Introduction
In my seminar, “Cancer Biology and Technology”, we were provided with the current thinking of how cancer begins and what technology exists to diagnose and treat it. During our cancer exploration we learned the basic science and classification of cancers by cell type, tissue and organ origin, the difference between normal and cancer cells, how the cells change to become cancer cells, progress, and metastasize. Cancer is extremely complicated. Our studies included cancer treatment, the science of treatment resistance and the use of body’s immune system to assist in treatment and cancer therapy. Our studies included the human elements of diet, exercise, and heredity on contracting cancer as well as cancer’s progress through discovery, treatment, remission, and reemergence from dormancy sometimes after hiding for years.
Problem Statement
I teach high school math at The Philadelphia High School for Girls, a special admission, magnet, high school with a rich history of academic excellence. Founded in 1848 to “prepare teachers for the common schools of Philadelphia,” Girls’ High, as it is affectionately known, was the first municipally supported secondary school for girls in the United States and was called the Girls’ Normal School. In 1893, the Philadelphia High School for Girls separated from the Girls’ Normal School, and the foundation for today’s college preparatory curriculum was laid. The school continues its legacy as a school for academically talented girls, providing young women with outstanding opportunities for scholarship, leadership, and service. Its motto, “Vincit qui se vincit” (He conquers who conquers himself), is a key centering point for our students maturing into young woman of purpose and honor (Cutler, 2013).
Girls’ High students represent the variety of schools in the City of Philadelphia having applied and gained admission from diverse neighborhoods and middle schools of every type (public, private, parochial, magnet and neighborhood) from all over the city. They are as ethnically diverse as the City of Philadelphia having students from African American, White, Latinx, Asian, Native American, and Mixed-Race households. Our diverse population while representative of Philadelphia neighborhoods across Philadelphia has become increasingly dominated by black and brown students of multiple economic levels, cultures, and ethnicities. Their demographics are also consistent with the poverty recorded for the City of Philadelphia and the School District of Philadelphia.
The Pew Charitable Trusts 2018 Report, “Philadelphia’s Poor: Experiences from Below the Poverty Line” indicates that Philadelphia is home to nearly 400,000 people living below the poverty line. These statistics are corroborated in the Princeton University report which says, “Poverty effects a quarter of Philadelphians, with the burden falling heaviest on the city’s minorities (particularly Hispanics) and children at 39% and 38% impoverished, respectively.
To be living in poverty means supporting a family of four on an income of $24,300. Raising a family in a city, below the level of poverty is difficult, but for 12.2% of Philadelphians these difficulties are compounded below the deep poverty line supporting a family of four on an income of less than $12,1502 . This means that 40% of the city’s poor, are living in deep poverty. To further put these statistics into perspective, the poverty rate for the United States is 15.8%, a rate Philadelphia has never approached in the past decade .
The 2020-2021 Girls’ High Economically Disadvantaged (CEP Rate)of 98% represents an 8% increase compared to a year ago. This measure is calculated by taking the percentage of students identified as economically disadvantages, multiplying by a USDA defined CEP factor of 1.6 and capped at 100%
The student population at Girls’ High as described by demographics, shares many of the challenges of the city overall. Students have excelled academically which makes them different than many Philadelphia students living in poverty.
| Girls’ High School Demographics |
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| Girls’ High Economically Disadvantaged
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Percent of Students with IEP’s |
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Source: School Profiles, The School District of Philadelphia, https://dashboards.philasd.org/extensions/enrollment-public/index.html#/demographics
The prevalence of English language learner has also increased at Girls’ High. The number of ELL students while unchanged between 2020-2021 and 2019-2020, has doubled since 2018- 2019.
Academic Rationale
Girls’ High students are a group of highly able learners who bring energy and the thirst for learning when they interact with each other and question both the content and relevance of the work they are given. Even with their strong educational background, my students struggle to apply and extend the mathematical concepts to real life word problems. According to Christopher Masullo in the December 6, 2017 Education Week article, “What’s the Problem with Word Problems?”, “Word problems tend to be complicated in part because of their descriptive language. Students often do not understand what exactly they are being asked, especially when the problem includes abstract concepts. Other issues arise when students lack the fundamentals of math and cannot formulate a plan for solving or separating an equation’s steps.”.
While strong math students in previous settings, across all grades, my students’ math skills are often rudimentary and reflect rote learning. Their math knowledge consists of algorithms and formulas of which they have little practical knowledge, and they do not know how to apply their math skills to real world applications. My students read the problem and focus narrowly on the question. They focus on remembering “the right way,” the “formula” or “algorithm,” or “the prior lesson” required to solve the problem. While my students have solid math skills and a strong grasp of math fundamentals, they lack the ability to apply these skills to new math content and to solve word problems. They fail to learn deeply with an ability to apply the recent skill later when it is applied to a more complex problem.
Tracing mathematical roots back to the earlier grade levels, teachers often use key words often help younger students create a problem-solving plan. The word “more” is taught as a clue for when to add and similarly, the word “fewer” indicates subtraction. The difficulties with word problem reoccur when there is a lack of key words to follow. In more advanced problems, each problem is unique and a single, overarching algorithm to solve them does not exist. (Masullo, 2017)
To engage my students so that they can apply math concepts to real world problems, I will use cancer as a background since most if not all my students have encountered cancer in their lives. Whether relatives, friends, neighbors, or celebrities their lives much like mine have been touched by cancer. My curriculum unit will focus on breast cancer. Following a description of the general development of cancer, this curriculum unit will hone into breast cancer its prevention, causes, development, treatment, prevalence, and advocacy in the African American community. Girls’ High student body is 78% black and brown and overall has a 96% non-white population.
My curriculum unit will focus on breast cancer and the disease among African American women. I am focusing on breast cancer because it has the highest incidence of cancer in PA and over the last 5 years is the 2nd highest cause of cancer-related death. Since 2007, the number of women aged 50 and over who have died of breast cancer has continued to decrease. The number of women under age 50 who have died of breast cancer has stayed steady. (Cancer.net, 2021) I want to both educate about the disease and its prevalence as well as highlight opportunities for advocacy, education and self-awareness or actualization as tools to help in their personal journey. Topics to be covered include how the disease begins and grows and why it is so prevalent and deadly. We will also discuss preventative screening approaches that allow women to screen for breast cancer using monthly self-breast exams. This will offer them some control over their health and bodies. I plan to talk about methods for testing for breast cancer. I understand from my own experience that effective diagnosis even with mammography within communities of color is problematic due to characteristics such as dense breasts. African American women die from breast cancer including an extremely aggressive strain, at higher rates. While there are some limits to mammography there is good evidence that across all populations that screening helps reduce mortality. Recent technology advances such a digital breast tomosyntheses and MRI help in screening women with dense breasts and /or higher risk populations.
According to the Breast Cancer Prevention Partners, an organization dedicated to reducing the risk of developing breast cancer, African American women face both disproportionate exposure to breast carcinogens and the highest risk of serious health impacts from the disease. Some products marketed to Black women, some used for generations and many popular with black women of all ages, contain ingredients known to contribute to developing breast cancer. Examples include skin lighteners, hair relaxers, Brazilian blowout treatments and acrylic nails, to name a few.
My curriculum unit will provide opportunities for students to experience the math content with real life data. My plan is to provide historical are well as local data that students can analyze while learning about the disease. I also hope to communicate when do they start? What are the trends and why it is important to young people?
Content Narrative
The content narrative will summarize the basic science and classification of cancers by cell type, tissue, and organ origin to provide a background on the biology of how cancer develops. This will include the difference between normal and cancer cells, how the cells change to become cancer cells, how they progress, and metastasize. The next section will discuss the diagnosis and treatment of cancer. Finally, the human elements of diet, exercise, and heredity on contracting cancer as well as cancer’s progress through discovery, treatment, remission, and reemergence from dormancy sometimes after hiding for years.
Introduction to Cancer Biology
To understand cancer, the first question to answer is, what is cancer? Cancer is a disease of ageing. While young children, even infants can develop cancer, their cancers are rare and often more treatable than cancers of the adult population. The median age of a cancer diagnosis is 66 years old. Women receive diagnoses at earlier ages than men but the male rate of diagnosis increases rapidly and surpasses that of women in the 55–64-year-old range.
Normal cells, shown below as the “normal epithelium”, exist for decades before becoming abnormal in the dysplasia/adenoma phase. These abnormal cells can remain in this form or can mutate and become carcinoma cells. When cancer progresses, abnormal cells progress to become cancer cells. On average, cancer is diagnosed later in life because it develops over time and is only discovered once the cells have grown to a point where they can be detected.
Cancer develops in stages. First a cell mutates generally without any knowledge of the person. The first cancer stage, hyperplasia, occurs when a cell begins to grow too much. At this stage, the cells still appear to be normal. The next phase, dysplasia, is the first stage where cells grow rapidly and appear to be abnormal. The cancer cells are present in an organ and as they grow to the in-situ cancer stage the tumor continues to grow within the organ. Finally, as more cancer cells grow, the tumor cells invade blood vessels and the malignant cells continue to grow.
Mutations in two classes of genes are responsible for beginning the cells’ change to cancer: oncogenes and tumor suppressor genes. This change happens between to normal epithelium and dysplasia/adenoma stages.
Oncogenes are mutated genes that contribute to cancer development. In their normal, unmutated state, oncogenes are called proto-oncogenes, and they regulate normal cell division and cell death When a proto-oncogene becomes mutated, the cell makes extra proto-oncogene copies leading to uncontrolled cell division and tumors. (National genome research)
Tumor suppressor normal genes that control cell growth and division. They repair DNA mistakes or tell cells when to die (a process known as apoptosis or programmed cell death). When tumor suppressor genes do not work properly, cells can grow out of control, which can lead to cancer. (American Cancer society).
The next question is what causes cells to mutate? There are two caused of gene mutation environment and genetic inheritance (BRCA12). Environmental factors are ultraviolot light from sunlight, radiation and carcinogens such are found in tobacco asbestos, nickel, cadmium, radon, vinyl chloride, benzidene, and benzene. Gene mutations caused by genetic inheritance (BRCA12) are specific to inherited DNA and may or may not develop.
As the tumor cells invade blood vessels and the malignant cells continue to grow the cells leave the organ and travel to other parts of the body via the blood or lymphatic system. People who die from cancer generally die from metastatic disease.
What Causes Cancer?
Many milestones in have been reached in the study of cancer.
The diagrams above show the current understanding regarding the progression of cancer. In the early 200’s after the cells began uncontrolled growth the prevailing thought was that tumor cells proliferated evading growth suppressor cells, evaded adjacent cells and metastasised. The cells continues to grow in new locations forming new blood vessels and continuing to grow and proliferate evading cell death signals.(left hand side) The diagram on the right-hand side is similar showing advances in the details of each of the previous steps from the left. The circa 2011 diagram identifies the cells the cancer evades, that would normally cause death, as T and B cells, the mechanisms for cell propagation and metastasis and more clearly defined and the molecular cell characteristics identify effects on the genome and cancer cell’s ability to reprogram normal cellular metabolism for growth.
Theories on how cancer develops are being studies around the world. The Alfred Knudson tow hit theory of cancer theorizes that a norml cell is hit twice before mutation. This represents another advance which provides more information to scientists studying caner and developing treatments.
How is Cancer Diagnosed and Treated?
Cancer Diagnosis
If a patients’ symptoms suggest cancer is present, tests are available many which are specific to different types of cancer. Three types of tests are discussed here: lab tests, imaging tests and biopsies.
Lab tests of bodily substances can be used to diagnose cancer. Testing of blood or tissue samples are looking for tumor markers, substances produced by cancer cells in response to the presence of cancer.
Imaging tests create pictures of areas inside the body and can include CT scans, MRI, nuclear scan, bone scan, PET scan, mammogram, ultrasound, and x-rays depending on the indicated type of cancer . CT scans use x-ryas to take pictures of organs and create a 3-D image. A dye or contrast material is frequently used for highlight areas in the study. MRI tests use powerful magnets and radio waves to take pictures of the body in slices. The slices are used to create detailed images of the body. Nuclear scans use radioactive material injected into the body to create pictures of bones or organs. Bone scans are specialized nuclear scans specific to bone cancer. (Berg, 2012) PET scan is another type of nuclear scan that uses radioactive glucose to look for glucose build up in the body. Cancer cells absorb more glucose than normal cells. Mammogram is used to find cancer in the breast tissue. , Ultrasound uses high-energy sound waves which echo off bodily tissue. A computer used the echoes to create sonograms, pictures of the area studied. A transducer covered with a warm gel moves on the skin over the part of the body that is being examined. X-rays create pictures of the body using low levels of radiation.
A surgical procedure called a biopsy can also be used to diagnose cancer. A doctor removes a sample of tissue using a needle, endoscopy, or surgery. (National cancer institute)
There are multiple types of biopsies. A standard biopsy samples tissue in an invasive surgical procedure where tissue is collected under anesthesia . A liquid biopsy is a minimally invasive surgical procedure where tissues are drawn into a liquid for a quick comprehensive tissue profile.
Cancer Treatments
There are four cancer treatment methods: surgery, chemotherapy, targeted therapy, and radiation. Surgery removes the tumors by resecting primary tumors. Chemotherapy kills rapidly dividing cells. Targeted therapy blocks a specific protein of pathway that identified as key to cancer growth. Radiation causes damage to the tumor’s DNA and triggers the cancer cells to die.
Health Factors Related to Cancer
The American Cancer Society (ACS) in their Guideline for Diet and Physical Activity for Cancer Prevention outlines their cancer prevention recommendations. The first and most important health risk factor is tobacco use. Tobacco is followed by body weight, diet, and physical activity.
Whether smoking cigarettes or using smokeless products, tobacco can cause cancer and then block your body from fighting it:(CDC). Poisons in tobacco weakens the body’s immune system which makes it harder to kill cancer cells. The body is then unable to stop the growth of cancer cells. Tobacco can also damage DNA and the natural cell coding that manages cell growth and duplication.
For people who do not use tobacco, changes are body weight, diet, and physical activity can help mitigate cancer development. At least 18% of all cancers diagnosed in the US could be prevented by reducing excess body weight, increasing physical activity, limiting alcohol consumption, and improving nutrition. Avoiding these risk factors throughout life may greatly reduce a persons’ lifetime risk of developing or dying from cancer. These actions are affected by the social, physical, economic, and regulatory environments where a person lives.
Curriculum Unit Background
Breast Anatomy
Breast anatomy involves an arrangement of ducts, lobes, ligaments, and muscle. Each breast contains 15 to 20 lobes of glandular tissue arranged in a circle pattern like the petals of a daisy. The lobes are further divided into smaller lobules that produce milk for breastfeeding. Small ducts that look like tubes move the milk to a reservoir beneath the nipple. The nipple is in the middle of the areola, which is the darker area surrounding the nipple.
The breast exchanges fluid in the lymph nodes. Lymph nodes are small, bean-shaped organs that help fight infection and are found throughout the body. They produce and filter a colorless fluid called lymph, which contains white blood cells known as lymphocytes (immune cells involved in defending against infections and such diseases as cancer). Lymph vessels filter and carry lymph fluid from the breast to the lymph nodes. Clusters of lymph nodes near the breast are in the armpit (known as axillary lymph nodes), above the collarbone, in the neck, and in the chest. Breast cancers can form in the ducts and the lobes.
(Memorial Sloan Kettering Cancer Center https://www.mskcc.org/cancer-care/types/breast/anatomy-breast)
Breast Cancer
Breast cancer starts when cells in the breast (such as cells lining the ducts and lobules) begin to grow abnormally. The abnormal cells could be in the lining of the milk ducts and/or lobules. These abnormal cells if mutated to cancer cells can grow out of control and invade surrounding cells. If it continues to grow it can spread beyond the breast to other areas of the body and become life threatening.
The figure shows a breast cancer tumor. During the development of cancer, the normal balance between cell division and cell loss is disrupted. The malignant cells divide far faster than new cells are needed. Since each division of a malignant cell results in 2 newly formed cells that retain their capacity to divide, there is an overall increase in the total number of dividing cells. (Winchester Hospital.com Breast cancer can invade through nearby tissue or spread through the body via the lymphatic system and blood.
Breast Cancer in Pennsylvania
Female breast cancer along with lung and bronchus and prostate cancer, are two of the three most commonly diagnosed invasive cancers in Pennsylvania. In 2011, nearly 10,600 women were diagnosed with invasive breast cancer. Detection and prevention have played a critical roll in stabilizing breast cancer in Pennsylvania. Here are some statewide data from the Bureau of Health Promotion & Risk Reduction report, The Burden of Cancer in Pennsylvania: Calculating Costs Understanding Impacts Exploring Interventions, which describes breast cancer in Pennsylvania.
The percentage of women in Pennsylvania who had a mammogram in the last two years was slightly lower than the national average in 2010 after m meeting or exceeding national trends prior to the year 2000 when it reached its peak at 77.8 percent. . Overall, the rate of mammography for 2010 is 73.9 percent for Pennsylvania and 75.2 percent for the United States.
The chart on the left shows that compared to the national average, Black women 40 years and over had the highest rate of mammography in 2012. Their rate of testing was 80.5 percent for Pennsylvania and 78.3 percent nationally. Multiracial women showed the lowest screening rates at 49.6 percent for Pennsylvania and 60.1 percent nationally.
The report on mammograms by eduction and income show that in Pennsylvania, the lowest rate of mammogram screening was found in women with lower incomes and less than a high school education at 63.8 percent, and those making less than $15,000, at 65.0 percent. These rates are slightly higher than the national rates where the rate for women with the lowest level of education was 62.7 percent and 60.4 percent for those with the least amount of annual income.
The map on the right shows the percentage of Pennsylvania women, 40 years and older, who had mammograms in the past year. Philadelphia women received mammograms during 2012 at a higher rate, 64%, compared to the statewide average 60 percent. Philadelphia County had the highest percentage of mammograms in the state.
Over the last 10 years, female breast cancer incidence rates for all women have remained steady without dramatic changes. However, until 2011, the black females breast cancer incidence rates remain higher than any other ethnicity.
Breast Cancer in Philadelphia
This section will provide an overview of breast cancer in Philadelphia: how the disease develops and grows and the technology available for its detection. The data source is, “Cancer and Cancer Health Disparities in Philadelphia – Urban Health Collaborative.” Drexel Urban Health Collaborative, 2020, The next, section provides personal preventative screening approaches that allow women to use monthly self-breast exams to begin a practice offering some control over their health and bodies.
The most common type of cancer deaths among women in Philadelphia in 2016 were from lung and bronchus disease. The second highest cause of death was breast cancer.
While the most common cancer death was from lung and bronchus disease, breast cancer had the highest number of new cases. By ethnicity, new cases among black women per 100,000 was almost equivalent to white women.
A summary of 2016 deaths by ethnicity show significantly more black women died from both all cancers and breast cancer than white or Hispanic women per 100,000 people.
Lowering the Risk of Developing Breast Cancer
What dos the term risk factor mean? A risk factor is “an activity or action that increases your chances of getting a disease, such as breast cancer” (American Cancer society). Having a risk factor, or even many risk factors, does not indicate you are certain to develop the disease. The risk factor categories related to developing breast cancer are similar to those outlined in the general cancer section: drinking alcohol, being overweight or obese, tobacco use and lack of physical activity. Emphasis is placed on some key concerns in each category.
Drinking Alcohol
Alcohol use is one of the most important preventable risk factors for cancer in general breast cancer specifically. Alcohol use accounts for about 6% of all cancers and 4% of all cancer deaths in the United States. Alcohol use has been linked with cancers of the mouth, throat, voice box, esophagus, liver, colon and rectum and breast. For each of these cancers, the more alcohol you drink, the higher your cancer risk.
Drinking alcohol is clearly linked to an increased risk of breast cancer. Even small amounts of alcohol consumption can increase breast cancer risk and the risk increases with the amount of alcohol consumed. Women who consume one alcoholic drink a day have a small (about 7% to 10%) increase in risk compared with non-drinkers, while women who have 2 to 3 drinks a day have about a 20% higher risk than non-drinkers. TO mitigate against developing breast cancer, women should not drink alcohol. For women who do drink, they should have no more than 1 drink a day.
Healthy Weight and Regular Exercise
The American Cancer Society recommends women stay at a healthy weight throughout their life and avoid excess weight gain by balancing your food intake with physical activity. Exactly how physical activity might reduce breast cancer risk has not been conclusive, but it is recommended that adults get 150 to 300 minutes of moderate intensity or 75 to 150 minutes of vigorous intensity activity each week (or a combination of these). Getting to or going over the upper limit of 300 minutes is ideal. It is important to adapt a healthy weight and regular exercise routine early in life so that it is a natural part of one’s lifestyle. It is much harder to change life-style practices later in life. (CA, 2020
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Exponential Growth and Disease
Cell division in a human cell is well documented in mitosis. One cell divides to create two new cells. This is the normal cell propagation process that repeats itself in abnormal cells as well. Each division of an abnormal cell whether bacterial or malignant also results in two newly formed cells and each cell retains its capacity to divide on its own. In each case, there is an overall increase from one cell to two after the first cell divides.
The mathematical category for numbers increasing in this manner is an exponential model. Students have been introduced to exponential equations in algebra 1. Exponential functions will be modeled using cancer, bacteria, and communicable diseases such as Covid-19.
