Black schoolgirl solving addition sum on white board during Covid-19 pandemic.

Arithmetic, Anxiety, and Reform Math

Key Concepts:

  • Educator’s math anxiety and beliefs about their students’ ability to learn math can impact their students’ math performance
  • Caregivers’ (parents, guardians, etc…) views and anxiety about math can impact their children’s math attitudes and achievement (Silver et al., 2021).

Math Anxiety

Many adults feel a rush of adrenaline and discomfort when required to perform math in their daily lives. It is often easier to open the calculator on our smartphones than to estimate or mentally calculate the cost of an item after taxes or a sale discount. If we feel the need to defend our actions to an observer, we may nonchalantly announce, “I’m bad at math,” with a careless wave of the hand, as though to brush away the sentiment. Over 90% of adults have some level of math anxiety (Blazer, 2011,) which may lead them to avoid math and dismiss its importance.

In addition to impacts on your own life, your feelings about math can affect learners around you. Whether you are an educator or caregiver, your attitudes about math can impact the math stress and success of the students who observe your interactions with math.

Your Beliefs Impact the Learners in Front of You

Several studies explore the myriad ways in which adults’ beliefs about math can impact learners of various ages. While a teacher or caregiver’s math anxiety can impair students’ math performance, there are also perceptions about math that can mitigate that effect. Each of the studies below asked about other beliefs that exacerbate or mitigate the effect of adult math anxiety on student learning.

This graph shows an interaction between caregiver math beliefs and math anxiety. Those with low math anxiety (solid line), did not greatly impact their children’s math achievement, regardless of the caregivers’ beliefs about the importance of math. Caregivers with high math anxiety (dotted line), on the other hand, impacted their children’s math achievement differently, depending on the caregivers’ beliefs about the importance of math.

One study hypothesized that the math anxiety of female teachers would negatively impact their elementary students’—especially girls’—math achievement (Beilock et al., 2010). They tested this hypothesis by measuring the students’ ability in the first and last few months of the school year, assuming that teacher anxiety would have an effect on students’ beliefs only after spending the school year teaching them. As predicted, the researchers found that there was no significant relationship between teacher beliefs and student performance in the beginning of the year. However, by the end of the school year, increased teacher math anxiety correlated with lower math achievement for the girls, but not the boys, in the classroom. Luckily, when controlling for the girls’ gender ability beliefs (i.e. the belief that boys are better at math,) the impact of teacher math anxiety disappeared. In other words, when girls hold the belief that they are equally as capable as the boys in math, teachers’ math anxiety does not negatively impact the girls’ performance.

A similar study examined the effect of teacher math anxiety on adolescent students and found that teacher anxiety was negatively correlated with performance of their students (Ramirez et al., 2018). Regardless of gender of teacher or student, higher teacher math anxiety led to lower GPA. Interestingly, students’ perceptions of their teacher’s fixed mindset beliefs partially mediated the relationship between teacher anxiety and student achievement. A fixed mindset belief holds that ability is inherent and cannot be changed by learners. In contrast, a growth mindset asserts that effort and practice can improve ability. In the context of this study, students perceived teachers who utilized less process-oriented teaching practices as having a fixed mindset.

This graph shows an interaction between caregiver math beliefs and math anxiety. Those with low math anxiety (solid line), did not greatly impact their children’s math achievement, regardless of the caregivers’ beliefs about the importance of math. Caregivers with high math anxiety (dotted line), on the other hand, impacted their children’s math achievement differently, depending on the caregivers’ beliefs about the importance of math.

A new study examined the relationship between math achievement in early childhood and the math anxiety of their caregivers (Silver et al., 2021). This longitudinal study investigated beliefs about the importance of math, in addition to math anxiety and discovered some complex relationships. Caregiver beliefs about the importance of math did predict the math performance of their children; children whose caregivers believed math was more important scored higher. Unlike the research on the effect of teacher math anxiety, caregiver math anxiety was not predictive of the children’s math abilities, at this young age. Interestingly, there was an interaction between caregiver math beliefs and math anxiety (shown in the graph on the right). Those with low math anxiety (solid line), did not greatly impact their children’s math achievement, regardless of the caregivers’ beliefs about the importance of math. Caregivers with high math anxiety (dotted line), on the other hand, impacted their children’s math achievement differently, depending on the caregivers’ beliefs about the importance of math. The children of those with high math anxiety and low importance beliefs had decreased math achievement, while the children of those with high math anxiety and high math importance beliefs had increased math achievement. In other words, if you are a caregiver of a young child, it is okay to have math anxiety, as long as you also believe math is important for your child to learn.

Your math anxiety can impact your children’s/students’ math achievement, but it is not the only aspect of your beliefs about math that can have an effect. You will positively affect a learner’s math achievement when you believe that people can improve their math skills, that people have an equal ability to learn math, that math is important, and that the process of solving math problems is valuable. In addition, there are a few powerful strategies that can help you boost learners’ math performance.

Strategies

  1. Consider how you might be communicating (explicitly or implicitly) your attitudes about math and math learning.
  2. Express positive beliefs about math and demonstrate positive uses.
  3. Work to neutralize gender ability beliefs in yourself and children/students. Dispel harmful misconceptions stemming from gender assumptions and fixed-mindset beliefs.
  4. Place more emphasis on process-oriented teaching practices (as opposed to correct answers and computational speed.)
  5. Use a variety of assessments, technology, manipulatives, and real-life or concrete examples.
  6. Other strategies for educators, students, and caregivers: Strategies for Reducing Math Anxiety

Bibliography

  • Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860-1863.
  • Blazer, C. (2011). Strategies for Reducing Math Anxiety. Information Capsule. Volume 1102. Research Services, Miami-Dade County Public Schools.
  • Ramirez, G., Hooper, S. Y., Kersting, N. B., Ferguson, R., & Yeager, D. (2018). Teacher math anxiety relates to adolescent students’ math achievement. Aera Open, 4(1), 2332858418756052.
  • Silver, A. M., Elliott, L., & Libertus, M. E. (2021). When beliefs matter most: Examining children’s math achievement in the context of parental math anxiety. Journal of Experimental Child Psychology, 201, 104992. https://doi.org/10.1016/j.jecp.2020.104992

Spatial Thinking in Science and Education

Key Ideas

  • Spatial thinking skills are a group of abilities that involve mentally picturing and manipulating objects.
  • Spatial thinking skills impact many school outcomes, such as science, technology, engineering, and mathematics (STEM) abilities. New research suggests it may also impact language arts skills.
  • Educators can identify components of lessons that require spatial thinking and add supports to their instruction with relative ease.

What is Spatial Thinking?

Eukaryotic cell diagram, top view of inside structures
Eukaryotic cell diagram, top view of inside structures (Credit: lvcandy)

When you picture how a room would look after rearranging furniture, plan a route from point A to point B, or imagine designing a craft or project, you are using spatial thinking skills. Spatial thinking skills are a group of abilities that involve mentally picturing and manipulating objects. These skills are utilized to learn topics at every level: Young children may picture a number line to add or subtract, while middle school students study diagrams of cell organelles, and high school physics students visualize the movements of individual particles to understand the motion of waves. Even at the postgraduate level, medical students’ understanding of anatomy is supported by their ability to imagine spatial relations of bones and muscles. As these examples demonstrate, spatial thinking skills are heavily intertwined with STEM.

What does the research say?

Years of educational neuroscience research have found a couple of interesting relationships between spatial thinking and STEM. First, there is a strong positive correlation between spatial thinking abilities and STEM success, where those who have stronger spatial thinking skills perform better in STEM1. Second, studies have shown that this connection is causal: improving spatial thinking skills improves success on STEM activities and assessments2. However, it is not yet time to advocate for the use of spatial thinking interventions in hopes of improving STEM scores. While a meta-analysis summarizing findings from other studies2 found that spatial thinking interventions can transfer effects to tasks in STEM domains, some individual studies have found only weak support for the impact of those improvements on school learning.

Furthermore, though the majority of studies focus on the relationship between spatial thinking and math or science, a recent study has found a similarly strong bond between spatial skills and Language Arts test scores 3. This may indicate that the relationship between STEM achievement and spatial thinking skills may be driven by a third factor that has not yet been found. If this were true, then it would be precipitous to create a curriculum solely for spatial thinking skills prior to discovering the external influence. While more research is needed to support what—if any—spatial thinking lessons should be created, it is clear that spatial thinking skills are essential for academic performance on a breadth of subjects throughout the years.

What can I do?

Eukaryotic cell diagram, vector illustration, text on own layer
Eukaryotic cell diagram, vector illustration, text on own layer (Credit:jack0m)

Educators can still support their students’ spatial thinking in the context of relevant topics, however. I can still remember my shock when I realized—embarrassingly late—that cells were three-dimensional sphere-like shapes, not pancake-shaped. Prior to seeing an illustration of a spherical cell from an outside angle, I took textbook illustrations of a cell slice at face value. My high school biology teacher’s choice to show a depiction that indicated the three-dimensional nature of the cell enabled me to more deeply (and correctly) understand cells.

To support your students’ spatial thinking, you can first identify when it might need to be utilized. Some subjects, such as geography and geometry, contain a plethora of concrete spatial information. Other subjects may represent spatial concepts abstractly: for example, most people imagine positive numbers to the right and negative numbers to the left (or up and down, respectively.) Spatial thinking skills help us understand these topics, so those with lower spatial abilities may be at a disadvantage. Whether spatial concepts in your topic are concrete or abstract, there are a number of ways to bolster your students’ spatial thinking skills and deepen their understanding.

Concrete spatial concepts (i.e. geometry, geography, biology, etc…)

  • Concretely represent the spatial relations (i.e. point to areas on maps or gesture to show relations, instead of only using language to describe relative locations)
  • Show multiple perspectives on the same object; bonus points if you include multiple types of media
    Ask students to imagine what an asymmetrical object (i.e. the heart) would look like from a different point of view

Abstract spatial concepts (i.e. number concepts, chemistry, etc…)

  • Think of ways to represent the concepts concretely (fraction manipulatives are a common example in 4th grade.) Such representations can be used physically with younger students and as analogies with older students.
  • Explicitly draw attention to spatial relationships and organization (in chemistry, e.g., describe not only bonds and attraction, but how those affect particles’ relative (hypothetical) positions)

Bibliography

1 Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial Ability for STEM Domains: Aligning Over 50 Years of Cumulative Psychological Knowledge Solidifies Its Importance. Journal of Educational Psychology, 101(4), 817–835.

2 Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402. https://doi.org/10.1037/a0028446

3 Rutherford, T., Karamarkovich, S. M., & Lee, D. S. (2018). Is the spatial/math connection unique? Associations between mental rotation and elementary mathematics and English achievement. Learning and Individual Differences, 62, 180–199. https://doi.org/10.1016/j.lindif.2018.01.014

Images retrieved from Unsplash

Image 1—Eukaryotic cell diagram, top view of inside structures; Credit: lvcandyLargest; size:Vector (EPS) – Scalable to any size; Stock illustration ID:488721125; Upload date:May 07, 2014

Image 2—Eukaryotic cell diagram, vector illustration, text on own layer; Credit:jack0m; Largest size:Vector (EPS) – Scalable to any size; Stock illustration ID:1155003177; Upload date:June 10, 2019

Image 3—Seeking solutions in the maze-shaped human brain, 3D – Computer generated image; Credit:syolacan; Largest size: 5250 x 3500 px (17.50 x 11.67 in.) – 300 dpi – RGB; Stock photo ID:1291371216; Upload date:December 22, 2020

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Christine Bresnahan is an educator with a passionate drive to reduce the barriers that impede the learning of students with diverse learning needs. She is currently a PhD student studying individual differences in spatial thinking skills, specifically in people with dyslexia, in the Behavior, Cognition, and Neuroscience program at American University. She first developed her interest in educational neuroscience and Universal Design for Learning (UDL) in the Mind, Brain, and Education program at the Harvard Graduate School of Education, where she earned her Master of Education. She then taught as a special educator at a public elementary school in Massachusetts for six years, and has presented for over a decade at schools and conferences about how educators can support the learning of students at various ages and abilities.

Teach with Your Hands

Key Ideas

  • Students must organize new and pre-existing knowledge to effectively learn new ideas.
  • Concepts can be organized into abstract structures—such as cause-and-effect, classifications, hierarchies, and generalizations.
  • Educators can support students’ understanding by demonstrating the conceptual structure of the content being taught.

Conceptual Structures in the Classroom

Students are required to understand increasingly complex material as they progress through school. Educators are responsible for teaching not only subject content, but also for developing students’ ability to comprehend new ideas with increasing levels of independence. One way to increase independent analytical ability is to teach about conceptual structures. Conceptual structures—such as cause-and-effect, classifications, hierarchies, and generalizations—describe relationships among concepts. People learn new ideas more deeply when information is organized—internally or externally—into conceptual structures.

While conceptual structures are abstract, they can be represented verbally and concretely.

Drawing of a cheeseburger with labels of parts of a paragraph on each layer (i.e. Topic Sentence on top bun)
The “paragraph hamburger” is a writing organizer that shows the key components of a paragraph.2

For example, transition words, such as “first,” “next,” or “in contrast,” signal how one topic will relate to others. Furthermore, diagrams, outlines, and concept maps  can help students arrange and integrate knowledge. In elementary school, it is common for teachers to provide graphic organizers, such as Venn diagrams for comparing and contrasting information, or character webs and plot diagrams for narrative stories. These visual organizers allow children to externalize concept organization and learn common conceptual structures. For example, “paragraph burgers” help structure expressive writing. As students progress through school, they receive direct instruction on conceptual organization less often, as students are expected to to employ previously learned skills independently.

A diagram of the Krebs Cycle shows two conceptual structures—a cycle & reactions—with arrows.3

However, as subject material grows in difficulty, it is often difficult for students to generalize and apply previously learned strategies. In order to integrate new knowledge, students still need support to understand relationships among topics. High level subjects often have conventions for representing relationships, such as Δ (delta) for “change” and  arrows for chemical reactions. Students at all levels benefit when structures and their representations are explained in relation to the content.

Conceptual structures can be represented visually, (i.e. graphic organizers and outlines,) signaled verbally (i.e. transition words,) and indicated with symbols. New research has investigated whether educators can also use hand movements to help students understand the organization of the material being taught.

New Research

In a recent study, Celeste Pilegard and Logan Fiorella investigated the effects of different types of hand gestures on student learning. Specifically, Pilegard and Fiorella hypothesized that teachers who indicated the structure of the material with gestures would see the largest increases in student learning.

To test this, a little over a hundred college students watched video lessons about two types of steamboats. The 123 students were split into four groups to watch different versions of the video. In the first, the instructor utilized structure gestures to demonstrate the compare/contrast conceptual structure of the information. In this case, she gestured on one side of her body when discussing characteristics of the first type of steamboat and on the other side of her body when discussing the second type of steamboat. In the second, surface gestures indicated concrete characteristics: for instance, holding hands further apart when talking about a steamboat with a larger hull. The third video combined structure and surface gestures, while the fourth video only contained small, meaningless hand movements.

After watching the videos, the participants answered recall and inference questions about the content. They also rated their engagement with the lecturer. The researchers found support for their hypothesis: the use of structure gestures during teaching significantly improved the students’ ability to make inferences about the material. Interestingly, structure gestures did not significantly impact memory performance. In contrast, surface gestures did not have a significant effect on either type of question. Students who watched any of the videos with meaningful gestures rated higher levels of engagement with the lecturer.

Implications for Education

This research supports the hypothesis that using structural gestures—those that indicate the relationships among concepts—support higher level learning, in a way that other gestures do not. However, there were some substantial differences between the videos and actual teaching. The videos were substantially shorter (about one minute) and less complex than typical lessons for that grade range. Also, in order to control for the impact of gesturing, there were no other visual aids, such as pictures, diagrams, outlines, or slides. While these glaring differences make it difficult to predict the extent to which gestures can aid learning in more realistic settings, the findings nevertheless indicate two important considerations for education.

First, students at all grade levels benefit from explicit support on the conceptual structure of lessons.

  • Organize lectures by topic and subtopic, and make the organization explicit through the use of outlines, heading hierarchy, and introduction and summary statements to each topic
  • Make connections between previously learned material and new material; use analogies to compare the structures of complex concepts to easy-to-understand equivalents
  • Utilize diagrams, flow charts, and other visual representations of text-heavy material

Second, structural gestures are one way to support students’ conceptual organization.

  • Consider what your hands are doing while you are teaching. Utilize gestures that help students visualize relationships.
  • When preparing the lesson, consider the interrelationships among or within topics. Ask yourself:
    • What type(s) of conceptual structure exists (cycle, hierarchy, compare/contrast, cause/effect…)?
    • How can I represent this structure visually, verbally, and/or via gesture?

Conceptual structures facilitate the organization of knowledge and the integration of new information. Teachers can support higher level understanding of material by explaining the conceptual structure of the material. Learners of all ages benefit from descriptions and representations of conceptual structures, such as diagrams, transition words, and gestures.

 


Christine Bresnahan is a PhD student and adjunct professor at American University, where she researches and teaches about Educational Neuroscience and Special Education, with a focus on dyslexia and ADHD. Previously, she was an elementary special educator in a Massachusetts public school.

References

CAST (2018). Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org

Pilegard, C., & Fiorella, L. (2021). Using gestures to signal lesson structure and foster meaningful learning. Applied Cognitive Psychology, 35(5), 1362–1369. https://doi.org/10.1002/acp.3866

  • Images:
    1. Teacher Photo courtesy of Dr. William Hobbs, Image Source
    2. Paragraph Burger retrieved from Reading Rockets
    3. Krebs Cycle diagram retrieved from learnbiology.com