Overview of Class #11 Comments on project #1 Remediating common student errors (whole number multiplication) Multiplication of decimals Teaching whole group (video) Choosing numerical examples wisely Comments on Project #1: Generally very good work. Most of you seemed to value and learn a lot from this and to gain some appreciation of the value and complexity of inquiring into student thinking. Most of you were extremely sensitive to creating an environment that was comfortable and non threatening for the student, and yourselves. Many of you were surprised, by what the student was able, or not able to do. Many gained an appreciation of the importance of manipulatives as a resource for clarifying and scaffolding student understandings. There was generally a strong appetite for doing as much of this as is feasible in your work as a teacher. Note that many of the skills involved apply to whole group teaching as well as to individual interviews. The framing of questions, and having flexible variations ready in your pocket as the need arises is important.

Multiplication Computation Go to next slide.

Modeling Multiplication Computation as Area Discuss the mapping between the rectangular model and standard procedures for multiplication. (Optional: slide #11) Focus on seeing the partial products and their combinations to equal the product. BREAK

Three Common Student Errors 1 2 3 What is the error? What produces this error, and what does that show about the student’s understanding? What does the textbook suggest you do to teach this? How adequate are the book’s suggestions? How would you use and also change these suggestions, and why? Here are three common errors that students make. What is going on in each case, and what might be the cause? How might you remediate this? Divide class into three groups. Provide sections from the teacher’s guide for a very common series. Can use as a resource. But should also use what we have learned to do a) Place value error; second row is not multiplied by 20, but only by 2 b) Regrouping error; adding in “carry” before multiplying c) “Straight down” multiplication (like addition These teacher’s guide sections from Harcourt Brace are not good. I want them to get the sense that they will need to think carefully about the mathematics –– that textbooks will provide support and resources, but will also be lacking in the same. These sections do not provide much at all in the way of conceptual explanations for the procedures. 10 min. Do another one for homework.

Remediating What DOES work? What does NOT work? Identifying carefully where the problem(s) lie(s) (Teacher works to locate) Teacher helps student to initially “see” the problem Teacher draws upon/helps to develop student estimation skill Using manipulatives or other representations to focus on the meaning and the procedure Sharing the talk with the student, scaffolding Offering a similar example to try Teacher reduces complexity in someway and then encourages connection of that work with original problem Teacher invites another student to explain how he/she understood the problem and mediates the interchange to highlight productive ideas or contrasting ones What does NOT work? Repeating the same things over again, slower, more loudly Re-teaching everything Teaching that “coerces” or forces students to get the “right answer” Teacher does the problem for the student Teacher refers the student to easier problems that they could handle and cuts off work returning student to work on basic concepts Teacher gives students more problems, thinking practice is the issue A few ideas about the additions: What does not work? The first point that I added might actually be what you meant by “Teacher coerces”. The second and third items that I added have to do with other forms of attribution that were not mentioned in other items on this list. In these items the teacher attributes student problems to the fact that they are low skilled and can’t do the problem. The first one from the original slide relate to the attribution of “students just didn’t hear or weren’t paying attention”. The second one could relate to that or that the lesson itself was a “bomb” for some reason. The third one has more to do with the attribution that “students need to be persuaded” but then the teacher chooses poor methods of persuasion. In the last one the teacher attributes student difficulty to lack of practice. In this case the teacher incorrectly thinks more of the same will suffice without any further intervention What does work? The first one I added is important in that many times the hardest thing to do when working on remediation is to help the student to realize that there actually is a problem. This is tricky work in that some students will pretend to see the problem when in actuality they don’t. I added the second one because number sense and proficiency with estimation are key in helping students to see that there is a problem AND also an important tool that students can use to self monitor as they work (the internal check as they go forward, seeing if they really do understand or if they still have a problem). I think this is a bit different than seeing meaning and knowing the procedure to use with exact numbers. I added the idea of other representations of the problem (like drawings of manipulatives, number lines, etc) because they may also work. I added the fourth one because it contrasts with one that I put in the “does not work” column. It is useful at times to simplify, but it is key to come back to where the problem happened after simplifying. I added this last one because it is a way of acknowledging that there may be other useful sources of ideas and strategies that help struggling students (they may be of even more use than one on one teacher intervention in some cases)

Drawing Decimals on Grid Paper Grid paper drawings instead of base-10 blocks. For the big 10 x 10 square, taken as the unit, Mark the side lengths and the area (using “units” or “cm” or “in”) What is the area of the little square? Why? What is the side length of the little square? Why? Relation of area to length. 1/100 = 1/10 x 1/10, or .01 = .1 x .1 .7 x .1 = .07

Drawing Decimals on Grid Paper Grid paper drawings instead of base-10 blocks. For the big 10 x 10 square, taken as the unit, Mark the side lengths and the area (using “units” or “cm” or “in”) What is the area of the little square? Why? What is the side length of the little square? Why? Relation of area to length. 1/100 = 1/10 x 1/10, or .01 = .1 x .1 .7 x .7 = .49 Note: product is smaller than the factors.

Modeling Multiplication of Decimals Calculate. Draw area model, matching each part of the problem to the drawing. Use grid paper in notebooks, or on hand outs. Work with a partner. Each partner does one of the problems, and explains it to the other.

Grade 4 Lesson on Place Value One hour Modeling numbers with base ten blocks; using place value to increase numbers by ones, tens, hundreds mentally; broken calculator problems Clip we will watch (~10 MIN) “Broken calculator” Set up movie clip

Teaching the Whole Group While Leading a Whole-Class Discussion Attention to equity: Holding and enacting high expectations of students to learn significant mathematics What specific types and difficulties of mathematical questions can you identify in this segment? What methods of getting student participation can you name in this segment? How does the teacher manage errors? Other observations or critique Watch video (~10 min.) Then small group discussion. Form class into 4 groups, for the four questions. Use colored unifix cubes to form groups.

CHOOSING NUMERICAL EXAMPLES WISELY HANDOUT

Choosing Numerical Examples Wisely Considering what happens with the particular numbers –– is it what you want? Considering simpler and more difficult cases and learning to distinguish the two Learning to “stage” or sequence from simple to more difficult Managing size of numbers, number of numbers –– learning to pay attention and make decisions based on purposes Protecting from confusion Provoking error on purpose

GROUP 1: Introducing concepts, materials, or procedures  What numerical examples would you select for these purposes and why?  Articulate your rationale for your choices, and try to generalize the principles for these. a) Introducing base-10 blocks with 4-digit numbers to 3rd graders b) Introducing area models of 2-digit multiplication to 4th graders c) Introducing ordering decimals in 5th grade d) Introducing 2-digit addition with regrouping in 2nd grade GROUP 2: Focused instances of special cases e) An example where, when multiplying by a multiple of ten, you end up with more than one zero in the product. f) An instance where the median exceeds the mean. g) Evidence that all rectangles with the same area don’t necessarily have the same perimeter. h) An example where, when you divide two numbers, the quotient is greater than both the dividend or the divisor, and one where the quotient is less than both the dividend and the divisor. i) An example of where the context of the problem supports reasoning ahead of where the students’ skills are. GROUP 3: Short problems for different purposes j) A warm-up problem for a lesson on subtracting two-digit numbers in 2nd grade. k) A boardwork problem for 1st graders in midyear. l) An end-of-lesson check problem for ordering large numbers in 4th grade. m) A problem to check whether students understand regrouping in multiplication computation in 5th grade GROUP 4: ?  What do these have in common that might form a category of teachers’ choices of numerical examples?  Identify four more in this category, one for each operation. Explain your rationale for including your new examples. n) .3 X .2 = o) 25 x 65 = p) 17 + 35 + 9 = q) 520 ÷ 5 =

GROUP 2: Focused instances of special cases  What numerical examples would you select for these purposes and why?  Articulate your rationale for your choices, and try to generalize the principles for these. e) An example where, when multiplying by a multiple of ten, you end up with more than one zero in the product. f) An instance where the median exceeds the mean. g) Evidence that all rectangles with the same area don’t necessarily have the same perimeter. h) An example where, when you divide two numbers, the quotient is greater than both the dividend or the divisor, and one where the quotient is less than both the dividend and the divisor. i) An example of where the context of the problem supports reasoning ahead of where the students’ skills are. GROUP 3: Short problems for different purposes j) A warm-up problem for a lesson on subtracting two-digit numbers in 2nd grade. k) A boardwork problem for 1st graders in midyear. l) An end-of-lesson check problem for ordering large numbers in 4th grade. m) A problem to check whether students understand regrouping in multiplication computation in 5th grade GROUP 4: ?  What do these have in common that might form a category of teachers’ choices of numerical examples?  Identify four more in this category, one for each operation. Explain your rationale for including your new examples. n) .3 X .2 = o) 25 x 65 = p) 17 + 35 + 9 = q) 520 ÷ 5 =

GROUP 3: Short problems for different purposes  What numerical examples would you select for these purposes and why?  Articulate your rationale for your choices, and try to generalize the principles for these. a) A warm-up problem for a lesson on subtracting two-digit numbers in 2nd grade. b) A boardwork problem for 1st graders in midyear. c) An end-of-lesson check problem for ordering large numbers in 4th grade. d) A problem to check whether students understand regrouping in multiplication computation in 5th grade

GROUP 4: ?  What do these have in common that might form a category of teachers’ choices of numerical examples?  Identify four more in this category, one for each operation. Explain your rationale for including your new examples. a) .3 X .2 = b) 25 x 65 = c) 17 + 35 + 9 = d) 520 ÷ 5 =

Wrapping Up Assignment can be done in notebooks Grading tool (like project #1) will be on the website