PATHFINDER SKILLS LIST SENIOR PROJECT REPORT MARCH 2020 - MAY 2020Â BY: Madalena Kombo & Rayane Taroua
Senior Project Report March -May 2020 Rayane Taroua Madalena Kombo
Table of Contents Project Purpose ……………………………………………………………………………..Page 1 Summary of Interviews …………….……………………………………………………….Page 2 Research Methods/How We Formulated our Lists………………………………................Page 3 How to Use this List………………………………………………………………………..Page 3 Sixth Grade Skill List……………………………………………………………………….Page 4 Seventh Grade Skill List…………………………………………………………………...Page 7 Eighth Grade Skill List……………………………………………………...……………..Page 10 Our Recommendation for the Future………………………………………………..……..Page 13
Kombo & Taroua 1 Project Purpose We felt compelled to begin this project because as Pathfinder teachers, we noticed that there were many disparities in skills within the group of nine students that we taught. We noticed that a couple kids were very well versed in the previous concepts they had been taught, and there were also a couple students on the opposite side of that spectrum, understanding very little of the material. Because of this, we found it hard in our classrooms to navigate between who was up to speed on the skills and who was not. The following is an anecdote based on Rayane Taroua’s tutoring experience: Last year, I worked one-on-one with a student trying to catch her up to her classmates. While teaching the Pythagorean Theorem to the small group, I realized that this student didn’t understand the principle of squaring a number. I further found that solving simple one step equations with unknown variables such as x was difficult. Through trial and error, I was able to reach an appropriate level for this student. Through this experience, I realized that there has to be a better way for me to support a student wherever they are on their math journey. But to help them, I have to first really understand their current level. With this skills list, teachers will be encouraged to consistently track students to better understand how much material they have actually grasped. The next teacher that has to teach this student will know exactly where to start. This skills list was also a way for teachers to actively recognize that not all of their students are necessarily at the math level that their math class teaches. We hope that the skill list will help teachers identify what students need one-on-one work to help them build the skills that they need to go to an enriched geometry program in ninth grade.
Kombo & Taroua 2 Summary of Interviews We began this project interviewing Dante Brito and Jennifer Lane to get a sense of what the current Pathfinder math curriculum is. Overall we wanted to identify the purpose of said curriculum. The following is a summary of our interviews broken down by question asked. ● What is the overall purpose of math at pathfinder? Provide enrichment through additional exposure and increase mathematical competence. ○ Ensure that the students will enter the honors level track in New Haven Public School system ○ Want all students to be able to enter geometry freshman year at private schools ● What do you want the students to gain by the end of their journey? ○ Want students to have confidence and develop a continuous love for math. We want them to look at a problem and not be intimidated by the challenge it presents. ○ The goal is to make math more accessible to students. By emphasizing small groups and interactive lessons, Pathfinder is less like school. We want the students to know they are cared about. By having younger tutors students are more inclined to learn because stronger relationships are created. ● What are the skills that they are currently teaching? ○ In the 6th grade there is an emphasis on manipulating fractions and gaining number sense. Rising 7th graders learn pre-algebra over the summer. In the 7th grade there is a shift to heavy SSAT preparation. Rising 8th grade summer and 8th grade spring is all about solidifying algebraic concepts. Rising 9th graders receive an introduction to geometry over the summer. ● How are teachers currently tracking student skills? ○ There is an entrance exam that students take in the 5th grade. It is mostly based on the SSAT and is out of 300 points. Most students score within a 50-100 range. ○ At the end of every spring, students have a benchmark test for placement of their summer classes. ○ Over the summer there are weekly meetings with head teachers looking at students' ability to comprehend the material. ○ SSAT Preparations: Select 6th grade students take the SSAT in September. All students take a SSAT practice test during their 7th grade spring. All rising 8th graders receive test prep over the summer. Select 8th graders take the actual SSAT in September, paid for by Pathfinder.
Kombo & Taroua 3 ● Where do you want Pathfinder to go? ○ Want to foster a sense of personal achievement for the students. We want students to be able to follow their progression possibly through a ‘report card.’ ○ Want the students in the “middle percentiles” to push themselves to become high achieving students. Research Methods/How We Formulated our Lists To create each skills list we used multiple resources. Our main research question was; what are the skills and concepts necessary for students to be successful on the SSAT, Pre-Algebra, Algebra I, and Geometry? We began by collecting lesson plans, projects, topic lists, and documents currently being used in Pathfinder. We met with Dr. Hartog to collect 6th grade spring information. We received documents from Rachel Hudelson (R6 Math teacher) and Richard Cordaway (R7 Math teacher). We also used information from the lesson plans we currently use as 7th and 8th grade spring tutors Mr. Brito and Ms. Chavez. When we understood what is currently being taught, we looked at New Haven Public School official curriculum maps for 6th through 8th grade. Next, we leaned heavily on current common core standards to provide the bulk of our list. We relied on Common Core because it is the current standard for public schools across the country. We determined which topics to bold based on these standards. Finally we cross checked our list with Khan Academy to ensure we were not missing any topics. To summarize: we used current Pathfinder lesson plans, N.H.P.S. Curriculum Maps, Common Core Standards, and Khan Academy to create our skills lists. How to Use this List The skills list that we have created is a working document based on our research. We highly recommend Pathfinder teachers edit this list before it can be officially used. Our hope is that a program wide spread sheet can be made, similar to the one made for R6 students during the summer of 2019. Every student will have their name on this list. Each column will be a separate skill that corresponds with our list. The skills should be listed in order from rising 6th grade to rising 8th grade. Teachers can then place a 1 or a 0 based on if the student has a strong grasp on said topic. For this project, we have just created a working skills list. This next step is to think about how to teach these skills and how to assess each topic. How can a teacher assess if a student has successfully mastered a topic, what warrants a 1. The purpose of this skills tracker is to be able to provide individualized learning to students and easily identify gaps. All Pathfinder teachers should have access to this list and consistently update it. With an updated tracker, students can have seamless transitions from Spring to Summer sessions and vise-versa because every new teacher/tutor they have will automatically know their skill level.
Kombo & Taroua 4 Sixth Grade Skill List: Note: -
Bold is for mandatory skills every student should know. Non-bolded skills are not as pertinent but still good for students to learn. For more information on how we chose
bolded skills see Research Methods. - * Indicates skills that are for the advanced group ● Ratios/ Fractions ○ Define a ratio conceptually ■ Relate to ratios to proportions ○ Ratios in proportional relationships ■ Examples from Common Core ● The ratio of wings to beaks in the birdhouse at the zoo was 2:1 because for every 2 wings there was 1 beak. ● For every vote candidate A received, candidate C received nearly three votes ○ Equivalent ratios ■ How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? ○ Part to part ratios ○ Part to whole ratios ○ Operations with fractions ■ Multiplication ■ Division ■ Addition ■ Subtraction ○ Converting ■ Mixed ↔ improper fractions ■ Decimal ↔ fractions ↔ percents ○ Percents and proportions ■ Find the whole given a part and the percent ○ Unit conversions ■ Understand/use a conversion factor ● Know and when to use it ■ Examples from Common Core ● This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. ● We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.
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If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? ○ Prime factorizations ■ Understand how to use factor trees ■ Greatest Common Factor ■ Least Common Multiple ● Decimals ○ Operations with decimals ■ Multiplication ■ Division ■ Addition ■ Subtraction ○ Students should be able to develop the concept of decimal representations of fractions using base 10 block ■ Students should be aware of how similar fractions are to decimals ○ Students should be able to complete problems using fractions in a decimal ● Expressions + Equations ○ Order of Operations (P.E.M.D.A.S./G.E.M.D.A.S.) ○ Understanding the use of variables ○ Using /evaluating/ writing algebraic expressions ■ Can set up equations based on word problems ○ Equivalent algebraic expressions → distributive property ■ Examples from Common Core ● 3 (2 + x) is equivalent to 6 + 3x ● y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for ○ Simplifying algebraic expressions → combining like terms ○ Solving simple one-step algebraic expressions ■ Adding/Subtracting/Multiplying/Dividing ○ Showing relationships ■ Dependent and independent variables ■ y=x+30 ○ Number lines ■ Understanding negative numbers ● Real-world examples from Common Core ○ Temperature, sea level, credit/debt, electric charge ■ Fractions on number lines ■ Introduction to inequalities with variables
Kombo & Taroua 6 ● For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. ■ Operations with Integers ● Multiplication ● Division ● Addition ● Subtraction ○ Location of numbers on coordinate plans ○ Introduction of mean and median to refer to measures of center ● Geometric Concepts ○ Classifying triangles ○ Triangle/polygon angle sums ○ Area of a triangle ○ Perimeter and area of a rectangle ○ Area of parallelograms ○ Classifying quadrilaterals ○ Area and volume of complex shapes by breaking them down
Kombo & Taroua 7 Seventh Grade Skill List: Note: - Bold is for mandatory skills every student should know. For more information on how we chose bolded skills see Research Methods. - * Indicates skills that are for the advanced group ● Basics, Fractions, Decimals, and Percents ○ PEMDAS ○ Convert a fraction to a percent ○ Convert a decimal to a percent ○ Operations with Integers ■ Addition ■ Subtraction ■ Multiplication ■ Division ○ Rational numbers ■ Operations using rational numbers ● Multiplication ● Division ● Addition ● Subtraction ■ Word Problems ● Multi-Step Equations and Word Problems ○ One- Step equations ○ Two-step equations ○ Solve two-step equations by working backward ○ Turn phrases into algebraic expressions or equations ○ Calculation of mean, median, and mode ○ Using substitution ○ Inequalities ■ Graphing ■ Multi-step algebraic equations ■ Modeling and writing Two- Step Inequalities ● Linear Equations/ Graphs ○ Plot ordered pairs ○ Slope ○ Convert table of values to a graph Proportions ○ Define proportions ○ Solving proportions
Kombo & Taroua 8 ○ Solving rate problems ■ With fractions ○ Measurement conversions ○ Equations using proportions ○ Calculate the percent change ● Exponents ○ Perfect Square ○ Operations ■ Multiplication ■ Division ■ Addition ■ Subtraction ○ Scientific Notation ■ Use to express smaller numbers ■ Use to express large numbers ● Percents ○ Words problems ○ Sales tax ○ Tips ○ Commission ● Factoring/ Introduction to Geometry ○ Mapping ■ Scale Drawings ● Create ● Use ■ Using Maps ○ Angles ■ Supplementary ■ Complementary ■ Vertical ○ Circles ■ Circumference ■ Area ○ Volume ○ Right Triangles* ■ Pythagorean Theorem ○ Quadratic equations* ○ Zero Product Property* ○ Factor by grouping*
Kombo & Taroua 9 ○ Factor an expression with a leading coefficient that is greater than 1* ● Statistics and Basic Probability ○ Basic probability ○ Comparing and sampling populations ○ Probability models ○ Compound events and sample spaces
Kombo & Taroua 10 Eighth Grade Skill List: Note: - Bold is for mandatory skills every student should know. For more information on how we chose bolded skills see Research Methods. - * Indicates skills that are for the advanced group ● Number Sense ○ Rational/Irrational Numbers ■ Estimating irrational numbers ● Understand irrational number in decimal form ■ Comparing rational and irrational numbers ■ Ex: the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. ○ Exponent/Radicals ■ Exponent properties ● Multiplying ● Dividing exponents ● Negative exponents ● Powers of zero ● Powers of powers ■ Square roots and cube roots ● Area of squares ● Volume of cubes ■ Scientific notations and Powers of 10 ● Multiplying /Dividing scientific notation ● Ex: estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger ● Expressions/Equations → Investigating patterns ○ Solve linear equations - use both models and algebra ■ Derive y = mx+b conceptually based on connections with proportional relationships ○ Systems of linear equations - graphing and algebraically ■ Solve real-world problems ○ Solving inequalities and absolute value equations * ● Functions ○ Definition of functions ■ Identifying functions
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■ Graphing functions ■ Representing functions ■ Analyzing functions ■ Constructing functions ■ Mapping inputs and outputs ■ Finding rates of change conceptually ○ Linear Functions ■ Rate of change ■ Representing linear relationships ○ Slope ■ Linear equations and slope ■ Identifying slope using tables and graphs ■ Comparing slopes of different lines ■ Modeling linear relationships using slope-intercept form Angle pair relationships ○ Complementary and Supplementary angles ○ Angles formed by parallel lines ○ Angles in a triangle ○ Exterior angles and angles in quadrilaterals Geometry ○ Properties and conditions of similar triangles and overall congruence ○ Stretching and shrinking geometric figures ■ dilations, translations, rotations, and reflections on two-dimensional figures using coordinates ○ Effects of scale factor ○ Surface area - prisms/cylinders ○ Volume - prisms/pyramids/cylinders/cones/spheres Pythagorean Theorem ○ Finding the missing length of triangles ○ Finding the distance of two points on the coordinate plane Exploring probability/statistics ○ Mean ○ Estimating probability ○ Scatter plots ■ Creating ■ Interpreting ○ Understanding patterns ■ Clustering ■ Outliers
Kombo & Taroua 12 ■ Positive association ■ Negative association ■ Linear association ■ Nonlinear association. ○ The implication of the line of best fit ○ Understanding slope-intercept form ■ Ex: in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. ○ Identify categorical data → frequency on two-way tables * ■ Ex: collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? ● Polynomials ○ Operations with polynomials ■ Adding polynomials ■ Subtracting polynomials ■ Multiplying polynomials ■ Dividing polynomials ● Quadratic function * ○ What is a quadratic function * ○ Graphing quadratic functions *
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Our Recommendation for the Future ● Teaching Methods ○ Further research into how to effectively teach these topics are the National Teachers Council of Mathematics and Professor Jo Bolar. Professor Bolar encourages alternative teaching methods that center around problem solving and reasoning around skills as opposed to passively practicing methods. This method of teaching breeds the enjoyment of math along with further retention and less anxiety for students. Both of these recommendations were given to use by current math teachers. ○ An important shift is to allow students to develop a conceptual understanding of a topic before the skill. This will help students retain concepts and gain confidence in their overall math abilities. This can be very difficult for Spring session tutors given we are not actual teachers and have usually only learned skills not conceptual understandings. ● How to Assess Students ○ In terms of assessment, improvements can be made to the placement test that students take at the end of each spring session. These placement tests can be redesigned to assess each student's current knowledge and mastery. These tests can also examine retention of previous topics. For example, the 7th grade test can have 6th grade topics to see what students still remember. Spring Session ● How to make Spring sessions most effective? ○ Based on our limited experience, we found that teaching new content can be challenging because we only see the students once a week for forty-five minutes. Given the amazing tutor to student ratio of about 1:3, the spring session is a great time to apply individualized learning and fill the gaps which can clearly be identified using the tracker. ● How does one make the Spring session fun? ○ We must look into creating lesson plans that keep students engaged beyond the standard classroom setting. As tutors it is often easier to create lesson plans that consist of worksheets and powerpoints, but this is too school like. Tutors can incorporate math games such as Kahoot and jeopardy to test students' mastery. Another possible solution is to introduce project based learning. This will allow students to go beyond their comfort zone and deeply grasp concepts in an interactive setting. ● Differentiated instruction
Kombo & Taroua 14 ○ With the new heterogeneous classes, it is imperative that students should be encouraged to challenge themselves, however they should still be able to understand the material. Students will essentially be taught the same concept but at varying levels. When creating lessons, make sure that you know what the skill is, ex. Operations with integers, and then the different levels will all feature the same types of problems, however the numbers may be three digits for the “most challenging” set of problems, two digits for the “relatively challenging” set of problems, and one digit for the “regular” set of problems. ● Be Aware ○ The role of a Pathfinder teacher, especially in the Spring, is to offer these students one- on-one time. Your obligation as a teacher is to ensure that you are focussed on the students well being and noting how they are understanding the concepts. Many of the students are not used to asking for help, so they most likely will not and instead listen for the answers and tell you they understand it. Be sure to come over to individual students and ask them to show you how they arrived at the answer, and using a white board or another piece of paper, write down a similar problem that resembles the problem they showed you and ask them to explain it to you. If you notice they are struggling, help them through the problem and give them another problem to make sure they understand. ○ We encourage you to adopt the “Do Now” strategy to incorporate skills that were taught in earlier weeks to ensure that the students still understand the concepts, and then tie it into that week's concepts. “Exit Tickets” are also useful to check for understanding after new material is introduced.