Reducing Algebra as a Calculus Pain Point

Time and time again, we hear from calculus professors across the country that one of the biggest issues in their classes is students remembering the building blocks of algebra.

Here are a few suggestions to help ensure students have the basics mastered:

Consider diagnostic testing.

Identify students who have skill gaps, then provide them with supplemental assignments in the first weeks of class for additional support.

Ask students who excel on the diagnostic test if they are willing to be course mentors, which will build classroom camaraderie.

If possible, host a 1-day algebra refresher workshop before the first day of class.

Begin the term explaining how algebra is foundational to calculus. Let students know they are not alone in struggling with algebraic concepts. Hosting an algebra refresher will help students feel more comfortable asking questions.

Remind students that you’re grading for accuracy.

Feedback is critical for students to realize they are struggling. If time permits, set aside a few minutes after passing back assignments so students can look over the feedback you gave them and ask questions.

Provide every student with technology resources as further help.

List out a few tech resources that are easy to access, such as YouTube videos or online interactive games, on your syllabus. Additionally, look for calculus materials that provide a brief algebra refresher as part of the text.

Hawkes Learning’s Calculus with Early Transcendentals textbook and NEW courseware offer exercises and diagnostic testing that target the key algebraic topics calculus students need to master. Request a complimentary exam copy.


5 Ways to Get Students Interested in Statistics

Creating a universally engaging classroom environment can be challenging, but having the right tools that make lesson content relevant to students helps! Below are 5 ways to get your students more excited about statistics:

1. Interesting Data
Finding data on topics students think are fun, like beers and breweries across the country, might pique interest. Use this spreadsheet from the U.S. Census to show them socioeconomic trends they may witness themselves in their own demographic (or age bracket).

2. Visualization Tools
Seeing is believing. The free online resource Gapminder offers a graphical simulator depicting 5 dimensions of real-world data in 2D. Students can change the relationships between demographic, economic, and societal variables animated over time to see some pretty neat relationships in motion.

3. Applications Challenge
Knowing the immediate value of the lesson they’re learning gives students more encouragement to commit the content to memory. Asking students to find their own data sets on their favorite sports team or something they connect with might engage their interest and help them truly grasp the concepts.

5 ways to makes stats more relevent

4. Games
You know statistics can (and is!) fun, and who doesn’t like to win? Interacting with a game and trying to win it make learning more exciting. View some examples of statistics games here.

5. Simulations
Help students grasp key concepts through simulations that hold their attention! Use simulations in class and encourage students to work through as a group to liven up the lecture time. Check out fun simulations here.




Research is influencing college remediation (including Coreqs)

[Originally published on Brookings]

Judith Scott-Clayton, an Associate Professor of Economics and Education at Teachers College, Columbia University, writes about the lack of evidence surrounding the effectiveness of traditional remedial placement and delivery practices in “Evidence-based reforms in college remediation are gaining steam – and so far living up to the hype.” She describes the calls for less collegiate remediation, the reforms that have occurred, and how those reforms are working.

Scott-Clayton has conducted research showing that “misplacement into remediation was far more common than misplacement into college-level courses.” She documents questions surrounding the quality and validity of entrance exams to determine placement. Additionally, her research indicates that an estimated “one-quarter to one-third of students assigned to remediation could have earned a B or better in college-level coursework, had they been given the chance.”

Scott-Clayton goes on to detail specific, state-level reforms that have been instituted because of research on remedial placement. She ends by describing ongoing research that, so far, has largely indicated the benefits of co-requisite support as opposed to the traditional pre-requisite model of remediation.

Read this article on Brookings

Scott-Clayton, Judith. “Evidence-based reforms in college remediation are gaining steam – and so far living up to the hype.” Brookings, Brookings, 29 March 2018, Accessed 29 August 2018.

Evidence continues to show corequisite effectiveness

[Originally published on Inside Higher Ed]

Alexandra Logue, a research professor at the Center for Advanced Study in Education at the City University of New York Graduate Center, describes an effective implementation of the corequisite model at City University of New York. This study adds to the growing body of research on the benefits of corequisite remediation.

According to Logue, “Currently, around 68 percent of new college freshmen in public community colleges and 40 percent in public four-year colleges take at least one remedial course in reading, writing or mathematics (somewhat more often in math), but most students assigned to remediation either never take a course or don’t complete it.” She cites several other studies that have shown higher course pass rates in corequisite remedial courses than in traditional remedial courses and argues that the educational community has a responsibility to look seriously at corequisite classes.

At City University of New York in a randomized controlled trial, students benefited from corequisite remediation over traditional remediation. Logue suggests some possible explanations, “including the incorrect assignment of some students to remediation, the demotivating effect of being assigned to traditional remediation, the extra time and cost to students if they must take traditional remedial courses, the greater number of potential exit points from traditional remediation course sequences, and so on.”

Read this article on Inside Higher Ed.

Logue, Alexandra W. “The Extensive Evidence of Co-Requisite Remediation’s Effectiveness.” Inside Higher Ed, Inside Higher Ed, 17 July 2018, Accessed 20 August 2018.

Hawkes Corequisite Model Course Strategies: Guest Post by Dr. Paul Nolting

Dr. Paul Nolting speaks to a group of tutors in the TILT building Tuesday morning. Nolting spoke to the tutors about strategies, particularly in math, to better aid in the learning of material when working with students.Dr. Paul Nolting is a national expert in assessing math learning problems, developing effective student-learning strategies, and assessing institutional variables that affect math success and math study skills. Over the last 25 years he has consulted with over 100 colleges, universities, and high schools to improve success in the math classroom. Dr. Nolting is the author of Winning at Math, which is the only math-specific study skills book to offer statistical evidence demonstrating an improvement in students’ ability to learn math and make better grades.

Below, Dr. Nolting provides his insight regarding how to incorporate the study skills that are crucial to student success into co-requisite course structures.


The math redesign movement has put more demand on institutions to have students complete developmental and first-credit math courses more quickly and with higher pass rates. Research and experts at the National Math Summits—conducted at AMATYC and NADE conferences—have indicated that this higher demand on students requires them to become improved learners.

National research indicates that student affective characteristics make up 25% to 41% of students’ math grades. Institutions can improve student success by teaching math study skills, math test-anxiety reduction, math test-taking skills, and math self-efficacy. Research conducted in dissertations, master’s theses, Title III projects, QEPs, and the classroom has shown that students who learn these skills from the Winning at Math text improve their grades.

The purpose of this document is to help instructors implement corequisite designs and integrate math study skills into the corequisite lab by teaching math study skills topics and then assigning Winning at Math homework to improve math learning and grades by having students practice these skills in the lab and classroom.

The corequisite model, which is becoming one of the most popular course designs, blends the content of two courses, usually one that is a developmental course and the other a credit course like College Algebra. The corequisite course has a support lab course, which is usually two hours. These courses have two sets of students, developmental students and non-developmental students. Depending on the state, possible corequisite courses could be Elementary and Intermediate Algebra, Intermediate and College Algebra, College Algebra and Pre-Calculus, or developmental courses with Quantitative Reasoning, Statistics, or Liberal Arts courses. The developmental students are required to take the lab course while the non-developmental students can opt to enroll in the lab course. Students in the lab course learn the pre-requisite math skills and become more effective learners through math study skills while mastering the lab course content.

Developmental students can lack both pre-requisite math skills and math learning strategies, which are essential abilities when taking two math courses at the same time—one of which is college-level. Assessing developmental math students is a must, measuring their pre-requisite math skills and math study skills to provide appropriate training. The lab course is a combination of math study skills instruction, remediation, just-in-time math learning, and tutoring. The credit course mainly has instruction and supports the math study skills. The lab and course instructors also need to coordinate with the Learning Resource Center/Math Lab to provide additional support for the students. In fact, the lab instructors, course instructors, and Learning Resource Center staff need to develop a plan for all students. If possible, both the course and the lab are taught by the same instructor. When properly designed the corequisite model can improve the success of developmental and non-developmental students.

Course Curriculum and Strategies

The curriculum of any mathematics course can be enhanced with math study skills. The first course strategy is to assess the students on their prerequisite math skills and math study skills. The students would be assigned to take both the Math Skills Assessment and the Math Study Skills Evaluation, both of which are provided in the Hawkes Learning courseware. In the case of a corequisite College Algebra, students can take the math skills assessment for Intermediate Algebra at no extra cost. Students should take these assessments during the first week of class. The math skills assessment results should be divided into two groups consisting of the developmental students (required to take the lab) and the non-developmental students. The individual results should be given to the students all at once so they know how many pre-requisite math skills they need to improve. The non-developmental group also needs their results given to them based on their assessment so, if necessary, they can be encouraged to take the lab course. Then, the two data groups can be separately aggregated to determine which pre-requisite math skills are the most needed to be taught in the lab course and the credit-bearing course. A comparison of the needed math skills may also bring additional insight. The class should then receive an overall view of the results, which will help the students understand the reasons for teaching the pre-requisite and new math skills.

All students should also take the free Math Study Skills Evaluation in the Hawkes Learning courseware to determine their math study skills needs. The results and a printout of the evaluation are sent to the students and the instructor or lab. The evaluation can be reviewed in the course and/or lab to help students understand their math study skills needs. A class average score can be given to the group and, if needed, broken down by developmental and non-developmental students. Reviewing the correct answers will help all students understand how to further develop their math study skills, and non-developmental students will be encouraged to take the lab course, which will teach them further math study skills. Note that on student surveys, the correctly answered questions are not listed. Remember, a low score on this evaluation should be framed as good news because this lack of math study skills may be the reason for previous poor math success that makes students a high risk for a corequisite course. Learning math study skills improves math learning and grades.

Pre-requisite Lab Curriculum and Strategies

The corequisite lab provides support for the credit course. This support is in the form of remediation, just-in-time instruction, math study skills, tutoring, and coordination with the Learning Resource Center. Based on the math skills assessment results, students are informed that lesson plans were developed to remediate the most commonly missed math pre-requisite skills. Instructors then teach these lessons along with math study skills. Since students’ entire needed pre-requisite math skills cannot be addressed in the lab, especially the low-level skills, students will individually need to learn these skills and be referred to specific Hawkes lessons for pre-requisite math skills development and/or to the Learning Resource Center for additional prescribed help. The lab instructors can work with the Learning Resource Center staff to develop these resources and understand how to help students use the courseware. Every effort should be made for the students to complete their basic skill learning at the Learning Resource Center during the first three weeks of the semester or before the first major test. Students need to have these skills learned before the first major test, and this is when the center has time to help them.

Students would also be informed that, based on the Math Study Skills Evaluation, they need to improve their math study skills. The instructor would go over the Math Study Skills Evaluation and indicate that poor scores are a good sign that students can improve their math success, and that also it is not their fault that they have not been taught how to learn math. Improving math study skills and reducing math/test anxiety have shown to improve self-efficacy and math grades. Instructors would lecture on math study skills using the Winning at Math text, and students would complete the assignments in the Winning at Math text. However, students would be encouraged to use the results from the Math Study Skills Evaluation and start learning math study skills on their own by reading the recommended chapters and pages and practicing these skills. The math study skills lectures would be followed by students demonstrating these skills in the lab and applying these skills in the course and on tests. The math study skills lectures could be one per week, ending in week seven. The math study skills need to be learned as quickly as possible in order to apply all of the skills by midterm. If possible, the lab needs a letter grade to make the work more creditable. Part of students’ lab grades would be tests on math study skills through short answer questions, multiple choice (provided in the courseware) activities, attendance, and/or projects. After about the seventh week, the remediation, math study skill lectures, and most of the just-in-time lectures would be completed. Then, the lab would be more of a resource for re-teaching course content, tutoring, applying math study skills, and continuing test anxiety reduction.

Syllabus/Class Schedule

Instructors can use the same syllabus/class schedule from the course by integrating the lab course requirements, or a separate syllabus/class schedule can be developed just for the lab. The Winning at Math chapters to be read are listed for every week. It is important to complete Chapters 1-3 before the first major test. Chapter 7 or 8 in Winning at Math-Concise (on test-taking) should be completed before the second major test. Instructors should switch around chapter orders to best fit students’ needs.

Students will not take a study skills text seriously unless they are required to turn in work or are tested on its material. Asking students to read chapters to prepare for a short discussion as part of the lecture will help them learn the skills. Instructors can divide Winning at Math homework into chapter activities and end-of-the-chapter assignments. Students can complete section and Chapter Reviews in the text. This involves emailing completed assignments directly to an instructor or turning in the assignments on lab test day. It is also easy to check off activities and end-of-the-chapter assignments while students are taking lab tests or working on group projects. Record the assignments as Complete or Non-Complete instead of grading them. Determine the amount of points for completing the assignments just like you may do for completing math homework. Lab instructors should count study skills homework separately or alongside participation points.

Testing and Assessment

Lab instructors can test math study skills as part of their regular lab grade or as part of the course grade. For at least the first two tests, lab instructors can use open-ended math study skills questions (Appendix A) or the already developed multiple-choice questions with feedback for incorrect answers in the Hawkes courseware.

It is very important to answer “yes” when students inevitably ask, “Is this going to be on the test?” In lab class, consider having students form groups and create ten open-ended questions they might want to answer on the test. Then, discuss the questions and tell your students that you will select five of these to be on the test. Do not worry about students creating “easy” questions. Almost every time, they come up with questions so difficult that they cannot be used on a test. Most students will learn the answers to the questions they came up with because this assures them that they can obtain a good grade or points. This encourages them to learn about math study skills, and thereby improves their grades. In addition to these five questions, instructors could also include a bonus question. In other cases, student take the multiple-choice questions in the lab on the computer. In any of these scenarios, indicate on the syllabus/class schedule that there will be math study skills questions on the tests.

Another way to test students is to assign readings and then reserve five to ten minutes during lab time for quizzes. This also encourages students to read about and remember math study skills. Lab instructors can issue these quizzes more frequently early in the semester, so students can then apply learning strategies throughout the remainder of the semester.

Decide which testing methods you want or combine these methods. Assessing the bulk of math study skills learning early enough in the semester makes an immediate difference. Students will learn the material that will be on the test.

When some students first see Winning at Math listed as part of the course, they may have questions. Explain that every student must take math; these skills are applicable to STEM courses and lead to improved grades in other courses. You should also explain that math study skills are important because students must become improved learners when they are taking two courses at the same time. Also, becoming successful in math allows students to choose from a broader range of majors that tend to be more financially successful. This is true for students who have struggled with math, those who suffer from anxiety, and those taking math for the very first time. Other students, especially those repeating math for the second, third, or fourth time, can use the math study skills to finally pass a troublesome course. It is worth devoting time to helping all students develop into effective learners.


The corequisite model is a new adventure in math learning. It was developed to have students complete their math courses in a shorter amount of time. When designed correctly this model can help both the developmental and non-developmental student become more successful. This effort involves the delivery of assessments, remediation, just-in-time learning, math study skills, and coordination with the Learning Resource Center. Research on the success rates of different types of students is also needed to determine which students are most successful and which are not. The last strategy is developing math success plans for students repeating the course. Part of the math success plan assesses the reasons for the non-completing students and then develops individual success plans for them. The success of the corequisite model depends on the teamwork of the course instructors, lab instructors, Learning Resource Center staff, and the students to blend in remediation, instruction, and math study skills.


Appendix A

Co-requisite Lab Math Study Skills Questions

Test One

Name: ________________________________

Number and answer the questions on the attached sheets of paper.  Read all the questions first.

  1. List and define three ways how learning math is different than other subjects. Provide an example for each of the three ways.
  2. Why is math considered a sequential learning pattern?
  3. How does previous/mass math knowledge affect your grade?
  4. Draw and explain the Bloom chart on page 37 in Winning at Math. How does each component of the chart apply to your learning? Use specific personal examples to illustrate.
  5. List and describe four of the anxious/stress behaviors. Provide an example for each of the four behaviors you select.
  6. Name and describe the two different types of test anxiety.
  7. Describe three relaxation techniques. Select one you use and describe the situation during which you use it.
  8. List and describe the components of the Math Learning System Overview. Select three of these components and explain how can you use each one to improve grades?

9 . List and explain the four basic college management concepts (EH).

  1. List three strategies to set up a positive study environment. How can you use these strategies?

Bonus  Questions  (5 points each)

  1. List the results from your surveys. Explain what these results mean as far as improving how you study math.
  2. List your most positive strength and describe three areas you need to improve.
  3. If you use complete sentences correctly to answer the questions, you will earn five points.




Co-requisite Lab Study Skills Questions

Test Two

Name: ________________________________                      Date: ____________________

Number and answer the questions on separate sheets of paper.

  1. List and describe each stage of the memory process. Which stage causes you the most difficulty in learning? How can you improve that stage? Provide an example for each improvement suggestion.
  2. Give four examples how memory and learning relate to each other.
  3. List and describe three ways to become an effective and active listener.
  4. List and describe the Seven Steps to Taking Math Notes. Draw and explain the note-taking system.
  5. List and describe the SQ3R. What is the extra R that I put in as an extra step?
  6. List and describe five general memory techniques.
  7. List and describe the Ten Steps to Doing Your Math Homework.
  8. List and describe five resources you could use to get through homework problems that you can’t solve on your own or when you are stuck.
  9. List and explain the Ten Steps to Taking a Test. Now, list your personalized test-taking steps.
  10. List and give examples of the Six Types of Test-taking Errors. What is the error you commit the most often and how can you correct it?

Bonus Questions (5 points each)

  1. Describe metacognitive learning.
  2. List two ways you could use metacognitive and group learning to improve grades, including the final exam.


Integrate Developmental Math with Statistics in Corequisite Course

Cover of Discovering Statistics and Data Plus Integrated ReviewDiscovering Statistics and Data Plus Integrated Review leads students through the study of statistics with an introduction to data.

It pays homage to the technology-driven data explosion by helping students understand the context behind future statistical concepts to be learned. Students are introduced to what data is, how we measure it, where it comes from, how to visualize it, and what kinds of career opportunities involve its analysis and processing.


This integrated course enhances curriculum-level statistics with applicable review skills to shorten the prerequisite sequence without compromising competency. Target specific remediation needs for just-in-time supplementation of foundational concepts.

Table of Contents:

Chapter 1: Statistics and Problem Solving

1.1-1.8: Introduction to Statistical Thinking

Chapter 2: Data, Reality, and Problem Solving

2.R.1: Problem Solving with Whole Numbers
2.R.2: Introduction to Decimal Numbers
2.1: The Lords of Data
2.2: Data Classification
2.3: Time Series Data vs. Cross-Sectional Data
Chapter 2 Review

Chapter 3: Visualizing Data

3.R.1: Introduction to Fractions and Mixed Numbers
3.R.2: Decimals and Fractions
3.R.3: Decimals and Percents
3.R.4: Reading Graphs
3.R.5: Constructing Graphs from a Database
3.R.6: The Real Number Line and Inequalities
3.1: Frequency Distributions
3.2: Displaying Qualitative Data Graphically
3.3: Constructing Frequency Distributions for Quantitative Data
3.4: Histograms and Other Graphical Displays of Quantitative Data
3.5: Analyzing Graphs
Chapter 3 Review

Chapter 4: Describing and Summarizing Data From One Variable

4.R.1: Addition with Real Numbers
4.R.2: Subtraction with Real Numbers
4.R.3: Multiplication and Division with Real Numbers
4.R.4: Exponents and Order of Operations
4.R.5: Evaluating Algebraic Expressions
4.R.6: Evaluating Radicals
4.1: Measures of Location
4.2: Measures of Dispersion
4.3: Measures of Relative Position, Box Plots, and Outliers
4.4: Data Subsetting
4.5: Analyzing Grouped Data
4.6: Proportions and Percentages
Chapter 4 Review

Chapter 5: Discovering Relationships

5.R.1: The Cartesian Coordinate System
5.R.2: Graphing Linear Equations in Two Variables: Ax + By = C
5.R.3: The Slope-Intercept Form: y = mx + b
5.1: Scatterplots and Correlation
5.2: Fitting a Linear Model
5.3: Evaluating the Fit of a Linear Model
5.4: Fitting a Linear Time Trend
5.5: Scatterplots for More Than Two Variables
Chapter 5 Review

Chapter 6: Probability, Randomness, and Uncertainty

6.R.1: Multiplication and Division with Fractions and Mixed Numbers
6.R.2: Least Common Multiple (LCM)
6.R.3: Addition and Subtraction with Fractions
6.R.4: Fractions and Percents
6.1: Introduction to Probability
6.2: Addition Rules for Probability
6.3: Multiplication Rules for Probability
6.4: Combinations and Permutations
6.5: Bayes’ Theorem
Chapter 6 Review

Chapter 7: Discrete Probability Distributions

7.R.1: Order of Operations with Real Numbers
7.R.2: Solving Linear Inequalities
7.1: Types of Random Variables
7.2: Discrete Random Variables
7.3: The Discrete Uniform Distribution
7.4: The Binomial Distribution
7.5: The Poisson Distribution
7.6: The Hypergeometric Distribution
Chapter 7 Review

Chapter 8: Continuous Probability Distributions

8.R.1: Area
8.R.2: Solving Linear Equations: ax + b = c
8.R.3: Working with Formulas
8.1: The Uniform Distribution
8.2: The Normal Distribution
8.3: The Standard Normal Distribution
8.4: Applications of the Normal Distribution
8.5: Assessing Normality
8.6: Approximation to the Binomial Distribution
Chapter 8 Review

Chapter 9: Samples and Sampling Distributions

9.R.1: Ratios and Proportions
9.1: Random Samples
9.2: Introduction to Sampling Distributions
9.3: The Distribution of the Sample Mean and the Central Limit Theorem
9.4: The Distribution of the Sample Proportion
9.5: Other Forms of Sampling
Chapter 9 Review

Chapter 10: Estimation: Single Samples

10.1: Point Estimation of the Population Mean
10.2: Interval Estimation of the Population Mean
10.3: Estimating the Population Proportion
10.4: Estimating the Population Standard Deviation or Variance
Chapter 10 Review

Chapter 11: Hypothesis Testing: Single Samples

11.R.1: Translating English Phrases and Algebraic Expressions
11.R.2: Order of Operations with Fractions and Mixed Numbers
11.1: Introduction to Hypothesis Testing
11.2a: Testing a Hypothesis about a Population Mean with Sigma Known
11.2b: Testing a Hypothesis about a Population Mean with Sigma Unknown
11.2c: Testing a Hypothesis about a Population Mean using P-values
11.3: The Relationship between Confidence Interval Estimation and Hypothesis Testing
11.4a: Testing a Hypothesis about a Population Proportion
11.4b: Testing a Hypothesis about a Population Proportion using P-values
11.5: Testing a Hypothesis about a Population Standard Deviation or Variance
11.6: Practical Significance vs. Statistical Significance
Chapter 11 Review

Chapter 12: Inferences about Two Samples

12.1a: Inference about Two Means: Independent Samples with Sigma Known
12.1b: Inference about Two Means: Independent Samples with Sigma Unknown
12.2: Inference about Two Means: Dependent Samples (Paired Difference)
12.3: Inference about Two Population Proportions
Chapter 12 Review

Chapter 13: Regression, Inference, and Model Building

13.1: Assumptions of the Simple Linear Model
13.2: Inference Concerning β1
13.3: Inference Concerning the Model’s Prediction
Chapter 13 Review

Chapter 14: Multiple Regression

14.1: The Multiple Regression Model
14.2: The Coefficient of Determination and Adjusted R2
14.3: Interpreting the Coefficients of the Multiple Regression Model
14.4: Inference Concerning the Multiple Regression Model and its Coefficients
14.5: Inference Concerning the Model’s Prediction
14.6: Multiple Regression Models with Qualitative Independent Variables
Chapter 14 Review

Chapter 15: Analysis of Variance (ANOVA)

15.1: One-Way ANOVA
15.2: Two-Way ANOVA: The Randomized Block Design
15.3: Two-Way ANOVA: The Factorial Design
Chapter 15 Review

Chapter 16: Looking for Relationships in Qualitative Data

16.1: The Chi-Square Distribution
16.2: The Chi-Square Test for Goodness of Fit
16.2: The Chi-Square Test for Association
Chapter 16 Review

Chapter 17: Nonparametric Tests

17.1: The Sign Test
17.2: The Wilcoxon Signed-Rank Test
17.3: The Wilcoxon Rank-Sum Test
17.4: The Rank Correlation Test
17.5: The Runs Test for Randomness
17.6: The Kruskal-Wallis Test
Chapter 17 Review

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