Association of Teachers of Mathematics in Maine
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Chapter 1 Response Choice 1

28 Nov 2018 5:06 AM | Anonymous member (Administrator)

Identify one important mathematics topic that you teach. Think about your goals for this topic in terms of the SOLO model discussed in this chapter.

1. Do your learning intentions and success criteria lean more toward surface (uni and multi-structural) or deep (relational and extended abstract)?

2. Are they balanced across the two?

3. What can you do to create a balance within this topic? Or do you think a balance isn't necessary? 


Comments

  • 29 Nov 2018 12:50 PM | Anonymous member
    Reading about the SOLO model reminded me of the learning progressions that my district (South Portland) has asked teachers to make for students. The progressions start off at the surface level and progress towards the deep. These are based on Jan Chappuis' work with Assessment for Learning.

    For example, this is a learning progression that we created for dividing decimals:

    I can...
    - divide non-decimal multi-digit numbers
    - estimate the quotient when dividing decimals
    - convert decimals to fractions with common denominators and divide the numerators to solve decimal division problems
    - divide a decimal by a power of 10 (i.e. 0.01, 0.1, 10, 100)
    - solve division problems with whole number divisor and dividend, but decimal quotients (i.e. 15 ÷ 2)
    - solve division problems with decimals only in the dividend (i.e. 1.5 ÷ 2)
    - solve division problems with decimals in the divisor and dividend (i.e. 1.5 ÷ 0.2)
    - recognize and solve word problems that involve dividing decimals
    - find and explain errors in given decimal division word problems

    I suspect that this progression leans more on the surface level of learning since only the last two bullets really emphasize deep learning. But, I feel the steps leading up to the deep learning are all necessary.
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  • 29 Nov 2018 1:39 PM | Anonymous member
    In Kindergarten most of the instruction I give leans towards surface learning. Students are being introduced to a topic for the first time. For example, Geometry begins by introducing a shape and describing its attributes. Once they can identify and name the attributes of the shapes they can then relate them to real life and be given the higher-order thinking questions to allow for a deeper understanding. However sometimes the surface learning takes longer for this age group because of their limited schema. There is more of a balance between surface and deep learning once the student has an understanding of the topic. I think a balance between the two is crucial to the students and their future learning. Perhaps deep learning can be promoted via whole group discussion or 1:1 conversations immediately after the surface learning to get them used to the vocabulary and thinking beyond the facts of the topic.
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  • 29 Nov 2018 2:56 PM | Michele Martin-Moore
    When thinking about the Quadratics Unit I am about to start, I see the surface level learning that will need to happen as the foundation FOR the deeper learning that will happen once they understand what a quadratic equation is, and how each term contributes to the graph/real life situation represented. Once my students explore/understand these basic concepts we will be able to dig deeper into more problems that can be represented/solved using quadratics.

    We begin each class with a challenge problem or two for the tables to discuss and formulate solutions to (The first day I put three parabolas on the board and ask for 2 similarities, 2 differences, and 1 question they want to ask). It is fun to see them put their heads together... even when earning the basics... and see them come up with various ways of looking at the situations and share their thoughts. The learning may be of basic facts that they need to build understanding of quadratics, but their discussions typically go deeper than I expect. The way you phrase a question can often times lead them to not only scratch the surface, but also tap into their "what if..." gene causing them to create more questions than answers to the challenge problems. Love it when they make those statements that lead them where you wanted them to go!
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  • 30 Nov 2018 9:12 AM | Anonymous member
    I feel that in my current work as a 6th grade teacher, our learning targets and success criteria lean more towards surface learning initially, and then delve deeper as we progress further in the topic. Right now they are dictated by the Learning targets connected with the math program that we use, but I do see the potential to "dig deeper" in development of more powerful success criteria. Initially we start with the basics and build from there ... and the "digging deeper" comes when we focus on the word problems/real world application and relevance, and explaining our answers, which many students struggle with if they have not used the reasoning strategies all along.

    There definitely should be a balance between the two, as the surface learning, I feel, provides the foundations for the abstract learning. Without the concrete skills, the students are not able to support their reasoning. The "surface" learning places the building blocks so that students can provide evidence of their reasoning. The "surface learning" is the "initiation to new ideas" (p.29). Therefore, creating a balance means that educators could spend the time on learning targets and success criteria that build the foundation with the "surface learning", and then take the time to really apply the skills using real-world problems to solve, so that students are able to make the connection that what they do matters and they can apply it outside the classroom in the "deep learning" (p. 30-31). The transfer comes when the students are able to take their learning and apply their thinking in different scenarios or situations (p.32).
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  • 01 Dec 2018 11:40 PM | Anonymous member
    When teaching Quadratics, I use different approaches depending on whether I am teaching Basic Math or Algebra 2. In Basic Math, I use more of an “applied” approach and do a lot of chunking with the material. The extent of the learning targets in Algebra 2 are more complex. The students, however, often struggle with the basics (lacking the pre-requisite knowledge to be more successful). To cover the required material on the rubric in the time allotted time, and include time to work on learning gaps, I often find that I only get to surface learning. To factor quadratics, Algebra 2 students need a solid understanding of working with signed numbers, greatest common factors, FOIL, monomial division, and distribution. I also work with students on strategies for coping with the signs of the factors. Pair activities are useful for getting students to review, do extra practice, and develop a deeper level of understanding.

    One of my goals is to work with my colleagues to adapt the “common assessment for this unit” to allow for projects that reflect proficiency for those students that learn best with a hands-on approach. In the meantime, giving examples and having discussions of how to connect quadratics to real-world models is about as far as I have gotten. I used to do a class project relating a dropped or thrown object and the effects of gravity and initial velocity, but this fell aside as we (the math department) worked developing rubrics and common assessments that reflected consistent expectations for standards (performance indicators)..
    I have considered the possibility of making up silly stories to help the Algebra 2 students remember the steps in rewriting quadratic equations, as model examples aren’t always enough to generate the results that students aspire towards. I want a better balance, but need more work on developing extensions that are manageable (in the time frame for the unit).

    Pam
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  • 06 Dec 2018 6:23 PM | Anonymous member
    One of the broad topics taught in our curriculum is exponents. Initially, and appropriately, the learning intentions and success criteria lean more toward surface learning as students need to first understand the foundational terms and concepts associated with exponents prior to being about to problem solve and use higher order thinking through application of real-world problems (scientific notation, large numbers and measurements, cross-curricular examples) which then leads to deep learning. I tend to think that the type of learning the students do does lean more towards surface learning. I think at the upper elementary and lower middle school levels, this may be appropriate, but that the deeper learning should definitely be more balanced with surface learning from 7th grade and beyond. One of the challenges faced at our school is when the middle school students have not attained enough surface learning to delve into deeper learning targets. They often come up to us with no fluency in computation with all operations. This challenge often requires us to focus on the standards from 2-3 years earlier before getting into grade level standards. After reading this chapter, I do feel that more discussion and talking through problems/solutions with students or students with each other can be a critical step in attaining deep learning even at lower levels.
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  • 06 Dec 2018 6:43 PM | Anonymous member
    One important topic in third grade is multiplication. A very few of my students were exposed to multiplication in second grade so most of my students have no schema for it. My very first and most important learning intention initially is that students understand that multiplication is about combining equal groups. This closely parallels the relational intention shown in the SOLO model for addition and subtraction. And, for the first few days of instruction, we do a lot of work with cubes to model various real world problems-such as placing bagels on a cooling rack in an array. This work really also brings in some surface level learning as we continue to model various problems by using other manipulatives and pictures. I think the balance early in this unit leans more toward surface learning. My students understand simple, real-life word problems but struggle with more abstract problems such as when they need to understand that finding the cost of four tickets priced at $6 each is also a multiplication problem.
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    • 10 Dec 2018 6:04 PM | Anonymous member
      Since I teach the same math program and teach 4th grade, I can say that they are competent by the end of the multiplication unit in using multiplication in real life situations, such as your example of 4 tickets that cost $6. Perhaps connecting to what they learned in 3rd and being more developmentally ready allows them to take this step a little more successfully.
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  • 09 Dec 2018 11:37 PM | Anonymous member
    After diving into all our assessment data, my fourth-grade team noticed our students struggle with fractions, even with 3 units of instruction and materials provided in our math program and supplemental materials. We made it our focus unit for the year to document growth and share with our administrator to measure growth in student learning. As we examined the objectives and learning standards, we noticed we needed more deep learning opportunities for our students. We made several adjustments to our instructional strategies, our formative and summative assessments, manipulatives, tasks, questions, discussions, and student reflection activities. It was a complete overhaul with much success. We continue to make improvements/adjustments. I continued this practice into other math units, but this chapter made me realize I need to analyze all the units and ensure more deep learning experiences/opportunities are presented to my students and not just surface learning, when appropriate.
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  • 10 Dec 2018 5:16 AM | Anonymous member
    We started using Eureka this year and I have been impressed with how the topics that start out as surface learning are brought back into the learning in later lessons. This connection allows the students to bring what was previously surface learning to a deeper level. It's also just a great way to not let learning that was mastered back in September become obsolete through lack of use.
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  • 10 Dec 2018 9:06 AM | Anonymous member
    I don't have much experience with the SOLO Model but it reminds me of the learning maps that I am incorporating into my Algebra II courses this semester. When I break down the chapter, I think I am breaking it down into those different parts. Students must first understand the basic vocabulary and big picture, then I begin to break it down into the deeper parts and then finish the unit with a more advanced application. I've just finished up Rational Functions and we began by reviewing the basics of fractions, then moved into manipulating algebraic fractions, solving rational functions, and finished by exploring the graphs of rational functions. I think there does need to be a balance in each unit, especially at the Algebra II level, and I strive to make that happen.
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  • 15 Dec 2018 5:56 AM | Anonymous member
    I thought about this question in terms of a geometry unit I will be teaching to fourth graders. The students have had limited exposure to geometry concepts. The vocabulary of the unit is often unfamiliar to them. This unit does require a greater allocation of time to surface learning. However, the unit also affords opportunities for deep learning through math problems, games, and art activities that ask the students to apply their understanding of the vocabulary and introduced concepts.

    The balance for this unit favors the surface learning. I work with my students in both fourth and fifth grade so I know that laying a sturdy foundation during this fourth-grade unit will allow the students to use these concepts later in our school year as well as next year when we revisit and then extend these geometry concepts.

    A balance between surface and deep learning is necessary but it can be achieved over time. It does not have to be achieved in every one or two-week unit. The ability to think deeply about a concept and make connections requires that the students have time to process the surface learning. The cyclical nature of math allows for this.
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