3 days 3 hours ago
1 week 6 days ago
Do you remember this formula from middle school?
a = (n x W) / d
I didn't either. I was watching a 5th-grade lesson the other day and this formula was taking up a good part of the whiteboard. Considering it my business to know middle school math (among other things), my curiosity was piqued.
It turns out that the formula can be used to multiply a whole number (W) by a fraction (n/d). If you knew that already, I'm impressed...and saddened.
What makes me sad is the broad effort to reduce mathematics to the study of information. The resulting set of facts and problem-solving steps, once acquired by students, helps them to appear competent in mathematics, all without ever reaching a functional level of understanding of the underlying concepts. Worst of all, this reduction results in an incredibly overwhelming set of seemingly unrelated bits of information. It's no wonder that most kids dislike math...or, more accurately, they dislike what they are exposed to in math lessons.
In case your curiosity is also piqued, here's how my concern related to the proffered formula. Consider the multiplication problem
2 x (3/5).
A conceptual understanding of math leads the solver to see that 3/5 is literally three of something, three fifths in this case (just like someone may be dealing with three dogs or three cars). For this solver it's quite reasonable that multiplying by two would result in six of these things, six fifths in this case. What exactly is the need for a special formula??
4 weeks 1 day ago
Classroom teaching is a unique profession. Nearly everyone has a great deal of experience in classrooms as a student, so it is a familiar environment. Conversely, we are not all so accustomed to hospitals and courtrooms, so we're less apt to make assumptions about the work of our doctors and lawyers.
Consider, however, that familiarity with classroom culture is not equivalent to familiarity with the teaching profession. This is a mistake that the public often makes, most visibly in moments when the profession is under fire.
This notion came to mind when I was visiting a 3rd-grade classroom yesterday. I was witness to a perfect storm of academic disruption incurred by a bee and a cat. The bee came first, flying around the room, prompting a few screams and a lot of eye contact. Then, in a moment of relative calm, one of the students noticed a cat outside the window. The student quickly convinced several classmates that the cat had belonged to her before it ran away two years prior. A growing chorus of students made dramatic pleas for permission to retrieve the "lost" animal. Then the bee began flying again...
We all know enough about schools, and standards, and mandatory testing to know that the teacher in this room needed to teach the students an academic lesson in the disrupted session. Thankfully every day is not peppered with bees and cats, but there is a steady stream of absences and student needs that need to be addressed, and all of this as teachers face wider and wider achievement gaps within like-aged groups.
The job cannot be done perfectly and it is tremendously difficult to do it well. The patience and tenacity possessed by good teachers is astounding. If you haven't taught, you'll need to take my word for this because, frankly, you don't understand the scope of the challenges they face. Just be amazed! #goodteaching
1 month 2 days ago
"Drill and kill" is alive and well is mathematics classrooms. In support of this approach, we often hear that "practice makes perfect" or "practice builds confidence." After all, basketball coaches will require their players to shoot hundreds of free-throws during a season of practice in order to develop proficiency. So why is math any different?
The trouble is that math isn't like shooting free-throws. Free throw shooters practice free-throws in order to get better at shooting free-throws. The goal of studying math, on the other hand, is to build an understanding of a rich landscape of interconnected concepts. Practicing one particular skill repeatedly needlessly ignores other contexts, and it sends students the wrong message about what mathematics is all about.
Every moment studying mathematics should be a moment of new challenge...more like a scrimmage against the varsity team than an hour at the free-throw line! #studymath #understandingmath
1 month 4 days ago
Me in a conversation with my HSA customer service rep today:
CSR: "Give me the last four digits of your work ID number or your social security number."
Me: "####" (I called off my SS# because I don't have any idea what my work ID# is)
CSR: "Okay, I've pulled up your account. In the future, give us the last four digits of your social security number. That works much better."
Me: "I did give you the last four digits of my social security number."
CSR: "That's fine. Just for in the future."
One of the wonderful things that students learn in the study of mathematics is logical flow. No, they won't need geometry to balance their checkbooks. Yes, they will need it to avoid frustrating people in conversation throughout their lives. #studymath #understandingmath
1 month 6 days ago
I stopped in to visit my son's class. After a nice reading session, the teacher started into a math lesson on repeated addition. The problem 5 + 5 + 5 showed up on the board. The teacher called on my son to identify the answer (I believe as a courtesy to me). He called off "15." Proud dad. The teacher then turned to place a new problem on the board, but my son jumped in with "Want to know how I got it?" "Well, yes" the teacher said. He told her "I added two of the 5s to get ten, then knew 10 and 5 is 15." (As much as he lets me, I like to talk with him about how he thinks.) Really proud dad!
Math isn't answers. We need to talk with kids about how they're thinking! #understandingmath
1 month 1 week ago
MATH LESSON (....and you thought that was all behind you!)
When you and I first learned about finding the perimeter of a rectangle (think way back) we were taught to add the four sides together. Then, like now, this stage quickly led to the exclusive use of the formula, P = 2L + 2W, which provides a seemingly quicker route to the answer.
Consider, however, the rectangle such that L = 17.3 m and W = 2.7 m. The perimeter, using the formula, is P = 2(17.3) + 2(2.7) = 34.6 + 5.4 = 40 m. Note, however, that this perimeter is more easily identified by using the 'adding' approach (17.3 + 2.7 = 20, 20 + 20 = 40 m). The student who understands the underlying mathematics can spot the two-side sum of 20 m, and from there the problem solves quickly.
No big deal? Well, yes it is. This example illustrates the limitations of the current state of mathematics instruction in our schools. Teachers are given a broad curriculum and little time, and the result for the students is a smorgasbord of algorithms and formulas...drill and move on. Mathematics has become an overwhelming diet of steps to be memorized despite the fact that there is little that needs to be memorized (e.g. the formula P = 2L + 2W is merely an observation derived from the properties of a rectangle). Mathematics has a beautiful internal consistency with very few arbitrary rules (unlike the English language!). Students who finish their educations without realizing this simply haven't learned mathematics.
Incidently, I didn't begin to "learn mathematics" until I began teaching it. Let's change how math is taught. #understandingmath