The Mathematics of Students

  • The Mathematics of Students

    By David Wees, posted July 6, 2015 –

    There is a tremendous drive in the United States, and in many other countries, to support and develop teachers’ use of formative assessment practices in mathematics education. I work for a project called Accessing Algebra Through Inquiry, in which one of our objectives is to support teachers’ formative assessment practices across 31 New York City high schools.

    But what knowledge does one need to be able to implement regular, responsive formative assessment in the classroom every day? Clearly, the better one understands the mathematics one is teaching, the easier it is to teach. For formative assessment practices, however, one also needs to understand the typical ways that students understand mathematical ideas.

    Science educators have known this about teaching science for a long time. Check out this amazing and well-organized database of the different ways students understand scientific concepts!

    Unfortunately, there is no such database for mathematics teachers. Until such a collection is created, each mathematics educator who wishes to incorporate formative assessment practices has to begin to develop his or her own understanding of the common ways that students conceive of and use mathematical ideas.

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    What does this formative assessment cycle look like in practice? First, anticipate what you expect students to do in response to a chosen task. Next, design a set of responses, perhaps by selecting a different task, choosing what feedback you will give students, or deciding how you will structure your classroom discussions. Once you implement your response in the classroom, reflect on its effectiveness and then revise your anticipation and response for the next formative assessment cycle.

    Observing student work and looking for trends in how students represent their thinking are incredibly valuable. Consequently, the strategy I suggested in my second blog post should, over time, lead to a better understanding of students’ mathematics acumen. It should also lead to stronger formative assessment practices.

    Here’s an example from Elizabeth Green’s book, Building a Better Teacher, from the knowledge and experience of Magdalene Lampert:

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    Take a moment and hypothesize why a student might think this is true.

    If you have no idea, based on this written sample, don’t despair. One excellent way to make sense of students’ ideas is to interview the actual students. If you had asked this student, he or she might tell you that 12 divided by 7 is 1, with a remainder of 5, so he or she wrote the answer as 1.5. In this case, the student has interpreted the fraction in an unconventional way and written the answer differently than perhaps expected. Knowing this information makes it easier to plan your response.

    The big idea from this series of blog posts is that making sense of student work is a valuable process that can help inform your teaching and support your developing understanding of student thinking. These qualitative approaches shouldn’t stop you from finding quantitative ways to describe student growth, but they can inform and expand your available information with which to make decisions.

    2015-05-25 Wees DavidDavid Wees,, is a Formative Assessment Specialist for New Visions for Public Schools in New York. He tweets at @davidwees and blogs at


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    Justin Butler - 11/18/2015 6:57:36 AM
    The link to the AAAS Science Assessment site is great. A similar site geared towards math standards would be an excellent resource. Like Sahid, I agree that no student should have been taught that 7/12=12/7, but in my experience it seems that most misconceptions were never taught. Instead they are usually a result of an over-generalization or a failure to understand the process that an algorithm represents. I believe the SMPs provide a wonderful guideline to create instruction that enables the student to learn the concept rather than strictly an algorithm.

    Abbie Arlinghaus - 11/7/2015 9:45:00 PM
    This is an excellent way to determine how you will assess students. It is smart to ask or interview students on how they got an answer. As you go through a problem, it is important to try to think of all of the different possibilities that students could come up with, in order to expand their creativity and critical thinking in math. I want my students to think outside of the box when working on a math problem, so that I know that they actually understand what the problem is asking, instead of just going through the motions with a formula of some sort. As you work through selected student answers as a class, to lead students in the right direction and to prove a learning point, ask students how they arrived at certain answers, so that you can clear up any misconceptions the students might have about a problem.

    David Wees - 7/8/2015 10:16:15 PM
    Sahid, Students often think that division requires a larger number of things divided into a smaller number of groups, often because when they are first introduced to division, problems are chosen where this is specifically always true. Basically, we should spend a bit more time introducing division in cases that don't work out to nice whole numbers but which students can easily visualize, like 2 applies divided into 4 pieces. The symbols are more useful if they are introduced as students have a need to talk about and describe fractions to each other and to their teacher, just like most formal language. As for how students understand the symbols, this depends on a lot of their experiences. For example, my son learned the symbols, words, and ideas for 1/2 and 2/4 via everyday experiences. He inferred from his experiences that the 2 in 1/2 told you how many quarters were in the fraction, leading him to decide that 1/3 = 3/4. After more experiences with fractions, he doesn't think this is true anymore (introducing fractions on a number line is useful here, so that relative sizes of fractions can be seen). Matt, I agree that we too often jump into formal experiences in mathematics, but even with carefully constructed instruction, these kinds of ideas crop up. So yes, procedure-focused mathematics is more likely to lead to students having less useful mental models (eg. their mathematical concepts are almost purely discovered rather than made explicit), students are likely to have other ideas come up with different types of instruction too. The key is that we need to figure out how students understand the ideas no matter how we are teaching them.

    Sahid MSc - 7/8/2015 2:42:56 AM
    I have a question: Why students can have a misconception such as you mentioned that they think 7/12 as 12/7? Of course they never been taught that 7/12 is the same as 12/7. Why could they have such misconception? Do they already have their own concept about fraction? Have they misread? Or, forgot the meaning of that symbol? What make students having difficulty in reading some mathematical symbols? What is the best way to introduce mathematical symbols to students? Do students have their own intuitive about mathematics symbolization instead of those used by mathematicians? There are many other questions to answer to know why students may have such kind or error in mathematics. I would like to hear from the experts here... :)