[LOA] What “The Literature” Says

If all of this ladder of abstraction material has seemed soft, fuzzy, and opinionated so far, I’ll offer up my summer project, A Literature Review of the Process and Product of Abstraction. Feel free to add comments or questions in the margins. I’ll try to get in there and chop it up with you. If you have written more than a handful of literature reviews yourself, I’d be grateful for your feedback on the format.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Hi, Dan. You asked about literature review format. On brief glance at your literature review, I noticed that it’s in somewhat of a “book review format” in that each paragraph appears to be devoted to a different author or article. (So-and-so says is the way that you seem to start many paragraphs.)

    To take it to the next level, paragraphs would be about connections between readings, not about individual pieces that you read. Each section of the paper would be instead devoted to a claim about what the literature says (in terms of groups of authors) about abstraction. (Use major headings to structure the paper for readers.) Make a point that cuts across a few pieces (so-and-so, such-and-such, and that other dude say…) rather than having each paragraph devoted to a separate author. The goal would be to integrate across pieces within each paragraph and then multiple paragraphs would be woven together to support one of the themes that you noticed across the readings.

    In other words, the paper would be structured very explicitly like: there are three significant themes about abstraction that are addressed in this literature. [Describe the themes.] Then the paper would be in support of those themes, and rarely would any one article, chapter, or study be featured in its own paragraph.

    Maybe you did all of this already and I did not look carefully enough on my first read. Also, you’re awesome for making your work public like this!

  2. Have you looked at Hans Freudenthal’s work on “mathematizing”? I haven’t read in much detail, but I wonder how the work in math ed on constructivism relates to what you’re working on (e.g., Catherine Twomey Fosnot). Seems like it could be relevant.

  3. I keep meaning to send you this and your literature review gives me the perfect opportunity. I’m in Chapter 3 of Lesh & Doerr’s book Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. It’s pretty great, and if you’d been wondering what a Modeling Math curriculum might look like after talking with people like Frank Noschese about Modeling Physics, it’s got some great ideas.

    Mostly I’ve been meaning to tell you about Chapter 3, which has a great story about a classroom of first graders who were studying plants and represented their heights by making strips of paper that were the same height as their plant. At first, the kids insisted on green strips of paper with paper flowers on top.

    Then, their questions evolved to be about which plant was taller, and the flowers got in the way. They removed the flowers and used the green strips to model the height quantity.

    They continued to measure the plants over time and began to wonder about how tall each plan would get, which was growing faster, etc. Now it became more important to think about a changing height quantity for each plant, and it was easier to see if each plant’s strip was color coded. Now the strips didn’t resemble plants at all, but they still represented the quantity plant height. They were a more abstract representation, and as the representation got more abstract, the children got better at thinking mathematically about the height quantity.

    Anyway, if you haven’t read the book, I highly recommend it (or at least the first 3 chapters and the introduction).


  4. Lovely anecdote, Max. I’ll be sure to pick up the collection.

    Lindsey, I’ve read some Freudenthal. I like how he treats the adjective “real” in the expression “real-world math.” As in, “real in the student’s mind,” which is subjective in the best possible way.

    Amanda, thanks for the feedback on the structure. (This is more than I’ve received from my advisers yet.) My goal was to find conflicts and consonance between different papers and use those as my segues but I can see how that came off disjointed.

  5. Not sure if this is the sort of thing that’s helpful, but I was reading a book about jazz piano and the piano players often talked in terms of abstractions, or removing the frills of the songs and refining them to their essence. I don’t know if music theorists have a particular theory of jazz abstraction and whether that might provide a helpful framework for the math theoretician, but I could conceive of that being fruitful.

  6. Dan,
    I agree with Amanda with regards to the structure and goal of a lit review.

    In general, I see a lit review as a product of a process of synthesis of the various sources and existing research through which you are learning (vs. just presenting) more about the topic you are reviewing. One of the challenging tasks when doing a lit review is to come up with relevant themes (to use Amanda’s word) that can shed more light on the topic beyond just summarizing what has been said by other people. If at the end of your review, you have not generated new insights, then it means you have missed the analysis portion of the review.

    One way of looking at a lit review is to think of it as a research project in which the different studies from the past are your data and you are now trying to make sense of these data. This also means that you should have a question that guides your review (not as rigid as a “research question” but you can think about it as an “inquiry question”) and the sole purpose of the lit review then becomes to answer that question using whatever is out there. When such a question exists, it is also much easier to show the gaps in the literature and raise important issues that need to be examined.

    I hope this is helpful,

  7. Freudenthal – Mathematics as an educational task
    Tall, Sfard – Procepts, reification. Do make sure something new is created, there are already plenty of litreviews.

  8. I’m with Arnon on this — it’s like treating the literature like your qualitative data and analyzing across it to see themes that are present across the texts.

    I am still super impressed that you’re making your work public like this.

  9. Met with my adviser (Boaler) today. She took exception to my use of the third person throughout the piece (ie. “This review initially searched Google Scholar ….”) She said the third person extracts me, my ideas, and my perspective out of the piece leaving behind the presumption of objectivity when in fact I’m not.

    She said I overstepped the literature with this line:

    Without being conscious of and explicit about her own proficiency with abstraction, a teacher may undermine her students’ nascent powers of abstraction.

    What I said might be true. It might not. Either way, that line doesn’t follow from the citation preceding it.

    She didn’t have any of the problems some of you mentioned here re structure or a disjointed city-by-city tour of the literature. It’s never been clearly to me who actually read the piece, though, and who’s just echoing Amanda Jansen’s first comment. In any case, conflicting feedback has been a consistent feature of my graduate education.

  10. Dan,
    From my experience (currently at the proposal stage of my Math Ed PhD), you will always have more than type of feedback, which may come from different styles of writing and perspectives, but that should not discourage you.
    I agree with Jo about the third form as well as the comment she made regarding the unwarranted claims.
    With regards to the synthesis – you may decide to apply it or not. I do believe that most people in the field would consider your lit review stronger and more valuable to the field that way.

    Looking forward to reading more of your works as you move forward.

    I did read the entire piece.

  11. Quick update here. I got feedback from my other adviser, Pam Grossman, who generally agreed with the assessment of my commenters: too much of a laundry list, not enough of an argument, too few connective threads between my citation. All of which is useful.

  12. Noteworthy, out of the way reference for you:

    Luria, A. R. (1976). Cognitive Development: Its Cultural and Social Foundations.

    In part of it he interviews farmers in early 20th century Russia that have a pre-scientific mindset — essentially, they lack abstraction.

    Q: What do a fish and a crow have in common?
    A: A fish — it lives in water. A crow flies. If the fish just lies on top of the water, the crow could peck at it. A crow can eat a fish but a fish can’t eat a crow.
    Q: Could you use one word for them both?
    A: If you call them “animals,” that wouldn’t be right. A fish isn’t an animal and a crow isn’t either. A crow can eat a fish but a fish can’t eat a bird. A person can eat a fish but not a crow.

  13. Jason, that example makes me wonder… is it fair to say they lack abstraction, or just that they are actively resisting/questioning the usefulness of the scientists’ abstraction for their needs. They were able to name and question the abstraction, just chose not to use it. It reminds me of students who are classified as not being able to do algebra because they persist at guessing and checking. But if you ask them they say, well yeah, I could write an equation, but I got there quickly and more easily with my more concrete strategy. There was no experience that drove them to need a more abstract strategy.

  14. Thanks Jason. I’m looking forward to reading the book, as I’ve read bits of Luria and Vygotsky in school and was always curious to know more of their thinking and the context it arose from. I think the article that got me curious about how to interpret their findings was this one: http://www.psy.cmu.edu/~siegler/35rogoffmor89.pdf. Specifically the anecdote from Glick (1975) about the Kpelle man who, when asked, how would a fool sort these objects, sorts them into the categories the researcher would have.

    Overall, this conversation seems to fit really nicely with the idea that abstraction isn’t a single state or skill (i.e. these people do or don’t/can or can’t abstract) but rather that the process of abstracting is one with many rungs. So, for example, looking at a picture of a log and seeing it as representing a wooden log is an abstraction. Grouping axes, hammers, and saws together as tools, or objects made of metal, and excluding a log from that group, is another abstraction. It seems important to think about which abstractions are culturally meaningful and valued, especially when the non-valued abstractions are also powerful in the world (e.g. the coordinate plane, algebra, etc.) and to think about abstractions and culturally determined rather than fixed, which Luria helps us see. Rejecting Luria’s results as racist or biased seems like throwing the baby out with the bath water. On the other hand it’s important to question whether the people in the studies were unable to engage in a certain sophisticated cognitive skill, or whether they didn’t value that skill, or didn’t value that particular application of that skill. (Another interesting source is the “What the Research Shows” section of this book: https://www.oralitystrategies.org/files/1/772/The%20Non-Literate%20Min.pdf)