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Gray Matter

Does Math Make You Smarter?

By Manil Suri

Dr. Suri is a professor of mathematics at the University of Maryland, Baltimore County.

• April 13, 2018

Does studying mathematics enhance your overall mental prowess?

Abraham Lincoln certainly believed so, embarking on the arduous task of mastering Euclid’s treatises on geometry to increase his cognitive capacities, in particular his linguistic and logical abilities. This idea — that mathematics strengthens your mind much as physical exercise strengthens your body, helping you negotiate a variety of mental challenges — goes all the way back to Plato. Alive and well in today’s world, it is one reason popularly given for why everyone should study mathematics.

So it can come as a surprise to learn that cognitive psychologists have a different take on the issue. Various studies point to the conclusion that subjecting the mind to formal discipline — as when studying geometry or Latin — does not, in general, engender a broad transfer of learning. There is no sweeping increase of a general capacity for tasks like writing a speech or balancing a checkbook.

But surely a narrower claim is true: that mathematics, so systematically built as it is on inference, must develop logical thinking. Right?

By “logical,” I mean the kind of thinking needed to solve the following problem:

Four cards are laid in front of you, each of which, it is explained, has a letter on one side and a number on the other. The sides that you see read E, 2, 5 and F. Your task is to turn over only those cards that could decisively prove the truth or falsity of the following rule: “If there is an E on one side, the number on the other side must be a 5.” Which ones do you turn over?

Clearly, the E should be turned over, since if the other side is not a 5, the rule is untrue. And the only other card that should be flipped is the 2, since an E on the other side would again disprove the rule. Turning over the 5 or the F doesn’t help, since anything on the other side would be consistent with the rule — but not prove it to be true.

[ ALSO READ: Standing Up at Your Desk Could Make You Smarter ]

This innocuous-looking puzzle, a variation of which was introduced by the British psychologist Peter Wason in 1966, has been called “the single most investigated paradigm in the psychology of reasoning.” If you answered E and 2, congratulations: You are among the roughly 10 percent of the public able to solve the puzzle. Many reasons have been advanced for this poor showing, including the lack of relevance of such an abstract exercise to people’s daily lives.

Most people reflexively eliminate the cards not explicitly specified in the rule (the F and the 2) and then continue with slower, more analytic processing only for the E and the 5. In this, they rely on an initial snap judgment about superficial similarity, a tendency that some scholars speculate evolved in humans because in most real-world contexts, quickly detecting such similarities is a good strategy for survival.

Interestingly, though, it turns out that if the puzzle’s abstract rule is translated into terms that are logically equivalent but grounded in real-world experience — as in, “If someone is drinking beer at a bar, she must be at least 21 years of age” — then the success rate jumps to 75 percent or more.

I learned about the Wason selection task and its intricacies from a fascinating recent book, “Does Mathematical Study Develop Logical Thinking?” by the education and cognition researchers Matthew Inglis and Nina Attridge. They conducted experiments that found that university students studying mathematics were just as likely as those studying history to quickly reject the F and the 2 cards. But differences emerged in the slower, more effortful cogitative phase that followed, leading to divergent success rates in the end: 18 percent for the mathematics students versus 6 percent for the history students.

Based on results from a slate of such reasoning tasks, Dr. Inglis and Dr. Attridge show that studying higher mathematics (at the advanced secondary and college levels) does lead to an increase in logical ability. In particular, mathematics students become more skeptical in their reasoning — they begin to think more critically.

But these gains, though significant enough to establish a causal relation between mathematics education and logical thinking, are too modest to settle the debate on how much mathematics should be prescribed as part of a general education, and for which students. (An 18 percent success rate is hardly compelling.) Moreover, there is the possibility of a self-selection effect: Students with the greatest potential to get a benefit in their logical reasoning might be disproportionately attracted to mathematics classes in the first place, so these gains might not apply to the entire population.

In any event, the most crucial finding of such research, in my view, is how much insight the psychological study of learning can contribute to the practical teaching of mathematics — two fields of endeavor that are too often pursued separately. It is sadly telling that while the Wason selection task is well known among psychologists, it is not familiar to most mathematicians and math teachers.

I propose we start to teach the Wason selection task in mathematics courses at the high-school level and higher. The puzzle captures so much that is essential to mathematics: the nuts and bolts of inference, the difficulty of absorbing abstract concepts when removed from the context of real-world experience, the importance of a slow, deliberative cogitative process and the pitfalls of instant intuitive judgments. I presented the puzzle to a recent college class of mathematics majors and they listened with rapt attention afterward — startled by their lowly 19 percent success rate.

Logical thinking may be promoted by mathematics, but it is a gradual and complex learning process. Psychological insight into learning, such as that offered by Wason’s puzzle, can give students a head start by educating them on the challenges they will face.

Manil Suri , the author of the novel “The City of Devi,” is a mathematics professor at the University of Maryland, Baltimore County, and a contributing opinion writer.

Follow The New York Times Opinion section on Facebook and Twitter (@NYTopinion) , and sign up for the Opinion Today newsletter .

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Open Access

Peer-reviewed

Research Article

Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors

Roles Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Resources, Writing – original draft, Writing – review & editing

Affiliation School of Mathematics and Statistics, The University of Sydney, Sydney, Australia

Roles Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliation School of Arts and Humanities, Edith Cowan University, Joondalup, Australia

• Clio Cresswell, 
• Craig P. Speelman

• Published: July 29, 2020
• https://doi.org/10.1371/journal.pone.0236153
• Peer Review
• Reader Comments

Mathematics is often promoted as endowing those who study it with transferable skills such as an ability to think logically and critically or to have improved investigative skills, resourcefulness and creativity in problem solving. However, there is scant evidence to back up such claims. This project tested participants with increasing levels of mathematics training on 11 well-studied rational and logical reasoning tasks aggregated from various psychological studies. These tasks, that included the Cognitive Reflection Test and the Wason Selection Task, are of particular interest as they have typically and reliably eluded participants in all studies, and results have been uncorrelated with general intelligence, education levels and other demographic information. The results in this study revealed that in general the greater the mathematics training of the participant, the more tasks were completed correctly, and that performance on some tasks was also associated with performance on others not traditionally associated. A ceiling effect also emerged. The work is deconstructed from the viewpoint of adding to the platform from which to approach the greater, and more scientifically elusive, question: are any skills associated with mathematics training innate or do they arise from skills transfer?

Citation: Cresswell C, Speelman CP (2020) Does mathematics training lead to better logical thinking and reasoning? A cross-sectional assessment from students to professors. PLoS ONE 15(7): e0236153. https://doi.org/10.1371/journal.pone.0236153

Editor: Jérôme Prado, French National Center for Scientific Research (CNRS) & University of Lyon, FRANCE

Received: January 13, 2020; Accepted: June 30, 2020; Published: July 29, 2020

Copyright: © 2020 Cresswell, Speelman. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Mathematics is often promoted as endowing those who study it with a number of broad thinking skills such as: an ability to think logically, analytically, critically and abstractly; having capacity to weigh evidence with impartiality. This is a view of mathematics as providing transferable skills which can be found across educational institutions, governments and corporations worldwide. A view material to the place of mathematics in curricula.

Consider the UK government’s commissioned inquiry into mathematics education “Making Mathematics Count” ascertaining the justification that “mathematical training disciplines the mind, develops logical and critical reasoning, and develops analytical and problem-solving skills to a high degree” [ 1 p11]. The Australian Mathematical Sciences Institute very broadly states in its policy document “Vision for a Maths Nation” that “Not only is mathematics the enabling discipline, it has a vital productive role planning and protecting our well-being” (emphasis in original) [ 2 ]. In Canada, British Columbia’s New 2016 curriculum K-9 expressly mentions as part of its “Goals and Rationale”: “The Mathematics program of study is designed to develop deep mathematical understanding and fluency, logical reasoning, analytical thought, and creative thinking.” [ 3 ]. Universities, too, often make such specific claims with respect to their teaching programs. “Mathematics and statistics will help you to think logically and clearly, and apply a range of problem-solving strategies” is claimed by The School of Mathematical Sciences at Monash University, Australia [ 4 ]. The School of Mathematics and Statistics at The University of Sydney, Australia, directly attributes as part of particular course objectives and outcomes skills that include “enhance your problem-solving skills” as part of studies in first year [ 5 ], “develop logical thinking” as part of studies in second year, which was a statement drafted by the lead author in fact [ 6 ], and “be fluent in analysing and constructing logical arguments” as part of studies in third year [ 7 ]. The University of Cambridge’s Faculty of Mathematics, UK, provides a dedicated document “Transferable Skills in the Mathematical Tripos” as part of its undergraduate mathematics course information, which again lists “analytic ability; creativity; initiative; logical and methodical reasoning; persistence” [ 8 ].

In contrast, psychological research, which has been empirically investigating the concept of transferability of skills since the early 1900s, points quite oppositely to reasoning skills as being highly domain specific [ 9 ]. Therefore, support for claims that studying mathematics engenders more than specific mathematics knowledge is highly pertinent. And yet it is largely absent. The 2014 Centre for Curriculum Redesign (CCR) four part paper “Mathematics for the 21st Century: What Should Students Learn?” concludes in its fourth paper titled “Does mathematics education enhance higher-order thinking skills?” with a call to action “… there is not sufficient evidence to conclude that mathematics enhances higher order cognitive functions. The CCR calls for a much stronger cognitive psychology and neuroscience research base to be developed on the effects of studying mathematics” [ 10 ].

Inglis and Simpson [ 11 ], bringing up this very issue, examined the ability of first-year undergraduate students from a high-ranking UK university mathematics department, on the “Four Cards Problem” thinking task, also known as the Wason Selection Task. It is stated as follows.

Each of the following cards have a letter on one side and a number on the other.

Here is a rule: “if a card has a D on one side, then it has a 3 on the other”. Your task is to select all those cards, but only those cards, which you would have to turn over in order to find out whether the rule is true or false. Which cards would you select?

This task involves understanding conditional inference, namely understanding the rule “If P then Q” and with this, deducing the answer as “P and not Q” or “D and 7”. Such logical deduction indeed presents as a good candidate to test for a potential ability of the mathematically trained. This task has also been substantially investigated in the domain of the psychology of reasoning [ 12 p8] revealing across a wide range of publications that only around 10% of the general population reach the correct result. The predominant mistake being to pick “D and 3”; where in the original study by Wason [ 13 ] it is suggested that this was picked by 65% of people. This poor success rate along with a standard mistake has fuelled interest in the task as well as attempts to understand why it occurs. A prevailing theory being the so named matching bias effect; the effect of disproportionately concentrating on items specifically mentioned in the situation, as opposed to reasoning according to logical rules.

Inglis and Simpson’s results isolated mathematically trained individuals with respect to this task. The participants were under time constraint and 13% of the first-year undergraduate mathematics students sampled reached the correct response, compared to 4% of the non-mathematics (arts) students that was included. Of note also was the 24% of mathematics students as opposed to 45% of the non-mathematics students who chose the standard mistake. The study indeed unveiled that mathematically trained individuals were significantly less affected by the matching bias effect with this problem than the individuals without mathematics training. However, the achievement of the mathematically trained group was still far from masterful and the preponderance for a non-standard mistake compared with non-mathematically trained people is suggestive. Mathematical training appears to engender a different thinking style, but it remains unclear what the difference is.

Inglis, Simpson and colleagues proceeded to follow up their results with a number of studies concentrated on conditional inference in general [ 14 , 15 ]. A justification for this single investigatory pathway being that if transfer of knowledge is present, something subtle to test for in the first place, a key consideration should be the generalisation of learning rather than the application of skills learned in one context to another (where experimenter bias in the choice of contexts is more likely to be an issue). For this they typically used sixteen “if P then Q” comprehension tasks, where their samples across a number of studies have included 16-year-old pre-university mathematics students (from England and Cyprus), mathematics honours students in their first year of undergraduate university study, third year university mathematics students, and associated control groups. The studies have encompassed controls for general intelligence and thinking disposition prior to training, as well as follows ups of up to two years to address the issue of causation. The conclusive thinking pattern that has emerged is a tendency of the mathematical groups towards a greater likelihood of rejecting the invalid denial of the antecedent and affirmation of the consequent inferences. But with this, and this was validated by a second separate study, the English mathematics group actually became less likely to endorse the valid modus tollens inference. So again, mathematical training appears to engender a different thinking style, but there are subtleties and it remains unclear what the exact difference is.

This project was designed to broaden the search on the notion that mathematics training leads to increased reasoning skills. We focused on a range of reasoning problems considered in psychological research to be particularly insightful into decision making, critical thinking and logical deduction, with their distinction in that the general population generally struggles with answering them correctly. An Australian sample adds diversity to the current enquiries that have been European focussed. Furthermore, in an effort to identify the impact of mathematics training through a possible gradation effect, different levels of mathematically trained individuals were tested for performance.

Well-studied thinking tasks from a variety of psychological studies were chosen. Their descriptions, associated success rates and other pertinent details follows. They were all chosen as the correct answer is typically eluded for a standard mistake.

The three-item Cognitive Reflection Test (CRT) was used as introduced by Frederick [ 16 ]. This test was devised in line with the theory that there are two general types of cognitive activity: one that operates quickly and without reflection, and another that requires not only conscious thought and effort, but also an ability to reflect on one’s own cognition by including a step of suppression of the first to reach it. The three items in the test involve an incorrect “gut” response and further cognitive skill is deemed required to reach the correct answer (although see [ 17 ] for evidence that correct responses can result from “intuition”, which could be related to intelligence [ 18 ]).

In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

Bat and ball

A bat and a ball cost $1.10 in total. The bat costs a dollar more than the ball. How much does the ball cost?

The solutions are: 47 days for the Lily Pads problem, 5 minutes for the Widgets problem and 5 cents for the Bat and Ball problem. The considered intuitive, but wrong, answers are 24 days, 100 minutes and 10 cents, respectively. These wrong answers are attributed to participants becoming over focused on the numbers so as to ignore the exponential growth pattern in the Lily Pads problem, merely complete a pattern in numbers in the Widgets problem, and neglect the relationship “more than” in the Bat and Ball problem [ 19 ]. The original study by Frederick [ 16 ] provides a composite measure of the performance on these three items, with only 17% of those studied (n = 3428) reaching the perfect score. The CRT has since been studied extensively [ 19 – 21 ]. Research using the CRT tends not to report performance on the individual items of the test, but rather a composite measure of performance. Attridge and Inglis [ 22 ] used the CRT as a test for thinking disposition of mathematics students as one way to attempt to disentangle the issue of filtering according to prior thinking styles rather than transference of knowledge in successful problem solving. They repeat tested 16-year old pre-university mathematics students and English literature students without mathematics subjects at a one-year interval and found no difference between groups.

Three problems were included that test the ability to reason about probability. All three problems were originally discussed by Kahneman and Tversky [ 23 ], with the typically poor performance on these problems explained by participants relying not on probability knowledge, but a short-cut method of thinking known as the representativeness heuristic. In the late 1980s, Richard Nisbett and colleagues showed that graduate level training in statistics, while not revealing any improvement in logical reasoning, did correlate with higher-quality statistical answers [ 24 ]. Their studies lead in particular to the conclusion that comprehension of, what is known as the law of large numbers, did show improvement with training. The first of our next three problems targeted this law directly.

• (a). the larger hospital
• (b). the smaller hospital
• (c). about the same (that is, within 5 percent of each other)

Kahneman and Tversky [ 23 ] reported that, of 50 participants, 12 chose (a), 10 chose (b), and 28 chose (c). The correct answer is (b), for the reason that small samples are more likely to exhibit extreme events than large samples from the same population. The larger the sample, the more likely it will exhibit characteristics of the parent population, such as the proportion of boys to girls. However, people tend to discount or be unaware of this feature of sampling statistics, which Kahneman and Tversky refer to as the law of large numbers. Instead, according to Kahneman and Tversky, people tend to adhere to a fallacious law of small numbers, where even small samples are expected to exhibit properties of the parent population, as illustrated by the proportion of participants choosing the answer (c) in their 1972 study. Such thinking reflects use of the representativeness heuristic, whereby someone will judge the likelihood of an uncertain event based on how similar it is to characteristics of the parent population of events.

Birth order

• (a). What is your estimate of the number of families surveyed in which the exact order of births was BGBBBB?
• (b). In the same survey set, which, if any, of the following two sequences would be more likely: BBBGGG or GBBGBG?

All of the events listed in the problem have an equal probability, so the correct answer to (a) is 72, and to (b) is “neither is more likely”. Kahneman and Tversky [ 23 ] reported that 75 of 92 participants judged the sequence in (a) as less likely than the given sequence. A similar number (unspecified by Kahneman and Tversky, but the statistical effect was reported to be of the same order as in (a)) reported that GBBGBG was the more likely sequence. Again, Kahneman and Tversky suggested that these results reflected use of the representativeness heuristic. In the context of this problem, the heuristic would have taken the following form: some birth orders appear less patterned than others, and less patterned is to be associated with the randomness of birth order, making them more likely.

Coin tosses

• (a). H T H T H T H T
• (b). H H H H T T T T
• (c). T T H H T T H H
• (d). H T T H T H H T
• (e). all of the above are equally likely

The correct answer in this problem is (e). Kahneman and Tversky [ 23 ] reported that participants tend to choose less patterned looking sequences (e.g., H T T H T H H T) as more likely than more systematic looking sequences (e.g., H T H T H T H T). This reasoning again reflects the representativeness heuristic.

Three further questions from the literature were included to test problem solving skill.

Two drivers

• (a). Driver A would win the race
• (b). Driver B would win the race
• (c). the two drivers would arrive at the same time (within a few seconds of one another)

This problem was developed by Pelham and Neter [ 25 ]. The correct answer is (a), which can be determined by calculations of driving times for each Driver, using time = distance/velocity. Pelham and Neter argue, however, that (c) is intuitively appealing, on the basis that both drivers appear to have the same overall average speed. Pelham and Neter reported that 67% of their sample gave this incorrect response to the problem, and a further 13% selected (b).

Petrol station

Imagine that you are driving along the road and you notice that your car is running low on petrol. You see two petrol stations next to each other, both advertising their petrol prices. Station A’s price is 65c/litre; Station B’s price is 60c/litre. Station A’s sign also announces: “5c/litre discount for cash!” Station B’s sign announces “5c/litre surcharge for credit cards.” All other factors being equal (for example, cleanliness of the stations, number of cars waiting at each etc), to which station would you choose to go, and why?

This problem was adapted from one described by Galotti [ 26 ], and is inspired by research reported by Thaler [ 27 ]. According to Thaler’s research, most people prefer Station A, even though both stations are offering the same deal: 60c/litre for cash, and 65c/litre for credit. Tversky and Kahneman [ 28 ] explain this preference by invoking the concept of framing effects. In the context of this problem, such an effect would involve viewing the outcomes as changes from some initial point. The initial point frames the problem, and provides a context for viewing the outcome. Thus, depending on the starting point, outcomes in this problem can be viewed as either a gain (in Station A, you gain a discount if you use cash) or a loss (in Station B, you are charged more (a loss) for using credit). Given that people are apparently more concerned about a loss than a gain [ 29 ], the loss associated with Station B makes it the less attractive option, and hence the preference for Station A. The correct answer, though, is that the stations are offering the same deal and so no station should be preferred.

And finally, a question described by Stanovich [ 30 , 31 ] as testing our predisposition for cognitive operations that require the least computational effort.

Jack looking at Anne

• (c). Cannot be determined

Stanovich reported that over 80% of people choose the “lazy” answer (c). The correct answer is (a).

The above questions survey, in a clear problem solving setting, an ability to engage advanced cognitive processing in order to critically evaluate and possibly override initial gut reasoning, an ability to reason about probability within the framework of the law of large numbers and the relationship between randomness and patterning, an ability to isolate salient features of a problem and, with the last question in particular, an ability to map logical relations. It might be hypothesised that according to degrees of mathematical training, in line with the knowledge base provided and the claims of associated broad and enhanced problem-solving abilities in general, that participants with greater degrees of such training would outperform others on these questions. This hypothesis was investigated in this study. In addition, given that no previous study on this issue has examined the variety of problems used in this study, we also undertook an exploratory analysis to investigate whether there exist any associations between the problems in terms of their likelihood of correct solution. Similarities between problems might indicate which problem solving domains could be susceptible to the effects of mathematics training.

• Introductory—First year, second semester, university students with weak high school mathematical results, only enrolled in the current unit as a compulsory component for their chosen degree, a unit not enabling any future mathematical pathway, a typical student may be enrolled in a Biology or Geography major;
• Standard—First year, second semester, university students with fair to good high school mathematical results, enrolled in the current mathematics unit as a compulsory component for their chosen degree with the possibility of including some further mathematical units in their degree pathway, a typical student may be enrolled in an IT or Computer Science major;
• Advanced1—First year, second semester, university mathematics students with very strong interest as well as background in mathematics, all higher year mathematical units are included as possible future pathway, a typical student may be enrolled in a Mathematics or Physics major;
• Advanced2—Second year, second semester, university mathematics students with strong interest as well as background in mathematics, typically a direct follow on from the previously mentioned Advanced1 cohort;
• Academic—Research academics in the mathematical sciences.

Participants

123 first year university students volunteered during “help on demand” tutorial times containing up to 30 students. These are course allocated times that are supervised yet self-directed by students. This minimised disruption and discouraged coercion. 44 second year university students completed the questionnaire during a weekly one-hour time slot dedicated to putting the latest mathematical concepts to practice with the lecturer (whereby contrast to what occurs in tutorial times the lecturer does most of the work and all students enrolled are invited). All these university students completed the questionnaire in normal classroom conditions; they were not placed under strict examination conditions. The lead author walked around to prevent discussion and coercion and there was minimum disruption. 30 research academics responded to local advertising and answered the questionnaire in their workplace while supervised.

The questionnaires were voluntary, anonymous and confidential. Participants were free to withdraw from the study at any time and without any penalty. No participant took this option however. The questionnaires gathered demographic information which included age, level of education attained and current qualification pursued, name of last qualification and years since obtaining it, and an option to note current speciality for research academics. Each problem task was placed on a separate page. Participants were not placed under time constraint, but while supervised, were asked to write their start and finish times on the front page of the survey to note approximate completion times. Speed of completion was not incentivised. Participants were not allowed to use calculators. A final “Comments Page” gave the option for feedback including specifically if the participants had previously seen any of the questions. Questionnaires were administered in person and supervised to avoid collusion or consulting of external sources.

The responses were coded four ways: A) correct; B) standard error (the errors discussed above in The Study); C) other error; D) left blank.

The ethical aspects of the study were approved by the Human Research Ethics Committee of the University of Sydney, protocol number [2016/647].

The first analysis examined the total number of correct responses provided by the participants as a function of group. Scores ranged from 1 to 11 out of a total possible of 11 (Problem 6 had 2 parts) ( Fig 1 ). An ANOVA of this data indicated a significant effect of group (F(4, 192) = 20.426, p < .001, partial η 2 = .299). Pairwise comparisons using Tukey’s HSD test indicated that the Introductory group performed significantly worse than the Advanced1, Advanced2 and Academic groups. There were no significant differences between the Advanced1, Advanced2 and Academic groups.

• PPT PowerPoint slide
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Error bars are one standard error of the mean.

https://doi.org/10.1371/journal.pone.0236153.g001

Overall solution time, while recorded manually and approximately, was positively correlated with group, such that the more training someone had received, the longer were these solution times (r(180) = 0.247, p = .001). However, as can be seen in Fig 2 , this relationship is not strong.

https://doi.org/10.1371/journal.pone.0236153.g002

A series of chi-squared analyses, and their Bayesian equivalents, were performed on each problem, to determine whether the distribution of response types differed as a function of group. To minimise the number of cells in which expected values in some of these analyses were less than 5, the Standard Error, Other Error and Blank response categories were collapsed into one category (Incorrect Response). For three of the questions, the expected values of some cells did fall below 5, and this was due to most people getting the problem wrong (Four Cards), or most people correctly responding to the problem (Bat and Ball, Coin Tosses). In these cases, the pattern of results was so clear that a statistical analysis was barely required. Significant chi-squared results were examined further with pairwise posthoc comparisons (see Table 1 ).

https://doi.org/10.1371/journal.pone.0236153.t001

The three groups with the least amount of training in mathematics were far less likely than the other groups to give the correct solution (χ 2 (4) = 31.06, p < .001; BF 10 = 45,045) ( Table 1 ). People in the two most advanced groups (Advanced2 and Academic) were more likely to solve the card problem correctly, although it was still less than half of the people in these groups who did so. Further, these people were less likely to give the standard incorrect solution, so that most who were incorrect suggested some more cognitively elaborate answer, such as turning over all cards. The proportion of people in the Advanced2 and Academic groups (39 and 37%) who solved the problem correctly far exceeded the typical proportion observed with this problem (10%). Of note, also, is the relatively high proportion of those in the higher training groups who, when they made an error, did not make the standard error, a similar result to the one reported by Inglis and Simpson [ 11 ].

The cognitive reflection test

In the Lily Pads problem, although most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, it was also the case that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 27.28, p < .001; BF 10 = 15,554), with the standard incorrect answer being the next most prevalent response for the two lower ability mathematics groups ( Table 1 ).

Performance on the Widgets problem was similar to performance on the Lily Pads problem in that most people in the Standard, Advanced1, Advanced2 and Academic groups were likely to select the correct solution, but that the less training someone had received in mathematics, the more likely they were to select an incorrect solution (χ 2 (4) = 23.76, p< .001; BF 10 = 516) ( Table 1 ). As with the Lily Pads and Widget problems, people in the Standard, Advanced1, Advanced2 and Academic groups were highly likely to solve the Bat and Ball problem (χ 2 (4) = 35.37, p < .001; BF 10 = 208,667). Errors were more likely from the least mathematically trained people (Introductory, Standard) than the other groups ( Table 1 ).

To compare performance on the CRT with previously published results, performance on the three problems (Lily Pads, Widgets, Bat and Ball) were combined. The number of people in each condition that solved 0, 1, 2, or 3 problems correctly is presented in Table 2 . The Introductory group were evenly distributed amongst the four categories, with 26% solving all three problems correctly. Around 70% of the rest of the groups solved all 3 problems correctly, which is vastly superior to the 17% reported by Frederick [ 16 ].

https://doi.org/10.1371/journal.pone.0236153.t002

Responses to the Hospitals problem were almost universally split between correct and standard errors in the Standard, Advanced1, Advanced2 and Academic groups. Although this pattern of responses was also evident in the Introductory group, this group also exhibited more non-standard errors and non-responses than the other groups. However, the differences between the groups were not significant (χ 2 (4) = 4.93, p = .295; BF 10 = .068) ( Table 1 ). Nonetheless, the performance of all groups exceeds the 20% correct response rate reported by Kahneman and Tversky [ 23 ].

The two versions of the Birth Order problem showed similar results, with correct responses being more likely in the groups with more training (i.e., Advanced1, Advanced2 and Academic), and responses being shared amongst the various categories in the Introductory and Standard groups (χ a 2 (4) = 24.54, p < .001; BF 10 = 1,303; χ b 2 (4) = 25.77, p < .001; BF 10 = 2,970) ( Table 1 ). Nonetheless, performance on both versions of the problem in this study was significantly better than the 82% error rate reported by Kahneman and Tversky [ 23 ].

The Coin Tosses problem was performed well by all groups, with very few people in any condition committing errors. There were no obvious differences between the groups (χ 2 (4) = 3.70, p = .448; BF 10 = .160) ( Table 1 ). Kahneman and Tversky [ 23 ] reported that people tend to make errors on this type of problem by choosing less patterned looking sequences, but they did not report relative proportions of people making errors versus giving correct responses. Clearly the sample in this study did not perform like those in Kahneman and Tversky’s study.

Responses on the Two Drivers problem were clearly distinguished by a high chance of error in the Introductory and Standard groups (over 80%), and a fairly good chance of being correct in the Advanced1, Advanced2 and Academic groups (χ 2 (4) = 46.16, p < .001; BF 10 = 1.32 x 10 8 ) ( Table 1 ). Academics were the standout performers on this problem, although over a quarter of this group produced an incorrect response. Thus, the first two groups performed similarly to the participants in the Pelham and Neter [ 25 ] study, 80% of whom gave an incorrect response.

Responses on the Petrol Station problem were marked by good performance by the Academic group (73% providing a correct response), and just over half of each of the other groups correctly solving the problem. This difference was not significant (χ 2 (4) = 4.68, p = .322: BF 10 = .059) ( Table 1 ). Errors were fairly evenly balanced between standard and other, except for the Academic group, who were more likely to provide a creative answer if they made an error. Thaler [ 27 ] reported that most people get this problem wrong. In this study, however, on average, most people got this problem correct, although this average was boosted by the Academic group.

Responses on the Jack looking at Anne problem generally were standard errors, except for the Advanced2 and Academic groups, which were evenly split between standard errors and correct responses (χ 2 (4) = 18.03, p = .001; BF 10 = 46) ( Table 1 ). Thus, apart from these two groups, the error rate in this study was similar to that reported by Stanovich [ 30 ], where 80% of participants were incorrect.

A series of logistic regression analyses were performed in order to examine whether the likelihood of solving a particular problem correctly could be predicted on the basis of whether other problems were solved correctly. Each analysis involved selecting performance (correct or error) on one problem as the outcome variable, and performance on the other problems as predictor variables. Training (amount of training) was also included as a predictor variable in each analysis. A further logistic regression was performed with training as the outcome variable, and performance on all of the problems as predictor variables. The results of these analyses are summarised in Table 3 . There were three multi-variable relationships observed in these analyses, which can be interpreted as the likelihood of solving one problem in each group being associated with solving the others in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. Training also featured in each of these sets, moderating the relationships as per the results presented above for each problem.

https://doi.org/10.1371/journal.pone.0236153.t003

The final “Comments Page” revealed the participants as overwhelmingly enjoying the questions. Any analysis of previous exposure to the tasks proved impossible as there was little to no alignment on participant’s degree of recall, if any, and even perceptions of what exposure entailed. For example, some participants confused being exposed to the particular tasks with being habitually exposed to puzzles, or even mathematics problems, more broadly.

In general, the amount of mathematics training a group had received predicted their performance on the overall set of problems. The greater the training, the more problems were answered correctly, and the slower the recorded response times. There was not an obvious difference between the Advanced1, Advanced2 and Academic groups on either of these measures, however there were clear differences between this group and the Introductory and Standard groups, with the former exhibiting clearly superior accuracy. While time records were taken approximately, so as to avoid adding time pressure as a variable, that the Advanced1, Advanced2 and Academic groups recorded more time in their consideration of the problems, may suggest a “pause and consider” approach to such problems is a characteristic of the advanced groups. This is in line with what was suggested by an eye-movement tracking study of mathematically trained students attempting the Four Cards Problem; where participants that had not chosen the standard error had spent longer considering the card linked to the matching bias effect [ 14 ]. It is important to note, however, that longer response times may reflect other cognitive processes than deliberation [ 32 ].

Performance on some problems was associated with performance on other problems. That is, if someone correctly answered a problem in one of these sets, they were also highly likely to correctly answer the other problems in the set. These sets were: (1) Lily Pads, Widgets and Petrol Station; (2) Hospitals, Four Cards and Two Drivers; (3) Birth Order and Coin Tosses. This is different with how these problems have been typically clustered a priori in the research literature: (I) Lily Pads, Widgets and Bat and Ball (CRT); (II) Hospitals and Two Drivers (explained below); (III) Hospitals, Birth Order and Coin Tosses (representativeness heuristic); (IV) Birth Order and Coin Tosses (probability theory). Consideration of these problem groupings follows.

Correctly answering all three problems in (I) entailed not being distracted by particular pieces of information in the problems so as to stay focused on uncovering the real underlying relationships. The Lily Pads and Widget problems can mislead if attention is over focused on the numbers, and conversely, the Petrol Station problem can mislead if there is too much focus on the idea of a discount. While the Lily Pads and Widget problems are traditionally paired with the Bat and Ball problem in the CRT, it may be that performance on the Bat and Ball problem did not appear as part of this set due to an added level of difficulty. With the problems in (I), avoiding being distracted by certain parts of the questions at the expense of others almost leads directly to the correct answer. However, with the Bat and Ball problem, further steps in mathematical reasoning still need to occur in answering which two numbers add together to give a result while also subtracting one from the other for another.

With the problems in (II) it is of interest that the Two Drivers problem was created specifically to be paired with the Hospitals problem to test for motivation in problem solving [ 23 ]. Within this framework further transparent versions of these problems were successfully devised to manipulate for difficulty. The Two Drivers problem was amended to have Driver B travelling at exactly 5 mph during the first half of the race and at exactly 95 mph during the last half of the race. The Hospitals problem was amended so the smaller hospital would have “only 2” babies born each day and where for a period of one year the hospitals recorded the number of days on which all of the babies born were boys. Could the association in (II) be pointing to how participants overcome initial fictitious mathematical rules? Maybe they reframe the question in simpler terms to see the pattern. The Four Cards Problem also elicited a high number of incorrect answers where, associated with mathematical training, the standard incorrect solution was avoided for more cognitively elaborate ones. Indeed, a gradation effect appeared across the groups where the standard error of the “D and 3” cards becomes “D only” ( Table 4 ). Adrian Simpson and Derrick Watson found a comparable result across their two groups [14 p61]. This could again be pointing to having avoided an initial fictitious rule of simply concentrating on items directly found in the question, participants then seek to reframe the question to unearth the logical rule to be deduced. An added level of difficulty with this question may be why participants become trapped in a false answer. The eye-movement tracking study mentioned above supports this theory.

https://doi.org/10.1371/journal.pone.0236153.t004

The problems in (III) fit naturally together as part of basic probability theory, a topic participants would have assimilated, or not, as part of various education curricula. While the equal likelihood of all possible outcomes with respect to a coin toss may be culturally assimilated, the same may not be as straightforward for birth gender outcomes where such assumptions could be swayed by biological hypothesis or folk wisdom [ 33 ]. The gradation of the results in terms of mathematical training does not support this possibility.

The effect of training on performance accuracy was more obvious in some problems compared to others, and to some extent, this was related to the type of problem. For instance, most of the problems in which performance was related to training (Four Cards, CRT [Lily Pads, Widgets, Bat and Ball], Two Drivers, Jack looking at Anne) could be classed as relying on logical and/or critical thinking. The one exception was the Birth Order problems, which are probability related.

In contrast, two of the three problems in which training did not appear to have much impact on performance (Hospitals and Coin Tosses) require domain-specific knowledge. The Hospitals problem requires a degree of knowledge about sampling statistics. This is a topic of quite distinct flavour that not all mathematically trained individuals gain familiarity with. On the other hand, all groups having performed well on the Coin Tosses problem is in line with a level of familiarity with basic probability having been originally presented at high school. While the questioning of patterning as negatively correlated with randomness is similar to that appearing in the Birth Order question, in the Birth Order question this aspect is arguably more concealed. These results and problem grouping (III) could be pointing to an area for improvement in teaching where the small gap in knowledge required to go from answering the Coin Tosses problem correctly to achieving similarly with the Birth Order problem could be easily addressed. A more formal introduction to sampling statistics in mathematical training could potentially bridge this gap as well as further be extended towards improvement on the Hospitals problem.

The other problem where performance was unrelated to training, the Petrol Station problem, cannot be characterised similarly. It is more of a logical/critical thinking type problem, where there remains some suggestion that training may have impacted performance, as the Academic group seemed to perform better than the rest of the sample. An alternate interpretation of this result is therefore that this problem should not be isolated but grouped with the other problems where performance is affected by training.

• The Introductory group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Introduction to Differentiation, Applications of the Derivative, Antiderivatives, Areas and the Definite Integral), Financial Mathematics, Statistical Analysis. The Introductory group then explored concepts in mathematical modelling with emphasis on the importance of calculus in their first semester of mathematical studies.
• The Standard group’s mathematics high school syllabus studied prior to first semester course entry covered: Functions, Trigonometric Functions, Calculus (Rates of Change, Integration including the method of substitution, trigonometric identities and inverse trigonometric functions, Areas and Volumes of solids of revolution, some differential equations), Combinatorics, Proof (with particular focus on Proof by Mathematical Induction), Vectors (with application to projectile motion), Statistical Analysis. In first semester their mathematical studies then covered a number of topics the Advanced1 group studied prior to gaining entrance at university; further details on this are given below.
• The Advanced1 group’s mathematics high school syllabus studied prior to first semester course entry covered: the same course content the Standard group covered at high school plus extra topics on Proof (develop rigorous mathematical arguments and proofs, specifically in the context of number and algebra and further develop Proof by Mathematical Induction), Vectors (3 dimensional vectors, vector equations of lines), Complex Numbers, Calculus (Further Integration techniques with partial fractions and integration by parts), Mechanics (Application of Calculus to Mechanics with simple harmonic motion, modelling motion without and with resistance, projectiles and resisted motion). The Standard group cover these topics in their first semester university studies in mathematics with the exclusion of further concepts of Proof or Mechanics. In first semester the Advanced1 group have built on their knowledge with an emphasis on both theoretical and foundational aspects, as well as developing the skill of applying mathematical theory to solve practical problems. Theoretical topics include a host of theorems relevant to the study of Calculus.

In summary, at the point of our study, the Advanced1 group had more knowledge and practice on rigorous mathematical arguments and proofs in the context of number and algebra, and more in-depth experience with Proofs by Induction, but the bulk of extra knowledge rests with a much deeper knowledge of Calculus. They have had longer experience with a variety of integration techniques, and have worked with a variety of applications of calculus to solve practical problems, including a large section on mechanics at high school. In first semester at university there has been a greater focus on theoretical topics including a host of theorems and associated proofs relevant to the topics studied. As compared to the Introductory and Standard groups, the Advanced1 group have only widened the mathematics knowledge gap since their choice of post-compulsory mathematics at high school. The Advanced2 group come directly from an Advanced1 cohort. And the Academics group would have reached the Advanced1 group’s proficiency as part of their employment. So, are specific reasoning skills resulting from this level of abstract reasoning? Our findings suggest this should certainly be an area of investigation and links in interestingly with other research work. In studying one of the thinking tasks in particular (the Four Cards Problem) and its context of conditional inference more specifically, Inglis and Simpson [ 15 ] found a clear difference between undergraduates in mathematics and undergraduates in other university disciplines, yet also showed a lack of development over first-year university studies on conditional inference measures. A follow up study by Attridge and Inglis [ 22 ] then zeroed in on post-compulsory high school mathematical training and found that students with such training did develop their conditional reasoning to a greater extent than their control group over the course of a year, despite them having received no explicit tuition in conditional logic. The development though, whilst demonstrated as not being the result of a domain-general change in cognitive capacity or thinking disposition, and most likely associated with the domain-specific study of mathematics, revealed a complex pattern of endorsing more of some inferences and less of others. The study here focused on a much broader problem set associated with logical and critical thinking and it too is suggestive of a more complex picture in how mathematics training may be contributing to problem solving styles. A more intricate pattern to do with the impact of mathematical training on problem solving techniques is appearing as required for consideration.

There is also a final interpretation to consider: that people in the Advanced 1, Advanced2 and Academic groups did not gain anything from their mathematics training in terms of their ability to solve these problems. Instead, with studies denying any correlation of many of these problems with what is currently measured as intelligence [ 30 ], they might still be people of a particular intelligence or thinking disposition to start with, who have been able to use that intelligence to not only solve these problems, but also survive the challenges of their mathematics training.

That the CRT has been traditionally used as a measure of baseline thinking disposition and that performance has been found to be immutable across groups tested is of particular interest since our results show a clear possible training effect on these questions. CRT is tied with a willingness to engage in effortful thinking which presents as a suitable ability for training. It is beyond the scope of this study, but a thorough review of CRT testing is suggestive of a broader appreciation and better framework to understand thinking disposition, ability and potential ability.

Mathematical training appears associated with certain thinking skills, but there are clearly some subtleties that need to be extricated. The thinking tasks here add to the foundational results where the aim is for a firmer platform on which to eventually base more targeted and illustrative inquiry. If thinking skills can be fostered, could first year university mathematics teaching be improved so that all samples from that group reach the Advanced1 group level of reasoning? Do university mathematics courses become purely about domain-specific knowledge from this point on? Intensive training has been shown to impact the brain and cognition across a number of domains from music [ 34 ], to video gaming [ 35 ], to Braille reading [ 36 ]. The hypothesis that mathematics, with its highly specific practice, fits within this list remains legitimate, but simply unchartered. With our current level of understanding it is worth appreciating the careful wording of the NYU Courant Institute on ‘Why Study Math?’ where there is no assumption of causation: “Mathematicians need to have good reasoning ability in order to identify, analyze, and apply basic logical principles to technical problems.” [ 37 ].

Limitations

One possible limitation of the current study is that the problems may have been too easy for the more advanced people, and so we observed a ceiling effect (i.e., some people obtained 100% correct on all problems). This was most obvious in the Advanced1, Advanced2 and Academic groups. It is possible that participants in these groups had developed logical and critical thinking skills throughout their mathematical training that were sufficient to cope with most of the problems used in this study, and so this would support the contention that training in mathematics leads to the development of logical and critical thinking skills useful in a range of domains. Another interpretation is that participants in these groups already possessed the necessary thinking skills for solving the problems in this study, which is why they are able to cope with the material in the advanced units they were enrolled in, or complete a PhD in mathematics and hold down an academic position in a mathematics department. This would then suggest that training in mathematics had no effect on abstract thinking skills—people in this study possessed them to varying extents prior to their studies. This issue might be settled in a future study that used a greater number of problems of varying difficulties to maximise the chances of finding a difference between the three groups with the most amount of training. Alternatively, a longitudinal study that followed people through their mathematics training could determine whether their logical and critical thinking abilities changed throughout their course.

A further limitation of the study may be that several of the reasoning biases examined in this study were measured by only one problem each (i.e., Four Cards Problem, Two Drivers, Petrol Station, Jack looking at Anne). A more reliable measure of these biases could be achieved by including more problems that tap into these biases. This would, however, increase the time required of participants during data collection, and in the context of this study, would mean a different mode of testing would likely be required.

Broad sweeping intuitive claims of the transferable skills endowed by a study of mathematics require evidence. Our study uniquely covers a wide range of participants, from limited mathematics training through to research academics in the mathematical sciences. It furthermore considered performance on 11 well-studied thinking tasks that typically elude participants in psychological studies and on which results have been uncorrelated with general intelligence, education levels and other demographic information [ 15 , 16 , 30 ]. We identified different performances on these tasks with respect to different groups, based on level of mathematical training. This included the CRT which has developed into a method of measuring baseline thinking disposition. We identified different distributions of types of errors for the mathematically trained. We furthermore identified a performance threshold that exists in first year university for those with high level mathematics training. This study then provides insight into possible changes and adjustments to mathematics courses in order for them to fulfil their advertised goal of reaching improved rational and logical reasoning for a higher number of students.

It is central to any education program to have a clear grasp of the nature of what it delivers and how, but arguably especially so for the core discipline that is mathematics. In 2014 the Office of The Chief Scientist of Australia released a report “Australia’s STEM workforce: a survey of employers” where transferable skills attributed to mathematics were also ones that employers deemed as part of the most valuable [ 38 ]. A better understanding of what mathematics delivers in this space is an opportunity to truly capitalise on this historical culture-crossing subject.

Supporting information

https://doi.org/10.1371/journal.pone.0236153.s001

Acknowledgments

The authors would like to thank Jacqui Ramagge for her proof reading and input, as well as support towards data collection.

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How To Encourage Critical Thinking in Math

By Mary Montero

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Critical thinking is more than just a buzzword… It’s an essential skill that helps students develop problem-solving abilities and make logical connections between different concepts. By encouraging critical thinking in math, students learn to approach problems more thoughtfully, they learn to analyze and evaluate math concepts, identify patterns and relationships, and explore different strategies for finding the solution. Critical thinking also involves a great deal of persistence. Those are critical life skills!

When you think about it, students are typically asked to solve math problems and find the answer. Showing their work is frequently stressed too, which is important, but not the end. Instead, students need to be able to look at math in different ways in order to truly grasp a complete understanding of math concepts. Mathematics requires logical reasoning, problem-solving, and abstract thinking.

What Does Critical Thinking in Math Look Like?

When I think about critical thinking in math, I focus on:

• Solving problems through logical thinking . Students learn how to break down complex problems, analyze the different parts, and understand how they fit together logically.
• Identifying patterns and making connections. Students learn how to identify patterns across different math concepts, make connections between seemingly unrelated topics, and develop a more in-depth understanding of how math works.
• Evaluating and comparing solutions. Students learn to evaluate which solution is best for a given problem and identify any flaws in their reasoning or others’ reasoning when looking at different solutions

Mathematician Posters

These FREE Marvelous Mathematician posters have been a staple in my classroom for the last 8+ years! I first started using a version from MissMathDork and adapted them for my classroom over the years. 

I print, laminate, and add magnetic stickers on the back. At the beginning of the year, I only put one or two up at a time depending on our area of focus. Now, they are all hanging on my board, and I’ll pull out different ones depending on our area of focus. They are so empowering to my mathematicians and help them stay on track!

A Marvelous Mathematician:

• knows that quicker doesn’t mean better
• looks for patterns
• knows mistakes happen and keeps going
• makes sense of the most important details
• embraces challenges and works through frustrations
• uses proper math vocabulary to explain their thinking
• shows their work and models their thinking
• discusses solutions and evaluates reasonableness
• gives context by labeling answers
• applies mathematical knowledge to similar situations
• checks for errors (computational and conceptual)

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

I love to incorporate challenge problems (use Nrich and Openmiddle to get started) because they teach my students so much more than how to solve a math problem. They learn important lessons in teamwork, persistence, resiliency, and growth mindset. We talk about strategies for tackling difficult problems and the importance of not giving up when things get hard.

This square of numbers challenge was a hit!

ALL kids need to feel and learn to embrace challenge. Oftentimes, kids I see have rarely faced an academic challenge. Things have just come easy to them, so when it doesn’t, they can lack strategies that will help them. In fact, they will often give up before they even get started.

I tell them it’s my job to make sure I’m helping them stretch and grow their brain by giving them challenges. They don’t love it at first, but they eventually do! 

This domino challenge was another one from Nrich . I’m always on the hunt for problems like this!!  How would you guide students toward an answer??

Fifteen Cards

This is a well-loved math puzzle with my students, and it’s amazing for encouraging students to consider all options when solving a math problem.

We have number cards 1-15 (one of each number) and only seven are laid out. With the given clues, students need to figure out which seven cards should be put out and in what order. My students love these, and after they’ve done a few, they enjoy creating their own, too! Use products, differences, and quotients to increase the challenge.

This is also adapted from Nrich, which is an AMAZING resource for math enrichment!

This is one of my favorite fraction lessons that I’ve done for years! Huge shout out to Meg from The Teacher Studio for this one. I give each child a slip of paper with this figure and they have to silently write their answer and justification. Then I tally up the answers and have students take a side and DEBATE with their reasoning! It’s an AMAZING conversation, and I highly recommend trying it with your students. 

Sometimes we leave it hanging overnight and work on visual models to make some proofs. 

Logic Puzzles

Logic puzzles are always a hit too! You can enrich and extend your math lessons with these ‘Math Mystery’ logic puzzles that are the perfect challenge for 4th, 5th, and 6th grades. The puzzles are skills-based, so they integrate well with almost ANY math lesson. You can use them to supplement instruction or challenge your fast-finishers and gifted students… all while encouraging critical thinking about important math skills!

Three levels are included, so they’re perfect to use for differentiation.

• Introductory logic puzzles are great for beginners (4th grade and up!)
• Advanced logic puzzles are great for students needing an extra challenge
• Extra Advanced logic puzzles are perfect for expert solvers… we dare you to figure these puzzles out! 

Do you have a group of students who are ready for more of a fraction challenge? My well-loved fraction puzzlers are absolutely perfect for fraction enrichment. They’ll motivate your students to excel at even the most challenging tasks! 

Math Projects

Math projects are another way to differentiation while building critical thinking skills. Math projects hold so much learning power with their real-world connections, differentiation options, collaborative learning opportunities, and numerous avenues for cross curricular learning too. 

If you’re new to math projects, I shared my best tips and tricks for using math projects in this blog post . They’re perfect for cumulative review, seasonal practice, centers, early finisher work, and more.

I use both concept-based math projects to focus on specific standards and seasonal math projects that integrate several skills.

Error Analysis

Finally, error analysis is always a challenging way to encourage critical thinking. When we use error analysis, we encourage students to analyze their own mistakes to prevent making the same mistakes in the future.

For my gifted students, I use error analysis tasks as an assessment when they have shown mastery of a unit during other tasks. For students in the regular classroom needing enrichment, I usually have them complete the tasks in a center or with a partner.

For students needing extra support, we complete error analysis in small groups.  We go step-by-step through the concept and they are always able to eventually identify what the error is. It is so empowering to students when they finally figure out the error AND it helps prevent them from making the same error in the future!

My FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

When you’re ready for more, this bundle of error analysis tasks contains more than 240 tasks to engage and enrich your students in critical thinking practice.

If you want to dig even deeper, visit this conceptual vs computational error analysis post to learn more about using error analysis in the classroom. 

Related Critical Thinking Posts

• How to Increase Critical Thinking and Creativity in Your “Spare” Time
• More Tips to Increase Critical Thinking

Critical thinking is essential for students to develop a deeper understanding of math concepts, problem-solving skills, and a stronger ability to reason logically. When you learn how to encourage critical thinking in math, you’re setting your students up for success not only in more advanced math subjects they’ll encounter, but also in life. 

How do you integrate critical thinking in your classroom? Come share your ideas with us in our FREE Inspired In Upper Elementary Facebook group .

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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Mary Thankyou for your inspirational activities. I have just read and loved the morning talk activities. I do have meetings with my students but usually at end of day. What time do you

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