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I’m a homeschool teacher of a nine-year-old, and we sometimes have one-on-one ‘number talks’ (a.k.a. 'math talks') similar to the activity used in primary school classrooms.

Part of this process involves writing down the thinking process used to solve a particular problem. For example, my child might explain the steps they used to solve 5 x 25 in their head. In turn, I write the process down on paper as they watch.

Is there a standard notation / recording technique for writing down thinking processes used in number talks? Or is there a resource that may provide some helpful ideas in this regard?

In my brief experience, I’m finding that my written diagrams or notations are not ideal. Admittedly, I will draw operators, numerals, and arrows on-the-fly, and they don’t always clearly communicate the process, especially when looking at them a few days later.

I’m particularly concerned about this because one of my longer term goals of these number talks is to encourage good written communication about math, and I’m not always being a good role model.

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The notation used on the board in math lectures should be the same as that found in any other mathematical writing. That is: Get a good book, read it very carefully, and follow that same notation precisely.

This strengthens connections and prepares the student to read outside texts and articles on their own (which should be one of the principal goals of any educational program, in my opinion).

If you don't have any other textbooks available right now, then I recommend the OpenStax Math series.

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    $\begingroup$ @SueVanHattum: OpenStax Prealgebra starts with sections on identifying whole numbers, place values, how to add whole numbers, etc., so arguably that's at an elementary level. Counter-argument would be it's meant as review for remedial college students. My guess is that it could be used as an example of the core writing and expanded by an elementary-level tutor. $\endgroup$ – Daniel R. Collins Dec 23 '19 at 0:42
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    $\begingroup$ @DanielR.Collins: I'm with you on using correct notation, but I'd like to aim for more flexibility when it comes to illustrating a thought process. For example, if a student solved 75+25 by thinking about how quarters (25-cent coins) relate to dollars (100 cents), it seems reasonable to record this by drawing four circles containing "25." Is there formal notation that could communicate the student's thinking as clearly as that? $\endgroup$ – EJ Mak Dec 23 '19 at 4:54
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    $\begingroup$ I don't think that "standard" mathematical notations captures what many mathematicians think/do. Simple things like simplifying an expression, substituting one term for another etc. are not made explicit with the usual notation. $\endgroup$ – Michael Bächtold Dec 23 '19 at 8:53
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    $\begingroup$ The question has nothing to do with teaching math, as in lessons. It has to do with: "Part of this process involves writing down the thinking process used to solve a particular problem. For example, my child might explain the steps they used to solve 5 x 25 in their head. In turn, I write the process down on paper as they watch. Is there a standard notation / recording technique for writing down thinking processes used in math talks? Or is there a resource that may provide some helpful ideas in this regard?" I.e., how to transcribe the flow of logic, student's thought processes in their... $\endgroup$ – amWhy Dec 23 '19 at 19:34
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    $\begingroup$ @Namaste Some clarification: The handwriting I'm asking about is not just about keeping records. It is part of the educational process. During a math talk — a.k.a. number talk — the student will often watch the teacher write down their thinking process. In a regular classroom, this would done on a board. When everything goes right, a student will be able to see of visual representation of what they verbally described. If you're interested in a video example, see: players.brightcove.net/5387496875001/default_default/… $\endgroup$ – EJ Mak Dec 24 '19 at 21:41
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As you realize, this

In my brief experience, I’m finding that my written diagrams or notations are not ideal. Admittedly, I will draw operators, numerals, and arrows on-the-fly, and they don’t always clearly communicate the process, especially when looking at them a few days later.

is part of your problem. I suspect you may have difficulty reconstructing your own trains of thought (let alone those of the kids) when recorded this way.

Writing down one's thinking process is very difficult. There is no standard model. I suggest words in short sentences, with lots of "so I know that" and "because" and "I remember". Keep the narrative glue. Keep symbols to the minimum necessary.

Focus the need for explanation to the hard parts of the current discussion. Don't make kids "justify their thinking" at every step. If you're teaching multidigit multiplication you can just assume that the kids know the multiplication table and can do the necessary addition routinely.

Welcome different ways to solve problems.

Only tangentially related, but a site you might enjoy: https://www.explodingdots.org/

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Most elementary school teachers in the USA get their teaching certification by majoring in "elementary education" as an undergraduate. Curriculum for such programs typically includes a sequence of courses that focus on both the content of mathematics taught in elementary school and how to guide children in their learning. I think this guidance is what you refer to as "math talks."

Perhaps you would find some answers by reading about what is called "Cognitively Guided Instruction." For an open-source textbook that might be helpful, look here

https://open.umn.edu/opentextbooks/textbooks/mathematics-for-elementary-teachers

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