Tips for Teachers by Craig Barton

Using AI to help convey purpose

Conveying purpose is one of the key building blocks of a lesson.

I think there are four ways we can convey the purpose of a new idea to out students:

  1. Connect prior, present and future knowledge
  2. Share hinterland knowledge
  3. Use real-life contexts
  4. Introduce a headache

AI can help us with this

The prompt

As ever, the key is in the prompt you write. After many iterations, here is the one that seems to yield the best results:

I am a maths teacher. I want your help to convey the purpose of various mathematical ideas to my students. I would like you to come up with ideas for how I can do this, breaking each response down into four categories:

  1. Connect prior, present and future knowledge

Create a flow diagram like the one attached to show students what ideas they have met before that feed into this new idea and what ideas they will meet in the future that build upon this new idea. The example I have attached is for the new idea of the area of a circle.

2. Share hinterland knowledge

Hinterland knowledge is the rich array of content, stories, and examples that give meaning to procedures. Please share interesting stories about how this idea was first discovered or created, and how it has been used, changed and applied over time.

3. Use real-life contexts

Where is the idea used in the real world? Please only provide genuine contexts that will be meaningful and interesting to students.

4. Introduce a headache

The idea is that we try to show our students that, without this new idea, their lives would be a bit painful at this very moment. In other words, before we provide the aspirin (the new idea), we need students to experience a headache.

Here are a few examples of headaches I use:

Asking students to calculate 693 + 693 + 693 + 693 + 693 + 693 + 693 + 693 before teaching long multiplication

Asking students to copy down the mass of the earth as 5970000000000000000000000 kilograms before teaching standard form

Winning at “rigged” games before teaching probability

Do you understand, and are you ready for the first idea?

Idea #1: Adding and subtracting fractions

Claude’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE
  1. SHARE HINTERLAND KNOWLEDGE
  • Ancient Egyptians used unit fractions (fractions with numerator 1) and had special symbols for them. They would break down other fractions into sums of unit fractions – essentially doing fraction addition but backwards! The Rhind Papyrus (c. 1650 BCE) shows problems like this.
  • In ancient Rome, they used a base-12 system for fractions because 12 has many factors, making it easier to work with parts. This is why we still have dozen (12) and gross (144) as units today.
  • The word “fraction” comes from the Latin “fractus” meaning “broken” – literally describing breaking something into pieces.
  1. REAL-LIFE CONTEXTS
  • Cooking: Combining 3/4 cup of flour with 1/4 cup for a recipe, or adjusting recipe quantities
  • Carpentry: Measuring wood lengths (7 1/2 inches + 3 3/4 inches)
  • Music: Adding note durations (1/4 note + 1/8 note in music timing)
  • Sports statistics: Adding up partial game scores or player statistics
  • Construction: Measuring and cutting materials to specific lengths
  • Medicine: Combining different portions of medicines or solutions
  1. INTRODUCE A HEADACHE

Here’s a suggested headache activity:

Tell students they’re going to be party planners. Give them this scenario: “You need to make punch for a party. The recipe calls for:

  • 2 1/3 cups of orange juice
  • 1 2/3 cups of pineapple juice
  • 3/4 cup of lime juice

You need to:

  1. Calculate the total amount of juice needed
  2. Figure out what size container to buy (containers come in whole and half cups)
  3. Calculate how much punch will be left if everyone drinks 1/2 cup each and there are 15 people”

Let them struggle with trying to add these fractions without a proper method. They’ll likely:

  • Try to add numerators and denominators separately (wrong method)
  • Try to convert to decimals (messy with recurring decimals)
  • Draw pictures (time-consuming and imprecise)

This creates the perfect opportunity to introduce the proper method of finding common denominators and adding fractions.

ChatGPT’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE
  • Prior Knowledge:
    • Understand fractions as parts of a whole (e.g., ½, ¾).
    • Compare and order fractions with the same or different denominators.
    • Simplify fractions (e.g., 6/8 = ¾).
    • Understand equivalent fractions (e.g., ½ = 2/4 = 3/6).
  • Present Knowledge (Adding/Subtracting Fractions):
    • Add and subtract fractions with the same denominator.
    • Find a common denominator to add and subtract fractions with different denominators.
    • Simplify answers where possible.
  • Future Knowledge:
    • Solve word problems involving fractions.
    • Perform operations with mixed numbers.
    • Use fractions in algebraic expressions.
    • Multiply and divide fractions.
    • Apply fractions in real-life contexts, such as probability, proportions, and ratios.

Flow Diagram:
Fractions Basics → Equivalent Fractions → Adding/Subtracting Fractions → Mixed Numbers → Multiplying/Dividing Fractions → Algebraic Fractions

  1. SHARE HINTERLAND KNOWLEDGE
  • Story of Fractions: The concept of fractions dates back to ancient Egypt and Babylon. The Egyptians used a system of unit fractions (fractions with 1 in the numerator) for trade and land measurement. The Rhind Mathematical Papyrus, dating back to around 1650 BCE, contains early fraction calculations.
  • Cultural Impact: Fractions became more standardized during the Islamic Golden Age, where scholars like Al-Khwarizmi advanced arithmetic. Later, during the Renaissance, fractions were popularized in Europe for use in trade and commerce.
  • Applications Over Time: Fractions have been critical in scientific advancements, from astronomy (calculating planetary orbits) to engineering (dividing materials into parts). For example, NASA engineers use fractions and ratios to divide fuel and calculate trajectories for space missions.
  1. REAL-LIFE CONTEXTS
  • Cooking and Baking: Recipes often require adding and subtracting fractions when doubling or halving portions. Example: If a recipe calls for ½ a cup of sugar and ¾ of a cup of flour, how much sugar and flour are needed in total?
  • Time Measurement: Time is often measured in fractions (e.g., ¼ of an hour = 15 minutes). Adding fractions helps calculate elapsed time or plan schedules.
  • Construction and Design: Architects and carpenters use fractions to measure and cut materials accurately. For instance, combining lengths like 1½ meters and ¾ meters requires fraction addition.
  • Sports: Fractions are used in scoring systems or distances. For example, in running, fractions help record times (e.g., 1¾ laps completed in a relay).
  1. INTRODUCE A HEADACHE
  • Headache Scenario: Ask students to add the following without teaching them about finding a common denominator:They might try to simply add the numerators and denominators and feel confused when the answer doesn’t seem right.
  • Follow-Up Question: Ask them to subtract:Without a method to find a common denominator, this can feel frustrating and disorganized.
  • Motivation: Show how fractions are used everywhere in life (cooking, measuring, time). To use them effectively, we need to be able to add and subtract them accurately. Then introduce the idea

Idea #2: Pie charts

Claude’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE
  1. SHARE HINTERLAND KNOWLEDGE
  • Pie charts were invented by William Playfair in 1801 in his “Statistical Breviary” – interestingly, he was a failed businessman and political economist who turned to statistical graphics!
  • Florence Nightingale revolutionized their use with her “rose diagrams” (a variation of pie charts) in 1858 to show seasonal deaths in the Crimean War. This helped convince government officials to improve hospital conditions.
  • The term “pie chart” wasn’t used until 1858 – before that, they were called “circular charts” or “circle graphs.”
  • During WWII, isotype pie charts (where each slice shows whole units as icons) were used to explain complex statistics to the general public.
  1. REAL-LIFE CONTEXTS
  • Population demographics in students’ local area/school
  • Screen time analysis on their phones (many phones now show this as a pie chart)
  • School budget allocation
  • Sports statistics (possession time in football matches)
  • Election results in their area
  • Food group proportions in their daily diet
  • Streaming service viewing habits (Netflix often shows this)
  1. INTRODUCE A HEADACHE

Set up a “School Improvement Committee” scenario. Give students this raw data about how 240 students voted on what to improve in school:

  • Library resources: 60 students
  • Sports equipment: 48 students
  • Computer facilities: 72 students
  • Canteen food: 36 students
  • Outside seating: 24 students

Ask them to:

  1. Present this data in a way that makes it easy to see:
    • Which option was most/least popular
    • How the votes compare to each other
    • What fraction/proportion each option represents
  2. Create a visualization that would be convincing in a presentation to the school board

Let them try various ways to display this. They might:

  • Make a list (hard to compare)
  • Draw a bar chart (works but doesn’t show the “whole”)
  • Try to draw a circular representation without knowing how to convert to angles

This creates the need for pie charts as a way to show parts of a whole, leading naturally into the method of:

  1. Converting to fractions (e.g., 60/240)
  2. Converting to degrees (× 360)
  3. Using a protractor to create accurate sectors

This is effective because:

  • It uses data they care about
  • Creates a genuine need for the visualization
  • Shows why other methods are less effective
  • Connects to their role as stakeholders in school improvement

ChatGPT’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE

Flow Diagram:

  • Prior Knowledge:
    • Understanding fractions, percentages, and proportions.
    • Calculating percentages of a quantity.
    • Basic geometry concepts like circles and angles (360° in a circle).
    • Interpreting and representing data using bar charts or tables.
  • Present Knowledge:
    • Representing data as fractions or percentages of a whole.
    • Converting percentages into angles (e.g., 50%50\%50% of a circle = 180∘180^\circ180∘).
    • Drawing and interpreting pie charts accurately.
  • Future Knowledge:
    • Using pie charts to compare datasets or visualize survey results.
    • Applying pie charts in real-world data visualizations (e.g., financial reports, election results).
    • Moving to more advanced data representation techniques like histograms and scatter plots.
  1. SHARE HINTERLAND KNOWLEDGE
  • Historical Context:
    • The pie chart was first used by Scottish engineer and economist William Playfair in 1801 to represent data. Playfair created pie charts to illustrate proportions in the British national debt.
  • Interesting Anecdotes:
    • Pie charts are a popular choice in the media and business world due to their visual simplicity. However, they are sometimes criticized by statisticians (like Edward Tufte) because they can be misleading if the proportions are too similar or if too many categories are included.
    • Pie charts are so iconic that they’ve inspired humor: people joke about how pie charts can be used to represent what percentage of a pie you’ve eaten!
  1. REAL-LIFE CONTEXTS
  • Business and Finance:
    • Companies use pie charts to represent market shares, sales distribution, or budget allocation. For example, “What percentage of the company’s budget is spent on salaries, marketing, and operations?”
  • Elections and Surveys:
    • Pie charts are used to show the distribution of votes or survey responses. For example, “What percentage of students prefer football, basketball, or swimming as their favorite sport?”
  • Daily Life:
    • Visualizing personal expenses: “How much of your monthly spending goes to food, rent, entertainment, etc.?”
  1. INTRODUCE A HEADACHE
  • Headache Scenario:
    • Provide students with a dataset in percentage form and ask them to visualize it (without guidance). For example:
      • “A survey shows that 40% of people like chocolate, 35% like vanilla, and 25% like strawberry. Draw a pie chart to represent this data.”
      • Many students will struggle to divide a circle into proportional sections without understanding how to convert percentages into angles

Idea #3: Quadratic graphs

Claude’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE
  1. SHARE HINTERLAND KNOWLEDGE
  • The parabola shape was first studied by Apollonius of Perga (c. 240-190 BCE). He discovered it by slicing a cone with a plane parallel to its side.
  • Galileo Galilei proved in 1638 that objects thrown through the air follow a parabolic path. The story goes that he observed cannonballs being fired and noticed this pattern.
  • Ancient civilizations like the Babylonians used quadratic relationships to calculate areas, but didn’t visualize them graphically.
  • The word ‘parabola’ comes from Greek meaning “to throw beside” – relating to how these curves can represent thrown objects.
  • Modern satellite dishes and car headlights use parabolic shapes because of their unique reflective properties – all parallel rays hitting a parabolic mirror reflect to a single point (the focus).
  1. REAL-LIFE CONTEXTS
  • Architecture: The Gateway Arch in St. Louis is an inverted quadratic curve
  • Sports: The path of a basketball, football, or tennis ball through the air
  • Physics: Motion of objects under gravity
  • Engineering: Cable-stayed bridges (the cables form parabolic curves)
  • Design: Water fountains
  • Phone signals: The shape of signal strength coverage from a cell tower
  • Spotlights: The reflection pattern from parabolic mirrors
  1. INTRODUCE A HEADACHE

“The Championship Game” Tell students they’re going to analyze a championship-winning basketball shot. Give them this scenario:

“A basketball player takes a shot from the 3-point line. You have these measurements of the ball’s height at different distances from the player:

  • At 0m (when shot): 2m high
  • At 2m: 3m high
  • At 4m: 3.5m high
  • At 6m: 3.5m high
  • At 8m: 3m high
  • At 10m: 2m high

Questions:

  1. Plot these points
  2. What’s the maximum height of the ball?
  3. Will the ball reach the hoop, which is:
    • 3.05m high
    • 7m away from the player?”

Let them struggle with:

  • Plotting the points (they should get something that looks roughly parabolic)
  • Trying to connect the dots (straight lines? curved lines?)
  • Estimating the maximum height (how can they be sure?)
  • Predicting if it will go in (how can they find the exact height at 7m?)

This creates several needs:

  1. A way to draw a smooth curve through the points
  2. A mathematical way to find the exact height at any point
  3. A method to find the maximum height precisely

When you introduce quadratic graphs, they’ll see:

  • Why the curve must be smooth
  • How to find exact values using the equation
  • How the vertex gives the maximum height
  • How they can use this to model real-world situations

This works because:

  • It’s a genuine real-world application
  • The context is engaging for many students
  • The need for precision is clear
  • It connects to their experience and interests

ChatGPT’s attempt

  1. CONNECT PRIOR, PRESENT AND FUTURE KNOWLEDGE

Flow Diagram:

  • Prior Knowledge:
    • Understanding coordinates and plotting points on a Cartesian plane.
    • Solving simple linear equations and graphing straight-line equations.
    • Recognizing the structure of quadratic equations
    • Calculating square numbers.
  • Present Knowledge:
    • Plotting quadratic graphs by substituting values i
    • Understanding the shape of a parabola (its U or inverted-U shape).
    • Identifying key features: vertex, axis of symmetry, roots, y-intercept.
    • Recognizing how the coefficients aaa, bbb, and ccc affect the graph.
  • Future Knowledge:
    • Solving quadratic equations graphically.
    • Understanding transformations of functions (e.g., translations, stretches).
    • Using quadratic graphs in physics (projectile motion) or optimization problems.
  1. SHARE HINTERLAND KNOWLEDGE
  • Historical Context:
    • Quadratic equations and their graphs date back to ancient civilizations. The Babylonians (around 2000 BCE) solved quadratic problems using geometric techniques.
    • The parabola was studied by Greek mathematician Apollonius of Perga around 200 BCE, but it wasn’t until René Descartes introduced coordinate geometry in the 17th century that we could plot them on a graph.
    • Parabolas are not just mathematical abstractions: they appear in nature and engineering (e.g., the trajectory of objects in freefall).
  • Interesting Anecdotes:
    • The St. Louis Gateway Arch in the USA is modeled on a type of parabola.
    • Satellite dishes and headlights use the reflective property of parabolas: they focus energy (light or sound) on a single point, the focus.
  1. REAL-LIFE CONTEXTS
  • Physics and Engineering:
    • Quadratic graphs represent projectile motion in physics. For example: “A ball is thrown, and its height over time is modeled by h=−5t2+10t+2h = -5t^2 + 10t + 2h=−5t2+10t+2. What is the maximum height?”
    • Parabolas are used in bridge and arch construction (e.g., suspension bridges and parabolic arches).
  • Economics and Business:
    • Profit or revenue models often involve quadratic functions. For example: “A company’s profit is modeled by P(x)=−2×2+20x−50P(x) = -2x^2 + 20x – 50P(x)=−2×2+20x−50. What quantity maximizes their profit?”
  • Daily Life:
    • Modeling things like the height of a fountain jet or the path of fireworks.
  1. INTRODUCE A HEADACHE
  • Headache Scenario:
    • Give students a problem involving a quadratic function without mentioning graphs:
      • “A ball is thrown into the air, and its height is modeled by h=−4t2+16t+5h = -4t^2 + 16t + 5h=−4t2+16t+5. What is the highest point it reaches?”
      • Without understanding how to visualize the quadratic function as a graph, students may struggle to interpret the problem or find the maximum height (the vertex).
    • Alternatively, present the equation of a parabola (y=x2−4x+3y = x^2 – 4x + 3y=x2−4x+3) and ask students to list all solutions without showing the graph. They’ll realize the difficulty of solving it algebraically (especially if the roots are not integers), motivating the need for a graphical solution.

Conclusion

What do you think?

I am still at the stage where this stuff blows my mind. Sure, it is not perfect, and the conclusion is much the same as when we looked at using AI to help uncover student misconceptions, write multiple-choice questions, and plan lessons – AI helps, but we cannot rely on it 100%.

However, there are some really nice ideas between Claude and ChatGPT. Once you have the prompt, you can generate similar ideas for any lesson you are teaching in less than 5 seconds. So long as you are willing to spend a few minutes reading the responses and thinking about the most suitable for that topic and your particular class, you have a tool that could save you time and help you do your job better.