be interested in learning (Sarama & Clements, 2006). Teachers are challenged to maintain that interest throughout the strands of the math curriculum.

The basic math curriculum is usually thought of in five strands. These components include: numbers and operations; algebra; geometry; measurement; and data analysis and probability (Lemlech, 2006).

The numbers and operations strand includes the number systems and how they are used. Strategies and techniques for computing with numbers are taught at this stage of learning, beginning with basic counting and advancing to activities that involve comparing numbers and sets. Fundamental addition and subtraction facts are also part of numbers and operations, and methods of computing are introduced and refined as well.

Children need to understand that, just as the letters of the alphabet represent parts of words, numbers represent ideas. When they have grasped this concept, they will be able to work with the counting process more readily (Lemlech, 2006). To work with young learners at this stage, teachers can instruct children to sort objects by shape and size, classify objects by their different characteristics, and fit objects inside of other objects. Children can also be taught to perform basic balancing activities (Lemlech, 2006). In their discussions with preschool children, teachers should also incorporate the use of small numbers. Instead of saying, for example, that there are chairs available, teachers can be more instructive by saying that four chairs are available. Inserting numbers across the curriculum will help children learn to attach meaning to them (Sarama & Clements, 2006).

As numbers become more a part of the curriculum, so should counting. Teachers can make counting part of the school day by inviting students to count small numbers that are part of their daily routine, like the number of doors they pass as they go out to the playground, or steps it takes to get to the front of the classroom. Later, they can instruct children to compare numbers. They can ask students to look at a pile of pencils and determine if there are enough for each child in the classroom. Children can also do a one-to-one match with items from two piles (e.g. plates and cups, pencils and paper) to figure out if there are enough of each group to form pairs (Sarama & Clements, 2006).

When students are under the age of six and still at a preoperational level of thinking, they often don't realize, for example, that despite the unfamiliar ordering of a specific set of numbers (e.g. <3.1.2>, <2,1,3>), the numbers themselves are still the same. Students will often have to count the numbers ordered in the original, left-to-right way and the new right-to-left way to discover that the numbers are the same even when listed both ways. As students develop, the concept of reversibility will begin to seem logical and automatic (Lemlech, 2006).

The study of algebra entails working with the language of variables. Important skills typically taught in the algebra strand are: performing operations within equations containing variables; working with functions; and manipulating symbols within equations. Even the youngest students can understand basic algebra. Number patterns and sentences using objects and manipulatives, for example, can help preschool students begin to think algebraically. Arranging blocks and objects in a simple pattern and inviting students to say which block would logically be placed next helps students begin to think algebraically (Lemlech, 2006).

In the geometry strand, students work with space and form to learn how these concepts are linked to numbers and math. Students are taught about figures, lines, points, lanes, polygons, geometric solids, and three-dimensional space. In geometry especially, manipulatives help students explore and discover; young students will likely grasp geometric concepts more clearly when links to real-life experiences are stressed (Lemlech, 2006).

Basic geometry concepts can also be introduced to young learners. Matching shapes is interesting and fun for preschool children, and putting shapes together within a puzzle is one way for children to learn how certain shapes can work together. Teachers can cut colorful basic shapes from construction paper and encourage the children to create pictures and then talk about what they have made (Sarama & Clements, 2006).

Since measurement is part of everyday life, it is a key strand in teaching mathematics. Within the measurement strand, students learn to gauge capacity, distance, and time as they are taught about units of measure, estimation, and the nature of measurement. Students should be encouraged to use an assortment of units of measure to understand the importance of using common and accepted units of measure. Estimation and approximation are also a part of this strand of math (Lemlech, 2006).

The data analysis and probability strand involves teaching the students how to plan and collect data, organize and infer conclusions from what they have collected, and share what they have learned. As with other mathematics strands, even very young children can gather and organize data. They can collect information about the color of leaves, how many birds are seen outdoors at certain times of the year, or how many hours of television people they know watch each day. Students can attempt to solve science, health, and social studies problems with the data they glean from themselves and their family members. In the process, they will reinforce counting techniques as they organize and interpret data (Lemlech, 2006).

Problem solving is an integral part of every strand of mathematics. When teaching math, instructors must be wary of introducing problem solving as simply another basic skill which can be solved in a step-by-step fashion. Many students work through problems without much thought, or by using a rule they believe the problem follows. If they aren't sure.

(The entire section is 3495 words.)

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Math/157

June-22-2014

Evan Schwartz

Reflective Paper

The course math for elementary teachers has taught me many concepts on how to work with students and to help them achieve success in the classroom. The objective of the course Math for Elementary teachers two is to assist the learner with understanding day to day applications of mathematics, and to give different ideas on how to differentiate learning. There were several ideas covered in this course but there are several of the major mathematical concepts that stand out to me. The National Council of Teachers of Mathematics principals and standards was a very big help in guiding me to understand what was required to teach specific grades based on the common core.

The major mathematical concepts that are in this course are, data analysis, probability, introduction to geometry, applications of geometry, application of measurement and mathematical connections. In data analysis we used appropriate statistical methods to analyze data such as detecting patterns, developing explanations and testing hypotheses. Teaching the children how to sort the data such as using tally marks for how many people like a certain object over something else. Bar graphs, pie graphs, line graphs and pictographs. Learning data is a very important concept that the students will need for the rest of the educational career.

Probability which is the measure of how likely something is was also a concept that was covered in the class. With probability you have the experiment which is the situation that involves the probability. The outcome is the result of the trial of the single experiment, the event is one or more outcomes of said experiment. You can teach probability using spinners, dice, and coins.

The concept of geometry in elementary school is closely involved with other topics in math like numbers and measurement. In the lower grade levels you are taught shapes and measuring angles. As you progress through.

*Teaching Tools and Learning Resources*

Introductory Algebra, by Lial, Hornsby, and McGinnis.

Lecture and tutorial notes; online resources.

*Regulations and Requirements of the Course*

Each student must fulfill the following requirements of the course:

(1) Take part in computer based activities on various topics by responding to a variety of questions and report to his/her tutor. Class participation is assessed.

(2) Use the learning tools and the Internet materials which are recommended by your tutor.

(3) Self-study (approximately 4 hours per week) includes searching for information, selecting relevant information and preparing PowerPoint presentations. Self-study also includes completion of all assignments given by tutors.

(4) Attend all classes regularly. If a student misses more than two classes without a valid reason she/he will not be allowed to continue the course without a special permission from the tutor.

(5) Attend all examinations and/or tests. Cheating at examinations is not tolerated and students who are caught cheating will be automatically considered as having failed themselves.

(6) Students are not allowed to walk around or chat loudly with each other during a lesson without a teacher’s permission. If a student ignores the teacher’s remarks, uses offensive words or offensive body language, insults other students and the teacher, behaves in an arrogant manner or in any other inappropriate manner, is late for classes, leaves the classroom early without the teacher’s permission, the student will be sanctioned or disciplined in accordance with college regulations, including expulsion from the College. Expulsion implies that the student should not consider the College for further education.

(7) Students are expected to comply with the college-wide requirements for academic integrity. The College is committed to academic integrity—the honest, fair, and continuing pursuit of knowledge, free from fraud or deception. This implies that students are expected to be responsible for their own work. Presenting another individual’s work as one’s own and receiving excessive help from another individual will qualify as a violation of academic integrity. Plagiarism is cheating. In this course, using another person’s words or ideas as your own without giving credit, producing a memorized piece (either your own or someone else’s), or having someone do any portion of your work is cheating. You are expected to complete your own, original work by using your own words.

(8) The Course Outline is the main document of the course and all the topics should be studied in class and at home every week. The duty of each student is to conduct Internet research, study the handouts, complete and submit all assignments on time and prepare for examinations and tests. The Course Outline is handed in to each student during the first week of the semester.

(9) Any additional questions or suggestions related to the above can be answered or discussed personally by your tutor either during office hours or by email. Each student who approaches their tutor by email must write their student ID number, name and the class number on the subject. Anonymous emails will not be replied.

Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001)

A guide to the teaching of learnercentred mathematics

B.K. Thekwane

The author of this Guide, Mr B.K. Thekwane, has published this article with the purpose of inviting comments for improving the draft from classroom teachers, facilitators, advisory teachers and other stakeholders. Your comments as an expert in the field will be highly welcomed. 1. Preface It is clear that we live in an ever increasingly scientific and technological world. Hence, changing demands from society and the continual development in the field of technology have led to a radical change in the aims and objectives of school mathematics since independence in Namibia. As a result of this, the Ministry of Basic Education has adopted a new approach to teaching school mathematics, namely the learner-centred teaching approach. In the new approach, greater emphasis is placed on: ✴ ✴ ✴ ✴ understanding, communication, problem solving and autonomy the ability to reflect on own methods and thinking creating a positive self-image and attitude among all learners accepting the responsibility for own work

In view of the complicated and challenging nature of the teaching, learning and application of Mathematics, the development of knowledge and skills should preferably take place within a flexible, safe and non-prescriptive environment. The teaching and learning environment should give learners the opportunity to give free expression to originality of thought and enable them to approach further studies and training purposefully and with dedication and self-confidence. 2. The purpose of the guide The purpose of this Guide is to provide Mathematics teachers with some ideas and guidelines as to how to teach Mathematics using a learner-centred approach. Teachers are encouraged to explore other approaches and should feel free to share them with other colleagues. This Guide will be updated from time to time as more ideas and.

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*Standards in Mathematics*

Standards in *mathematics* are crucial to the process of instruction as they improve methods of instruction in several ways. In any subject, standards help to regulate the things that the student learns in each classroom, limiting the content to what the student is required to know at that point in his or her education. Standards give instructors a proper framework to operate in terms of assessment, curriculum, evaluation, and professionalism. Standards called curriculum focal points give insight as to what are the most important topics that need to be covered at each level of mathematics instruction. The National Council of Teachers of Mathematics (NCTM) provide resources such as the handbook entitled “Principles and Standards for School Mathematics” (PSSM), which was published to help instructors create an effective program for learning. Principles covered in this handbook include equity, curriculum, teaching, learning, assessment, and technology. These principles are covered in depth to help understand how students’ learning and knowledge should grow throughout the education process.

*Traditional vs. Constructivist Instruction*

The traditional method of mathematics instruction refers to the most common form of instruction of mathematical concepts. The idea behind traditional mathematics, also known as classical math education, is to teach concepts using direct instruction and standard methods. Concepts are taught using isolated formulas, one by one. A text book that follows traditional mathematics methods would have, in print, the formula for the area of a triangle, which students could copy and apply to the specifics of a math problem. There have been many criticisms of traditional math programs, under the basis that this type of instruction gives too much emphasis on repetition and memorization while neglecting the inventive and imaginative aspects of mathematics. Tests have shown that students who experience traditional mathematical methods have a harder time with problem solving and conceptual understanding.

Constructivist mathematicians believe that creating situations where a student can develop his or her own constructions does not require lecturing, explaining, or transferring knowledge. This method gives the student the freedom of choice of how they want to process the information. The focus here is on determining how people learn and on engaging the learner in his or her own application of mathematical models. Critics of this method of instruction say that techniques such as learning by doing are not suitable for novice learners. Many believe that those with no prior knowledge to any mathematical concept cannot excel in unstructured constructivist teaching methods.

*Objective of Lesson Plan*

The objective of the lesson that was observed was to teach the students the equivalent forms of fractions and decimals by constructing models that would represent these fractions and decimals.

*National Council of Teachers of Mathematics Standards*

The NCTM curriculum standards were addressed and followed in many aspects. The students were expected to have prior instruction and knowledge of concepts such as percentage, decimals- one, tenths, hundredths, and proper fractions- tenths and hundredths. The instructor benefited greatly from having the confidence that the students already knew the basic principles of fractions, decimal, and percentages because time was not wasted on unnecessary repetition of these topics. Teaching and learning standards were exemplified as well. The PSSM states that while learning the basics is important, procedural facility and conceptual understanding is vital as well. The teacher of this class was able to choose a high-quality and hands-on method of instruction by utilizing professional expertise and intimate knowledge of the students. When it came to assessing the students’ understanding, the teacher used a personal and formative approach, evaluating and attending to each student’s level of understanding of the topic. The topic, numbers and operation with regards to understanding decimals and fractions, is a focal point for the fourth grade level, according to NCTM standards. The students in this class were aged 9-10 and the subject matter was appropriate in theory and in practice.

*Methods of Instruction*

The use of several instruction methods by the teacher were observed, including brainstorming, the Inductive-Deductive method, and the Learning by Doing method. In order to introduce the subject, the teacher asked the class to brainstorm about their ideas of what the Base 10 Models that were distributed to them represented. The ideas gathered by the instructor, written on the board, and were used to explain the idea of equivalence with words and relate it to the idea of equivalent numbers.

Aspects of the Inductive method were visible when the teacher asked the pupils to measure the amount of strips it took to cover one mat, in order to communicate the concept of tenths. The process was repeated when the purpose was to have the class show a representation of 27/100 or .27 using the Base 10 Models. Later on, the teacher used the board and an overhead projection to help the students formulate a rule based on their experiences and observations. The teacher utilized the ‘Learning by Doing’ method throughout the class by instructing the pupils to use Base 10 models. It was evident by the students’ participation that they felt interested in the simple experiment. The exercise had elements of the Deductive method also, as the students were encouraged to learn individually and not as a group.

*Instruction Differentiation*

The instruction was standard across the whole class and the students did not require much special treatment. However, the teacher provided extra assistance during the assessment phase to those students who were having difficulty making the models and understanding the concepts. Although the class was filled with students who have different learning styles and interests, the various methods of instruction, which included writing on the board, using the overhead, and self-administered exercise, provided a rainbow of activity. Therefore the students were able to consume the information in the way they felt most comfortable. If the pupils had varying abilities on a larger scale or suffered from some sort of handicaps, then instruction differentiation would have been necessary. For instance, if the students had physical handicaps, the instruction could therefore have involved audio and video programming including animations and text to teach the concepts and a less hands on approach to application of the topic.

*Use of Technology*

Technology was used in the instruction of the math lesson. The instructor used an overhead projector near the end of the exercise in order to solidify the principles being taught. The physical Base 10 Models had an overhead counterpart which was displayed on the overhead projector. The teacher used drawings of squares, lines, and smaller squares to represent the mats (one), strips (tenths), and units (hundredths). The students followed the projection by comparing it to the models they had made individually. They were able to identify resemblances to the work they had completed correctly and differences in the parts they had misunderstood.

Technology that wasn’t utilized was the use of an electronic presentation to provide a demonstration of the activity. If the pupils saw the physical model prepared and saw a short video clip of other pupils performing the same exercise they were about to perform, it may have instilled more self-confidence in the class and more enthusiasm for the activity. This could have been an effective way to explain the process of making a model before distributing the Base 10 Models.

*The Use of Concrete Objects and Manipulations*

The Base 10 Models are concrete objects which can be manipulated via organization into various shapes. They were the focus of the lesson and were used effectively. The students perceived them as fun playful objects and received the instructions well. However, some students started to play with the items before any instructions were given. The teacher distributed the models before giving instructions and it was harder for her to get the message across because of this.

*Assessment*

The assessment of the learning in the classroom was done using two assessment methods. During the class exercise, the instructor circled the classroom to track progress of each individual student. Anyone who required extra assistance in making the models of the different fractions and decimals that the teacher requested received help at that point. This allowed the teacher to understand whether or not the class understood the concepts as they proceeded through the exercise. The lesson plan also included an activity sheet that requires the students to draw models similar to the ones they made, which represent a corresponding fraction or decimal given to them within the activity sheet. After they draw the models, they must answer whether or not the numbers are equivalent to one another.

*Recommended Changes*

Overall, most of the pupils of the class were successful in achieving the objective. However, it was evident that some students did not attain the same level of understanding as others. Some students felt demotivated because they did not understand the instructions, so they passively played with the Base 10 Models instead of participating. The main reason for this was the distribution of the models before instruction was given. The class could have benefited from an introduction video and would have had more enthusiasm if the tasks were executed in a group. Moving from teacher-student assessment and individual work to peer-review and group work seems like a plausible modification. Furthermore, the instructor could still circle the class and help when needed, as well as assess the knowledge and comprehension using the activity sheet.

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Architects design buildings and other structures. They make sure buildings are functional, safe, and economical. They draw plans of every part of a building, including the plumbing and electrical systems. They also help choose a building site and decide what materials to use. Most architects today use computers in their work, and many are self-employed.

*Low End Salary:* $46,080/yr *Median Salary:* $76,100/yr *High End Salary:* $125,520/yr

There are three main steps in becoming an architect. First is the attainment of a professional degree. Second is work experience through an internship, and third is licensure through the passing of the Architect Registration Exam.

College Algebra, Trigonometry, Calculus I and II, Probability and Statistics, Linear Programming

Mathematics is used by architects to express the design images on a drawing that can then be used by construction workers to build that image for everyone to see. Mathematics is needed to analyze and calculate structural problems in order to engineer a solution that will assure that a structure will remain standing and stable. The sizes and shapes of the elements of a design are possible to describe because of mathematical principles such as the Pythagorean Theorem.

Approximately 7 out of 10 jobs are in the architectural, engineering, and related services industry—mostly in architectural firms with fewer than five workers. A small number work for residential and nonresidential building construction firms and for government agencies responsible for housing, community planning, or construction of government buildings, such as the U.S. Departments of Defense and Interior, and the General Services Administration. About 1 in 5 architects are self-employed. Employment of architects is projected to grow 24 percent from 2010 to 2020, faster than average for all occupations.

Architects serve in a variety of capacities in their practice. they are primarily responsible for creating a functional and appealing design for their clients. As the project proceeds to the construction phase, architects may work closely with contractors to execute their design, or they may leave the design-build function in the contractor's hands. In recent decades, architects have become to specialize in certain project types, such as retail, health care, and housing.