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Impact of Quick Review of Math Concepts
Math proficiency is a vital skill for mastering concepts in General Chemistry courses. In this article, the author discusses simple yet powerful, pedagogical interventions implemented in General Chemistry courses to assist students with math. (a) A quick review of math concepts essential for solving chemistry problems has led to positive learning experiences in General Chemistry courses. This includes topics such as rearranging equations, exponents, etc. (b) A major challenge in General Chemistry courses include improper use of calculators. A quick discussion emphasizing the importance of parentheses/various function keys on calculators has led to efficient problem solving sessions. (c) Problem-solving skills are one of the most important skills acquired in these courses. Students learn how to read a problem, identify the given content, and then proceed to solve the problem. To alleviate stress during the problem solving sessions, pedagogies such as pause method were explored. Sample math review content, tips for using a calculator effectively, and problem solving strategies used in General Chemistry courses will be presented in this article.
Introduction
Math is an integral part of STEM courses. Applying math concepts to chemistry is extremely important while mastering problem solving skills in chemistry.1,2 Math hinders some students from appreciating chemistry in chapters such as unit conversions, stoichiometry, kinetics, equilibrium, etc. Students struggling with math quickly develop a fear of these concepts and eventually lose interest in chemistry.3 For some students, math makes problem solving a stressful experience in General Chemistry courses. Many books have served as valuable references to assist students with math for chemistry.3,4
There are multiple math concepts such as simple algebra, logarithms, etc. used in General Chemistry courses where students face challenges. Sample situations include use of calculators to plug in exponents correctly, use of calculators to divide numerators and denominators, rearranging equations, using logarithms to determine pH values, etc. When students solve chemistry problems using the aforementioned math topics, they determine incorrect values due to simple math mistakes. As an instructor, I have always explored ways to address math concerns in my General Chemistry courses.
Initially, I tried giving math quizzes/handouts before the start of the course/chapters, with limited success. Most students wanted to excel on these quizzes, and some of them actually did excel by seeking extra help during my office hours. This inspired me to run quick math concept reviews in my classes. Currently, I review relevant math content just before the start of chapters or during the problem solving sessions. This has alleviated math phobia during problem solving. I have realized that this exercise is an investment, rather than an expenditure of my class time. The net result is, students are more confident about math content during the problem solving sessions.
Details regarding some of the helpful interventions in my classrooms are discussed in the next section. Interestingly, past studies have suggested that extended math courses might provide extra time for students to master math contents required for future STEM courses.5 Here is an attempt to provide a quick refresher on math concepts before the introduction of chemistry concepts, thus preparing students for problem solving.
A survey was given to students in a General Chemistry course (Spring 2017). One of the questions asked during the survey was related to challenges in the course.
Question: “Describe the biggest challenge in the class” (read as course).
Some of the student responses are presented below.
“Use of calculator with larger problems.”
“Math”
“I found the first few sections to be a challenge because of the rearrangement of formulas. If i rearranged it wrong, I got the answer wrong”
“The biggest challenge would be trying to remember what equation to use and how to completely understand the concept of it.”
“My biggest challenge would have to be using the correct amount of sig figs”
Valuable feedback from students over the years has prompted me to invest my class time in reviewing math concepts. Various concepts reviewed during these courses are presented in the next section.
Review of math concepts in General Chemistry courses
As an instructor, I felt math reviews were a valuable use of my class time. Student misconceptions were identified either (a) through formative assessments and/or (b) during problem solving sessions.
(a) Formative assessments were simple multiple-choice problems on Canvas (Learning Management System). Students completed these problems (multiple choice questions) before the start of the chapter. They had one attempt to complete the simple homework problems and could not see the questions again. Based on the results from the homework, challenges/misconceptions were identified and addressed during class time. Problems similar (but not identical) to homework problems were discussed. The same problems were then assigned after the class to assess students’ learning from class. Simple problems used during formative assessments were correlated to short-term learning.
(b) Problem solving sessions: Class lectures were mostly problem solving sessions. Students were assigned problems and asked to work in teams. As teams performed the calculations, I walked around the classroom. If students were struggling with math or calculators, I presented the calculation or the use of calculator to the whole class.
Key advantages of these math review sessions included:
(a) The review sessions took about five minutes of class time.
(b) Math review concepts were carefully picked based on student needs. The concepts were not predetermined. The concepts were identified from formative assessments before the start of the chapter or during problem solving sessions in class.
(c) Students incorporated their learning into their problem solving right away.
A number of instances when math reviews were helpful to students are presented below. In each of these cases, challenges faced by the students, the context of the chemistry content, steps presented to students as a part of math review, and impact on student learning through formative assessments (if available from Spring 2016) are presented. Towards the end of the semester, long-term learning was assessed using final exam scores (Summative assessments).
Case 1: Exponents (and Powers)
Math challenge: In problems similar to the given example, some students struggle how to input exponents into calculators. Which buttons should they use?
Problem: Solve 10-2 * (10-5)2 = ????
Intervention: During lectures on unit conversions, kinetics and/or equilibrium, I demonstrate the use of a calculator. I stress the (a) use of parentheses, and (b) how to plug in exponents etc. using TI 30X calculators (a common type of calculator among my students).
Steps:
- Use Shift or 2nd button in TI30X calculator
- Algorithm: (2ndEE-2)*((2ndEE-5)^2) =
- Highlight the importance of parentheses and Shift or 2nd button on the calculator
Formative Assessment Results: Before the lecture, 10% of the students were confused about exponents. After the lecture, almost all the students had a better understanding of exponents.
(a) 
(b) 
Figure 1: Formative assessment results related to exponents (a) before and (b) after the lecture during Spring 2016.
Case 2: Unit Conversions
Math challenge: In problems similar to the given example, my experience shows that students forget to plug parentheses into their calculators. This leads to wrong results due to incorrect handling of numerators and denominators.
Problem:
= ???
Intervention: During lectures on unit conversions, I demonstrate the importance of parentheses and isolation of exponents using colors. Elsewhere, I have discussed the use of colors chalks or pens during my teaching, which aids better visualization.6
Steps:
-
1 mg = 1 * 10-3 g, 1 ng = 1 * 10-9 g
-
Pool numbers and exponents separately as demonstrated below
-
Add parentheses
-
Calculate numerator, then denominator, separately (especially helpful when numbers are not 1)
-
Divide numerator by denominator
Case 3: Choosing the correct equation
Math challenge: When problems are presented to students, based on the given information, students find it difficult to identify the correct equation.
Problem: Choose the correct equation to solve the problem.
Intervention: During in-class activities, I designed something I call the pause method.6 Students take a pause for a moment when they see a number, write it down, and continue working on the rest of the problem. Finally, they look for what needs to be calculated. Based on this exercise, they identify what is given and what they need to calculate, and determine the relevant equation to use.
Steps:
- Identify what is given and what needs to be calculated
- Find an equation that connects the given and to be determined parameters
Formative Assessment Results: Before the lecture, only 63% of the students were able to identify the correct equation. After the lecture, ~ 96% got the right answer.
(a) 
(b) 
Figure 2: Formative assessment results related to choosing the correct equations (a) before and (b) after the lecture during Spring 2016.
Case 4: Rearranging equations
Math challenge: How should the equations be rearranged to determine unknown parameters?
Problem: Calculate n1 in the given problem.
Intervention: I use colors to distinguish between given and unknown parameters.
Steps:
- Rearrange equation as V1 n2 = V2 n1
- Next, rearrange the equation with known and unknowns on each side
- Plug in values
Case 5: Logarithms, inverse logarithms and natural log
Math challenge: In the chapter on acids and bases, when working with the pH of acids and bases, there is a lot of confusion regarding the use of logarithms and inverse logarithms.
Problems:
1. Calculate the pH of 1.0 x 10-2 M HCl.
pH = - log [H3O+]
2. If pH = 3.3, Calculate [H3O+]
[H3O+] = 10-pH
Intervention: I quickly demonstrate how to run log and inverse log using calculators. I highlight the power of the shift button on the calculator.
Steps:
Problem 1:
- Algorithm -log (2nd EE-2) =
- Highlight the importance of parentheses
Problem 2:
- For inverse log use the shift button
- Algorithm is 2nd 10x (-3.3) = OR 10 ^ (-3.3)
- Highlight the importance of parentheses
Note: When we discuss thermodynamics, some students get confused between ln (natural logarithm) and log (common logarithm). I demonstrate how to use LN button on calculators.
Natural logarithms (ln): use LN function on your calculator.
Case 6: Word problems
Math challenge: Word problems! When problems are presented to students as sentences, students feel overwhelmed by the problems. Students struggle to break down the problem and proceed with the calculations.
Problem:
My car travels 40 miles per gallon. I drove from Boston to New York and back in my car. Distance between Boston and NYC is about 220 miles. If the gas price is $2.30/ gallon of gas, calculate the gas cost for the round trip. Choose the closest answer!
(a) $ 25.3
(b) $ 0.41
(c) $ 3800
(d) $ 12.7
Intervention: Pause method was used in class. Students pause for a moment when they see a number, write it down, and look for what needs to be calculated.
Steps:
When this problem was assigned before the lecture, a significant number of students did not bring me back from NYC! Once the pause method was discussed in class, they started paying attention to the problem and the right answer was obtained.
Formative Assessment Results: Before the lecture, only 59% of the students got the right answer. After the lecture, 96% of the students got the right answer.
(a) 
(b) 
Figure 3: Formative assessment results related to word problems (a) before and (b) after the lecture during Spring 2016.
Case 7: Significant figures in General Chemistry II
Math challenge: Significant figures and retention of content from General Chemistry I during General Chemistry II courses.
Problem:
The pressure in a boiler at a chemical plant is 2.00 atm. Express the pressure in psi.
(a) 29.40 psi
(b) 760 psi
(c) 29.4 psi
(d) 29 psi
Intervention: In class we quickly review the rules for significant figures for various mathematical operations.
Steps:
- For addition and subtraction, the least number of significant figures after the decimal place decides the number of significant figures.
- For multiplication and division, the least number of total significant figures decides the number of significant figures.
Formative Assessment Results: Before the lecture, 16% of the students did not pay attention to the number of significant figures. After the lecture, ~ 97% of the students got the correct answer.
(b) 
Figure 4: Formative assessment results related to significant figures (a) before and (b) after the lecture during Spring 2016.
Summative assessments:
Summative assessments were challenging, cumulative, and open format problems (not multiple choice questions) as opposed to simple problems assigned during formative assessments. Students were assessed during the final exam of the course. Student performance during the final exam in the aforementioned topics is presented below. As one would expect, many parameters influence the performance of students during the final exam. Regardless, good retention of material was observed at the end of the semester. Similar results were observed in teacher developed CPT (Chemistry Progress Test) in 10th and 11th grade chemistry courses.7
Figure 5: Summative assessment results during the final exam in the aforementioned topics (Data from Spring 2016).
Student feedback on Math reviews:
A survey was sent out to students regarding the usefulness of math review in class. Students were asked the following questions:
Question 1: “Rate the helpfulness of quick review of math topics and use of calculators during lectures.”
From 25 responses, 100% of the responses indicated these quick reviews to be useful or very useful during the course. The data is very small, as the class size was small during Spring 2017.
Question 2: Describe how math review helped you with learning chemistry.”
Student responses are presented below:
“prior to reviewing these topics I didn't know how to use the calculator correctly.”
“Since the last time I took a math course was a couple of years ago, it was really helpful just to touch on the quick math topics that we need to know for this course. “
“I myself am proficient at math and use of calculators, but I have learned some new calculator tips from class so far this semester”
“I feel like the topic I really got used to using from your class was using parenthesis more while plugging full equations into our calculators. It made solving problems quicker and more accurately.”
“Very helpful she gave me pointers on how to rearrange the formulas. I had an issue before on how to do it.”
“Made math concepts less challenging and less scary overall”
“It was helpful because we need to know the stuff before we start doing harder things that stem from this basis. This was definitely extremely helpful.”
Conclusions:
Simple and quick math concept reviews have led to the transformation of a stressful math phobic classroom to an engaged-learning environment. One of the key limitations with this endeavor is the time spent during class reviewing the math content. Alternatively, an instructor can create videos and post them before the start of chemistry content in class. Also, these topics can be presented to students during SI sessions (Supplemental Instruction) or help/review/discussion sessions. As students report, after the math reviews they are more proficient with the use of calculators, feel less overwhelmed about math, and more ready to apply math to chemistry problems. It is a worthwhile experience to see students performing calculations related to chemistry problems more confidently without worrying about the math. In sum, the math reviews are a valuable investment of my class time to engage students in learning chemistry.
Acknowledgements:
The author especially thanks students from General Chemistry courses for providing their valuable inputs through surveys and helping me promote an active learning environment in my classrooms. The author acknowledges the support of the Department of Chemistry & Physics, Salem State University, and Salem State University for supporting my passion for teaching.
References:
- R. D. Perkins, Do community college introductory chemistry students have adequate mathematics skills? J. Chem. Educ., 1979, 56 (5), 329.
- A. Ozsogomonyan and D. Loftus, Predictors of general chemistry grades, J. Chem. Educ., 1979, 56 (3), 173.
- R. Britten, Book Note: Essential math for chemistry students (Ball, David W.), J. Chem. Educ., 1998, 75 (9), 1098.
- D. P. Pursell, Review of calculations in chemistry: An introduction, J. Chem. Educ., 2015, 92 (8), 1286-1287.
- F. Ngo, and H. Kosiewicz, How extending time in developmental math impacts student persistence and success: Evidence from a regression discontinuity in community colleges, Rev. High Ed., 2017, 40 (2), 267-306.
- J. S. Ranga, Using color in lectures to aid student learning, Chemistry Solutions, 2016, 3, Nuts & Bolts.
- A. M. Preininger, Embedded mathematics in chemistry: A case study of students’ attitudes and mastery, J. Sci. Educ. Technol., 2017, 26, 58-69.
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Comments
Exponential Math Without a Calculator
Dr. Ranga,
In my old age, I work part-with other first-year instructors to write assignments that try to transfer a part of chemistry content to homework. As you do, we include a “just in time” review of math fundamentals needed for the next chem topic, and we too find it improves first-year retention and achievement.
On calculations, we encourage “estimate” teaching some of the strategies that Drs. Penn and Leopold have recommended, “then calculate,” using your approach of teaching calculator data entry rules for numbers -- including exponential terms -- on TI, RPN, and graphing calculators.
But after reading the papers in this conference, including the Texas MUST results, I’ve come to the view that whenever possible, we should be encouraging students, toward the end of calculations, to group exponential terms separately from the numbers in front and units after them, and to then solve the exponential math using mental math, not a calculator.
The following are my prejudices. The exponential math rules are easy to learn. Solving requires only whole number simple arithmetic. Practice in that arithmetic helps to keep arithmetic fundamentals “automated,” as cognitive science says they need to free up working memory. Putting in exponential terms only increases the complexity and likelihood of error in calculator use. Exponential mental math speeds estimation calculations, and students need to be able to check by estimation their calculator use.
My questions are:
1. In your “Case 1,” wouldn’t it be easier for students to solve using the rules of exponential mental math than by using a calculator?
2. I am arguing for teaching a bit more math (exponential math) to improve understanding and problem solving success. Would the time spent be a good investment?
-- rick nelson
Related to Exponential Math Without a Calculator
Hello Rick:
I agree, I don't discuss the ball park estimation part in this paper. But that is an important part of problem solving in my classes.
Responses to Questions:
1. In your “Case 1,” wouldn’t it be easier for students to solve using the rules of exponential mental math than by using a calculator?
Yes, these are homeworks before the class. They can do it without or with calculator (their choice, I am hoping they do it without a calculator). However, in class I do demonstrate the use of calculator with exponents, so that everyone can punch in numbers correctly if they use calculators (truth is - manual calculations are lot easier). I 100% agree, regarding "But after reading the papers in this conference, including the Texas MUST results, I’ve come to the view that whenever possible, we should be encouraging students, toward the end of calculations, to group exponential terms separately from the numbers in front and units after them, and to then solve the exponential math using mental math, not a calculator. " That is what I promote in Case 2. Manual calculations are recommended.
2. I am arguing for teaching a bit more math (exponential math) to improve understanding and problem solving success. Would the time spent be a good investment?
In my experience, it is a total investment of my time! We revisit math on need basis. For example with exponents in Case 1 + Case 2, they should be comfortable with them. Just in case they are not, the whole chemistry class feels meaningless to them. In such cases, a simple math review helps them pick up math concepts. The question on the summative assessment was related to conversion of mm3 = ________ km3. I am happy with the Fig 5 performance on unit conversions (will be happiest if it was 100%), but still some mistakes prevail such as not raising the whole exponent to the cubic term etc. When I run math reviews, the struggling ones do pay attention and they progress a lot with chemistry! It is a pleasure at that point to move on with chemistry with completely (almost!) engaged class.
I would like to be explicit about not using calculators for some sections. Would be interesting to see how students repond.
Thanks,
Jay
Regarding Case #1:
Jay and Eric,
Regarding Case #1, might I make these suggestions from my Chem-Math Project research?
1) In my collaboration with my Algebra II partner, he recommended using the language of the mathematics teachers when we teach chem-math. Therefore, instead of “solve,” we should say “evaluate.” This allows students to access previous learning from their mathematics instruction and lessens our burden.
In point of fact, when we say “solve,” we should say, “distribute terms, combine terms, and solve.”
2) When you teach the “use of parentheses,” you are going to access the “order-of-operations” students have been taught. So we should remind them of that.
3) Both you and Doreen Leopold of Paper #4 discuss procedures for manipulating exponents and logs. In these cases, you might rather reference their understanding of the “Law of Exponents” (or Powers) and the “Law of Logarithms” that they have (hopefully) learned in Algebra II.
4) In teaching better use of their calculators, you might also mention the value of using the reciprocal or x-1 button. I constantly find my student dividing denominators into “1” when they encounter them!
5) On page 3 at the bottom you discuss students gaining “a better understanding of exponents.” Might I suggest that what you mean to say is that they gained “a better understanding of the mechanics of exponents, since research has shown that students in their advanced mathematics classes are often able to use their mathematics procedures (the mechanics of the mathematics) with great facility, yet not truly understand them conceptually, a big problem for mathematics instructors that lingers over to us!
Regarding Case #1
I agree with Jay that it's very useful to remind students to use the "EE" key! (Actually in Case 1, if the EE key is used as shown, it isn't necessary to also use the parentheses.) This is a nice example too because students can easily predict the correct answer by "mental math." As Cary noted, they can be reminded of one of the "laws of exponents", that to multiply numbers (10A x 10B), we add the exponents (10A+B). This example also illustrates that squaring a number doubles the exponent.
As Cary also noted, this is also a good opportunity for students to review the "order of operations", which they may have learned using the acronym PEMDAS (parentheses, exponents (and logs), multiplication or division, addition or subtraction). A fun exercise might be to predict in which cases the use of parentheses is required, and in which it's unnecessary.
I like to give this example of the importance of using the "EE" key in my general chemistry class (e.g., in the context of using a calculator to determine the pH of a 3 x 10-4 M solution of a strong acid, after we have predicted the answer using "mental math"):
Do not use “x 10 ^ (-) 4” to enter a number in scientific notation !!!
(This can lead to incorrect answers (unless one remembers to add extra parenthesis (to specify the order of operations)).)
Example: We know that 3000 / 300 = 10
but keying this in on a calculator as
3 x 10^3 (divided by) 3 x 10^2 = 1x105 wrong!
However,
3 2nd EE 3 (divided by) 3 2nd EE 2 = 1 x 101 right!
Related to Regarding Case #1
Hi Doreen:
"but keying this in on a calculator as
3 x 10^3 (divided by) 3 x 10^2 = 1x105 wrong!"
This is such a common mistake. That is why " Actually in Case 1, if the EE key is used as shown, it isn't necessary to also use the parentheses." I always make them use exponents with parantheses, they get confused when to use/not to use. One they start using it, they are on autopilot with its use. In this particular example, as Eric pointed out, manual estimate is lot easier.
I liked this idea. I am going to try this next semester to guage their understanding of PEMDAS. "As Cary also noted, this is also a good opportunity for students to review the "order of operations", which they may have learned using the acronym PEMDAS (parentheses, exponents (and logs), multiplication or division, addition or subtraction). A fun exercise might be to predict in which cases the use of parentheses is required, and in which it's unnecessary."
Thanks,
Jay
Order of operations
Another issue with order-of-ops is that different calculators and manufacturers actually use different systems. Thus, it would be important to stress to students that they actually LOOK in their operating manual. They should find out if their system agrees with PEMDAS, because some calculators do not.
related to Order of operations
I agree Cary.
Lot of them are surprised when we discuss SHIFT option on their calculators.
A fun exercise
"A fun exercise might be to predict in which cases the use of parentheses is required, and in which it's unnecessary." I opine that while it may be presented as a fun exercise (I'm always in favor of a lighter tone in class...) this is actually a specific and fundamental skill, and should be presented this way, reviewed occasionally in some form (un/graded quizzes?) and tested on exams.
Use Shift or 2nd button in TI30X calculator
Just got back in town and getting to these papers...I saw in Case 1, "Use Shift or 2nd button in TI30X calculator", and true to my roots wondered why a scientific calculator would put EE on a shifted key. Not familiar w TI calculators, but I looked up "TI-30Xa", and it seemed both less expensive and has the EE as a main key. Is is reasonable to "recommend" certain features to parents and students. Certainly at the undergrad level, I expect a minimum calculator performance and interface.
Related to Use Shift or 2nd button in TI30X calculator
Hi Rich:
Every calculator has its own style of placing the keys. In fact, on the first day of my class or first day of math calculation, I ask people to raise their hands with a variety of calculators. So that they can chat with each other in case they have calculator based questions. Every calculator is different. This conversation came up somewhere earlier in these conference discussions. This makes things a bit more difficult for students/everyone.
Thanks,
Jay
ask people to raise their hands with a variety of calculators
Thanks for the note. Much clearer. And I think that your comment "chat with each other in case they have calculator based questions" is a very useful point, to be encouraged. One of my classroom "planks" is strongly encouraging, or directing if need be, collaboration and discussion. One of the most useful skills that students can take to the workplace is the ability to consult with others across a broad range of backgrounds. As an engineer, I used to discuss some material with the department secretary, because even though she thought herself unqualified to chat, she often had very useful insights. And I often find "smart" students actually saying that they should not "waste" their time by chatting with those who are not as far along in the material, an attitude I find completely unacceptable. Hence my active focus on collaboration and consultation.
ask people to raise their hands with a variety of calculators
Hi Rich:
"And I often find "smart" students actually saying that they should not "waste" their time by chatting with those who are not as far along in the material, an attitude I find completely unacceptable. Hence my active focus on collaboration and consultation." This is so common. So, I have my problem solving and class actitivies in groups of 3 atmost 4. After the completion of activities, I randomly pick one and grade. All the team members receive the same points. This is so rewarding, there is so much more participation every member. I like this strategy (has worked very well in my classes), when the quick ones take time and explain it to the slow ones. This is some of my most cherished moments as a teacher. Something about peer learning, they learn so fast from their peers. To bring in humor, I make them play rock paper scissors and I select one of the activitiy sheets.
Thanks,
Jay
humor, I make them play rock paper scissors
+1 for active engagement with students! I, too, think that a bit of levity encourages better performance. It's not for everyone...each to his own style...
Related to Regarding Case #1
Hi Cary:
Thanks for the pointers.
"4) In teaching better use of their calculators, you might also mention the value of using the reciprocal or x-1 button. I constantly find my student dividing denominators into “1” when they encounter them!"
This is such a useful key. I wish students would use it much more.
5. "On page 3 at the bottom you discuss students gaining “a better understanding of exponents.” Might I suggest that what you mean to say is that they gained “a better understanding of the mechanics of exponents, since research has shown that students in their advanced mathematics classes are often able to use their mathematics procedures (the mechanics of the mathematics) with great facility, yet not truly understand them conceptually, a big problem for mathematics instructors that lingers over to us!"
I agree with your suggestion. Actually I have had positive experiences. When I teach, Gen Chem I, I visit these topics. It is rewarding to see same students in Gen Chem II (after few semesters or next semester) still remembering these tips. This brings back to our conversations, may be they should see math concepts often. My sincere hope is they will carry it to upper level courses. I will be mindful in future to use jargons from math classes.
Thanks,
Jay
Manipulating exponentials
It just occurred to me during this discussion that maybe we should emphasise that 1.02 x 10-5 is a representation of a NUMBER in scientific notation, and that entering numbers represented this way is NOT the same as entering the product 2 x 4.
The problem with Maths and Chemistry is a world-wide one. Several years ago I wrote an activity for BestChoice on powers of 10 and scientific notation. I have included links to the activity below - pardon for the scary-looking links.
https://www.bestchoice.net.nz/?c=25&hide=wseh&demo=1#cs=6022&p=11956 (for laptops and desktops)
https://www.m.bestchoice.net.nz/?c=25&hide=wseh&demo=1#cs=6022&p=11956 (for phones)
I have no idea of what percentage of the 8 000 or so students who have done this activity used calculators. My data shows that they are able to cope with positive exponents (multiplication and division), but we are down to the 70% first right for problems like 10-14/10-6 (lots of 10-20 answers) and 10-9/10-11 where they divide by an exponent.
There are also problems with more complex operations (like (10^-2/10^2)^2. 1200/8000 of them put in 0 as the answer instead of -8. On the other hand when both exponents were positive (like (10^8/10^2)^1/2) 80% got it right despite the unusual looking exponent.
I also put in some conceptual problems where they had to choose from a set of options, for example, two numbers that differ by a factor of 1000, two numbers that are the inverse of one another, two numbers that are the square of one another..... They don't do very well on those at all (actually they are all over the place).
For the next to the last problem where they had to estimate the exponent (the example just required adding and subtracting the exponents), we are down to 57% first right.
I too am a great believer in just-in-time math - namely re-introducing the maths required when required in teaching chemical concepts. They do forgot stuff that they do not use. We all do. This has all been very interesting to listent to other people's attempts to sort this out and their experiences.
Related to Manipulating exponentials
Hello:
"I have no idea of what percentage of the 8 000 or so students who have done this activity used calculators. My data shows that they are able to cope with positive exponents (multiplication and division), but we are down to the 70% first right for problems like 10-14/10-6 (lots of 10-20 answers) and 10-9/10-11 where they divide by an exponent." I have noticed a lot of this as well. For a questions 10^9 vs 10^5, which number is greater, most of them get it right. When I change to 10^-9 vs. 10^-5, which number is greater, I have sometimes had 50% of them answering it wrong. "Negative number phobia" Happens a lot while doing pHs as well.
"(10^-2/10^2)^2" In this case I have seen lots of errors, I really wish my students did this calculation manually with no calculators. With calculators they make so much mistakes with parenthesis. I like Cary's note related to pointers from math courses.
" choose from a set of options," I am bit nervous about multiple choice (even though I have used MCQ examples in this article). I try not to have any multiple choice during finals. I wonder if they start guessing answers from MCQ when they are not sure?
"https://www.bestchoice.net.nz/?c=25&demo=1#cs=6022&p=11974" I agree, that is why I tried Case 2. When the numbers and exponents are separated out, math gets a little less overwhelming. Calculators without parenthesis makes things more hard.
"I too am a great believer in just-in-time math - namely re-introducing the maths required when required in teaching chemical concepts." I 100% agree. Moreover, I am also mindful of my students' background. When they come back to college after a break from high school, these small just-in-time math tips greatly aid their learning.
Thanks,
Jay
Multiple choice questions
The most important thing about assessing is to use a question style appropriate to the assessment. One size does not fit all.
My focus is using multichoice questions in a teaching sense because that is what I am trying to do on the web. We have a big variety of types of questions in BestChoice, and many of them are multichoice. The standard multichoice question with distractors set out in a table underneath are very good, in my view, for questions that can be answered on the basis of "one thought". Certainly there is guessing, but if there is feedback given for every guess, then students are learning at the same time as they are guessing. I find that there is a lot more guessing for a multitick question (more than one right answer) than for a multichoice (one right answer), but there are sometimes things that I would like to find out which are best assessed by multitick. And in the case you refer to I would have never guessed from my other data that students would have so much trouble picking out numbers expressed as simple powers of 10 that differ by a factor of 1000.
Related to MCQ
Hello:
"Certainly there is guessing, but if there is feedback given for every guess, then students are learning at the same time as they are guessing." I like your idea of asking for reasoning with respect to their choices. That will probably prompt them to not guess! Perhaps I should ask them to reason even during open format questions.
"My focus is using multichoice questions in a teaching sense because that is what I am trying to do on the web. We have a big variety of types of questions in BestChoice, and many of them are multichoice. The standard multichoice question with distractors set out in a table underneath are very good, in my view, for questions that can be answered on the basis of "one thought"". Distractors are great! I have seen when students are asked to perform [2 x 10-5] [3 x 10-5] , some of them give me an answer of 5 x 10-5. Especially in kinetics and equilibrium problems. When I discuss that [][] indicates multiplication in between the two parts as oppose to addition, there is a bit surprise in class.
Thanks
mental math, calculator math, review
My experiences are that these 3 are fundamental approaches. I prefer to issue some quizzes and exams where no calculators are used, and rounding, estimating, and mental math are emphasized. Other quizzes and exams specifically call for the use of calculators. Why both? Because students need both sets of skills, and by separating the testing formats, I can see more clearly where strengths and weaknesses lie. As for review, I often put review questions on quizzes and exams, because my experiences indicate strongly that students need the expectation of regular review to encourage them to commit to acquiring skill sets. Testing once, or even a few times right after material is covered, seems to show short-term proficiency, but not long-term competency, and math skills are vital, everyday skills for STEM students. Even comprehensive final exams, while providing pressure to review material that is weeks or months old, is not adequate. I see too many students cramming for finals, doing well on them, then returning from a break with a significant loss of skills. The downside to repetitive testing is the man-hour cost up front; the benefits are that review material is easier to grade, and less time is needed to support lagging students as classes progress. We regularly see that TAs actually boost their own skills, precisely, I opine, because they are constantly reviewing material.
Related to mental math, calculator math, review
Hi Rich:
I am directing your to Doreen's message. She put it so well here (Topic:Calculator Use in High Schools in Other Countries?) One is silver, and the other is gold! I echo her teaching sentiments. I agree, with your statement "We regularly see that TAs actually boost their own skills, precisely, I opine, because they are constantly reviewing material." In fact, if I have a students aspiring for med school or grad school, I suggest them to be a TA to stay on top of content for their GRE or MCAT exams. It brings us back to 10,000-hour rule (I read this as lot of practice) to master something!
Thanks,
Jay
They do forget stuff that they do not use. We all do.
+1
Allow Only 4 Function Calculators?
Drs. Leopold and Ranga,
Permit me ask you both the same question.
I see the value of estimation, and agree with Dr. Leopold that it takes at least occasional tests without calculators allowed to encourage students to learn those skills.
At the same time, given that typically 30% to 40% of current students, as shown by the MUST results, likely do not know their times tables, I think we may realistically need to gradually build those skills to avoid losing too many of the “bio kids” who don’t have the preparation of the pre-meds and prospective physical science and engineering majors.
So here’s my question. For cases when precise answers are sought, could we start by simply requiring that the exponential math be done by mental arithmetic, and do so with a calculator that does not accept exponential notation?
If students are taught the simple rules for exponential math, what calculations are there in the two semesters of general chemistry that could not be solved to needed precision using a simple, cheap 4 function calculator? Such calculators add, subtract, multiply, and divide, usually include square root, but do not accept exponential terms or have log buttons.
Logs are usually not reached until 2nd semester, and if students are given (or better, commit to memory) the logs to 2 places of numbers 0 to 10, they can interpolate to get most answers to proper sig figs without the log button.
Higher roots of the number (non-exponental component) of values might need to be estimated by trial and error in Ksp and maybe other K calculations, but that would be a valuable numeracy exercise as well.
In the Texas MUST test, one of the easiest questions was Q5, which was two simple powers of 10 divided by two powers of 10, and according to Texas Figure 2, 65% to 90% missed it, of students in both semesters of gen chem.
Exponential math requires addition, subtraction, and multiplication of generally small whole numbers. Those are the skills that most need to be automated, because they are the foundation for numeracy. Requiring exponentials be solved separately from the numbers on a calculator would encourage overlearning of at least those fundamental skills.
If they can’t add, subtract, and multiply simple whole numbers, can we say they know chemistry?
So, to mix requiring estimation and precise answers, throughout gen chem, for which topics would a precise answer require more than a 4 function calculator?
-- rick nelson
Related to Allow Only 4 Function Calculators?
Hi Rick:
I agree with all the points brought up by Doreen here. I am glad she brought up the time aspect. Students are in general very nervous about chemistry classes. I teach mostly Gen Chem II. When I tell them they can take the exam as long as they want, insterstingly most of them finish on time. A few semesters, when I had classes back to back and I say I have class right after. More students feel extra might have helped them. May be students being relaxed is important to run problems more efficiently. It might have been just a coincidence, but time seem to make a difference in these two instances.
I have some students who come back to chemistry after a break from high school etc. Calculator-less problem solving sessions might be worth exploring in a during regular classtime might be worth investing (rather than exams). Would be interesting to ask students to put their calculators away and run some problems. Then compare another problem (similar one) with the use of a calculator.
I just had an exam. 2^n = 8, was a problem they had to solve in the context of orders of reaction. Clearly, this is calculator-free math scenario. A majority of them got it as 3 (I am very happy about that!), a few students still presented an answer of n = 4. I keep wondering, where did they get confused? They saw similar math in MCQ based HW + problem solving sessions with multiple open format problems in class + quiz. In class, I presented a similar misconception multiple times. I wonder if it is just exam anxiety?
May be I am opening up a new conversation here. Calculators, are they really helping our students? Infact, improper use of calculators leads to lot more confusion in these courses. I have also noticed interesting learning process in my classrooms. Intitially they are glued to calculators. Eventually, they switch to manual mode. I am guessing, your idea of promoting calculator free course might be a way to go. As discussed elsewhere, do calculators offer placebo effect? "I have noticed something amazing. I wonder once they understand how math works, calculators have a placebo effect. I have noticed students saying "my calculator is acting up". That is a 'aha' moment for me with respect to their learning. When I ask them to check, they run estimates and start performing math in a manual mode. Often times it is parenthesis issue on the calculator. So, I agree we should "keep them in shape" by revisiting every now and then."
Rick, thanks for your great ideas. Would like to try out calculator-free problem solving in my class.
Thanks,
Jay
Calculators, are they really helping our students?
I agree with all the points brought up by Doreen here. Props for getting so many succinct points in that post. I am a strong advocate for two seeming opposed concepts: mental math skills and math tool literacy. There are two major reasons for this, and they are related: students' time is precious, and so is an employer's. I opine that the time spent on building mental math skills as early as possible in a student's career pays off in dividends rapidly because the student handles the math portion of problem-solving more quickly, AND has increasing confidence, which I seem to observe translates directly to less time spent second-guessing oneself. Furthermore, it is my experience that employers have, for the past few decades, been very clear that STEM students entering the workforce are not prepared WRT math and math tool skills. So rather than limit them to a 4-func calculator, I suggest that there be a strong emphasis (including time out of class and quizzes) on mental math, and equally, pushing them to build math tool skills (advanced calculators, spreadsheets, Mathcad, MATLAB, etc.), as appropriate. The question to me is not whether calculators are helping our students, but are we teaching them effectively to use math tools effectively? My observations of "advanced placement" first year STEM students suggest to me that we are not.
the need (or not) for significant figures
Just throwing this idea out: What if lecture was limited to estimation/approximation and the significant figure math was done in the lab? This might allow for more emphasis on the concepts in lecture? I don't mean to go off on a tangent or start another thread I'm just wondering what would happen if the calculators were totally stashed in the classroom and the more exact calculations using student data were limited to the lab.
Sig figs
I agree with you, I think that teaching significant figures at the very beginning of a chemistry course is not very value added. Students find sigfigs confusing, frustrating, and boring and they retain very little of it. I find students will even round numbers like 103.5 to one signficant figure by just saying it is 1, I'd rather have too many significant figure than truncated numbers like this. I think we would get much more success with leaving basic sigfig concepts in the laboratory and teaching the estimation skills in the lecture course. I simply try to get my students to recognize if the number should be big or small in magnitude and + or - is signage, but it would be nice to have time to incorporate some of the estimation concepts that have been presented in these papers.
Related to sig figs + the need (or not) for significant figures
Hello Lou and Janna:
"103.5 to one signficant figure by just saying it is 1" This is so common. I joke around saying, ok then I will borrow $103 from you and return back $1, looks like based on your one sig fig rule, you are ok with that. It is a wake up call. No that doesn't sound right... the discussion goes on, conversation clearly suggest something is not ok etc. In my view, based on lot of nice ideas discussed here, perhaps the entire chapter on unit conversions in gen chem can be discussed w/o calculators.
"What if lecture was limited to estimation/approximation and the significant figure math was done in the lab? This might allow for more emphasis on the concepts in lecture?" I really like to idea of emphasizing estimations/manual calculations in these courses.
Thanks,
Jay
Sig fig in the lab
We have, for decades, used titration experiments in the lab to teach about stoichiometry, uncertainty and significant figures. There are no lectures on these topics at all.
From a personal perspective, even though I know very well the importance of significant figures in communicating with other scientists, all of the emphasis that is placed on significant figures for beginners drives me crazy. There is so much involved in beginners doing, for example, a titration calculation which usually involve three steps at least: converting the given quantities to the amount in moles, adjusting the amount in moles on the basis of the stoichiometry and then converting the amount in moles to the requested unit of the outcome. To me the important thing is that beginners get an answer consistent with all of that expressed to ANY number of significant figures. Using the correct number of significant figures is the icing on an already significant cake.
I very well know, from my long experience with providing on-line calculations in BestChoice, how frustrating students find systems where their otherwise correct answer was not accepted due to significant figure differences. The frustrated ones click on the comment button on that page and vent their wrath.
In recent times I had an idea which seems to have worked well in BestChoice. The final answer is two parts. The first is called the calculated answer which has great leeway with sig figs, and the second part (the reported answer) is revealed when the calculated answer is correct. In this part they choose from a dropdown which has the "answer" to a variety of significant figures. Often in the first few problems the reported answer is done poorly in comparison to the calculated answer, but then with feedback they get the idea.
Significant figures is an important part of experimental science but I think it is sad when it is used in a punitive way especially when so much other thought has to go into getting the calculated answer. I am particularly hyped up about this today because recently I was working through a calculated on a well-known free-access on-line system and was told I was wrong because I had four rather than three sig figs.
Regarding sig-figs
Sheila,
Yes, I agree – beating up students with sig-figs is counter-productive. But if we approach the subject gradually and gently, with plenty of practice and reminders, I find that students come around and actually take pride in showing they can follow correct sig-figs.
In the capstone Titration Project I wrote, and that we ran for years at Exeter HS (JCE 65(1):80, 1988), I kept all solutions and titrations to 3 sig-figs, insisting on agreement in all multiple titration runs to +/- 0.003 M. Some students caught on quickly to both the mechanical procedure and thus the agreement and were pleased. Others got frustrated but, upon seeing their peers accomplish this, came around. I found that this value was replicable with careful work, but was intolerant of loose work. Each student had two burets and thus had complete responsibility for his or her result, and could back-titrate (which was absolutely necessary for efficient practice and learning). However, I was lucky to have inherited ~60 nice glass 50 mL burets, half with stopcocks that could be removed and half with jet-tips with the little glass bead in a rubber tube, thus accommodating 20-26 students in each double lab-period.
I realize that this is at the end of the year, not the beginning when we typically present sig-figs, but I present it as an example of how I reinforced the use of sig-figs. Most students understood the concept (or at least the mechanics) by then, but some took all year to come around.
Which misconceptions were identified
Dr. Ranga,
I would like to try your strategy.
Could you share the misconceptions that were identified for each rubric of the multiple choice questions?
Did you focus in a different way to different answers? Could you elaborate on that?
Thank you,
Malka Yayon
Related to Which misconceptions were identified
Hi Malka:
Example 1: In this particular case, 2^n = 8, students get confused between "powers" with "multiplication", they think it is the same operation. In this case, I try various methods.
Method 1. Here we try n value by trial and error:
let us give a value of 0 to n, then the right hand side has to 1, which is not the case, hence n=0 is not the correct option.
let us give a value of 1 to n, then the right hand side has to 2, which is not the case, hence n=1 is not the correct option.
let us give a value of 2 to n, then the right hand side has to 4, which is not the case, hence n=2 is not the correct option.
let us give a value of 3 to n, then the right hand side has to 8, which is the case, hence n=3 is the correct option.
Method 2: How many times should I multiply 2 by 2's to get an answer of 8. They do it manually, 2 x 2 x 2, 3 times, hence an answer of 3.
Power is how many times you multiply a number by itself, while multiply is just x once. They mostly get it, but as I discussed earlier there are always a few who miss it.
Example 2: While dividing denominators with exponents (As Doreen discussed in one of the other posts - negative numbers are intimidating).
Method: Pool all number and exponents (there is lots of discussion in this conference on that) . Number is 50 (really simple), while handling exponents when you bring a denominator to numerator the sign changes. Misconception: Students change sign to negative numbers, but not for positive numbers. I emphasize that both positive or negative, when the exponents come up they change sign. It kind of gets better by the time we wrap up unit conversion chapter. I run sample exponent problems with positive and negative powers.
Example 3: They don't read problems completely, they start attempting them right away
Method: I call it PAUSE METHOD. They see a number they have to scream pause in class. Eventually it is a silent pause in their mind. They read the problem and actually highlight what is asked for. In my paper, they forget to bring me back from NYC. The real misconception is, it is okay not to read a problem carefully, which is not the right approach in science. I ask tricky questions, for example in gases chapter, I ask for the amount of gas lost as opposed to gas remaining in the vessel. Unless they read a question carefully, they can't get the correct answer.
Thanks,
Jay
Case 3, Figure 2: Is the question the same after lecture?
If so, as the figure seems to suggest, then I wonder if the students who moved from wrong answers to the correct one actually learned the material, or whether they memorized the correct reply as a result of the lecture, even if the question were not specifically adressed? I'm not assuming this is the case, but wondering how it all is addressed.
Reg Case 3, Figure 2: Is the question the same after lecture?
Hi Rich:
That is my hope! I hope they learned something from my class. Actually, these on online HW questions which are shuffled. 4-5 questions related to the topic appear and they have one chance to answer the question and the question gets locked. So, they don't see the answer. They come to class, and I DO NOT discuss the same question. However, I discuss similar scenarios during lectures. I give the same questions to really compare the same questions just out of shear curiousity. Shuffling the questions + locking the questions + seeing the answers before coming to class, my hope is they are learning something. After class, when they rerun these problems, they can see the answer. But the good news is (hopefully) they have learned something, so often times it is right.
Thanks,
Jay