At the end of the day for math education, it comes down to the simplest idea — what kind of math problems are we giving students…and why? So, that is why it is important to always burrow right down to the most micro idea of mathematics. A problem.
A problem that is uncluttered by language/instructions. A problem that is rich enough to drive pedagogical discussions — not be driven by them. Again, good content will always direct you to good pedagogy. Starting with good pedagogy, the result is not guaranteed.
Notice I haven’t even offered the question yet for the picture above. But, I am sure there is curiosity, and yes, even bewilderment how this question spilling over with variables could be a question that intersects the lowest elementary grades. How?
But, the larger embedded question is if it is possible that this question is challenging for every grade level from elementary to high school? Can a first grade student be mucking around with this question with the same excitement as a senior student who is in the throes of learning calculus.
Yes. And, that is why this question is so unique and, why I feel this is the greatest K to 12 question ever constructed. The author of the question is Peter Harrison. He is now well into his retirement. But back in the 80’s, in the middle of his career, he was a math education giant in Canada. He was ahead of his time. He still is. He spent a great deal of time thinking about the bridge between arithmetic and algebra. Here he is in his own words — written over 30 years ago.
I was fortunate to teach with Peter for 4 years back in the early 2000’s, at the tail end of his career. He even visited my classroom — in Switzerland — when I taught at an International IB School there in 2005/2006. Needless to say, our friendship has gone well beyond the classroom.
So much of who I am as a math educator is owed to him. I saw him teach. His prowess for mathematics might have only been eclipsed by his empathy and kindness he showed towards all his students. His message, implicit as it was, was simple and to the point — care about the mathematics and care about the students. Nothing else will matter. And so it…didn’t…
SO! You have been waiting patiently for the question! Here it is:
Peter understood that disarming students with algebraic thinking early on was critical. Here are 8 unknowns! But, the context is perfect, as they have to be labeled as such due to the nature/quality of the question.
K to 2 students should know their single digit numbers. With some basic fluency of single digit addition, they will happily plug and play, getting sections of the question correct. If A = 1 and B = 2, then x = 3. Now you just can’t use those numbers again! It will be mind-boggling fun for them!
Middle School students, still not wholly comfortable with algebraic expressions, will plug and play, but with perhaps more sophistication. What numbers can’t the bridges be? What numbers can’t the fields be? Can all the bridges be even? Can all the bridges be odd? There are 8 unknowns and 9 numbers. Does this mean anything?
The problem becomes Sudoku-like. And, while the numbers are single digits, and the operation only addition, the solutions to the problem will not come easily. This is a good thing. And, this is why this problem, for me, is the greatest K to 12 math question ever.
The question, while amusing to flail around with using logic/trial and error, opens the door that Peter has wanted all students that he has ever taught to walk through. That would be algebra.
The algebraic solution is elegant. This should not be surprising, as “elegance” is the defining characteristic of algebra — it cleverly generalizes arithmetic to beautifully argue for a clear path to the solution. And so, even though elementary students are a decade or so removed from the formalization of this problem with algebra, they are thoroughly enchanted with the problem and have no qualms about seeing the problem raining variables.
The best math questions have a journey that dwarfs the destination. The journey being the process of understanding and deep thinking embedded in rich problems. The destination, in this case, is just a structured set up of single digit numbers — not terribly interesting. The interest/fascination is the patient trek of playing, conjecturing, questioning, and solving a question with a variety of tactics.
The crowning tactic being algebra.
This is the reason why those who propose to get rid of algebra, do not understand the inextricable thread — and the length of it — that algebraic thinking plays in K to 12 mathematics. If it is as seen, as it currently is, as some disconnected topic/appendage awaiting teenagers, then by all means, contemplate its removal.
But, if we value — at least I hope we value — that algebraic thinking is a mathematical idea that stretches across the domain of learning from elementary to high school, then we have a moral obligation to not just keep algebra in our curriculum, but to expand upon it enthusiastically — and not apologize for its intersection with math students.
Should we change the language of Shakespeare from Early Modern English to something else? Of course not. The only question that should be asked is if learning about Shakespeare is important.
Similarly with mathematics. Algebra is the language of mathematics. I would have actually less problem about getting rid of mathematics than I would with getting rid of algebra. To deform an art form to satisfy misunderstood and superficial whims of those not in math education is not tolerable — especially when those people see algebra as a high school topic to find out how old Mary’s brother will be in 5 years. IF that was algebra, I would want it removed as well.
Algebra is like Spartacus. You want to take it out. The rest of mathematics will stand up and shout “I am algebra”!