Comments on: Speaking Mathematically

By: Helene Matte

Helene Matte — Tue, 21 Jan 2014 11:42:32 +0000

Knowing why we teach algebra is key. Algebra is the language we speak, we read and interpret as well as communicate with. Younger Ss should be exposed to growing patterns (The queen of this has to be Fawn Nguyen-Math talks). Kids translate what they see into algebra. They communicate in words what they see. They translate it to expressions that make sense to them. The opposite can also be done, if p=2s + 2 what could I be describing? Older Ss make other types of data match to algebraic expression. Algebra comes from the matching of this data to a particular function type. The expressions are the sentences with which we communicate and these sentences can sometimes be written in simpler forms. We then lose the visual of what we saw but is allows us to do other things. Solving for x at this point has meaning.
Transitioning from reading, to writing, to translating, to interpreting, to using this language is a beautiful thing!

By: Joe Schwartz

Joe Schwartz — Sun, 19 Jan 2014 11:58:05 +0000

Well said Cathy. Here’s a quote from an interview with Howard Gardner that I’ve been thinking about recently:

“Another obvious implication, one that only a few people have begun to take seriously, is that we’ve got to do a lot fewer things in school. The greatest enemy of understanding is coverage. As long as you are determined to cover everything, you actually ensure that most kids are not going to understand. ”

http://www.ascd.org/publications/educational-leadership/apr93/vol50/num07/On-Teaching-for-Understanding@-A-Conversation-with-Howard-Gardner.aspx

That’s from 20 years ago.
I worked on our fourth grade curriculum last summer, getting into alignment with the common core. We took out almost 15 days worth of lessons and replaced them with 15 days to dig deeper and explore fewer things in greater depth, and it’s a little better but it’s still not enough time.

By: Cathy Yenca

Cathy Yenca — Sat, 18 Jan 2014 15:05:17 +0000

Clara’s comments resonate with me so very much. At the start of my career, the 7th graders I taught received this type of exploration-based experience with little worry from me about the #1 enemy (TIME) because, at that point in history, 7th graders did NOT have state-mandated standardized testing in Pennsylvania – only grades 3, 5, 8 and 11 had to learn a year’s worth of math by the big test in March.

It saddens me that now I feel arms pulling in opposite directions constantly, between s l o w i n g down to let students explore, smell the mathematical roses and construct their own meaning, versus “getting through the content in time”. All this to say, perhaps some teachers are nervous about taking this type of teaching plunge because of the time factor and pressures to “get through curriculum”…? After all, it sure is quicker to just tell students what to do, isn’t it? (tongue-in-cheek)

I wish I could say that I always make the “right” choice here for my own students, but sometimes time-pressures win.

Thanks for sharing, Dan, and best wishes on your dissertation!

By: Clara

Clara — Mon, 13 Jan 2014 10:12:38 +0000

I too feel as Eric and many of you, that we should let students encounter the process first, then the formula. I am currently teaching at a high needs school and have many students struggling to learn math. Both the students and the other math teachers are caught up in teaching math formula first, pushing the memory tricks, rushing through the lessons (2-3 days per standard) and testing to death. They moan over the lack of progress. The students are slow to explore, ask questions or get curious. I know my colleagues think I’m addled as I push lessons that explore, expect the students to become engaged, and take a little more time in exploration with my class. And it’s not that they aren’t good, caring teachers. They are! They are simply stuck teaching how they were taught and teaching the way they learned to teach. And it is hampering progress for our students. Thank you Dan for continuing to spread this information- and for helping us recognize that these ideas are not new to us – and that they are really better for our students!

By: Eric Aispuro

Eric Aispuro — Mon, 13 Jan 2014 07:20:05 +0000

I agree with you all! :) As a middle school teacher of 10 years, primarily to remedial students, they are so much more capable of working with patterns when you take away the “math”. If you asked remedial students to write the algebraic expression for a linear toothpick pattern, less than half would succeed. If you asked them simply to write a paragraph describing how they would find the next figure, then the 100th figure, most would succeed. Fighting this culture of math-phobia is hard, but I’ve found that teaching the variables and “correct” math form should come way at the very end, after the students have found their voice in describing a concept and become convinced of it in their own mind.

By: Dan Meyer

Dan Meyer — Sun, 12 Jan 2014 02:41:00 +0000

@Joe, we're all obliged for the elementary ed perspective. You should bring it around here more often.

By: Joe Schwartz

Joe Schwartz — Sun, 12 Jan 2014 00:34:43 +0000

Dan,
Thanks for the post. There is quite a bit in there to think about and process, and I’ve left some comments in the google doc. One quote which has resonance:
“The teacher may be too concerned with the form of what is being said at the expense of the meaning which the pupil is trying to convey.”
Coming at it from an elementary school perspective, I see teachers who are uncomfortable with the concepts and the material themselves. They become overly dependent on their manuals and keys to provide them with the “correct” vocabulary and language and answers. When what the student says and what the “book” says does not match, then the student is wrong. Sometimes they cannot interpret what the student is trying to say, and they do not have the confidence and the experience (and experience is a big problem in this new age of attrition) to go “off message” and let the students take the lead expressing mathematical concepts in age-appropriate language.

By: Dan Meyer

Dan Meyer — Sat, 11 Jan 2014 13:43:19 +0000

Alex:

These students in this context do not need the symbolization. They need a voice. They need to explain how they got their solution. They need to understand that they can do math.

Thanks for your thoughts here, Alex. I hope you find a minute to dip into the Pimm quotes. I think you'll find a lot that resonates. He's pretty emphatic about the power of symbols, though. Both because you can manipulate them but also because it can be faster and clearer to express "well I have to subtract one and then divide by 3" as "(x - 1)/3". What do you think would happen if you showed your student the symbolic form after she wrote it in words?

By: Alex Overwijk

Alex Overwijk — Sat, 11 Jan 2014 12:46:11 +0000

Hi Dan,

Been reading your blog for a long time. Amazing journey.

Comment on “premature symbolization of school mathematics”

I have been teaching a grade 10 course for applied learners (many at risk for various reasons) for years now through activities only. Early in the course students are given linear relation situations to solve in a context. Think linear patterns. Something like a string of toothpicks in the form of a square with a common side. T=3S+1 where T is the number of toothpicks and S is the number of squares.

Most of these students, when left to their own devices (this means we have not discussed linear relations-not even seen them in the course yet), will look at the pattern of numbers and extend it until they reach the answer. Even the students that can symbolize the context tend to answer the questions using logic rather than the symbols. Knowing the number of toothpicks they would say” well I have to subtract one and then divide by 3″.

What am I getting at?
These students in this context do not need the symbolization. They need a voice. They need to explain how they got their solution. They need to understand that they can do math.

They do not need the symbols. I do not force it on them like I might of years ago. (I would not of even introduced linear relations like this years ago – I would of “taught” them it)

By: Mark D. James

Mark D. James — Fri, 10 Jan 2014 00:54:13 +0000

Great excerpts, great thoughts, and great comments by so many. Late to the MTBoS party though I am, and though mostly still lurking…I am really feeling inspired and energized by what has become the most meaningful PLC in my 14 years of teaching mathematics. Thanks Dan!