Children should be led to make their own investigations and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible.
-David Eugene Smith (1904)
I will never forget the day I finally became a mathematician.
Was it when I was placed in the advanced math pathway in middle school? Nope.
While passing the AP Calculus test? Nope.
When I graduated with a B.S. in Mathematics from Cal State Hayward? Not even close.
Not even while teaching 9th grade Algebra in Oakland Unified School District…my first teaching job in 1989.
None of those important milestones made me feel like a mathematician. I didn’t understand it at the time, but during those milestones, my relationship with mathematics was founded upon my ability to mimic what was taught to me by my teachers. I was very good at plugging-and-chugging my way through whatever homework was assigned to me. But that did NOT make me a mathematician.
The day I became a mathematician occurred early in my teaching career when I was preparing for the following day’s lesson on the Pythagorean Theorem. I was preparing a typical lesson that would have begun like this…
“Alright students. Whenever there is a right triangle, the the two legs and the hypotenuse of that triangle are in a special relationship that can be written as \(a^2 + b^2 = c^2\). Let’s practice on this triangle…”
Goodness…what a snooze-fest of a lesson I was planning for my students. I was going to inflict on my students, the same plug-and-chug style of math that was inflicted upon me. I might have been teaching math to my students, but I was definitely NOT teaching them to become mathematicians!
Fortunately, I suddenly stopped my lesson planning when I realized that I couldn’t quite remember any proof of the Pythagorean Theorem. There are well over 300 proofs of the theorem and I couldn’t recall any of them. Perhaps it was because I was never shown any of the proofs. Regardless, I decided to prove the theorem myself.
I began by drawing the classic figure…
Then I outlined the entire figure inside a rectangle…
I labeled the rectangle with the lengths that I knew…
I can now express the area of the large rectangle in two ways:
- The length times the width of the large rectangle
- Finding the sum of all the areas of the squares, rectangles, and triangles that make up the large rectangle
Since both of those methods should give me the same result, I can set them equal to each other…
Now I just do some simple algebra…
And there it was. Proof that the Pythagorean Theorem is true.
I can’t express enough the absolute JOY I felt looking at my work and knowing it was MY thinking that proved the Pythagorean Theorem. I was not mimicking something a teacher told me to memorize. Rather, for the first time, I BECAME A MATHEMATICIAN!
For the first time, I was something more than a mimicker who merely practices computations and answer-getting. I was an authentic mathematician who could invent his own understanding.
It didn’t matter that my proof was not the most efficient method possible. Nor was I the first person to come up with this proof. None of that mattered. What mattered was I came up with this solution technique on my own. It was MY invention.
I have since learned a variety of other, more efficient, methods of proving the Pythagorean Theorem, but none mean as much to me as the one I invented.
Don’t worry if you didn’t understand all that algebra goobledy gook. Just know the lesson I learned from this experience is that it is essential to let our students INVENT as much of their learning as possible. We must teach less, so our students can learn more. By being less helpful, we are allowing our students to become the mathematicians they desperately want to become.
It turns out that we have long known the joy students feel when given the opportunity to explore and develop their own understanding prior to more formal approaches from the teacher. From 1906 and 1823…
It is unwise to ask children always to solve examples by the method of the book or by that of the teacher. Often children see through problems in ways distinctly individual. If their solutions are logical, correct, clear, and brief, we ought not merely to accept, but even to welcome them.
-William Chancellor (1906)
To succeed in this [teaching math to students], however, it is necessary rather to furnish occasions for them [students] to exercise their own skill in performing examples, than to give them rules.
-Warren Colburn (1823)
Findings from a number of research studies show that when students discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas. In a nutshell, teachers should regularly allow students to build new knowledge based on their intuitive knowledge and informal procedures.
I like the phrase “explore first, formalize later”. MathMedic calls it “experience first, formalize later”. Regardless of the wordsmithing, the point is to allow students the opportunity to develop their own intuition and understanding first. THIS IS WHEN STUDENTS BECOME MATHEMATICIANS.
Formalizing – learning the algorithms – can come later. After the student has done the authentic sense-making work that mathematicians do.
What does this look like in your grade? Really it isn’t very exotic…and is very accessible. Just provide a word problem from your lesson and let students work on it PRIOR to you teaching the steps. Follow the 5 Practices Framework to prepare and deliver your lesson. I think the most important step of the 5 Practices is the Anticipation stage, but they are all important…
- Anticipate: Think about solution strategies students might use and misconceptions students might have. Anticipate which two or three methods you will highlight during your lesson.
- Monitor: While students are working on the problem, look for students to be using any of the solution strategies you anticipated. Also look for students exhibiting the misconceptions you predicted.
- Select and Sequence: Select two or three groups to share their thinking. Sequence those groups to share in an order that serves your purpose. For example, you might ask groups to share beginning with the least sophisticated solution strategy and moving on to more sophisticated strategies. Alternatively, you might ask the first group to share the most common misunderstanding and then the second group will share a successful solution strategy.
- Connect: After the groups have shared, this is your time to connect their thinking to the learning objective you have for the day. This is essentially a short direct instruction mini-lesson.
Give it a try with your students…whatever their grade level!
Look at the next lesson you would have taught. Find a good word problem in that lesson and give it to your students FIRST. Let students explore the problem.
It is very likely students will solve the problem. But – like my Pythagorean proof – it might not be the most efficient way to solve the problem. But, it will be the result of authentic math thinking of your students. You will have given them the chance to feel the JOY OF BECOMING MATHEMATICIANS.
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