Day 1 of Math as a Second Language (7/18/2016)

This has been an  exciting first day!

We started with the usual introductions, where you were greeted by Andrew Baxter, George Andrews, Yuxi Zheng, Matt Katz, Andrea McCloskey, and Fran Arbaugh.   We talked about expectations and our objectives for this week, and then dove into the Kayak Problem.

In discussing our solutions to the Kayak Problem we discovered a few things about mathematics and mathematics teaching:

  • There is more than one way to solve a problem, although often the ways to solve are related.  Here we saw a list of +2s and +4s to represent the hours, a organized chart, an algebraic formula, and a graphical representation.
  • There are advantages and disadvantages of these different methods, as some solutions make certain subtleties and patterns clearer than others.  Ultimately students should develop proficiency in the efficient methods, but there is danger in knowing only those methods at the exclusion of mathematical understanding.

After a break, the participants took a pre-assessment.  This was followed by a lunch hour.

After lunch Fran led a discussion about the reading you did before coming to UP: “Smarter Than We Think: Helping Students Grow Their Minds” by Cathy Seeley. She then shared three frameworks that ground our pedagogy work for PMI:

  1. What does it mean to know mathematics? We will draw on work presented in Adding it Up: Helping Children Learn Mathematics, which presents a definition of “mathematical proficiency” as being comprised of the following five strands:

Conceptual Understanding – comprehension of mathematical concepts, operations, and relations.

Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.

Strategic Competence –  ability to formulate, represent, and solve mathematical problems.

Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Adding it Up can be downloaded for free at:

2.  What kind of activity do students need to engage in to become mathematically proficient?  We will draw on CCSS-M Standards for Mathematical Practice (found on page 8 of Principles to Action).

3. What kind of teaching practices support students to engage in the SMPs and become mathematically proficient? We will draw on the content of Principles to Action.

Next Matt kicked off Unit 2 on the Adjective-Noun Theme and Addition.  We saw how in arithmetic situations, numbers are adjectives and we often omit (or forget) about the nouns.   This can shed light on why our standard algorithm for addition works as it does, as well as give insight into why fractions need common denominators when adding and subtracting.

Last Andrew discussed models for numbers, their relative strengths and weaknesses, and how those models for numbers yield models for addition.  We started on exploring even and off numbers, but we will pick up with that more tomorrow

HOMEWORK: One clarification: homework is not collected and graded or for anything other than  deepening and solidifying your own learning. Homework problems are discussed the following day.


Page 1.5, Variations on the Kayak Problem. Solve problems 3 and 4. One you solved it one way, seek out another way to solve it.

Page 2.11, number 7. Woodchuck Valley Tennis Association Trophies. Make an attempt, although we will look at this in class in the morning.


Read Principles to Action: Sections titled Progress and Change; Effective Teaching & Learning (Pages 1-12) and section titled “Implement Tasks that Promote Reasoning and Problem Solving” (pages 17-24).

  • Reflect on a typical in-class math lesson (use your textbook to refresh your memory). Using the descriptions of Levels of Cognitive Demand in Figure 3 (p. 18), describe the types of mathematical thinking your students are required to engage in during a typical lesson.
  • Reflect on a typical homework assignment that your students complete (use your textbook to refresh your memory). What level of cognitive demand do most of the tasks on a typical homework require of your students?

3 thoughts on “Day 1 of Math as a Second Language (7/18/2016)

  1. Dan Lichtenwalner

    I have had the privilege of exposure to various pedagogy for teaching mathematics. A typical math lesson for me is multifaceted. I try my best to expose my students each day to the various levels of demand. The curriculum we use lends itself to this. Each day begins with sprints or warm up problems that focus on lower level memorization or procedures without connections. These styles of problems are designed to reinforce known concepts and give students a feeling of success to start the day. Next, the lesson moves into a staged progression of procedures with connections. In these problems, which are often solved together through use of visuals, such as tape diagrams, or manipulatives students are challenged to build upon a given concept in smaller sections towards a deeper understanding. There then is always two practice problems which generally are on the procedures with connections level, and then an exit ticket to monitor understanding formatively. The homework assignment is always of this level too and models itself directly from the styles of questions given and practiced during the lessons.

  2. Jess Swatsworth

    An in-class math lesson requires both lower level and higher level demands. Students are usually given a task or word problem to complete, but step-by-step instructions are somewhat provided in order to guide the students through the desired procedure. Then another task or the same task is given and another method is practiced to find the answer. Students then complete a “Share and Show” section where they are given the opportunity to practice the methods taught. Next, they have an “On your Own” section which is typically lower-level demands. There is always a “Problem Solving” section at the end of each lesson that does require students to engage in higher order thinking. The questions are often multiple steps (where the students are given complete freedom to arrive at the answer), and require the use of multiple pieces of information (often from a graph or chart). I think these questions, which are labeled “GoDeeper” and “ThinkSmarter” are along the lines of problem solving that we’ve talked about in class already, however, my students often reach frustration level before trying to solve the problem on their own.
    Homework at my level is always a section which is mostly procedural practice at the lower level demands, but there is also a “Problem Solving-Real World” section with 2 questions. I find these are sometimes similar to ones that we’ve completed in class and allow the students to apply the skills learned, but are not as challenging as the “GoDeeper” questions. There are two lesson check problems that are presented in word problem form, and there is a review section that again is usually lower level-based.

  3. Tina Fleming

    In a typical math lesson that I currently teach, my students mostly use higher-level demands and lower level demands. The lessons usually start off with a basic procedure for the students to follow. They are given a few examples of different ways to complete a task and then provided time to choose the way they prefer to solve a problem and are asked to focus on getting a correct answer, without delving too much into the mathematical understanding of the process. There are questions at the end of each lesson, though, that do go more into higher level demands. A strategy or approach is not clearly laid out for them and they need to gather the information and relate it to topics covered in recent days to solve the problem. This pretty much applies to typical homework that I assign, as well. The back pages of our homework sheets have a “review” section that I do like, though, It focuses on topics previously learned and allows students the opportunity to connect what they’ve done in the past, to a current topic they are studying.

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