SD 10 Worskshop Lesson Plan: Positive Mathitude: Getting a Grip on Math at Kingsborough

Goal

The goals of this workshop are: a) to address math anxiety in students; b) promote positive attitudes toward math; c) to reinforce the importance of critical reasoning in everyday life; and d) to provide students with information about the developmental math sequence at Kingsborough Community College

Objective

The objective of the workshop is to determine the causes of math anxiety in students at Kingsborough. Students will be given the opportunity to state their feelings toward math, and the QR fellow will teach them the benefits of having positive attitudes toward math. They will be guided through an activity on the usefulness of math in everyday occupations, and why statistical comprehension is important in everyday life. Finally, students will be able to understand the developmental math sequence at Kingsborough, and will learn to navigate the Kingsborough Catalog to determine the math courses required (developmental and credit-bearing) for their major.

Materials 

Powerpoint slides—Positive Mathitude: Getting a Grip on Math at Kingsborough (Math Workshop), Developmental Sequence Planning Worksheet (Math Sequence Planning Worksheet), Kingsborough Catalog (Kingsborough Catalog), SMART classroom (room with computer and projector)

Audience

Initially created for the SD10 classes of incoming spring-semester first-year students, this workshop has been modified for a workshop for the Men’s Resource Center.

Workshop Lesson:

Total Runtime: 50 minutes

Coin/Age Prediction (10 mins): After introduction, this icebreaker is to ease the tension about preconceived notions about math. It begins by asking students to take some coins out less than $1.00. Then students are supposed to do some basic calculations, either by hand, or by using a calculator (in the interest of time, calculator use is preferable). The formula is as follows: Multiply age by 2, Add 5, Multiply by 50, Add pocket change, Subtract 250. The answer reveals 4 digits: where the first 2 digits reveal the age of the person, and the last 2 digits reveal the amount of coins in pocket. (Students are usually surprised by these results). Lead students into a discussion, and ask how does this “trick” work? Is it really a trick? Proceed with the actual formula, which is really a math formula that uses particular principles of math to conceal and reveal these 4 digits.

Math Attitudes (10 mins): Proceed by asking students what are their attitudes about math. This is the opportunity for students to vent freely, without being judged. Many students will express negative attitudes about math, so be understanding. Allow the few students that express having positive attitudes about math to explain their feelings too. Lead students in a discussion about how having negative attitude toward math is not conducive to learning. Then explain how having positive attitude toward math can help to alleviate the anxiety.

Specialness of Math/Math in Everyday Occupations (10 mins): Explain the “special” nature of math because it is a general subject that permeates many fields. Proceed with activity about math in everyday occupations, and ask students to explain how math is essential for particular occupations (i.e. nursing, architecture, music, computer science), and reveal the many types of math required for each occupation.

Critical Thinking and Math Activity (5 mins): Students will be shown a series of misleading graphs. The graphs vary in terms of ways in which they are misleading (i.e. scaling issues, misrepresentation of data). For each one, ask students what are some elements of the graph that makes it misleading.

Math Sequence at Kingsborough (15 mins): This section is to help students navigate through the Kingsborough catalog in order to determine how many math courses they would need in order to graduate (developmental and credit-bearing courses for individual majors). Describe what a math sequence is, and emphasize that the order of taking these courses is important. Show students how to find this information in the catalog. Describe the developmental math courses (M1: Arithmetic; M2: Introductory Algebra). Guide students through a typical math sequence scenario for Liberal Arts majors. Also consider students who are thinking about attending a 4-year CUNY college after graduating from Kingsborough, and explain what additional math courses would be required in order to be eligible for applying to these CUNY colleges. Explain that for students who are STEM (science, technology, engineering, and mathematics) majors, upper level math courses beyond the developmental courses are required. End with some useful tips on how to be successful in completing these courses and graduating on time. Pass out the handout (Planning your Math Sequence to students), and let them know that if they had any specific questions about prerequisite courses, to ask their advisor for more information.

 

QR Fellow comments: Active engagement with the students is very important. It is quite easy to lose students’ interests if there are no activities or discussion about the application of math in everyday life. Students typically have the sentiment that math is not useful, and therefore, is not worth the time or effort studying. The idea that math is everywhere seems necessary to reinforce. Students seem to open up more if you allow them to vent about math while not having feeling judged in doing so.

 

SD10 Learning Community Workshops

{aim} 

The overarching goals of this workshop was to connect with incoming college freshmen and express the importance of completing their math requirements in a timely fashion. Most students have a goal of completing their program of study within 2 years. Based on how they fare on their math placement exams, and their major requirements, they may have two years of math to complete. This workshop was presented to the students to reinforce the importance of beginning the math sequence early in their college career.

{objective} 

Students will learn why math is important beyond simple arithmetic. There are many applications of math beyond simple every day arithmetic. We make decisions and solve problems on a daily basis, and many of the skills that we use to do so (weighing consequences, prioritizing, determining probability) stem from higher math classes.

As part of this workshop students learn why math is important in their daily lives for complex problem solving and for critical thinking. In addition, students will learn about the math that goes into determining their grade point average (GPA), and how this may affect their college career. Finally, students will learn about the math sequence at Kingsborough and the potential pitfalls (failed classes, full classes) they may experience along the way.

{materials} 

For this workshop a smart classroom helps to facilitate the discussion, and allows for examples during the GPA Worksheet (file attached) through the use of the GPA GPI and Math Sequence presentation (file attached).

{audience} 

This workshop is generated for incoming freshmen.

{workshop lesson}

The workshop begins by asking students about their attitudes towards math. Prompts may be: how do you feel about math? How did you feel before/during/after the math placement exam? How was your math experience in high school? This is followed by some humorous math memes that each describe a common negative attitude towards math. Explain them and ask students if they agree/disagree.

The workshop continues by discussing the importance of math in our lives. Ask students to provide examples of how they use math. They will probably use examples of counting change, paying bills, etc. Introduce the idea of higher math and problem solving, and how math is related to critical thinking. A good example is geometry proofs (if they remember them). It’s something they will likely never have to do, but the action of forming logical arguments to prove a statement is something they probably do on a regular basis. This discussion of the relevance of math leads into the GPA worksheet. Have students pick imaginary grades and work through the sheet with them. Make sure to engage students, answer questions, and ask them what GPA they’ve calculated. The point is to show them that complex math can be broken down into simple steps (as the big reveal of the math notation for an average shows), and that the weighted class credits will affect their grade. This exercise is supposed to empower and educate the students, demystifying how a GPA is calculated, and making them aware of it’s importance to their college careers. End it by reinforcing the bigger picture math importance discussion.

The workshop ends with discussion of the math sequence at Kingsborough. Students take a math placement exam upon entering Kingsborough. Depending on their score (or other criteria – SAT, Regents, ACT, etc) they are placed in the developmental sequence (M1 or M2), or are eligible for a credit baring math class. All students must complete a credit baring math class to earn their associates degree, and most majors require at least one math class. Many students at Kingsborough put off this sequence due to underlying math phobia or anxiety. This can lead to problems if a student has completed all their other major requirements but have yet to take math and are placed at the M1 level. That is at a minimum 3 classes that must be completed sequentially, so it can take 3 class sessions or longer for students to graduate, assuming all classes are passed. This adds time and money onto their experience. This portion of the lecture discusses various scenarios and tips for success. The goal is to motivate students to register now because they don’t know what may impede their math progress (full classes, personal issues, class failure), and it’s better to start now than wait and have a math class be the only thing between them and a degree.

{teacher comments} 

Be aware of your population. Try to have an upbeat and positive attitude. Try to avoid terms like ‘remedial math’, and ‘senior college’, even if students and caseworkers use these terms. Be prepared with back-up prompts if your students aren’t interested in participating. Personal anecdotes are helpful as well.

 

Quantitative Reasoning Assessment

Aim

The main goal was to evaluate students’ math attitude and aptitude at the beginning and the end of the semester.

Objective

The main objective of this QR assessment was to assess math attitude and basic math knowledge of students enrolled in five learning communities at Kingsborough College and to evaluate benefits of the learning communities. The test of math knowledge was designed to test students in ten areas of math that were identified as important by several instructors of math in learning communities:

1. Ratios and proportions
2. Percentages and Decimals
3. Basic Fractions
4. Converting Fractions to Percentages
5. Multi-step Arithmetic
6. Basic Algebra
7. Conversions: Metric vs. US system
8. Estimation
9. Scientific Notation
10. Reading Charts and Tables

The tests were graded in three categories: strategy, accuracy and presentation, except for problem #10, which was graded only in accuracy.  The three categories were designed to be as independent of each other as possible.

In the strategy category, we graded student’s approach to solving a problem, i.e. whether the student properly identified the type of the problem and applied the appropriate method(s).

In the accuracy category, we graded student’s ability to accurately perform mathematical operations.

In the presentation category, we graded student’s familiarity with mathematical notation and ability to use it to show his/her solution.

Materials

Quantitative Reasoning Assessment Instrument: contains a survey about attitude toward math and a test of math skills. Each student receives a copy of it at the beginning of the assessment. Students should write their answer on it and hand it back at the end. Students are not supposed to use calculators.

Quantitative Reasoning Rubric and Grading Sheets: contains a rubric with instructions for grading the test and a grading sheet.

Audience

The assessment was designed for students in learning communities at Kingsborough, but it could be applied to other students.

Administration

The assessment was administered at the beginning of a math class in each of five learning communities. The intervention requires around 20min. After a brief introduction, the instruments were distributed to students. They were normally given around 15min for both the survey and the test. The intention was to give them 5min for the survey and 10min for the test, but this didn’t always work that way. Students first filled the survey. Some of them were faster than others and thus had more time for the test.

It might be a better strategy to have students first solve the test (10min) and then answer to survey questions. This would allow the test administrator to control the amount of test time and ensure it is the same for all students. There is another benefit of this strategy. The survey contains questions about student’s attitude toward math and their self-image. If they are negative, taking the survey may condition students in a way that reduce their performance on the test. Taking the test first would eliminate this problem.

One challenge we faced was student identification. Even though the assessment was anonymous, it was necessary to somehow identify students because the assessment consisted of two waves and it was important to match the pre- and post- assessments. To do that, we asked students to enter last five digits of their CUNY First ID. However, some students did not know their CUNY First ID, so they were asked to use last five digits of their social security number. The problem with this was that in the second wave, these students did not remember what they used for their ID in the first wave. This resulted in a small number of mismatched assessments, but we managed to match them based on other characteristics from the demographics questionnaire.

Grading

The tests were graded by three graders. To achieve consistency, we first performed several rounds of trials. In each round, all three graders would grade three randomly selected tests. Then we compared the grades. After three or four iterations this process led to consistent grading across the graders. It also helped us fine tune the grading rubric.

 

Workshop Lesson Plan

Goal

The main goals of this workshop are: (1) helping students deal with math anxiety, (2) advising them about effective math studying strategies, and (3) informing them about math requirements at Kingsborough.

Objective

The workshop will help students better understand the sources of math anxiety and counterproductive effects it has on their studying. They will learn the importance of a positive attitude toward math and how they can achieve it. One way is to be aware of usefulness of math, which is illustrated in the workshop by examples of applied math.

Students will also learn an efficient way of studying math, which includes learning how to use the textbook efficiently.

Finally, students will learn about math courses at Kingsborough, both remedial and credit-bearing ones, how to find out which courses they need to take and in what order.

Materials 

Materials used in the workshop are PowerPoint slides and a Math Sequence Planning Worksheet. A smart classroom (i.e. a classroom with a computer and a projector) is necessary for displaying the slides.

Audience

The workshops was initially developed for the SD1000 class of incoming spring-semester students. With minor modifications, the same material was used in a workshop at the Men’s Resource Center.

Workshop Lesson

Describe the workshop including discussions and activities. Include temporal breakdown for each section of workshop.

Math attitude (10 min). After an introduction, the workshop starts with a brief discussion about students’ attitudes toward math. As most students express negative attitudes, this is a good introduction to the math anxiety topic. It starts with listing (negative) effects of a negative attitude toward math, and (positive) effects of a positive attitude. Then it gives tips for changing the attitude (e.g. “I want to study”, not “I have to study”; math as a puzzle, regular studying, asking for help).

Challenges of math and how to overcome them (10 min). This part points at the generality, a feature unique to math, that makes math applicable to “everything”, but it also makes math abstract and sometimes difficult to grasp. To overcome it, students are advised to adopt an appropriate way of studying math: (1) use textbook, (2) study “theory”, in addition to problems, (3) identify “rules” and “restrictions” in the textbook and memorize them, (4) do problems after understanding rules and restrictions. Knowledge of theory should clarify relationships between abstract concepts they often encounter in math.

Applied math example (5 min). This parts shows a short video of a presentation on wealth and health of nations in the past 200 years. The presentation uses statistics to tell a story of global economic development in an intuitive and visually attractive way. The main goal is to demonstrate how, by representing phenomena of interest by variables and measuring them, math allows us to see a big picture, to make comparisons that would otherwise be impossible to make.

Calculating GPA and course grades (15 min). This part teaches students how to calculate their GPA and the course grades. Among the students whom I interacted with, virtually no one knew how to do that. The goal is not necessarily to have students do the calculations; rather it is to help them develop intuition about how additional grades affect their GPA, what it takes to maintain a good GPA, and what it takes to improve a bad one. During this segment, it is emphasized that their GPA affects their academic life in many ways, especially in making them eligible for graduation.

Math sequence at Kingsborough (15 min). This part explains math requirements at Kingsborough, with emphasis on the remedial math courses. One goal is to make students aware of these requirements; another is to help students understand the sequential nature of the requirements, importance of planning and starting taking math classes early. This is done by showing them a diagram of the sequence, and showing them various scenarios and the number of semesters that each of them takes.

Teacher Comments

It is important to keep the workshop interactive, even if that means straying away from the plan or leaving out some slides. Students showed surprising willingness to participate, to discuss their problems with math and ask questions about how to overcome them. A looser format of the workshop was conducive to such interaction and, in general, led to a better outcome than a more rigid format, in which case students would start losing interest relatively quickly.

One of the most acute problems regarding students’ math skills, in my opinion, is their approach to studying math. Almost no students reported using a textbook for studying math. They commonly describe their studying method as solving the problems that instructor solved in the classroom. This way of studying produces inadequate kind of knowledge. Even when they succeed in memorizing the solution methods, they rarely understand them and are unable to apply them to different types of problems.

 

Speeding down the highway…

Who needs algebra when they’ve got a speedometer?

 

Adding and Subtracting Fractions

When we perform operations with whole numbers, adding and subtracting is easy. We can practically do it on our fingers (especially if the sum is small!)

With fractions adding and subtracting is a little different. We need to be careful, or else the numbers won’t make sense. When adding and subtracting fractions we need to make sure that each fraction has the same common denominator. Remember what the denominator is? It’s the bottom half of the fraction, which tells us how many pieces there are in total. If the two (or more) fractions being added don’t have the same denominator it’s like we’re adding apples and oranges!

Adding half an apple and half an orange does not make one whole anything!

There are several ways to find a common denominator. One is the criss-cross method. It’s pretty straightforward. You simply multiply the bottom denominators by each other, and then each numerator by the opposite denominator. Huh? That wasn’t clear enough? Check out the image below.

So now we have our two fractions written with the same denominator. They are the same as before, because they are equivalent fractions. From this point, adding is pretty straightforward. The denominators are the same, so we don’t need to do anything with them. Why is that? Because we wanted to make sure we are adding the same sized pieces, and we don’t want to change those sizes again. But, we want to add the number of those pieces that we have, so we add the numerators straight across, left to right, just like addition with whole numbers. See picture below.

The final step is determining if the fraction can be reduced. In this case, 17/21 is the lowest this fraction will go.

This method will always work, however sometimes it makes fractions clunky because the numerators and denominators get so large. Imagine adding 1/8 + 1/16? Using this method you’d be adding up  16/128 + 8/128! Another way to find the lowest common denominator is by listing the multiples of the denominators. In this case multiples of 8 and multiples of 16

• 8 = 8(8*1), 16(8*2), 24(8*3), 32(8*4), etc.
• 16 = 16(16*1), 32(16*2), 48(16*3), 64(16*4), etc

Here I’ve put the lowest denominator that is the same in a bold font (that’s the lowest denominator the two fractions have in common!)

Since the second fraction (1/16) is already in 16ths, we only need to change the first fraction. And how do we change it? Well if we are multiplying the denominator by 2 to get it in 16ths, we need to multiply the numerator by 2 as well so we don’t change the fraction. Below see the problem worked out in steps.

 

In this version we only had to change one fraction, and we ended up with numbers that didn’t get super huge, making the addition process easier for us. This was two ways to find the lowest common denominator, often the most difficult part of adding two fractions. After this, multiplication and division are going to be a breeze!

One excuse for not answering a test question…

Watch out for that pesky bear!

Everyday uses of percentages

Percentages are frequently used in everyday life.  The symbol, %, is used to compare values, quantify changes, and to calculate amount represented by an increase or decrease.  Let’s look at some examples:

discounts:

in nutrition:

and checking cell phone/laptop battery life:

Percent means, “per hundred,” or “parts out of 100” and is the same as the numerator of a fraction whose denominator is 100 (e.g. 37% is equivalent to 37/100).  Percentages are simplified ways of expressing size, scale, or value.  They are also useful for comparing information where the totals are different.  By converting different data to percentages, you can readily compare them.

Math is all around you!

Another Geometry Joke…

Where’s Samuel L. Jackson when you need him?

 

What’s a Fraction?

It may seem like a simple question, after all a fraction is a FRACTION…but it’s always a good idea to understand what something means before you memorize all the rules about how to manipulate it.

A fraction occurs when a larger object is split or divided into several smaller equal parts. Each of those equal parts is a fraction. As you can see in the picture below, all shapes that are divided into smaller equal parts are fractions (check mark), and all shapes that are divided into smaller but unequal parts would not be fractions (red x).

 

 

A fraction always has two parts, the numerator and the denominator. The numerator indicates how many parts there are in the specific fraction, and the denominator indicates how many parts the whole has been equally divided into.

 

Now that you understand what a fraction is, and what it expresses, you can learn more by checking out our online resources, or reading our posts on fraction operations.

Remember that it’s always ok to go back to basics. If you can’t visualize a fraction, draw it! It’s as simple as creating a shape and then dividing it into equal parts! Or, try using a number line. This is a great tool for visualizing equivalent fractions on a number line. What are equivalent fractions? Fractions that represent the same amount! 1/3 = 2/6 = 3/9, etc. They are all the same! Don’t believe me? Visit that number line tool and see for yourself! Or, draw it out!