Association of Teachers of Mathematics in Maine
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Chapter 3 Response Choice 2

07 Jan 2019 7:17 PM | Anonymous member (Administrator)

Identify two or three mathematics tasks you've asked your students to work on recently. Think about each task in light of its difficulty and complexity (see Figure 3.1 on page 77). In which quadrant does each task fit? 


Is each the right kind of task given your learning intentions? 

Comments

  • 07 Jan 2019 7:39 PM | Anonymous member (Administrator)
    This year I have been trying new routines as part of our daily warm up exercises. For example, number of the day, number talks, alike and different, which one doesn't belong, and I've even made some routines of my own. I've noticed how much vocabulary actually helps my students understand the concepts and helps them communicate their thinking. In the past week, I have given students a list of words that we have written in our math dictionary (at the back of the math journal) and then have the students draw something or give examples and label the vocabulary. I would say this falls in Strategic Thinking (Low Difficulty and High Complexity) because it is just naming a concept, but by drawing a visual the students need to determine what to draw or write that will show that concept. For example to name a remainder, they need to create a problem that doesn't work out evenly to create the remainder to label. It has caused students to think and rethink. It also allows for creativity as some students use arrays, others use the standard algorithm, some use a place value chart, and some the area model. As they look at each other's examples their own understanding gets much deeper.

    I think many of the other routines I have been following also land in the Strategic Thinking quadrant. On the other hand, we have also been practicing some skills that would end up in the Fluency quadrant. For example, finding a missing factor helps them connect multiplication and division, and strengthens some of the lesser known multiplication facts, but it is easy and less complex. My goal is to have them practice and become familiar with those facts- which in turn helps with long division and long multiplication. I notice an increase in success as we practice multiple days.

    So I think it's appropriate to be in different quadrants for different purposes. I remember having a conversation with Tracy Zager about how sometimes it's good to practice easy things, because it makes math fun. Having a positive Math attitude is important too!
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  • 09 Jan 2019 8:58 AM | Anonymous member
    Our district (South Portland) has done a lot of work with aligning rigor and tasks. We use two versions of Marzano's Taxonomy - one for "skill" tasks (https://drive.google.com/open?id=0B16vUZdj87rzbi1Kdi11YmI2WVU) and "understands" task (https://drive.google.com/open?id=0B16vUZdj87rzWlhBdnFkV01SZzg).

    We are currently using a 1-4 scale, with 3 being proficient, 4 exceeding the standard and 2.0 and 2.5 approaching the standard. Many of our 2 level tasks tend to be exercises - computational task without any context. Our 3 and 4 level task are word problems with the 3 usually being "in context" and the 4 being in "complex context". Our Level 2 tasks are usually in the "Fluency" quadrant and our 4's try to trend towards the "Expert" quadrant. I'd say our 3's are a mix of the "Stamina" and "Strategic Thinking" quadrants.

    For example, on a recent assessment on finding percents, we had these questions:
    Level 2 - What is 85% of 120? and 56 is 75% of what number?

    Level 3 - Ms. Ngyen has a total of 150 students in her classes. Of these students, 30% eat during the first lunch period, 20% eat during the second lunch period, and the rest eat during the third lunch period. How many of her students eat during each lunch period?

    Level 4 - Terrific Toys is offering an additional 25% discount on board games that have already been reduced by 30%. Fantastic Fun is offering the same board games for a discount of 55% on the original price.Will the final cost be the same at each store? Work through an example and explain your answer.

    We make these rigor decisions collaboratively as team, so it can be time consuming and a bit exhausting! But overall I think we do a pretty accurate job!
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  • 09 Jan 2019 9:26 AM | Anonymous member
    I found the thinking this chapter inspired, fascinating.
    I teach three different math groups daily- I have three grades (6,7, and 8) in my classroom. The students are grouped by level not necessarily age. I realize the group that is working on the skills that are lowest on the continuum are most frequently being asked to work on fluency tasks. These are students who have arrived in middle school with out their multiplication facts, and are missing many of the algorithms for computation. The students who are farther along the continuum are more frequently given tasks that fall in the strategic thing or expertise quadrant.
    This realization has prompted me to be sure to introduce some problems that at least require strategic thinking. I have been providing calculators for these tasks for the students who lack the fluency to face these problems. This is increasing their confidence and their deeper understanding of concepts that is a huge part of their frustration and slow growth.
    In addition, I have lowered the difficulty and increased the complexity of some of our daily number talks. Freeing the students from the "grind" of calculation has increased the confidence of all of the students and increased levels of participation.
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    • 21 Jan 2019 8:09 AM | Anonymous member (Administrator)
      I love using routines like number talks! One way I have found to reduce the difficulty of number talks using multiplication is just using numbers like 2,3, 4, and 5 since students are most confident with these. However, I keep it complex by using four-digit by 1-digit problems (4th grade). What have you done to increase complexity while maintaining a lower difficulty?
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  • 20 Jan 2019 6:45 PM | Julie Nugent
    My class has been working on multiplying with multi-digit numbers. They have been doing just practicing multiplying and also multiplying with word problems that are multi-step word problems.
    I would say the just practicing of multi-digit multi-digit multiplication problems would fit in the Fluency quadrant even though it has been difficult for many of my students.
    As far as the multi-step multiplication word problems, I would place that in the Stamina category as the student has to really think about what to do and in what order for each step of the problems. They math part is not that hard, but choosing which math to do, and in what order is higher level thinking.

    Yes, I think each is the right kind of task for my learning intentions.
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    • 21 Jan 2019 8:12 AM | Anonymous member (Administrator)
      I think when our tasks involve a number of steps, that in itself poses challenges for students to keep their own thinking organized. What models and strategies have you given them to attack these. Are they allowed to choose whatever method works best for them?
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      • 27 Jan 2019 4:56 PM | Anonymous member
        It has been difficult for my fourth graders to do multi-step problems. I have been mainly just demonstrating how to do them on the board. Being really explicate in my thinking. I try to show them how to do a number model with parenthesis and that you do the what's in the parenthesis first. I have shown them how to draw pictures to represent each part and how to pull the problem apart into steps. Doing one at a time. Do you have any other methods or suggestions for doing multi-step math problems with 9 year olds?
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  • 02 Feb 2019 4:37 PM | Anonymous member
    The tasks below are from my remedial freshman math class....

    Task 1
    1. A baby born at 7 lbs gains 0.8 lbs each week. What algebraic rule models the baby's weight? Define your variables.
    2. Use your model to estimate the baby's weight at 12 weeks.
    3. When will the baby weigh 14 lbs?
    4. When would you stop trusting your model to give you realistic answers?

    Task 2
    A summer job you're interested in pays like this: they give you $0.01 as a signing bonus, and then each day they double your pay. How much do you make on the 30th day? What rule could you use to figure this out in a snap?

    The first half of task 1 is not complex or difficult for the kids. #1 and 2 are answered correctly by almost everyone with very little thinking - some kids need help in the form of a table to see how the baby is growing. I thought #3 would be the same - wrong. Kids wrote down "14 = 0.8x + 7" and then proceeded to solve it using guess-and-check! Great learning for me - I hadn't anticipated they wouldn't bring along their algebra toolbox to a problem that, to me, so clearly calls for it. Once I pointed out to them they had the algebra skills to solve this, there were multiple, surprised "oh!"'s in the room as the re-interpreted the problem as an equation to solve using algebra. So I guess I classify #3 as low difficulty, high complexity. #4 of task 1 is the same, low D, high C. Kids need to be expert enough at modeling to know it's merely a tool, and that like all tools, it can only be used under the right conditions.

    Task 2 is high D and high C. We do that one as a class so I can lead them along with questions, showing them how a mathematician would tackle the problem of finding an algebraic model for the situation.

    Is each task right given my learning intention? Hmm. I think so? In the first, my intention was to see if they can use linear models. The kids in this class need lots of success and have a low tolerance for frustration in math, so it seems appropriate to keep things low D. I have more leeway with Task 2 because the delight and surprise of seeing how much money they'll be making keeps most of them happily punching away at the calculator (they all solve the problem by brute force the first time around) through all 30 days. That's where the high D comes from. My learning intention is to make them learn the algebraic form for exponential models - not something they've ever done before, so I think high C is inevitable as they're forced to combine numbers in a way that's unlike the linear model they've become accustomed to.
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  • 18 Feb 2019 3:28 PM | Jenny Jorgensen
    My students were recently working with data and needed to determine percents. I started the class with a chart on the board that had 3 different numbers and students had to find 10%, 20% and 5% of each number. Students struggled with this at first. During a whole group discussion, one student indicated that he saw the pattern but couldn't explain the math. I was happy he'd seen a pattern and wish he'd recognized that the pattern he noticed "was" doing math. Students were about to see a relationship from 10% to 20% . After discussion they could also see using half of the 10% to figure out 5%. I'd like to think that this task falls in the Fluency part of the Difficulty and Complexity chart on pg. 77 and yet for my students it does not -- yet.
    Another task that my students (a different group of students from above example) recently worked on was writing a story problem that had a fraction (less than one) divisor. Students struggled a lot with this task and I would therefore put it in the Expertise part of the chart on pg. 77. In order to write a story problem with a fraction divisor students really needed to understand fractions and what kinds of problems lend themselves to fraction divisors. It was difficult for many of the students due to their lack of fraction knowledge and understanding which then made it a complex task for them.
    I've used the Pancake Problem on pg. 33 in the Hattie book. I would put this problem and the work involved for the students at the Stamina level. Students were able to solve the problem. When asked, "Is there another possible solution?" they were surprised. They figured that they had an answer they were done. They needed to re-engage in the problem and persevere to find another solution. When they found a second solution I again asked, "Is there another answer? How do you know you have all the possible solutions? Developing perseverance was definitely a part of this task and did meet my learning intentions.
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