We know that misconceptions don’t actually leave pupil’s heads, even when corrected. They’re notoriously sticky, and require the same, if not more, practice in correcting/avoiding them than it takes to embed them in the first place.
For my year 10s, ‘perimeter’s add, area’s times’ has been stickier than a post-it. And I can’t blame them. It’s pithy, easy to remember, but falls apart quickly – also like a post-it (if you pick it up and stick back down too many times).
[As an aside – “Hey diddle diddle the median’s the middle…” falls into this trap too. It’s great for discrete data presented in a list, but too simplistic for anything else – of which there’s a lot!]
This is an example of a type of question that was on their most recent assessment. They needed to find the area.
Of course this wasn’t the response I got from them during teaching, where we used the amazing Mathspad demonstration (here) to show very clearly what’s happening with the different lengths. It wasn’t the case during the questioning and mini whiteboards and all the usual maths teacher teaching strategies we use day to day in our practice.
But this was the case in the assessment at the end of a unit a few weeks later.
I’ve spent years seeing things like this in assessments and coming to the conclusion that it’s obviously my teaching that’s lacking, or I’m not doing enough retrieval practice, or the curriculum’s wrong. All of which could be true, but ultimately I also reckon that after a certain point, the misconception is stronger than the teaching.
I’ve wanged on about What I Reckon needs picking apart in perimeter (here) before, whereas below I’ll share what I did to try and sort out ‘area’s just times’.
In assessment feedback, I displayed the following slide, animated in gif form below, and questioned students about the changing in area.
I’ve put the (for some reason grainy and wobbly) gif below, then listed the questions I asked (with what I was getting it in brackets next to it).
- What’s the area of this rectangle then? (Area’s just times isn’t incorrect, you’re not wrong! You’re just narrow!)
- So what about the shaded area now? (SUBTRACTION?! But area’s just times!)
- Why is the shaded area the same? (Preservation of area)
- Why is the shaded area the same? (Labouring a point)
- Wheeeeeee. (Labouring a point. Still.)
- So what about now? (Still a subtraction, even though the 10 is moving, it’s not changing the shaded area)
- And what about this? (So I’m adding AND subtracting from a multiplication?! So area’s definitely not JUST TIMES IT then?)
The whole slide without animations is shown below:
In doing this, I kept the numbers simple, and used the same 10×9 to focus thought just on what’s changing, or most often what’s not actually changing.
I think it’s important to differentiate between misconceptions and errors. It’s not that area isn’t times as such, it’s just that area’s isn’t just times. So by beginning at a point where the misconception has meaning helps pupils see that it isn’t me stood at the front dictating and yawping that they’re wrong I’m right etc.
I then displayed these questions.
I’ve used this with other classes since, and used it differently. For example with my year 9s I used it cold for them to just have at it and discuss on their tables, whereas with year 10 we discussed the first two together to explore why the 32 isn’t always relevant to the shaded area, as a well as reinforcing triangle area, followed by a think, pair, share for the top right before working independently on the bottom 4.
I like this task (he says) because the numbers of it aren’t as important as the relative position and the effect on comparative answers. Pupils can really struggle with the layering of images in geometry, and this gives them opportunity to explore that unique to calculation. It worked really well (he says). Their justification for when we do and don’t have ‘halves’ of the square and/or the halves of the 32 where lovely.
After this, we did some practice using the 4 grids below to build up L-shapes as subtracting the area of a small rectangle from the bigger one, but starting with some squiqqlies (not a mathematical term) to really hammer home this subtraction relationship.
Of course, the fact that students were successful at this means nothing. It doesn’t actually tell me anything about whether I’ve fixed the misconception, because it’s sticky. Chances are, further down the line we’ll need to revisit this again.
The final task in the slides is this one below:
I’ve included it in these slides because it hints at the same idea or moving away from ‘Area’s Just Times’, into more reasoning with “simple” maths.
I didn’t use this with my year 10s, but will probably chuck one in to their starters/do nows/bell tasks over the coming weeks. This was designed for my year 9s when we looked at area. We’re a year 9 entry school, so our kids come from all over and have experienced completely different curricula. And as we’re really small, attainment ranges in all classes are really wide. I found this was a great way to reinforce reasoning with areas of rectangles and triangles, while I could circulate and intervene with pupils that needed extra input on the foundational concepts.
Taking the top left question:
Pupils came up with different approaches. While most added 45 and 18 before dividing by 9, some found 45/9 and 18/9 and added their results – which provided a nice little link to order of operations and the distributive property. Interestingly, one pupil formed the equation 9x=45+18 and then solved that, which would never have entered my head to do! (I’ve more of a geometry ‘default setting’ mathematically rather than algebra, myself.)
These different approaches all speak to the myriad prior experiences they’ve had up until this point, and showing them that as long as it’s valid they can do what they like, is a powerful example of the richness even something “simple” like rectangle area has bubbling under its surface.
All tasks are in the slides at the top of this post, with brief notes in the notes section.
1. Understanding Mathematical Misconceptions
1. What are mathematical misconceptions?
Answer:
Mathematical misconceptions are incorrect beliefs or understandings about mathematical concepts. They often arise from oversimplified rules, prior experiences, or misunderstandings, leading students to apply flawed reasoning in problem-solving.
2. How do misconceptions differ from errors?
Answer:
Errors are typically random mistakes or lapses in calculation, whereas misconceptions are systematic and stem from incorrect foundational beliefs that influence a student’s overall understanding and approach to mathematics.
3. Why are misconceptions so persistent in students?
Answer:
Misconceptions are deeply rooted in a student’s prior knowledge and experiences. They become “sticky” because they are reinforced over time, making them resistant to change even when corrected.
4. At what educational stage do misconceptions most commonly form?
Answer:
Misconceptions can form at any educational stage but are most prevalent during early learning when foundational concepts are being established. However, they can persist into higher grades if not addressed properly.
5. How can teachers identify misconceptions in their students?
Answer:
Teachers can identify misconceptions through formative assessments, observing problem-solving methods, listening to students’ explanations, and analyzing patterns in incorrect answers.
2. Common Misconceptions in Mathematics
6. What is the misconception “Perimeter’s add, Area’s times”?
Answer:
This phrase oversimplifies the concepts, suggesting that calculating the perimeter involves only addition, while area calculation involves only multiplication. It neglects scenarios where more complex operations, such as subtraction or division, are necessary.
7. Why is “Area is just times” misleading?
Answer:
While multiplication is fundamental in calculating area (length × width), this phrase ignores situations where area calculations require adding or subtracting areas of different shapes, especially in composite figures.
8. What misconception do students have about the median?
Answer:
Students often think the median is simply the middle number in a list, without understanding that it represents the middle value in a data set, which can be affected by the distribution and not just the order of numbers.
9. How do students misunderstand fractions in multiplication?
Answer:
Students might believe that multiplying fractions always results in a larger number or simply apply cross-multiplication without understanding the underlying proportional relationships.
10. What is the misconception regarding negative numbers in multiplication?
Answer:
Students may think that multiplying two negative numbers results in a negative product, not understanding that the product of two negatives is actually positive.
11. How do students misconceive the distributive property?
Answer:
Some students may apply the distributive property incorrectly, such as misunderstanding how to distribute multiplication over addition or subtraction within parentheses.
12. What misunderstanding do students have about exponents?
Answer:
Students often confuse exponents with roots or believe that exponents always indicate repeated multiplication without grasping their role in scaling and exponential growth.
13. How do students misinterpret ratios?
Answer:
Students may struggle to understand that ratios represent a relationship between two quantities, not just a comparison of their sizes.
14. What is the misconception about the order of operations?
Answer:
Some students may not follow the correct sequence of operations, leading to incorrect results, especially when dealing with parentheses, exponents, multiplication, division, addition, and subtraction.
15. How do students misunderstand probability?
Answer:
Students may believe that probability is the same as chance, not recognizing it as a measure of the likelihood of an event occurring, often leading to incorrect calculations.
16. What is the misconception about geometric shapes and their properties?
Answer:
Students might incorrectly assume that all quadrilaterals have equal sides or angles, not recognizing the diversity of properties among different geometric shapes.
17. How do students confuse data representation methods?
Answer:
Students may not understand when to use bar graphs, histograms, or scatter plots, leading to inappropriate data representation and interpretation.
18. What is the misunderstanding regarding linear equations?
Answer:
Students often believe that linear equations only represent straight lines without understanding their application in various real-world contexts.
19. How do students misconceive area versus perimeter in real-life applications?
Answer:
Students may apply perimeter and area calculations interchangeably in real-life situations, not understanding their distinct purposes, such as fencing versus covering a surface.
20. What is the misconception about the concept of infinity?
Answer:
Students might view infinity as a specific number rather than an abstract concept representing unboundedness, leading to confusion in calculus and advanced mathematics.
3. Strategies to Correct Misconceptions
21. What is retrieval practice and how does it help?
Answer:
Retrieval practice involves regularly recalling information from memory, which strengthens understanding and helps identify and correct misconceptions through repeated exposure and reinforcement.
22. How can interactive demonstrations help correct misconceptions?
Answer:
Interactive demonstrations, such as using visual aids or hands-on activities, allow students to explore concepts dynamically, providing concrete experiences that can challenge and replace incorrect beliefs.
23. What role does questioning play in addressing misconceptions?
Answer:
Effective questioning encourages students to articulate their understanding, exposing misconceptions. It also guides them to think critically and re-evaluate their reasoning.
24. How can mini whiteboards be used to address misconceptions?
Answer:
Mini whiteboards allow students to demonstrate their thinking process in real-time, enabling teachers to identify misconceptions instantly and provide immediate corrective feedback.
25. Why is it important to differentiate between misconceptions and errors in teaching?
Answer:
Understanding whether a student made an error or holds a misconception informs the teaching approach. Misconceptions require conceptual correction, while errors might need procedural guidance.
26. How can animated visual aids help in correcting misconceptions?
Answer:
Animated visual aids can illustrate changes and processes over time, making abstract concepts more tangible and helping students visualize and understand the correct principles.
27. What is the benefit of using think-pair-share in addressing misconceptions?
Answer:
Think-pair-share encourages collaborative learning, allowing students to discuss and challenge each other’s ideas, which can help surface and correct misconceptions through peer interaction.
28. How can educators use formative assessments to identify misconceptions?
Answer:
Formative assessments provide ongoing insights into student understanding, allowing teachers to detect misconceptions early and address them before they become entrenched.
29. What is the importance of feedback in correcting misconceptions?
Answer:
Constructive feedback helps students recognize and understand their misconceptions, guiding them towards the correct concepts and reinforcing accurate knowledge.
30. How can real-world applications help correct mathematical misconceptions?
Answer:
Applying mathematical concepts to real-world scenarios helps students see the practical relevance and correct application of principles, thereby dispelling misconceptions rooted in abstract understanding.
4. Differentiating Misconceptions from Errors
31. How can teachers distinguish between a misconception and a simple error?
Answer:
Teachers can analyze the nature of the incorrect response. If the mistake reflects a consistent misunderstanding of a concept, it’s likely a misconception. If it’s an isolated mistake, it may be a procedural error.
32. What strategies can help address both misconceptions and errors?
Answer:
Using a combination of targeted questioning, individualized feedback, and varied instructional methods can address both misconceptions and procedural errors effectively.
33. Why is it crucial to address misconceptions promptly?
Answer:
Addressing misconceptions promptly prevents them from becoming foundational barriers that hinder future learning and understanding of more complex concepts.
34. How does student self-reflection aid in identifying misconceptions?
Answer:
Encouraging students to reflect on their problem-solving processes helps them recognize inconsistencies and gaps in their understanding, making them aware of potential misconceptions.
35. What role does classroom discussion play in correcting misconceptions?
Answer:
Classroom discussions allow students to express their thinking, listen to others, and collectively identify and correct misconceptions through shared insights and collaborative reasoning.
5. The Role of Retrieval Practice
36. What is retrieval practice and why is it effective?
Answer:
Retrieval practice involves actively recalling information from memory. It strengthens neural pathways, enhances long-term retention, and helps identify and correct misconceptions through repeated reinforcement.
37. How can teachers implement retrieval practice in the classroom?
Answer:
Teachers can incorporate activities like quizzes, flashcards, low-stakes testing, and regular review sessions to facilitate retrieval practice and reinforce correct understanding.
38. What are the benefits of spaced retrieval practice?
Answer:
Spaced retrieval practice involves spreading out practice sessions over time, which improves memory retention and reduces the likelihood of misconceptions becoming permanent.
39. How does retrieval practice differ from traditional study methods?
Answer:
Unlike passive review methods, retrieval practice requires active engagement and effort to recall information, leading to deeper understanding and better retention.
40. Can technology aid in retrieval practice?
Answer:
Yes, educational technology tools like digital flashcards, interactive quizzes, and learning apps can facilitate retrieval practice by providing varied and accessible ways for students to recall information.
6. Using Visuals and Interactive Tools
41. How do visual aids help in correcting mathematical misconceptions?
Answer:
Visual aids make abstract concepts concrete, helping students visualize relationships and processes, which can clarify misunderstandings and reinforce correct concepts.
42. What are some effective visual tools for teaching area and perimeter?
Answer:
Tools like grid paper, geometric shapes, interactive software (e.g., GeoGebra), and animated demonstrations can effectively illustrate the differences and calculations for area and perimeter.
43. How can manipulatives be used to address misconceptions?
Answer:
Manipulatives like blocks, tiles, and measuring tools allow students to physically engage with mathematical concepts, promoting hands-on learning and correcting misunderstandings through tactile experiences.
44. What role do interactive whiteboards play in teaching?
Answer:
Interactive whiteboards facilitate dynamic teaching by allowing teachers to display and manipulate visual content, engage students in interactive problem-solving, and provide immediate feedback.
45. How can technology enhance understanding of mathematical concepts?
Answer:
Technology can provide simulations, interactive tutorials, and instant feedback, making learning more engaging and helping students explore concepts in a deeper and more meaningful way.
7. Engaging Teaching Methods
46. What are some effective teaching methods for correcting misconceptions?
Answer:
Methods include Socratic questioning, flipped classrooms, collaborative learning, differentiated instruction, and incorporating real-world applications to engage students and address misconceptions directly.
47. How can storytelling be used to teach mathematics?
Answer:
Storytelling can contextualize mathematical concepts, making them more relatable and easier to understand. It can also highlight common misconceptions and guide students through correct reasoning.
48. Why is it important to create a safe learning environment?
Answer:
A safe learning environment encourages students to express their thoughts and mistakes without fear of judgment, facilitating open discussions that can uncover and address misconceptions.
49. How can peer teaching help in correcting misconceptions?
Answer:
Peer teaching allows students to explain concepts to each other, reinforcing their own understanding and providing alternative perspectives that can help identify and correct misconceptions.
50. What is the impact of a growth mindset on correcting misconceptions?
Answer:
Fostering a growth mindset encourages students to view challenges and mistakes as opportunities for learning, making them more receptive to correcting misconceptions and persisting through difficulties.
