SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

28 Sep

We want every learner in our care to be able to say

I can reason abstractly and quantitatively.


I wonder what happens along the learning journey and in schooling. Very young learners of mathematics can answer verbal story problems with ease and struggle to translate these stories into symbols. They use images and pictures to demonstrate understanding, and they answer the questions in complete sentences.

If I have 4 toy cars and you have 5 toy cars, how many cars do we have together?

If I have 17 quarters and give you 10 of them, how many quarters will I have left?

Somewhere, word problems become difficult, stressful, and challenging, but should they? Are we so concerned with the mechanics and the symbols that we’ve lost meaning and purpose? What if every unit/week/day started with a problem or story – math in context? If learners need a mini-lesson on a skill, could we offer it when they have a need-to-know?

Suppose we work on a couple of Standards of Mathematical Practice at the same time.  What if we offer our learners a task, Running Laps (4.NF) or Laptop Battery Charge 2 (S-ID, F-IF) from Illustrative Math, before teaching fractions or linear functions, respectively? What if we make two learning progressions visible? What if we work on making sense of problems and persevering in solving them as we work on reasoning abstractly and quantitatively. (Hat tip to Kato Nims (@katonims129) for this idea and its implementation for Running Laps.)


Level 4:

I can connect abstract and quantitative reasoning using graphs, tables, and equations, and I can explain their connectedness within the context of the task.

Level 3:

I can reason abstractly and quantitatively.

Level 2:

I can represent the problem situation mathematically, and I can attend to the meaning, including units, of the quantities, in addition to how to compute them.

Level 1:

I can define variables and constants in a problem situation and connect the appropriate units to each.


You could see how we might need to focus on making sense of the problem and persevering in solving it. Do we have faith in our learners to persevere? We know they are learning to reason abstractly and quantitatively. Are we willing to use learning progressions as formative assessment early and see if, when, where, and why our learners struggle?

Daily we are awed by the questions our learners pose when they have a learning progression to offer guidance through a learning pathway. How might we level up ourselves? What if we ask first?

Send the message: you can do it; we can help.


[Cross-posted on Experiments in Learning by Doing]


Posted by on September 28, 2014 in Standards for Mathematical Practice


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4 responses to “SMP2: Reason Abstractly and Quantitatively #LL2LU (Take 1)

  1. Rudolph, Heidi

    September 28, 2014 at 7:00 pm

    I just posted a TI-nspire file for Laptop Battery Charge 2….

  2. howardat58

    September 28, 2014 at 8:02 pm

    The traditional approach has one fundamental flaw: We teach them a new idea, some theory, some techniques and some numeric examples. and then do “word problems”. So their minds are focused on how to use the techniques they have just learned, and do hot look at the problem as a real problem (I don’t mean real in the dan meyer sense here).
    Particularly with arithmetical problems there are often many ways to go at them. Some kids can “see” the “answer” but will never get the hang of the x’s and y’s. many problems can yield up approximate solutions by looking/thinking, with no use of formal methods.
    I say, go for it !

    I do think that the word “abstractly” is a bit heavy in these situations. Mathematicians do not think “abstractly”, but they do present the results in an often very formal and abstract form.
    Engineers too, although that is more understandable. Ask an engineer to explain Fourier series and you will, mostly, get a very down to earth explanation.

  3. howardat58

    September 28, 2014 at 8:09 pm

    Sorry, there is a bit more !
    I quote “Use symbols to represent a situation…”
    This is not abstract, it is a coding method, to avoid having to use the full description of the “thing”.
    Doing algebraic stuff before algebra was invented was just extremely hard and tedious – ask an ancient Roman. It is rather fun to get the kids to write their equations out in full, without x and y …
    I used to tell mine to stop complaining, this is how it was done before 1400AD


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