Recap: Math as a Second Language Day #3 (August 2, 2017)

Andrew Baxter joined us today for our session.  Before he left he shared with us that there is a free book called Nix the Tricks that helps us get out of bad habits of short cuts that we might use in our classroom.  The book Be the Teacher you Wish You Had is also a book he recommended.  It is not free though!

Today we resolved the handshake problem after watching a video of a 3rd grade teacher doing a similar problem with 20 people.  We noted that she used zero in her series to make the number of pairs an integer which helped with the mathematics.  Then we had someone explain the answer to our problem with 100 people.

We spent some time going over the problems we created last night for homework and will continues with these later.

In the course of our math instruction we talked about the ladder method of prime factorization.  Here is a video of the method in case you want to look further into it.  Sometimes students lose their factors in a factor tree.

There was good discussion about comparing fractions that are close to one half.

We discussed the Iceberg article.  Sherri felt that this might be a good activity to students to engage in as they encounter a new concept.  We spent time creating Icebergs for topics in our grade level. We will have time tomorrow to engage with them during breaks and add comments with post its.

HW tonight – read Principles to Action 42-52:

In your notebook – Identify a procedure or skill that you consider essential for students at your grade level to learn.  List the conceptual understandings that support students’ learning or the procedure or skill.

Review the beliefs chart on page 11 and list the beliefs that are evident in Ms. Flahive’s and Ms. Ramirez’s classrooms – figure 21 on page 1.  What are the impacts of these beliefs?

Recap Math as a Second Language Day #2 (August 1, 2017)

We began with the burning questions and explained that more practice with fractions is to come and that the idea of raising the expectations for our classwork and homework will be addressed later today with a Ted Talk.

In working through Block #2 we had some good discussion about adding positive and negative numbers – when the commutative property works and how to use compensation when adding.  The adjective and noun theme also helped ensure that we were adding quantities with the same nouns or units.

There was some discussion about the level of homework and the ability of students to complete our homework successfully.  The idea that maybe the collaboration in the classroom is helpful in the completion of the work and that the grit and time to focus at home could be an issue.  We wanted to make sure though that we are preparing them sufficiently in the classroom.  We watched a Ted talk by Dan Meyer about rethinking the types of questions we ask and how too much scaffolding and building a pathway for our students can hinder their success in persevering in problems that the real world presents.  There was an acknowledgement that his method requires more time that is often available in our days.

We ended class with the unresolved problem called the handshake problem.  It will be resolved tomorrow – I promise.  Maybe sleeping on it will allow people more time to struggle productively with this problem.


Read the reference pages and record any questions.

Read Beneath the Tip of the Iceberg.

Write the problems explained on page 8 of Block 2.

And I found these interesting websites about improving our students’ ability and mastering of the number line – it is a concept that we continue to build on:

Recap: Math as a Second Language Day 1 (July 31, 2017)


We were welcomed by Cynthia Lightfoot, our Director of Academic Affairs.

We then began the Kayaking on the Susquehanna River problem, and discussed the different approaches taken by members of our group including tables, equations, and graphical representation.  It was noted that graphing the example of a line was not quite accurate because of the way parts of hours were handled by the kayak company but because the answer to the third part of the problem was an integer the linear approach as a graph or as solving system of equations successfully produced the solution.  However a table worked just as well.

After a break we began the math content in Block 1.  There was good discussion about the use of the equal sign, the number pi and how to teach students the importance of place value.  The pre-assessment was finished before lunch.

Here is the lnk to download the book “Adding It Up”:

After lunch we continued in Block 1 discussing the number line.  One participant stated that negative numbers are explained as “something that you don’t have yet.”  There was some good discussion of the meaning of the ———-> arrow on the end of problem C. on page 11.  Charles explained that he reads it as a notation of direction, not that the line is continuing.

We watched a video about conflict in mathematical solutions and then discussed the “Smarter Than We Think” article and the ideas of grit and growth mindset.  Sherri shared this video with me that really explains the idea of grit.

It was also mentioned the Class Dojo has some great videos to use.

Here the schedule for the week:

PMI 2017 Workshops at Penn State  Brandywine

Daily Schedule, Week 1 (subject to change as needed)

Mathematics as a Second Language

Monday 7/31/17

8:00 – 9:00                  Registration, Welcome.

9:00 – 9:15                   Opening Remarks.

9:15 – 11:00                The Kayak Problem. Math Content – Block 1.

11:00-11:10                 Break

11:10 – 11:55               Pre-Test.

12:00 – 1:00                 Lunch.

1:00 – 3:00                   Math Content – Block 1.

3:00 – 3:10                  Break

3:10 – 3:45                   Pedagogy Discussion

3:45 – 4:00                  Summary; Conclusion; Daily Evaluation.

Tuesday 8/1/17

9:00 – 9:30                   Homework discussion

9:30 – 10:30                 Math Content – Block 2

10:30 – 10:40               Break

10:40 – 12:00               Math Content – Block 2, continued.

12:00 – 1:00                Lunch

1:00 – 2:35                   Math Content – Block 2, continued.

2:35 – 2:45                  Break

2:45 – 3:45                  Pedagogy Discussion

3:45 – 4:00                  Summary; Conclusion; Daily Evaluation.

Wednesday 8/2/17

9:00 – 9:30                   Homework discussion

9:30 – 10:30                 Math Content – Block 3

10:30 – 10:40               Break

10:40 – 12:00               Math Content – Block 3, continued.

12:00 – 1:00                Lunch

1:00 – 2:35                   Math Content – Block 3, continued.

2:35 – 2:45                  Break

2:45 – 3:45                  Pedagogy Discussion

3:45 – 4:00                  Summary; Conclusion; Daily Evaluation

Thursday 8/3/17

9:00 – 9:30                   Homework discussion

9:30 – 10:30                 Math Content – Block 3, continued.

10:30 – 10:40               Break

10:40 – 12:00               Math Content – Block 4

12:00 – 1:00                Lunch

1:00 – 2:35                   Math Content – Block 4

2:35 – 2:45                  Break

2:45 – 3:45                  Pedagogy Discussion

3:45 – 4:00                  Summary; Conclusion; Daily Evaluation


Friday 8/4/17

9:00 – 9:30                   Homework discussion

9:30 – 10:30                 Math Content – Block 4 or tying up loose ends.

10:30 – 10:40               Break

10:40 – 12:00               Block 4 or tying up loose ends.

12:00 – 1:00                Lunch

1:00 – 2:35                   Math Content – Block 4 or tying up loose ends.

2:35 – 2:45                  Break

2:45 – 3:45                  Pedagogy Discussion

3:45 – 4:00                  Summary; Conclusion; Daily Evaluation



For Tuesday:

  • Read through the Reference pages in our material.
  • Read in Principles to Actions:
    • Sections titled “Progress and Change” and “Effective Teaching & Learning” (pages 1-12 – stop reading at the middle of page 12)
    • Section titled “Implement Tasks that Promote Reasoning and Problem Solving” (pages 17-24).
  • Respond (in your notebook) to the following prompts:
    • Reflect on a typical in-class math lesson (use your textbook to refresh your memory) that you have taught. 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?
  • Bring your math textbooks to PMI on Tuesday.

Recap: Functions and Algebra, Day 5 (July 22, 2016)

The discussion on our reading in P to A on professionalism did not go far.  I am not sure everyone had a chance to read it – it has some very good information on how we can work toward collaboration, increasing our knowledge of mathematics and other things that were emphasized this week.  We tackled the last two standards (7 and 8) and watched videos of them being demonstrated in different classrooms.  Larry made a good point that the standard of structure could also relate to the structure of the classroom environment that he witnessed in the 5/6th grade room.

When the discussion was opened up as a wrap up there was talk of the benefits of hearing from teachers from different grade levels and being able to interact and discuss how math fits across the grades, it was mentioned that productive struggle came alive this week, we felt we could relate better to our students in terms of the rigors of the struggle and having long days of learning,   There was a recommendation that an opportunity for application would be valued in terms of time to spend creating a lesson using feedback from participants.  We are leaving with buckets full of new ideas, but it would be helpful to apply this to at least one lesson before leaving would cement things and allow for collaboration.

We took the post test and after lunch began our Bungee Barbie capstone activity.   We did our evaluations and then headed across campus for the Barbie drop.

Please make sure you use this space to communicate with us or each other as you process all you have learned these two weeks.  We are happy to be in community with you all.  Have a great rest of the summer.

Recap: Functions and Algebra, Day 4 (July 21, 2016)

We began talking about the vocabulary gallery walk and the benefits of using this type of activity both before and after covering a certain topic.  We then moved into talking about the concept of skills that translate with the ideas of Larry’s son at college and some of the number games in the power point that relate to input/output, functions (processes), and inverse operations.  We discussed the idea that these games would play out differently in the different grades and had two very different representations on the board for the idea of the inverse operations problem – “get me where I started”.

Charles then began with the pool membership problem and once we agreed on the additional information we needed, we worked on finding a “break even” point.  We had a great discussion about the 40% off versus the 10% then 30% and the 30% and the 10%.  There were several methods of approaching this idea.

Marina had us work in pairs on explaining the thinking behind the students on page FA5.6 and being precise in our explanations.  We saw benefits in all the types of thinking.

We ended sharing the website which has some great examples of mistakes students make but also affords the opportunity for collaboration about students’ mistakes.  We watched a video about a student’s misconception of the number 18 and place value behind it although she understood the idea in a different context.  We then discussed the benefit of thinking through our thinking and the thinking of others and watched some videos of what that can look like.

Math hw – do the three act problem Dandy’s Candies on FA pg 5.8

Pedagogy HW read P to A pages 99-108 on professionalism.

Recap: Functions and Algebra, Day 3 (July 20, 2016)

We began today with a funny video on how mot to use questioning in the classroom.  It elicited a lot of good discussion about funneling versus focusing and how we use that in our classroom.  We also talked about Standard 4 and looked at an example video.

Charles began with the material on functions.  We spent a lot of time understanding what makes something a function.  We learned the difference between range and co-domain. We connected the idea of function to the previous concept of process and then learned function notation.   There was a lot of good discussion about what makes a function one-to-one, onto and finally what one-to-one correspondence is.  Before lunch we began with the idea of functions relating to counting but got a bit hung up on the idea of what the domain and co-domain were and whether the domain changed when we skipped a stone when counting.

After lunch Marina began a discussion about measurement relating to functions but many found the idea a little too abstract for them to understand.    When we started the pattern problems many participants realized they related to last night’s homework.  The discussions were great and the presentations showed many ways to think about the pattern.  We revisited the idea of (n/2)(n+1) from the Days of Christmas problem and reinforced the reason that the formula makes sense.

The perimeter problem brought three very different and powerful approaches to y=4n and reinforced the idea that it could be repeated addition of n four times or repeated addition of 4 n times. There was also a very interesting question about why the candle burning problem was written as y = 12 -2x instead of in slope intercept-form of y = -2x + 12.  We ended with a good discussion of how to think about the f(x) notation.

We ended with a recap of our thoughts about questioning listed below:

  • A focusing question doesn’t lead to a one word answer.
  • A focusing question doesn’t just relate to this particular problem.
  • Ask a student to rephrase their thinking because that’s the basis for our questioning.
  • Let the students defend their positions (Socratic method)– issues and errors may just fall out.
  • Keep in mind the offense and defense model of adjusting on the fly.
  • It is important to know where the kids are going mathematically beyond what we are teaching them so we can question effectively.
  • It is also important to have thought about questions ahead of time so that we don’t have to come up with them on the spot. Keeping notes from year to year is helpful.

We also talked about the think-pair-share idea:  How do we deal with the fact that we can’t attend to every group at the same time, ways to ensure students remain on task and ways to reassemble them after.  We watched a video of a first grade class and Standard #5 – we saw that the paper plate activity modeled the idea of algebra but discussed other ways the teacher could have set it up and had the students more involved.  We ended with a gallery walk putting definitions, floating capacities, pictures, questions, etc. on the words – slope, y-intercept, x-intercept, proportion, and linear relationship.

By request I am listing some of the ideas here but please continue the gallery walk tomorrow morning in case things were added after you saw it.

slope:  y = kx,(underline k) y = mx+b (underline m), rise/run, change in y/change in x, + or -, horizontal line has zero slope, constant rate of change, think of a straight line, daredevil compare the steepness of a hill for sledding or comparing ski slopes with diamonds.

Linear relationship: predict where a point will be on a graph, constant rate, graphs like a straight line, one-to-one, not quadratic, + or – slope, comparing fees for phone companies, y = mx+b, each x has only one y, pairs and how they relate to form a line.

x-intercept: where the line crosses the x axis, picture drawn, phone bill minimum with no extra minutes, reserve a kayak(0 hours rented), ?? posed – what is its relation to domain??

y-intercept: Deposit on kayak, y = mx+b (circle the b), picture drawn, f(x)-range, point where x=0, monthly charges for phone bill,

Proportion: a:b, a/b, a to b, 2 equivalent ratios, line going through the origin, “parts of”, “k value”, one ration = to another, graphing lines

Math HW – finish page 4.18

Pedagogy – read P to A pgs 59-69 and in your journal use your textbook or curriculum guide to list the floating capacity for : doing and undoing process, rate, proportional reasoning, linear relationships.


Recap: Functions and Algebra, Day 2 (July 19, 2016)

We began today discussing our charts for the bus stop problem.  We talked about the benefit of titles, noting slope, noting “not drawn to scale”, pictures, color coding and whether or not scales on the x and y axes were helpful or not.  The graphs were amazing and I was so impressed with the process all the groups went through in drawing them.

We then discussed the idea of piece-wise functions and their use and moved into Standard 2 of CCSSM and watched a video of what this standard could look like in a 5th/6th classroom.

Charles picked up with Unit 3  material and reviewed the ideas of linear and proportional.  We had some good discussion about the exchange rate and how it works with rounding.  There as a question about when topics like this are handled in school.  Unit rates are in 6th grade and it builds to proportional statements in 7th grade.  One of the issues is using 10/9 or 1.11 and which is the numerator and which is the denominator.  There was also discussion about having students learn how to express ideas mathematically like “five plus seven” is 5+7.  Also we can talk about conversions younger grades discussed converting cups vs. ounces and understanding that both are valid representations of an amount and how we talk about how to decide on a scale when graphing. These are important building blocks to get to what they will be learning later on and we can use these ideas to connect to what they already know.   We also discussed the benefit of younger grade teachers seeing what types of skills seem to be stumbling blocks later on in order to inform their teaching of these concepts.  I was so excited by these questions and how participants were trying to think through problems like this that their age students could relate to.

Marina began talking about what makes a relationship linear and what makes it proportional.  We also talked about the equation of a line.

We ended the day by talking about Standard #3 and then discussed the concept of funneling and focusing questions.  We than moved into a questioning activity to practice the use of effective questions.  We will follow up with this in our homework tonight.


Math p. 3.16 #8, 10, 11, 12   Also determine why ax + by + c = 0 is more versatile than y = mx + b.

Pedagogy Read P to A 37, 39-40  Listen to your audio recording or think about the questions you asked in our activity.  Using pages 37, 39-40 to examine the questions you asked while playing teacher and respond to these prompts:

  1.  Were your questions funneling or focusing
  2. How might you change it now?
  3. What type of questions might be more productive for eliciting student thinking?  Write a few new questions – refer to pgs 36-37.

Percolating assignments:  When it is “ok” to use funneling questions.

Recap: Functions and Algebra, Day 1 (July 18, 2016)

We began today talking about norms for our time together.  We came us with:

  • Don’t be afraid to try a problem even if you think it is too hard
  • Be respectful of the speaker
  • Give space for trying problems on one’s own
  • Enjoy the productive struggle
  • Be patient with yourself

We then watched a video of a first grade class and watched the class explain the thinking of the presenter and talked about the importance of listening to other ways of solving problems and being able to understand them.  It is important that we pay attention to the speaker and be able to rephrase it.

Charles began the discussion of our materials with everyday examples of inverse processes.  There was a good discussion about why squaring is not an invertible process.  The idea of writing the composition processes was a new idea to some but we discussed using boxes for the operations and thinking of them as verbs and the input and outputs were nouns and not boxed.

After a break Marina talked about the idea of equivalent expressions.

After lunch we began the next unit and introduced graphing of ordered pairs.  We discussed the idea of inverse functions as well.  We liked the video representation of the chocolate milk with the extra scoop of chocolate and how can we fix it.  Charles wrapped up the graphical representation of the proportionality and approximation.

At the end of the session we discussed some of the standards from CCSSM and then worked on a problem of graphing distance versus time.  We will pick up with those graphs tomorrow.


Math:  Finish FA1.10 and challenge FA1.12 – pick one

Pedagogy:  Read P to A pages 35-41, 53-57  and the article Questioning our Patterns of Questioning.  In your journal respond to the following prompts:

In your notebook write a response to these 2 prompts:
1. in questioning small groups of students working on a problem, a teacher noticed that when she
asked “focusing” questions , the students continued to look at their own work and continued to
engage in gtheir own dialogue.  When she asked “funneling” questions, the students looked up at the
teacher.  Comment on these observations.
2.  Look at Maddie’s work and Gabe’s work shown in fig. 22 on pg. 55 of P to A.  How could Ms. Lewis
leverage the student’s representation to develop Maddie’s understanding of the problem?

Recap: Math as a Second Language, Day 5 (July 15, 2016)

We began today by discussion about what we want to praise in our classrooms and what our goals are for our students.  We watched a video about the power of praise when we are praising the effort and not the “intelligence”.  We brainstormed words we can use to do this in our classrooms.

We then continued with the study of fractions by talking about multiplying fractions in terms of the adjective noun theme (and related it to the idea of rates) and geometric representations for the multiplication.  This last topic yielded a lot of productive struggle when we jumped from proper fraction multiplication to improper fraction multiplication.  Several models were presented from the participants and we spent time at tables figuring out how they related to each other and why some did not seem to be valid.

After lunch we began the concept of division and we discussed the multiple ways of approaching it.  Before going on to the estimation topics we had each table become an expert at a particular problem from the problem set in the material.  We did a think, pair, share and then the groups created posters to represent their answers.  The presentations gave participants a chance to practice their vocabulary and mathematical reasoning and gave observers a chance to ask probing questions about the material.

The estimation page was a good way to look at ordering fractions and we worked through some issues with the list that was close to one.  The one fraction 26/51 made the pattern a challenge to find but we talked it through and came to a consensus.  Charles wrapped the section up with a discussion about inverse and the importance of not dividing by zero.

Have a great weekend.